Article pubs.acs.org/IECR
Crystallization Kinetics of Ampicillin Using Online Monitoring Tools and Robust Parameter Estimation Luis G. Encarnación-Gómez, Andreas S. Bommarius, and Ronald W. Rousseau* School of Chemical and Biomolecular Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0100, United States ABSTRACT: Crystallization kinetics of ampicillin were estimated from induction-times and seeded and unseeded experiments. The concentration of ampicillin in solution was monitored by polarimetry and refractometry, while crystals were analyzed by focused beam refractive measurements and light microscopy. Variations of solute concentration with time, along with measured induction times were used to estimate parameters in kinetic expressions for primary nucleation, secondary nucleation, and crystal growth rate. Experimental data were obtained in runs involving different rates of change of pH, mass of seed crystals, and initial supersaturation. These data were used in parameter-estimation routines consisting of a stochastic minimization to localize a set of parameters spanning all the experiments, followed by a deterministic minimization to refine the parameters. The result was a set of parameters that fit a range of conditions of seeded and unseeded crystallizations and gave good estimates of desupersaturation rates.
1. INTRODUCTION β-Lactam antibiotics can be manufactured using the enzymatic synthesis by penicillin G acylase.1 This enzyme catalyzes the reaction of phenylglycine methyl ester (D-PGME) and 6aminopenicillanic acid (6-APA) to form ampicillin. However, as shown in Figure 1, the enzyme also catalyzes the hydrolysis of D-PGME (primary hydrolysis) into phenylglycine (D-PG) and the hydrolysis of ampicillin into 6-APA and D-PG (secondary hydrolysis). The reaction network is a classic example of kinetically controlled synthesis in which an intermediate (ampicillin in this case) must be removed from the system to prevent its consumption (by secondary hydrolysis). Because the crystallization must be integrated with the enzyme-catalyzed reaction it is essential that the conditions for each do not interfere with progress of the other. Crystallization kinetics of ampicillin and the effects of reactants and byproducts on the induction time for nucleation of ampicillin have been reported.2 The present work expands the range of experimental conditions examined and proposes a model and associated parameters that are applicable over this broader range of variables. Included are pH, rate of change of pH, initial supersaturation, and the use of seed crystals. All were explored over a pH range that is suitable for an enzymatic reactive system; that is, the crystallization kinetics were obtained over a pH range of 6.00 to 7.50. Additionally, most of the work was performed using online process analytical technology tools that have become common in model development and control of crystallization processes.3 This has the advantage of producing large data sets adding robustness to the models developed. The strategy followed was as follows: First, necessary equilibrium data were obtained from the literature and fit with a thermodynamic model that allowed estimation of the solubility as a function of pH (section 2). Second, results of induction-time experiments were used to assess the relationship of initial supersaturation to primary nucleation, and the resulting data were fit to a model relating nucleation kinetics to supersaturation (section 3). Third, a combination of seeded © 2016 American Chemical Society
and unseeded crystallization experiments was used to obtain additional information about growth kinetics and both primary and secondary nucleation kinetics (section 3.4). Finally, results of these sets of experiments were combined to obtain a single set of model parameters (section 3.5), and these were used in the model to estimate the final crystal mean sizes in all of the runs, which were then compared to the experimental results (section 3.6).
2. MATERIALS AND METHODS 2.1. Chemicals. Anhydrous ampicillin was purchased from Alfa Aesar (Ward Hill, MA) and used without further purification. Ammonia was purchased in a 30% solution from Ricca Chemical Co. (Arlington, TX), and 5 M hydrochloric acid was purchased from Sigma-Aldrich (St. Louis, MO). 2.2. Equipment. Experiments to determine crystallization kinetics were performed in a 1-L OptiMax Synthesis Station from Mettler Toledo (Columbus, OH). A dosing unit, which is a part of the OptiMax apparatus, was used to add 2 M HCl (diluted from a 5 M HCl solution) to the reactor, and a pH probe unit from Mettler Toledo was used to monitor the pH value in solution with a precision of ±0.001 pH units. Solution concentration was monitored using a MCP 500 polarimeter and an Abbemat 500 refractometer, both from Anton Paar by using signal-concentration calibrations where the corresponding regression was y = 3.428x + 0.002 (R2 = 0.99) and y = 4737x −6314 (R2 = 0.99) for the refractometer. As shown by the picture in Figure 2, the solution was withdrawn from the reactor using a peristaltic pump and circulated through the refractometer and polarimeter. A 1-μm filter was installed at the inlet to prevent crystals from flowing through the recirculation loop. Chord-length distributions of crystals were monitored Received: Revised: Accepted: Published: 2153
October 15, 2015 January 27, 2016 January 29, 2016 January 29, 2016 DOI: 10.1021/acs.iecr.5b03880 Ind. Eng. Chem. Res. 2016, 55, 2153−2162
Article
Industrial & Engineering Chemistry Research
Figure 1. Enzymatic synthesis of ampicillin by penicillin G acylase.
of crystals was approximated by measuring the width and length of multiple crystals using the scales on pictures taken with the microscope. Seeded crystallization runs were initiated by adding 1 M HCl to reduce the pH in the system from approximately 7.85 to 6.00. Different amounts of seed crystals in the size range 20 to 250 μm were added to the crystallizer and the pH was gradually changed to 6.00 at a predetermined rate using the acid dosing unit (specific conditions reported later). The solution concentration, evolution of the chord-length distribution, and the recovery and analysis of crystals were accomplished as described in the preceding paragraph. Figure 2. Crystallization system.
3. RESULTS AND DISCUSSION 3.1. Crystallization Kinetics. If agglomeration and breakage are neglected, a population balance on a batch, constantvolume crystallizer can be written as
using a focused beam reflectance measurement (FBRM) apparatus from Mettler Toledo. Further details on this unit can be found elsewhere.4 2.3. Crystallization Experiments. The initial solutions in all crystallization experiments were obtained by charging the 1L vessel with 500 g of DI water and adding 2 M NH3 solution until the pH value was approximately 7.75. (Exact values of pH were used to estimate the solubility.) The desired amount of ampicillin was then dissolved in the solution. The FBRM was used to monitor the appearance of new crystals and the evolution of the chord-length distribution throughout the length of the run. In each run, counts were normalized by the total count to minimize run-to-run variations in the signal. Induction-time experiments were begun by abruptly adjusting the pH in the system to the range of 7.85 to 6.00 by adding 1 M HCl solution. The time at which the FBRM reported a rapid increase in chord counts was defined as the induction time. Unseeded crystallization runs were initiated by gradually changing the pH in the system from the initial value of around 7.85 to as low as 6.00 by adding 1 M HCl at a predetermined rate (specific conditions reported later). The solute concentration was measured by recirculating a small portion of the solution through the polarimeter and refractometer as described in section 2.2, and the evolution of the chord-length distribution was monitored using the FBRM. At the end of each run (after stabilization of the polarimetric signal), crystals were filtered, dried, and examined microscopically. The aspect ratio
∂n(t , L) ∂[n(t , L)G(t )] =− ∂t ∂L
(1)
where n is the population density function, t is time, L is crystal characteristic dimension, and G is the crystal growth rate dL/dt. The boundary and initial conditions are given by n(t , 0) =
J (t ) G (t )
n(0, L) = n0
(2) (3)
Alternatively, the system dynamics can be described by the method of moments, which transforms the population balance from a partial differential equation to an ordinary differential equation. The moments of the distribution are defined by μi =
∫0
∞
n(t , L)Li dL
(4)
and differentiation of the moments leads to expressions that contain nucleation and growth kinetics: dμ0 dt 2154
=J
(5) DOI: 10.1021/acs.iecr.5b03880 Ind. Eng. Chem. Res. 2016, 55, 2153−2162
Article
Industrial & Engineering Chemistry Research
dμi dt
= iGμi − 1
λij =
(6)
In the present work the following constitutive equations are used to model the rates of primary nucleation (B1), secondary nucleation (B2), and crystal growth where [Amp] is the concentration of ampicillin, [Amp*] is the corresponding concentration at equilibrium, and S is the system supersaturation (S = [Amp]/[Amp*]). ⎛ −Bo ⎞ B1 = kB1 exp⎜ ⎟ ⎝ ln(S)2 ⎠
(7)
B2 = kB2M bM(S − 1)s
(8)
J = B1 + B2
(9)
G = k G(S − 1)g
(16)
2πNAρσ 3 3 ε Bij = −Aij kB Aij =
(17) (18)
and Aij and Bij were used as adjustable parameters within the model. These parameters take into account nonidealities in solution. Note that regardless of the complexities of the equation only two parameters were used to correlate pH with solubility. Data from the literature7 were fit using the preceding procedure. There were differences between the two data sources, and our experiments tended to equilibrate at nearaverage values of the data. Figure 3 shows experimental and fitted solubilities of ampicillin as a function of pH; Table 1 provides the parameters
(10)
Additionally, the rate of change of the third moment is related to the change in solution concentration through a mass balance. dμ3 d[Amp] = −ρc k v = −3ρc k vGμ2 dt dt
2πσ 3NAρ ⎛ ε ⎞ ⎟ ⎜1 − 3 kBT ⎠ ⎝
(11)
3.2. Solubility. All crystallization protocols followed in this work were based on pH-induced supersaturation. Ampicillin is an ionizable compound that can exist in solution in three different forms (anion, cation, zwitterion). The distribution among the charged species depends on pH and can be related to the concentration at the isoelectric point with the equilibrium constants KA1 =
[Amp−][H+] [Amp± ]
KA2 =
[Amp± ][H+] [Amp+ ]
(12) Figure 3. Modeled and experimental solubility of ampicillin at different pH values.
(13)
and the Henderson−Hasselbach equation pKA = pH + log
[Amp± ] [Amp−]
Table 1. Extended Pitzer Model Parameters for Ampicillin at 298 K (14) Aij (kg/mol) 1/2
where pKA = −log KA. Assuming that the solubility of the zwitterion [Amp±] is constant throughout the pH range of interest, the solubility of ampicillin ([Amp] = [Amp±] + [Amp−]) can be estimated using the Henderson−Hasselbach equation. Nonidealities in solutions can be included by using the extended Pitzer model5 for electrolyte systems. This approach was used by Franco et al.6 to model the solubility of multiple ionizable compounds as a function of pH and temperature. However, in the present case we will focus on the pH as the primary variable affecting the solubility of ampicillin. With the approach just described, solubility data were fit with the expressions
11.05
σ (m)
ε/kB (K)
λij (kg mol−1)
−3.71 × 103
5.42 × 10−10
336.18
−1.42
obtained by fitting experimental data. The fitting procedure determined either the adjustable Pitzer parameters Aij and Bij or the thermodynamic parameters σ and ε/kB, which give equivalent results. As shown in Figure 3, when the pH is close to the isoelectric point there is little difference between ideal and nonideal solution behavior, but the effects of nonideal behavior increase with increasing pH. The deviation from ideality is controlled by ε/kB, where a value greater than the experimental temperature renders the interaction parameters negative. Therefore, the solubility exhibits a negative deviation from ideality and activity coefficients are less than unity (i.e., solubility is greater than that in an ideal solution). This is expected as charged species should exhibit greater interactions with a polar solvent than their zwitterion counterparts. 3.3. Primary Nucleation and Induction Time. The rate of primary nucleation can be related to induction time, which is defined as the time between a solution becoming supersaturated and detection of crystals.8 Hence, induction time can
⎡ 1 + 10 pH − pKA1 ⎤ [Amp*](pH) = pI − pH + log⎢ ⎥ [Amp*](pI) ⎣ 1 + 10 pI − pKA1 ⎦ ⎡ 1 + 10 pH − pKA2 ⎤ 2 + log⎢ λij([Amp*](pI) ⎥+ ln 10 ⎣ 1 + 10 pI − pKA2 ⎦
log
− [Amp*](pH))
Bij (kg/mol)1/2
(15)
where 2155
DOI: 10.1021/acs.iecr.5b03880 Ind. Eng. Chem. Res. 2016, 55, 2153−2162
Article
Industrial & Engineering Chemistry Research be considered as the combination of the times required to create a nucleus and that nucleus growing to a detectable size: t ind = t N + tG (19) Assuming that nucleation is the limiting step (i.e., tN ≫ tG), the induction time can be related to the rate of nucleation J by t ind = J −1
(20)
where J = B1
(21)
Finally, by substituting eq 7 into eq 20 and rearranging, the induction time can be related to supersaturation by ⎡ −B ⎤ 0 ⎥ t ind = kB1 exp⎢ ⎣ (ln S0)2 ⎦
(22)
or ln t ind =
B0 (ln S0)2
Figure 5. Fit of eq 23 to data from induction-time experiments.
− ln kB1
(23)
Figure 4 shows the total counts in the range of 0−10 μm for an induction-time experiment in which rapid supersaturation was
Figure 6. Induction time vs initial supersaturation.
parameters to have a full description of the batch crystallization process. Accordingly, a set of experiments consisting of seeded and unseeded crystallization runs were performed in which initial concentration, rate of change of pH, initial mass of seed crystals (wt % = mass of seed crystals/mass of product crystals), and initial supersaturation were changed as shown in Table 2. The idea was to train the model with multiple conditions, such that it could be used to predict system behavior under a variety of crystallization protocols. Figure 7 panels a to c show the variation of ampicillin concentration with time for unseeded crystallization at three different rates of change of pH change and initial concen-
Figure 4. Detection of induction time by FBRM for an initial supersaturation of 3.02 ([Amp)/[Amp]*) at 298 K.
induced by the addition of HCl into the crystallizer and crystals in solution were monitored using FBRM. The number of crystals in the range of 0−10 μm begins a rapid increase at around 8 min, and this was taken as the induction time for the given conditions. Following the procedure just described, induction periods at multiple initial supersaturations were obtained and the results fit with eq 23, as shown in Figure 5. The slope of the regression represents the exponential primary nucleation constant B0, which in this case was 1.27. Finally, using this value, the plot in Figure 6 was constructed. These results show that an initial supersaturation of at least 1.75 is necessary to achieve nucleation within a practical time range. Further details on the physical meaning of induction periods can be found elsewhere.8 3.4. Crystallization Experiments. In the previous section experiments were utilized to estimate the primary nucleation constant B0. However, eqs 7 through 10 still require six
Table 2. Experimental Conditions of Crystallization Runs
2156
run
seed (wt %)
[Amp]0 (g/kg H2O)
S0 [Amp]0/[Amp]
pH/h
pH0
pHF
a b c d e f
0 0 0 1.8 3.0 15.0
18.09 17.93 15.94 11.30 11.36 14.00
0.54 0.47 0.98 1.07 1.18 1.64
1.0 3.1 6.0 1.5 1.4 0.6
7.84 7.89 7.47 7.09 7.01 7.06
6.45 6.41 5.97 6.25 6.34 6.51
DOI: 10.1021/acs.iecr.5b03880 Ind. Eng. Chem. Res. 2016, 55, 2153−2162
Article
Industrial & Engineering Chemistry Research
Figure 7. Unseeded ampicillin crystallization at rates of pH change (a) 1.00 pH units/h, (b) 3.10 pH units/h, and (c) 6.00 pH units/h and crystallization of seed crystals whose mass was (d) 1.8%, (e) 3.0%, and (f) 15.0% of the final crystals mass.
system undergoing extensive nucleation followed by rapid growth of the resulting crystals. Figure 7 panels d to f show variations of concentration with time for three seeded crystallization experiments (runs d, e, and f) using different
trations. The data were obtained by transforming signals from the polarimeter to concentration as described in section 2.2. As seen in Figure 7 panels a to f, more rapid changes in pH led to steeper desupersaturation curves, which is characteristic of a 2157
DOI: 10.1021/acs.iecr.5b03880 Ind. Eng. Chem. Res. 2016, 55, 2153−2162
Article
Industrial & Engineering Chemistry Research
Figure 8. Final chord-length distribution of unseeded (left) and seeded (right) runs.
masses of seed crystals, initial concentrations, and rates of change of pH. Also shown in all six panels are solubilities for each case. For the seeded experiments, initial supersaturations were selected so as to be within the metastable limits. The data in Figure 7f show that a significant change of seed crystals was necessary to induce a rapid reduction in supersaturation. Although not tested experimentally, it is expected that this requirement depends on both mass of seed crystals and their size distribution. Details on how to approximate such a critical mass for different sizes can be found elsewhere.9 The effect of the rate of change of pH on crystal size distribution is reflected in the chord-length distribution (CLD) from these runs. Figure 8 shows that the final CLD for unseeded runs a, b, and c varied, but not greatly. Additionally, the final CLDs of seeded runs d and f in Figure 8 show slight deviation. These CLDs will be used later to test the capabilities of the model on predicting crystal properties. Mass-fraction and number-fraction histograms of seed crystals and the corresponding CLD are shown in Figure 9. The former were determined from sieve analysis, while chordlength distributions were determined by adding the seed crystals to a saturated solution and recording the initial CLD. Photomicrographs, as illustrated in Figure 10, were used to determine the area and volume shape factors of the seed crystals. This information was used to determine the properties of the seed crystals shown in Table 3. The mean crystal size was a number-based arithmetic average n
L̅CSD =
n
∑ NL i i / ∑ Ni i=1
i
(24)
mi/kvρcLi3,
Figure 9. (a) Size distributions of sieved ampicillin seed crystals and (b) chord-length distribution of seed crystals.
where Ni = mi is the mass on sieve i, and Li is the sieve midsize. The average crystal density ρc was obtained from the literature.2b 3.5. Crystallization Kinetics Parameter Estimation. In the previous sections the data necessary to extract crystallization kinetics were obtained from seeded and unseeded crystallization experiments. In this section, eqs 4 to 11 will be used to fit the experimental data and thereby estimate kinetic parameters for crystal nucleation and growth. Values of the exponential factor B0 were fixed prior to the fitting based on the induction-time results described in section 3.3. Equation 11
represents the rate of change of the concentration of ampicillin in solution, which was related to the third moment of the population density by a mass balance. Estimation of parameters in the kinetic expressions was performed using the following objective function: 2158
DOI: 10.1021/acs.iecr.5b03880 Ind. Eng. Chem. Res. 2016, 55, 2153−2162
Article
Industrial & Engineering Chemistry Research Spyi =
dyi dp
(27)
P = (SpTSp)−1
(28)
V = sR2 P
(29)
Using the previous equation, confidence intervals can be calculated; in this case, a t-test evaluation for a 95% confidence was used to estimate the intervals. p = p* ± tα diag(V )
Table 4 shows the results of the parameter estimation routing along the respective absolute values of the confidence intervals for one experiment, unseeded experiments, and for all the experiments. As shown, using a single experiment for the calculation of confidence intervals gives broad ranges; in some cases the error is bigger even than the actual parameter, a situation without any physical meaning (i.e., negative nucleation rates). In contrast, as the number of experiments is increased, the confidence interval gets so narrow that it appears that the calculated set is very close to the true set. It is obvious that this analysis does not provide any proof that the obtained set possesses a global minimum. However, it certainly demonstrates that the multiexperiment parameter-estimation routine provides superior estimates to those of singleexperiment routines or, in other words: the greater is the number of experiments, the smaller is the range of probable error. In this case, six experiments were sufficient to constrain the number of possible solutions of the parameter estimation problem. 3.6. Approximations of Crystal Mean Size. Since the parameters in the crystallization kinetic expressions were extracted from concentration data, we recognize that the model may predict desupersaturation but not the corresponding crystal size distribution. This concern was tested by examining the final number-based average chord length of each run; that is,
Figure 10. Light microscopy picture of ampicillin seeds (scale = 100 μm).
Table 3. Properties of Ampicillin Seed Crystals L̅seed (μm)
ρc (kg/m3)
ka
kv
74
1500
0.09
0.03
Nexp Nx
min p E =
2 ∑ ∑ ([Amp]i(t ,p) − [Amp]iexp (t ) ) j=1 i=1
(25)
where [Amp]i(t,p) represents the modeled concentration at exp some set of parameters p = [kB1 kB2 b s kg g], [Amp]i(t) represents the set of concentrations determined experimentally, Nx is the number of data points, and Nexp the number of experiments (Nexp = 6) . Usually, estimation of crystallization kinetic parameters leads to multiple solutions (combinations of parameters) that provide equally good fits to the experimental data. The probability of having the parameter estimation trapped in a local minimum was reduced by performing two consecutive minimizations of eq 25. First, a stochastic minimization (MATLAB simulannealbnd) was used to identify the best set of parameters among a wide range. (This minimization is especially useful when there is no guidance on initial parameter estimates.) Then, parameters estimated from the stochastic minimization were used as the starting point of a deterministic minimization (MATLAB f minsearch), which refines the parameters. The utilization of a large data set, where multiple operational conditions were included, allows evaluation of a single set of parameters that have capabilities for prediction over a wide range of conditions. The results of the multiexperiment approach are shown in Figure 11, which illustrates that the resulting model is capable of providing reasonably good fits for all of the conditions tested. The robustness of the estimated parameters was tested by evaluating confidence intervals using a parameter perturbation method described by Nagy et al.10 and Miller et al.11 Additional information on the effect of parameter uncertainty and how to minimize the effects of that uncertainty were reported by Ma et al.12 and Nagy et al.13 In this case, let Sp represent a measurements matrix that contains the sensitivity matrices Syip , P is the precision matrix, and V is the covariance matrix.
Sp = [Sp1...SpN exp]T
(30)
n
L̅CLD =
∑ niLi /nTotal i=1
(31)
Since the mean crystal size is unlikely to be the same as the mean chord length, we used a simple empirical proportionality relationship between the two number-based means. A proportionality constant was determined by comparing the number-based mean size of the seed crystals with the numberbased mean chord length: k = L̅CSD,seed /L̅CLD,seed
(32)
After determining the constant k, it was then used to estimate the mean crystal size at the end of the run from the final mean of the CLD. L̅CSD,run = kL̅CLD,run
(33)
Finally, to assess the utility of the model in estimating CSD properties, the mean of the CSD at the end of each run was estimated from the moments of the population. Model L̅CSD,run = μ1,end /μ0,end
(34)
Table 5 shows comparisons of the final CSD estimated from eq 33 (i.e., experimental results from chord-length measurements)
(26) 2159
DOI: 10.1021/acs.iecr.5b03880 Ind. Eng. Chem. Res. 2016, 55, 2153−2162
Article
Industrial & Engineering Chemistry Research
Figure 11. Fitting of seeded and unseeded crystallization experiments.
with values obtained from the moments of the distribution (eq 34). The third column was obtained using parameters estimated as described above, while the fourth column resulted from the model using parameters obtained from fitting data from only one run (run d). When all the data were used, the modeled mean size is relatively close to that observed, but this was not the case using the data from a single run. Values
obtained using runs other than run d resulted in higher deviations from the experimental values; in some of these, the estimation for some runs was not possible because the integration using parameters obtained from a single run was not accurate. The results from the above analysis support our hypothesis that the multiexperiment methodology employed in this work 2160
DOI: 10.1021/acs.iecr.5b03880 Ind. Eng. Chem. Res. 2016, 55, 2153−2162
Article
Industrial & Engineering Chemistry Research Table 4. Parameters Obtained from Multiexperiment Parameter Fitting Routinea runs
kB1 (#/(g/kg) min)
B0
× × × ×
1.27
p ±(run d) ±(runs d−f) ±(all runs) a
5.00 9.24 3.95 1.77
1010 1010 1010 109
kB2 (#/(g/kg) min)
b
s
× × × ×
0.60 70.1 0.12 0.03
1.37 635.4 0.10 0.04
2.20 3.46 3.18 5.69
109 1011 108 107
kG (m/min) 8.95 1.33 3.96 1.57
× × × ×
10−6 10−5 10−6 10−7
g 1.87 1.83 0.07 0.02
First row has the parameters p, while second, third, and fourth rows are respective confidence intervals for the indicated runs.
■
Table 5. Final Mean Crystal Size Obtained from ChordLength Measurements (eq 33) and Model Results for Final Population (eq 34) run
experiment (μm)
model all (μm)
model run d (μm)
a b c d f
71.15 74.00 80.40 64.03 81.83
86.74 82.86 66.24 63.77 68.28
49.30 46.09 45.04 76.61 45.55
can be used to obtain a set of parameters from concentration data that provide relatively good estimates of the average crystal size. Moreover, they support the simple approximation embodied in eq 32 to obtain mean crystal size from the CLD mean. The utility of the approximation is especially apparent when working with crystals that have small aspect ratios, as in the case of needles, where CLD data is best used as a qualitative indicator of the direction of the process.14 In a more robust approach, the characteristic chord length at each crystal size can be obtained and used to estimate crystal size distributions throughout the process.15
4. CONCLUSIONS Crystallization kinetics of ampicillin were determined from a large set of experimental data obtained from experiments using online monitoring of solute concentration and the chord-length distribution of crystals formed in the process. Data from all the experiments were combined to determine crystallization kinetics using stochastic and deterministic minimizations to evaluate model parameters. The data span different rates of change of pH, initial supersaturations, and masses of seed crystals, and therefore are expected to provide useful estimates of model parameters and, as shown, much better estimates than would be the case if the parameters resulted from fitting data a single set of conditions. An analysis of confidence intervals using data from different numbers of experiments confirmed that the confidence interval gets narrower as the number of experiments is increased. Additionally, the utilization of a wide range of conditions allowed us to obtain parameters that not only predict variations of solute concentration with time, but also provide good relative estimates of the average crystal size.
■
NOMENCLATURE Aij = Pitzer’s model adjustable parameter (kg/mol)1/2 b = secondary nucleation mass of crystals exponent Bij = Pitzer’s model adjustable parameter (kg/mol)1/2 B0 = primary nucleation constant g = growth rate exponent G = crystal growth rate (m/s) J = nucleation rate (#/(g/kg)) ka = area shape factor kBi= primary/secondary nucleation rate constant (#/(g/kg) min) kB = Boltzmann’s constant (m2 kg s−2 K−1) kG = growth rate constant (m/min) kv = volume shape factor l = crystal length (m) L̅ = average crytal length (μm) M = mass of crystals (g) n = number density function (#/m·g/kg) ni = number of crystals of length i approximated through FBRM Ni = number of crystals of length i estimated from sieving NA = Avogadro’s number (mol−1) p = set of parameters P = precision matrix s = secondary nucleation supersaturation exponent Si = sensitivity matrix t = time (min) V = covariance matrix xi = mole fraction (g/kg H2O·min)
Greek Letters
■
ε = van der Waals attractive interaction parameter (m2 kg s−1) λij = Pitzer’s binary interaction parameter (kg mol−1) ρc = crystal density (kg/m3) σ = van der Waals diameter (m) μi = ith moment of number density function
REFERENCES
(1) (a) Deaguero, A. L.; Blum, J. K.; Bommarius, A. S., Biocatalytic synthesis of beta-lactam antibiotics. In The Encyclopedia of Industrial Biotechnology: Bioprocess, Bioseparation, and Cell Technology, 2nd ed.; Flickinger, M. C., Ed.; Wiley: 2010; pp 535−567. (b) Wegman, M. A.; Janssen, M. H. A.; van Rantwijk, F.; Sheldon, R. A. Towards biocatalytic synthesis of beta-lactam antibiotics. Adv. Synth. Catal. 2001, 343 (6−7), 559−576. (2) (a) Ottens, M.; Lebreton, B.; Zomerdijk, M.; Rijkers, M. P. W. M.; Bruinsma, O. S. L.; van der Wielen, L. A. M. Impurity effects on the crystallization kinetics of ampicillin. Ind. Eng. Chem. Res. 2004, 43 (24), 7932−7938. (b) Ottens, M.; Lebreton, B.; Zomerdijk, M.; Rijkers, M. P. W. M.; Bruinsma, O. S. L.; van der Wielen, L. A. M. Crystallization kinetics of ampicillin. Ind. Eng. Chem. Res. 2001, 40 (22), 4821−4827. (3) (a) Fujiwara, M.; Nagy, Z. K.; Chew, J. W.; Braatz, R. D. Firstprinciples and direct design approaches for the control of pharmaceutical crystallization. J. Process Control 2005, 15 (5), 493−
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS Financial support from the Cecil J. “Pete” Silas Endowment and the Georgia Research Alliance is gratefully acknowledged. 2161
DOI: 10.1021/acs.iecr.5b03880 Ind. Eng. Chem. Res. 2016, 55, 2153−2162
Article
Industrial & Engineering Chemistry Research 504. (b) Nagy, Z. K.; Fevotte, G.; Kramer, H.; Simon, L. L. Recent advances in the monitoring, modelling and control of crystallization systems. Chem. Eng. Res. Des. 2013, 91 (10), 1903−1922. (4) Shi, B.; Frederick, W. J.; Rousseau, R. W. Effects of calcium and other ionic impurities on the primary nucleation of burkeite. Ind. Eng. Chem. Res. 2003, 42 (12), 2861−2869. (5) Pessoa, P. D.; Maurer, G. An extension of the Pitzer equation for the excess Gibbs energy of aqueous electrolyte systems to aqueous polyelectrolyte solutions. Fluid Phase Equilib. 2008, 269 (1−2), 25−35. (6) Franco, L. F. M.; Mattedi, S.; Pessoa, P. D. A new approach for the thermodynamic modeling of the solubility of amino acids and betalactam compounds as a function of pH. Fluid Phase Equilib. 2013, 354, 38−46. (7) (a) Rudolph, E. S. J.; Zomerdijk, M.; Ottens, M.; van der Wielen, L. A. M. Solubilities and partition coefficients of semi-synthetic antibiotics in water+1-butanol systems. Ind. Eng. Chem. Res. 2001, 40 (1), 398−406. (b) Santana, M.; Ribeiro, M. P. A.; Leite, G. A.; Giordano, R. L. C.; Giordano, R. C.; Mattedi, S. Solid-Liquid Equilibrium of Substrates and Products of the Enzymatic Synthesis of Ampicillin. AIChE J. 2010, 56 (6), 1578−1583. (8) Sohnel, O.; Mullin, J. W. Interpretation of Crystallization Induction Periods. J. Colloid Interface Sci. 1988, 123 (1), 43−50. (9) (a) Tseng, Y. T.; Ward, J. D. Critical seed loading from nucleation kinetics. AIChE J. 2014, 60 (5), 1645−1653. (b) Doki, N.; Kubota, N.; Yokota, M.; Chianese, A. Determination of critical seed loading ratio for the production of crystals of uni-modal size distribution in batch cooling crystallization of potassium alum. J. Chem. Eng. Jpn. 2002, 35 (7), 670−676. (10) Nagy, Z. K. A population balance model approach for crystallization product engineering via distribution shaping control. Comput.-Aided Chem. Eng. 2008, 25, 139−144. (11) Miller, S. M.; Rawlings, J. B. Model Identification and Control Strategies for Batch Cooling Crystallizers. AIChE J. 1994, 40 (8), 1312−1327. (12) Ma, D. L.; Chung, S. H.; Braatz, R. D. Worst-case performance analysis of optimal batch control trajectories. AIChE J. 1999, 45 (7), 1469−1476. (13) Nagy, Z. K.; Braatz, R. D. Open-loop and closed-loop robust optimal control of batch processes using distributional and worst-case analysis. J. Process Control 2004, 14 (4), 411−422. (14) Leyssens, T.; Baudry, C.; Hernandez, M. L. E. Optimization of a Crystallization by Online FBRM Analysis of Needle-Shaped Crystals. Org. Process Res. Dev. 2011, 15 (2), 413−426. (15) Li, H. Y.; Kawajiri, Y.; Grover, M. A.; Rousseau, R. W. Application of an Empirical FBRM Model to Estimate Crystal Size Distributions in Batch Crystallization. Cryst. Growth Des. 2014, 14 (2), 607−616.
■
NOTE ADDED AFTER ASAP PUBLICATION This paper was originally published ASAP on February 15, 2016. Several errors in Figure 1 were corrected, and the revised version was reposted on February 16, 2016.
2162
DOI: 10.1021/acs.iecr.5b03880 Ind. Eng. Chem. Res. 2016, 55, 2153−2162
