Model-Based Optimization of Particle Size Distribution in Batch

Model-Based Optimization of Particle Size Distribution in Batch-Cooling Crystallization of Paracetamol Jo¨rg

Worlitschek†

and Marco Mazzotti*

ETH Swiss Federal Institute of Technology Zurich, Institute of Process Engineering, Sonneggstrasse 3, CH-8092 Zurich, Switzerland Received September 26, 2003;

CRYSTAL GROWTH & DESIGN 2004 VOL. 4, NO. 5 891-903

Revised Manuscript Received January 12, 2004

ABSTRACT: The control objectives of batch crystallization processes are often defined in terms of particle size distribution (PSD), or properties related to the PSD, viz. average particle size, product filterability, dry solids flow properties, etc. To achieve these control objectives, a constrained nonlinear model-based optimization strategy has been adopted. This involves the detailed modeling of batch crystallization including model validation and parameter estimation, on-line monitoring of supersaturation and PSD, and the application of optimization strategies. A deterministic population balance model accounting for solution thermodynamics, crystal growth, and nucleation has been developed. State estimation is achieved by the on-line monitoring of temperature, concentrations in the liquid phase, particle density, and PSD. For this purpose, the focused beam reflectance measurement (FBRM) provides an on-line, in-situ information of crystal size and particle concentration in the form of a chord length distribution (CLD). A method using a three-dimensional geometrical CLD model and an inverse technique based on projections onto convex sets (POCS) has been introduced to calculate PSDs from CLD raw data. These concepts are applied to the batch cooling crystallization of paracetamol in ethanol. 1. Introduction Batch crystallizers are used extensively in chemical industry, often for small-scale production of high-value specialty chemicals. The control objectives of batch crystallization processes are defined in terms of product purity, crystal habit or morphology, average particle size, particle size distribution (PSD), bulk density, product filterability, and dry solids flow properties. In this paper, a mathematical model to describe batch cooling crystallization is developed, and it is applied to the crystallization of paracetamol from ethanol. The model accounts for solution thermodynamics, secondary nucleation, dissolution, and growth. Ad hoc designed experiments allow for the determination of solubility, and of particle formation and growth kinetics. The model allows for an open-loop optimization of the final PSD in batch cooling crystallization. Starting from a crystallization with a linear cooling profile, where significant nucleation was observed, a temperature trajectory is calculated that leads to a final PSD rather close to a defined “optimal” monomodal PSD. The main limitation hindering further development in the field of particulate processes lies in the difficulties in state estimation, where a sound observation of the process includes the properties of both the liquid and the solid phase.1,2 Various methods have been proposed to measure the concentration of the solute in the liquid phase. Among others, ATR-FTIR and densitometry have been successfully applied in batch crystallization.3-8 While ATR-FTIR allows for measurements of multiple component systems, densitometry is the least expensive measurement technique for binary systems, and provides a rather robust concentration measurement * To whom correspondence should be addressed. E-mail: mazzotti@ ivuk.mavt.ethz.ch. Tel: ++41-1-6322456. Fax: ++41-1-6321141. † Present address: Mettler-Toledo GmbH, Sonnenbergstrasse 74, CH-8603 Schwerzenbach, Switzerland.

throughout the entire course of the crystallization process. Densitometry has been used in this work, and has been implemented with a bypass that ensures a thermostated, bubble-free, and solid free flow through the measurement cell. In contrast, the on-line monitoring of the solid phase in general and of the PSD in particular remains challenging. Laser diffraction has been used in most previous studies, but in this case sampling and dilution of the suspension are necessary.9-11 More recent studies suggest the adoption of ultrasound attenuation, where rather spacious in-line probes have been developed, but a calibration is still required.12-14 On the contrary, the focused beam reflectance measurement (FBRM) is carried out using a cylindrical probe that is easily applicable in crystallization vessels ranging from laboratory to production scale.15-18 The FBRM provides on-line data of particle size and concentration in the form of a chord length distribution (CLD). Although useful in general, CLD data cannot be converted directly into a PSD, i.e., they cannot be directly integrated into population balance modeling approaches. Therefore, most applications deal with qualitative means of optimizing particulate processes using FBRM. In this paper for the first time, the measurement of PSDs and particle density during the process is obtained by using the Lasentec FBRM technique coupled with a modelbased method to calculate PSDs from FBRM CLD raw data.19,20 The drug paracetamol, also called acetaminophen, crystallized from its solution in ethanol was used to study the monitoring, modeling, and optimization of batch cooling crystallization. Several studies on different crystallization aspects of this well-known drug have been published. Paracetamol exists in three polymorphic forms, but only two can be readily isolated.21 Form I (monoclinic) is the most stable under ambient temper-

10.1021/cg034179b CCC: $27.50 © 2004 American Chemical Society Published on Web 02/17/2004

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Crystal Growth & Design, Vol. 4, No. 5, 2004

Worlitschek and Mazzotti

ature conditions, and is commercially available. A systematic search for the rigid body lattice-energy minima has proven that this crystal structure is the global minimum.22 However, form II (orthorombic) showed advantages in its tabletting properties due to its plasticity, which enables a direct compression without binders.23 Form I is considered in this work and is the one discussed in most studies. The investigations include solubility data of paracetamol in different solvents,24 effects of supersaturation and temperature on the morphology,25,26 and investigation of singlecrystal growth in water.27 Fernandez studied the effect of different solvents on the crystallization, including also the solubility of paracetamol in ethanol. Crystallization and primary nucleation in different acetone-water mixtures are investigated by Granberg et al.28,29 Several studies deal with the effect of structurally related substances such as p-acetoxyacetanilide.30-32 A feedback control strategy on the basis of a liquid-phase concentration measurement using ATR-FTIR was implemented in a crystallization of paracetamol from water.3 The paper is organized as follows. First, the applied experimental methods are described. Then, the model equations for batch cooling crystallization are presented along with the models for the particle rate equations to describe secondary nucleation, dissolution, and growth. Then, the solubility of paracetamol in ethanol is determined using both static gravimetric analysis and dynamic concentration data. Thereafter, properly designed measurements allow for model validation and kinetic parameter estimation. Finally, a temperature trajectory to optimize the final PSD of a batch crystallization is calculated based on a nonlinear open-loop optimization scheme, and tested experimentally. 2. Experimental Section 2.1 Experimental Setup. A 1-L batch cooling crystallizer has been used for the experimental investigation. The jacketed curved-bottom glass reactor has an inner diameter of 100 mm. A three-blade turbine stirrer with an upper and lower diameter of 50 and 45 mm, respectively, is placed 10 mm above the reactor bottom. Temperature control is achieved using a thermostat with a cascade controller for the temperatures in the reactor and in the circulating loop, resulting in maximum reactor temperature deviations of ( 0.2 K from the set-point. The setup is equipped with a FSC 402 turbidity meter (Mettler-Toledo, Greifensee, Switzerland), an ultrasound emulsion analyzer PGR4 SEA (E. Romagnoli, Milan, Italy), a temperature probe, a loop for the on-line measurement of the concentration in the liquid-phase including a DPRn 422 densitometer (Anton Paar GmbH, Graz, Austria), and a FBRM M400 probe (Lasentec, Redmond, WA). Alternatively to the FBRM probe, a Lasentec PVM 800 L videomicroscope (Lasentec, Redmond, WA) can be used. More details of the concentration measurement using a densitometer can be found in section 2.3. The experimental setup is fully automated using LabView (National Instruments, Austin, TX). The thermostat, the stirrer, the densitometer, and the ultrasound velocity probe are connected via serial ports; the turbidity meter and the magnetic valve for the pneumatic control of the filter switch within the concentration measurement loop are connected using a National Instruments PCI-8035 data acquisition card. Figure 1 shows a photograph of the glass reactor when equipped with the turbidity meter, FBRM probe, concentration measurement loop, and temperature sensor. 2.2 On-Line Monitoring of PSD and Particle Concentration. PSD and particle concentration were calculated from on-line CLDs obtained by FBRM data. The Lasentec FBRM

Figure 1. Photograph of the batch cooling crystallizer; turbidity probe (front); FBRM probe in the reactor (right); devices for on-line measurement of concentration in the liquid phase: peristaltic pump, pneumatic valve, thermostat, and densitometer (left). is carried out using a cylindrical probe, which can be easily located within the suspension to be characterized. An infrared laser beam rotates at high velocity, propagating into the suspension through a sapphire window on the probe tip. When the beam hits a particle, it is reflected back to the probe window. This optical signal is then processed by the device electronics and the corresponding chord length s is calculated as the product of the measured crossing time ∆t, i.e., the duration of the reflection, and the beam velocity vb. Chord length counts are recorded during a specified measurement period tm and summed up into a finite number of chord length intervals. This yields on one hand the total number of chord measurements mc in the specified time interval tm, and on the other hand a normalized CLD. The measurement data were processed by the Control Interface for FBRM Version 6 Build 9. The Lasentec device offers the possibility to move the focal point in the axial direction. Its distance with respect to the outer surface of the window was set to the standard value of df ) - 20 µm, i.e., inside the sapphire window.33 The probe is located with the probe tip 10 mm above the stirrer tip and with an angle of 12° to the radial axis. A measurement period of tm ) 2 min was chosen for all crystallization experiments. The measured CLD is recorded at the end of each measurement period, and the measurement is referred to tm/2, i.e., to the middle of each measurement period. This leads to a measurement time lag of 1 min.

Crystallization of Paracetamol

Crystal Growth & Design, Vol. 4, No. 5, 2004 893

Figure 2. Calibration curve for concentration on the basis of temperature, Tc, in the U-tube and oscillation period, P, of the U-tube. Paracetamol solutions with 0, 140.0, 180.0, 200.0, and 230.0 g/kg of ethanol were used for calibration. The units for quantities defining the horizontal coordinate are seconds for P and °C for Tc. The PSDs are restored from FBRM CLD data using a twostep procedure:19,20 (i) the computation of a matrix that converts the PSD of a population of particles with a given shape into the corresponding CLD using a three-dimensional geometric model;34 (ii) the solution of the resulting linear matrix equation for the PSD.35 Either the method of Tikonov35 or the method of projections onto convex sets (POCS)36 can be used to solve this mathematically ill-posed problem for the PSD. In this work, the latter method is applied. Measured CLDs qln with a discretization in 90 equally spaced logarithmic channels ranging from 1 to 1000 µm are transformed into PSDs having the same discretization. Within the discretized distribution, all particles are assumed to have a size corresponding to the midpoint of the corresponding channel. Alternative discretization methods have been recently proposed for spherical particles, where a linear size distribution is assumed within a given channel;37 in this case, the resulting inverse algorithms are based on solving nonlinear constrained problems. The shape of the paracetamol crystals is assumed to be that of an octahedral particle defined by the equation:34,36

|x| + |y| + |z| ) a

(1)

where a is the semi-axis, and the characteristic length L of the octahedron is defined as L ) 1.152a. This leads to a volume shape factor of kv ) 0.866, with the volume of the octahedron Vp ) kvL3, and a surface shape factor ka ) 5.196, with the surface of the octahedron Sp ) kaL2, i.e., the characteristic length L of the octahedron is chosen such that ka/kv ) 6. 2.3 On-Line Monitoring of Concentration in the Liquid Phase. The concentration in the liquid phase is measured using a U-tube densitometer DPRn 422 with the evaluation unit mPDS 1000 (Anton Paar GmbH, Graz, Austria). The device measures the oscillation period of the U-tube, P, and the temperature in the measurement cell, Tc. The concentration in the liquid phase is then calculated using a twoparameter (2-D) calibration. Figure 2 shows a result of the calibration for the paracetamol/ethanol system. The calibration is based on measurements with solutions of known concentration. For a given concentration, the oscillation period is a linear function of the cell temperature. This allows determining a relationship between the concentration and a linear combination of P and Tc, which represents the calibration curve of the instrument. Such a curve is plotted in Figure 2 and applies to operations in which the temperature in the densitometer cell is T ) 37.75 ( 0.25 °C. The procedure yields a calibration curve exhibiting a maximum deviation from the experimental values of the paracetamol concentration of ( 0.5 g/kg of EtOH. The on-line monitoring of the concentration with this device is implemented using an external circulation loop. A solid-free, bubble-free, and thermostated flow through the loop is obtained using the apparatus shown in Figure 3. The loop

Figure 3. On-line measurement of concentration using a U-tube densitometer; B1: bubble separator, C1-C7: tube connections, F1, F2: filters, H1: heat exchanger, PVC1: computer controlled pneumatic four-connection-valve. consists of two filters (F1 and F2, 5 µm metal filters) in the reactor, a four-way valve (PVC1) used to switch inlet and outlet of the liquid periodically between the two filters, a bubble separator (B1), a heat exchanger (H1), and the U-tube densitometer. The four-way valve, the bubble separator, the heat exchanger, and all connection tubing are placed in a thermostat. A peristaltic pump with two pump heads sucks the liquid out of the reactor, conveys it to the bubble separator, and recycles the bubble-rich part of the stream. A constant volumetric flow of 25 mL/min (peristaltic pump at 30 rpm with a tube of inner diameter 3.1 mm) is pumped through the loop and a partial flow of approximately 6 mL/min (peristaltic pump at 30 rpm with a tube of inner diameter 1.6 mm) is pumped from the top of the bubble separator to the reactor. The resulting measurement time lag is 60 s. The temperature of the thermostat is set to 40 °C leading to a temperature of the liquid in the U-tube of 37.75 ( 0.25 °C throughout the experiments. A start up of approximately 3 h is required to reach these constant temperature conditions, which on one hand prevent crystallization in the loop and on the other hand improve the accuracy of the concentration measurement. For the sake of accuracy, this temperature is kept constant independent of the process temperature in the crystallizer. During the start up, the loop is operated with solvent in the reactor at the initial temperature of the crystallization process. The switching between filter F1 and filter F2 prevents blocking of the filters, since the hot liquid stream is pumped back through the filter, that might have been blocked partially during the previous cycle. Every 300 s inlet and outlet filters are switched. The dead time of the measurement leads to nonutilizable records after each switching. A software filter eliminates disturbances in the data immediately after the switch. 2.4 Solubility Measurement Using On-Line Concentration Data. The solubility was determined by on-line measuring the paracetamol concentration in the ethanol solutions in the presence of paracetamol precipitate in excess. The temperature was raised to the equilibrium temperature, and kept constant for 1 h. It is worth noting that the method

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can be combined with experiments to determine crystallization kinetics or with the calibration procedure. 2.5 Solubility Measurement Using a Gravimetric Method. A waterbath shaker GFL 1086 (GFL mbH, Burgwedel, Germany) with a temperature control of ( 0.1 °C was used to establish phase equilibrium conditions. The temperature measurement was calibrated using a precision calibration thermometer (Thermo-Schneider, Wertheim, Germany). Flasks (50 mL volume) containing paracetamol in excess were agitated 12 h in the shaker at 150 rpm; samples of 5 g were taken and filtered with 0.45 µm membrane syringe filters; the samples were dried at 50 °C for 24 h. 2.6 Materials. Paracetamol, purum: >98%, and ethanol, pro analysi: 99.9%, were purchased by Fluka Chemika (Fluka, Dietikon, Switzerland).

where x and γ are the mole fraction and activity coefficient of the solute, R is the gas constant, Tm and ∆mh are the melting temperature and enthalpy (per unit mole) of the solute, respectively.40 It is worth noting that the solution composition, namely, the solvent utilized, is accounted for through the activity coefficient γ. A suitable solution model, in this case the NRTL model, is applied to describe the activity coefficient for the binary system under consideration. The activity coefficient of the solute in the liquid phase, γ, is given by:40

[(

ln γ ) (1 - x)2 τ21

G21

)

2

x + (1 - x)G21

τ12G12

3 Modeling 3.1 Crystallizer Model. We consider a well-mixed cooling crystallizer with constant volume. Assuming that only nucleation and size-independent crystal growth may occur, the time-dependent PSD n(t, L) obeys the following population balance equation:38

∂n ∂n +G )0 ∂t ∂L

(2)

where G denotes the mean crystal growth rate. The population density function is defined with reference to the volume of the liquid phase in such a way that n(t, L)dL represents at time t the number of crystals per unit volume of the solution in the size range between L and (L + dL). The molar concentration c of the solute in the liquid phase fulfills the following balance equation:

1 F dc )kG dt 2M a

∫0∞nL2dL

(3)

where F is the solid-phase density of the solute, and M is its molar mass. The initial and boundary conditions for the population and mass balance equations are given by

n(0, L) ) n0(L) n(t, 0) )

(4)

N G

(5)

c(0) ) c0

(6)

with c0 being the initial concentration of the solute in the solvent, n0(L) is the PSD of the seed population, and N is the nucleation rate per unit volume. It is worth noting the implicit assumption that newly formed particles through nucleation have L ) 0. The software package PARSIVAL is used to solve the population balance equations. The program uses a Galerkin h,p method based on a generalized finiteelement scheme with self-adaptive grid.39 3.2 Thermodynamics. 3.2.1 Binary Solid-Liquid Equilibrium. The solubility of a solute in a solution at the temperature T can be calculated using the Schroeder-van Laar equation:

ln(xγ) )

(

∆mh(Tm) 1 1 R Tm T

)

(7)

+

]

((1 - x) + xG12)2 G12 ) exp(-Rτ12) τ12 )

∆g12 RT

G21 ) exp(-Rτ21) τ21 )

∆g21 RT

(8) (9)

(10)

where R is related to the nonrandomness of the mixture, and ∆g12 and ∆g21 are assumed to be temperature independent. Therefore, the following three independent parameters must be assigned to characterize the behavior of binary solutions of paracetamol and ethanol, namely, R, ∆g12, and ∆g21. The estimation of the NRTL parameters is carried out by fitting the experimental solubility data using a Nelder-Mead simplex (direct search) minimization routine.41 3.3 Particle Rate Equations. The fundamental driving force for crystallization from solution is the difference between the chemical potential of the solute in the liquid and in the solid phase, ∆µ. The supersaturation, S, may be defined accordingly as

∆µ γc (RT ) ) ff ) γ*c*

S ) exp

(11)

s

where f and fs are the solute fugacities in the liquid and the solid phase, respectively; γ* and c* are the activity coefficient and the concentration of the saturated solution at the temperature of the system. Assuming γ ≈ γ*, the relative supersaturation can be defined on a concentration basis as

S)

c c*

(12)

which is the expression mostly used in crystallization studies. It is worth noting that for the system under consideration the approximation γ ≈ γ* holds true in the investigated supersaturation range (e.g., for S ) 1.05, γ/γ* ≈ 0.998 at all temperature values of interest). The absolute supersaturation can also be used, which is given by

∆c ) c - c*

(13)

Several relationships have been proposed and used to describe particle growth and solid dissolution.42,43 Prevalent growth models include diffusion-limited growth, spiral dislocation growth, and surface polynucleation.



Here, the following correlations have been adopted:

{

( )

Ea (∆c)g (S > 1) G) RT 2kd∆c/cc (S e 1) kg exp

(14)

where kg, Ea, and g are parameters. For growth an empirical correlation has been adopted, whereas dissolution is assumed to be mass transfer limited, cc being the molar density of the solid and kd is the mass transfer coefficient that can be predicted as reported in the appendix. In seeded crystallization, secondary nucleation may occur in the presence of crystals of the product. The surface of the crystals is assumed to be smooth if the supersaturation is low. With increasing supersaturation, the probability of surface nucleation also increases. In the following, secondary nucleation is described using the relationship:43

N)

{

kaµ2D

E

d4m

[ ( ) ]

γsld2m exp -π kT

2

(15)

(S e 1)

with E (0 < E < 1) representing the fraction of nuclei being detached from the surface and forming new particles, D is the diffusion coefficient, dm is the molecular diameter, γsl is the interfacial tension, and µ2 is the second moment of the particle size distribution PSD

µ2 )

∫0∞ L2n(L)dL

(16)

Further details on the determination of D and dM are given in the appendix. Various studies deal with the determination, prediction, and temperature dependence of the interfacial tension γsl.29,44 However, these do not lead to an unquestionable predictive relation for γsl as a function of temperature. Therefore, in this study, the simplified approach of considering γsl temperature independent is adopted. Its value is calculated together with the other parameters through experiments as discussed below. The parameters in the above equations to be estimated from experiments, i.e., Θ ) {kg, g, Ea, E,γsl}, are calculated by minimizing the objective function Ne Nm

Φ(Θ) )

i

T(i) ) Ts -

1 (S > 1) ln S

0

tion, the Nelder-Mead simplex algorithm (direct search) is used again.41 3.4 Optimization. The model developed above with the parameters estimated as discussed is used to simulate the process and to optimize its performance. In a batch process, this involves the determination of open-loop time profiles of the manipulated variables that optimize a proper measure of the process performance. In batch cooling crystallization, the temperature of the slurry is the manipulated variable of choice. The vector T defines the temperature profile during the batch cooling crystallization, where T(i) is the temperature at time step i, and the temperature is interpolated linearly between T(i) and T(i + 1). The constant time intervals ∆t are given by ∆t ) tf/Nt, where tf is the duration of the experiment, Nt is the number of time steps; hence, i ) 0, ..., Nt. In the following, the vector T is defined through the normalized vector x as

Nt,i

wij ∑ (yi,j,k - y˜ i,j,k)2 ∑ ∑ i)1 j)1 k)1

(17)

where Ne is the number of experiments, Nm is the number of measured variables, and Nt,i is the number of sampling points of the ith experiment. The variables yi,j,k and y˜ i,j,k represent the measurement and the model prediction of the kth data point of the jth measured variable during the ith experiment; wi,j is the weighting factor for the jth measured variable during the ith experiment. The measured values of the relative supersaturation, S, of the zeroth moment relative to the zeroth moment at the beginning of the crystallization, i.e., µ0/µ0(t ) 0), and of the normalized logarithmic volume-weighted PSD, n3,ln, are used for the parameter estimation. For the minimization of the objective func-

x(j)(Ts - Te) ∑ j)0

(18)

where Ts and Te are the start and end temperatures of crystallization, respectively. The problem of choosing the optimal temperature profile is then formulated as the following optimization problem:45

min φ x

(19)

subject to the constraints enforced by the model of the crystallizer and by the following relationships: Nt

x(i) ) 1 ∑ i)0

(20)

x(0) ) 0

(21)

δl < x(i) < δu

(22)

where δl ) 0 if no heating is allowed during the crystallization and δu is fixed by the maximum cooling rate in the crystallizer, with 0 < δu e 1. In the majority of crystallization processes, the PSD plays an important role either regarding the performance of further downstream processes or the specifications of the final crystalline product. When filterability of the crystals is the key issue, the flow of the liquid through the filter cake has to be optimized. It has been shown that many factors including PSD, particle shape, surface texture, and cake compressibility affect the filterability. However, under the assumptions that other variables remain constant, a narrow monomodal PSD with rather large average crystal size is usually desired.46 In the case of the final crystalline product, i.e., when the crystallization operation under consideration is the final process step yielding the final product, typically, a specific, defined size is required. To achieve these specifications, various definitions of the objective of the optimization have been proposed in the literature. These include either maximizing the integral of the growth rate over the course of crystallization,46 or minimizing the ratio of the mass of newly formed nuclei and that of the seeds at the end of the process,45 or minimizing a proper measure of the difference between

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Figure 4. Solubility data of paracetamol in ethanol in g/kg of EtOH determined by the gravimetric method (triangles), and by on-line concentration measurement (solid squares). The line is computed using the model and parameters reported in the text.

predicted and desired PSD in the case of model predictive control.47 In this case, φ can be defined in such a way that the normalized, volume-weighted PSD n3,ln(Nt, i), where i is the bin index, at the final time step Nt of the batch crystallization approximates in a least squares error sense a specified optimal distribution f(i):

φ)

∑i (n3,ln(Nt, i) - f(i))2

(23)

This formulation of the optimization problem and of the objective function is meaningful in two cases. On one hand, it is relevant where the PSD of the final product of the process is specified, and this is obtained through the crystallization step under examination. On the other hand, it is significant where postprocessing issues, e.g., filterability, are of concern, and there exists an established correlation, based on experience, for instance, between such postprocessing features of the particles and their PSD. It is worth noting that using the volume weighted PSD reduces the sensitivity of the model to variations in smaller particles, which might be important in determining the filterability of the product. Despite that, the results presented below are rather satisfactory; hence, in this work we have not explored alternative approaches. The nonlinear constrained minimization problem is solved by using the Fortran IMSL routine DCONF, which applies an algorithm proposed by Powell based on sequential quadratic programming.48 4 Results 4.1 Solubility of Paracetamol in Ethanol. The solubility of paracetamol in ethanol was determined using both the gravimetric method and on-line concentration measurements. The data are listed in Table 1 and plotted in Figure 4, where triangles refer to the data obtained through the gravimetric method and boxes indicate those measured on-line. The solubility of paracetamol in ethanol between 10 and 50 °C is in the range between 147.5 and 383.4 g/kg of EtOH. The two data

Figure 5. Comparison of measured and calculated solubility as in Figure 4 (solid line) with literature data, i.e., experimental data (triangles),24 and correlation (dotted line).49 Table 1. Solubility Data of Paracetamol in Ethanol Given as a Mass Ratio in g/kg of EtOH as Determined by the Gravimetric Method temp

solubility (g/kg)

95% confidence interval

10 ( 0.1 15 ( 0.1 20 ( 0.1 25 ( 0.1 30 ( 0.1 35 ( 0.1 40 ( 0.1 45 ( 0.1 55 ( 0.1

0.1475 0.1608 0.1796 0.1966 0.2160 0.2447 0.2772 0.3029 0.3834

( 0.0010 ( 0.0009 ( 0.0007 ( 0.0004 ( 0.0001 ( 0.0008 ( 0.0008 ( 0.0003 ( 0.0031

sets exhibit satisfactory agreement. The melting temperature and melting enthalpy of paracetamol are Tm ) 169.4 °C and ∆mh ) 26030 J/mol, respectively.49 Using these values, the solid-liquid equilibrium calculated with eq 7 fits the experimental data satisfactorily, when the parameters of the NRTL model are ∆g12 ) -486.41, ∆g21 ) 200.51, and R ) 19.98 (see the solid line in Figure 4). A comparison of the obtained results with literature data is shown in Figure 5. The solubility values obtained by Granberg et al. are larger than those determined in this study.24 The difference in solubility can be attributed to differences in the purity of ethanol and paracetamol used (paracetamol from Astra Production Chemicals AB, 100.3% pharmaceutical grade, no specifications about ethanol24). On the contrary, the solubility reported by Fernandez is rather close to the values of this study.49 In the last work, the solubility of paracetamol in kg/kg of ethanol is described by the equation C1 exp(C2T), where C1 ) 2.955 × 10-4 kg/kg, C2 ) 2.179 × 10-2 K-1, and T is in K; no range of validity of the equation is given.49 4.2 Kinetic Parameter Estimation. The kinetic parameters were estimated by minimizing eq 17, based on experiments of seeded crystallization runs, where different temperature input sequences have been applied. The values of the estimated parameters are kg ) 21.0 m s-1 (m3/kmol)g, g ) 1.9, Ea ) 4.16 × 107 J/kmol, E ) 7.0 × 10-20, and γsl ) 2.5 × 10-3 J/m2. It is worth noting that the value of the parameter Ea accounting



Figure 6. Temperature steps applied for nonlinear kinetic parameter estimation.

for the temperature dependence of the growth rate is in the range given by Mullin as typical for integration limited growth, i.e., between 4 × 107 and 6 × 107 J/kmol.50 An example of an input temperature sequence is illustrated in Figure 6, where the temperature is decreased from 30 to 8 °C in 10 h. The temperature profile consists of plateaus separated by linear variations. The duration of the temperature steps and of the decreasing ramps is kept constant, while the extent of change at every temperature variation increases during the course of the experiment. Applying such temperature profile leads to a measured supersaturation profile as shown in Figure 7a (solid line), which exhibits peaks at the same time where a temperature change occurs. While the first temperature step (carried out at the rate of 1 °C in 900 s) results in a maximum supersaturation, S, of 1.04, the second last temperature step with the highest temperature gradient (4 °C in 900 s) leads to a supersaturation of 1.08. The simulated supersaturation profile calculated by the model using the estimated parameters is drawn also (dashed line), and follows rather closely the measured profile. In Figure 7b, the course of the measured zeroth moment of the PSD (given as µ0/µ0(t ) 0)) during the same crystallization run is plotted as solid line. Steps with increasing zeroth moment corresponding to the temperature steps are observed. As it can be readily observed, the model (dashed line) is able to describe these steps, although with steeper changes than the experiments. This difference might be because only the zeroth moment of the PSD is used for parameter estimation in eq 17, and that such moment can be significantly underestimated by the FBRM during the occurrence of nucleation since the instrument has a lower detection limit of 0.8 µm, i.e., larger than the critical nucleus size.51 Two experiments having slightly different characteristics are shown in Figure 8, where the temperature is varied between 25 and 30 °C by imposing steeper and steeper temperature gradients for the heating and cooling sequences. The corresponding measured supersaturation profiles are given as solid lines in Figure 9. The simulated supersaturation profiles based on the estimated parameter set are plotted as dashed lines.

Figure 7. (a) Relative supersaturation S and (b) zeroth moment of the PSD (given as µ0/µ0(0)) as a function of time during batch crystallization where the temperature profile given in Figure 6 was adopted. Experimental data used for parameter estimation (solid lines) and corresponding simulated values (dashed lines).

The dissolution, i.e., the region where S e 0, is described rather well as shown in Figure 9, although the description of the dissolution rate in eq 14 is based on predictive calculations, only. It is worth noting that temperature profiles of the type presented in Figure 8 can be used for the determination of kinetic data as well as of solubility data. In the case of the experiment shown in Figures 8 and 9, it is obvious that the system equilibrates after each heating step. Therefore, the solubility at 30 °C for instance results from concentration data, where the system is assumed to be equilibrated at 30 °C. 4.3 Optimization of PSD. In the following, the strategy presented in section 3.4 is applied to optimize the final PSD for the seeded batch cooling crystallization of paracetamol from ethanol. The crystallization is optimized to approach a specified monomodal final PSD. The starting point for the optimization is a seeded batch crystallization, where a linear cooling profile has been applied. Such cooling policy leads to the significant formation of newly formed nuclei. This motivates the search for a better, optimal cooling profile that, starting

898


Figure 8. Temperature profiles adopted for nonlinear kinetic parameter estimation.

from the same initial conditions as linear cooling crystallization, allows minimizing secondary nucleation. The results of the linear cooling crystallization are illustrated in Figures 10-12, where the linear temperature profile is plotted as the dashed line in Figure 10. The temperature sequence has an overall time of 14 h and consists of a linear temperature decrease from the initial temperature of 30 to 10 °C in 12 h and a constant temperature plateau at 10 °C for 2 h. The measured and simulated profiles of the supersaturation, S, are shown in Figure 11, whereas the measured and simulated PSD, n3,ln, are drawn in Figure 12. It is worth noting that the experiment constitutes an independent experiment with respect to parameter estimation, i.e., it has not been used for the estimation of the kinetic parameters. The temperature profile of the experimental run, i.e., the crystallizer temperature throughout the crystallization, has been used for the simulation. The initial concentration is c0 ) 220.65 g/kg, i.e., slightly above the saturation concentration at 30 °C. It can readily be observed in Figure 11 that supersaturation follows a rather typical behavior for crystallization with linear cooling profile. The supersaturation increases at the beginning of the process and exhibits a maximum at S ) 1.04 after 2.5 h. Then, it decreases in 3 h to a slightly


Figure 9. Relative supersaturation S of seeded crystallization where the temperature profiles given in Figure 8 were adopted. Experimental data used for parameter estimation (solid line) and corresponding simulated supersaturation (dashed line).

Figure 10. Linear temperature profile (dashed line) and optimized temperature profile (solid line).

lower level between 1.03 and 1.035, and stays almost constant at this value until the end of the cooling phase. When the temperature in the crystallizer is kept con-



Figure 11. Measured (solid line) and simulated (dashed line) supersaturation profiles during the crystallization where the linear cooling in Figure 10 was adopted.

Figure 13. (a) Optimization of the final PSD n3,ln: measured final normalized PSD, n3,ln, of linear cooling crystallization as reported in Figure 12 (dotted line); optimal distribution f used for the optimization in eq 19 (dashed line); final PSD obtained with optimized cooling profile given in Figure 10; (b) Particle size distributions over time for the cooling crystallization where the optimized cooling profile of Figure 10 was adopted.

Figure 12. Normalized volume weighted PSDs, n3,ln, over time for the crystallization where the linear cooling profile of Figure 10 was adopted: (a) experimental and (b) simulated data.

stant at 10 °C, the supersaturation decreases rapidly and reaches a value of S ) 1.01 after 2 h. The measured and simulated supersaturation almost coincide. The fluctuations of both simulated and experimental super-

saturation reflect the oscillations of the reactor temperature around the set-point, i.e., the corresponding fluctuations of the saturation concentration. The dominance of the nucleation is obvious in Figure 12, where the measured and simulated normalized PSDs, n3,ln, are plotted as functions of time. The fraction of the particle population consisting of newly formed nuclei increases significantly, while crystallization proceeds while the fraction constituted of the seed particles decreases in the normalized representation. Therefore, in a next step, a temperature profile to obtain a monomodal PSD dominated by the seed particles is calculated by solving eq 19. The desired PSD, f, in eq 21 is set to the monomodal PSD given in Figure 13a by the dashed line. The temperature at the beginning and the end of the process are Ts ) 30 °C and Te ) 10 °C, respectively. The overall time of the optimized cooling was set to tf ) 50 400 s, i.e., the overall time of the linear cooling experiment. The temperature vector, T, has been chosen to have Nt ) 15. After the cooling step, the temperature is kept constant at 10 °C for 2 h. The resulting optimal temperature trajectory is shown in Figure 10 as the solid line and the corresponding simulated supersaturation profile is drawn as a dashed line in Figure 14. A comparison of the final PSDs of linear and optimized cooling with the desired distribution, f, is shown in Figure 13a and the measured

900



Figure 14. Relative supersaturation profiles: Simulated supersaturation S corresponding to optimized temperature profile of Figure 10 (dashed line); applying the optimized temperature profile to a crystallization run leads to the measured supersaturation profile given by the solid line.

normalized PSD, n3,ln, during the whole process for the optimized crystallization is reported in Figure 13b. With reference to Figures 10 and 14, it can readily be observed that at the beginning of the optimized crystallization the temperature is decreased rapidly to build up supersaturation quickly. Then, it follows a profile that leads to a slow increase of supersaturation until the end of the cooling phase (see Figure 14). When the temperature is kept constant at 10 °C at the end of the batch, the supersaturation (that has reached the value 1.035 at the end of the cooling) drops to a value of 1.01 in 2 h. Comparing the supersaturation profiles during the two crystallization runs underlines that the linear cooling leads to an inverse profile with a maximum of the supersaturation at the beginning of the process and a decrease toward the end of the cooling. The comparison of the final normalized PSDs, n3,ln, obtained through linear and optimized cooling with the desired monomodal distribution illustrates well the effect of the optimization (see Figure 13a). While the final PSD of the linear cooling is dominated by newly formed nuclei, significantly less nucleation is observed in the case of the optimized crystallization run. This is emphasized in Figure 13b, where the normalized PSD, n3,ln, of the optimized cooling is shown as a function of time. A growth of the seed crystals leading to a shift of the population toward larger particle sizes can be observed. With respect to the same plot for the linear cooling run in Figure 12a, the absence of the peak corresponding to the abundant formation of new nuclei is remarkable. Finally, the optimized cooling profile is compared with two different cooling profiles calculated according to Mullin,52 i.e., in such a way that the absolute supersaturation ∆c ) c - c* is kept constant at a fixed value throughout the crystallization, where it is assumed that only growth of the seed crystals occurs. Here, the profiles are calculated based on the initial conditions of the optimized crystallization run, and the final temperature is 10 °C in both cases. The initial temperatures are selected in such a way that the initial

Figure 15. Comparison among simulations for the optimized cooling prifle and for the cooling profiles calculated according to Mullin52 for fixed values of constant absolute supersaturation ∆c ) 6 g/kg (dashed-dotted line) and ∆c ) 7 g/kg (dotted line): (a) temperature profiles, (b) profiles of relative supersaturation S, (c) final PSDs n3,ln.

supersaturation values equal the absolute supersaturation levels chosen for the design of the cooling profiles. Such a comparison is shown in Figure 15; in Figure 15a



the optimized temperature profile of Figure 10 (solid line) is shown together with two profiles calculated according to Mullin for ∆c ) 6 g/kg (dashed-dotted line) and ∆c ) 7 g/kg (dotted line), respectively. The calculations lead to temperature profiles with cooling times larger and smaller than that of the optimized cooling for ∆c ) 6 g/kg and ∆c ) 7 g/kg, respectively. The corresponding simulated profiles of the resulting relative supersaturations S are plotted in Figure 15b. It can be observed that the supersaturation levels are rather constant at the beginning, but tend to increase toward the end of the cooling. The optimized supersaturation profile is close to the ∆c ) 6 g/kg profile at the beginning of the crystallization and gets closer to the ∆c ) 7 g/kg profile toward the end. The final PSDs (volume weighted) obtained in the optimized cooling crystallization and in the two at nominally constant supersaturation are compared in Figure 15c. The fraction of fines is larger in the case of the fast cooling at ∆c ) 7 g/kg than in the other two cases. The optimized cooling leads to a PSD rather close to that of the slow cooling at ∆c ) 6 g/kg, but in the latter case this requires about 25% more time.

uration slightly until the end of the cooling cycle. Applying the calculated optimal cooling profile to the crystallizer leads to a measured supersaturation that is indeed rather close to the calculated profile. The optimized experiment is compared to a crystallization, where a linear cooling profile has been applied for approximately the same overall time. The comparison shows that linear cooling leads to a supersaturation profile with inverse characteristics, i.e., it exhibits higher supersaturation at the beginning of the process, and lower supersaturation toward the end of the cooling phase. Both simulations and experimental results emphasize the sensitivity of the process to small changes in the applied temperature profiles. Although the overall time is almost the same for the linear and the optimal temperature profile, the resulting final PSDs exhibit completely different characteristics. While the final PSD after linear cooling is dominated by the fraction of newly formed nuclei, virtually no nucleation is observed in the optimized crystallization leading to a final PSD rather close to the desired monomodal distribution.

5 Conclusions

Appendix

The main issue limiting process performance in the field of particulate processes lies in the difficulties in state estimation.1,2 An on-line observation of the process includes the properties of both the liquid and the solid phase. Various methods have been proposed to measure the concentration of the solute in the liquid phase. Densitometry has been used in this work, since it provides a robust concentration measurement throughout the entire crystallization process. In contrast, the on-line monitoring of the solid phase in general and of the PSD in particular has always constituted a major challenge. Here, the measurement of PSDs and particle density during the process is obtained using the Lasentec FBRM technique and a method to calculate PSDs from FBRM CLD raw data.19,36 A deterministic population balance model accounting for thermodynamics, crystal growth, and secondary nucleation has been developed to describe the seeded batch cooling crystallization of paracetamol from a solution in ethanol. Proper crystallization experiments were designed to determine solubility, as well as kinetics of nucleation, dissolution, and growth. The resulting solubility data are in good agreement with literature data and with data obtained by gravimetric measurements of the solubility in the same temperature range. Kinetic parameters are determined by nonlinear parameter estimation, where supersaturation, zeroth moment of the PSD, and normalized volume weighted PSD are used as variables in the objective function. A rather significant dependence of the growth rate on temperature has been observed which is anyhow in the range of typical values reported in the literature. The model presented here allows for an optimization of the final PSD in batch cooling crystallization. An optimal temperature trajectory that minimizes the difference between the resulting final PSD and a given “optimal” monomodal PSD is calculated using a nonlinear constrained minimization algorithm. The supersaturation profile corresponding to the optimized tem-perature trajectory follows a profile to increase supersat-

The diffusion coefficient and the mass transfer coefficient can be estimated using proper predictive methods. The methods adopted for this purpose are presented in this section. The diffusion coefficient D is estimated from the Stokes-Einstein equation:

D)

kT 2πηldm

(24)

where k is the Boltzman constant, T is the temperature, and ηl is the kinematic viscosity of the liquid assumed to be the one of the pure solvent. The molecular diameter dm entering in the Stokes-Einstein equation is estimated from the relation:43

x

dm ) 3

1 ccNA

(25)

where cc is the molar density of the solid, and NA is the Avogadro number. The mass transfer coefficient, kd, is predicted from the relationship:43,44,53

kd )

( ( ) )

L h 4F3l D 2 + 0.8 L h η3l

1/5

Sc1/3

(26)

where the Schmitt number is defined as

Sc )

ηl DFl

(27)

For stirred vessels, the mean specific power input can be estimated from

) Ne v3s d5s /Vr

(28)

Typical values of Newton numbers in stirred vessels with axial flow pumps or propellers (Re < 104) are in the range of 0.3-0.7,53 thus an intermediate value of Ne ) 0.5 is used in our calculations.

902

Crystal Growth & Design, Vol. 4, No. 5, 2004 Table 2. Model Parameters

cc ) 8.573 dm ) 7.18 × 10-10 ds ) 0.05 k ) 1.381 × 10-23 L h ) 1 × 10-4 M ) 151.17 Ml ) 46.07 NA ) 6.023 × 1026 Ne ) 0.5 Vr ) 0.001 vs ) 10 ηl ) 1201 × 10-6 F ) 1296 Fl ) 789

molar density of solid paracetamol [kmol/m3] estimated diameter of paracetamol molecule [m] stirrer diameter [m] Boltzman constant [J/K] particle size applied for calculation of kd [m] molar mass of paracetamol [g/mol] molar mass of ethanol [g/mol] Avogadro number [1/kmol] Newton number [-] total reactor volume [m3] stirrer speed [1/s] kin. viscosity of ethanol [Ns/m2] solid density of paracetamol [kg/m3] liquid density of ethanol [kg/m3]

A list of the numerical values of the model parameters in the calculations is given in Table 2. Nomenclature D

diffusion coefficient of the solute in the liquid phase [m/s] E secondary nucleation rate parameter [-] G growth rate [m/s] Gij coefficient in NRTL model [-] L particle characteristic length [m] L h particle size applied for calculation of kd [m] M molar mass of solute [g/mol] molar mass solvent [g/mol] Ml N nucleation rate [no./(m3 s)] NA Avogadro number () 6.023 × 1026) [1/kmol] number of experiments [-] Ne Nm number of measured variables [-] Nt,i number of sampling points of ith experiment [-] Ne Newton number [-] R universal gas constant, R ) 8.3144 J/(mol K) [J/(mol K)] Re Reynolds number [-] S supersaturation [-] Sc Schmitt number [-] particle surface [m2] Sp T temperature [K] T temperature vector [°C] final temperature of crystallization [°C] Te melting temperature [K] Tm starting temperature of crystallization [°C] Ts Vp particle volume [m3] crystallizer volume [m3] Vr c molar concentration of solute in liquid phase [kmol/ m3] c* molar saturation concentration of solute in liquid phase [kmol/m3] initial molar concentration of solute in liquid phase c0 [kmol/m3] molar density of solid solute [kmol/m3] cc ∆c absolute supersaturation [kmol/m3] diameter of solute molecule [m] dm stirrer diameter [m] ds molecular diameter [m] dm f desired final PSD in optimization objective [-] f fugacity of solute in liquid phase [kmol/m3] fugacity of solute in solid phase [kmol/m3] fs g exponent in the growth model [-] coefficient in NRTL model [-] gij ∆gij coefficient in NRTL model () gij - gjj) [-] ∆mh melting enthalpy per unit mole [J/mol] k Boltzmann’s constant, k ) 1.38 × 10-23 J/K [J/K] surface shape factor [-] ka liquid side mass transfer coefficient [m/s] kd kg rate constant in the growth model [m s-1 (m3/kmol)g] volume shape factor [-] kv n population density function [m-4] initial population density function [m-4] n0 n3,ln discretized logarithmic volume-weighted PSD [-]

Worlitschek and Mazzotti qln t tf vs wij wi x x yi,j,k yi,j,k Φ R ηl φ γ γsl µj F Fl τij

discretized logarithmic CLD [-] time [s] duration of crystallization experiment [s] agitator speed [rev/s] weighting factor for the j-measured variable in the ith experiment [-] mass fraction of component i [-] mole fraction of solute in the liquid phase [-] normalized vector for description of T [-] kth date point of jth measured variable during ith experiment [-] kth data point of jth simulated variable during ith experiment [-] objective function for parameter estimation [-] coefficient in NRTL model [-] kin. viscosity of solvent [Ns/m2] objective function for optimizing temperature trajectory [-] activity coefficient of solute in liquid phase [-] solid liquid interfacial tension [J/m2] jth order moment of the PSD [mj/m3] density of solute [kg/m3] density of solvent [kg/m3] coefficient in NRTL model [-]

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CG034179B

Model-Based Optimization of Particle Size Distribution in Batch

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