Crystallization as a Separations Process - American Chemical Society

Chapter 8

Light-Scattering Measurements To Estimate Kinetic Parameters of Crystallization 1

W. R. Witkowski, S. M. Miller, and J. B. Rawlings

Downloaded by UNIV OF GUELPH on January 22, 2018 | http://pubs.acs.org Publication Date: September 21, 1990 | doi: 10.1021/bk-1990-0438.ch008

Department of Chemical Engineering, The University of Texas at Austin, Austin, TX 78712 This paper presents further development of the ideas presented by Witkowski and Rawlings (1) for improving kinetic parameter estimation of batch crys tallization processes using nonlinear optimization techniques and particle size distribution measurements. As shown by Witkowski and Rawlings (1), desu persaturation information from a seeded, batch crystallization contains only enough information to determine the growth parameters reliably. It is shown that the addition of the obscuration, a light scattering measurement, along with the solute concentration data, allows accurate identification of all of the kinetic parameters. The obscuration is related to the second moment of the crystal size distribution (CSD). A Malvern 3600Ec Particle Sizer is used to obtain the obscuration measurement. Through numerical and experimental studies it is verified that both growth and nucleation parameters are identifiable using these measurements. It is also shown that accurate knowledge of the shape of the crystals is essential when analyzing CSD information from light scattering measurements.

Batch crystallizers are often used in situations in which production quantities are small or special handling of the chemicals is required. In the manufacture of speciality chemicals, for example, it is economically beneficial to perform the crystallization stage in some optimal manner. In order to design an optimal control strategy to maximize crystallizer performance, a dynamic model that can accurately simulate crystallizer behavior is required. Unfortunately, the precise details of crystallization growth and nucleation rates are unknown. This lack of fundamental knowledge suggests that a reliable method of model identification is needed. Identification of a process involves formulating a mathematical model which prop erly describes the characteristics of the real system. Initial model forms are developed fromfirstprinciples and a priori knowledge of the system. Model parameters are typ ically estimated in accordance with experimental observations. The method in which these parameters are evaluated is critical in judging the reliability and accuracy of the model. The crystallization literature is replete with theoretical developments of batch crys tallizer models and techniques to estimate their parameters. However, most of the schemes are constrained to specific crystallizer configurations and model formulations. 1

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Myerson and Toyokura; Crystallization as a Separations Process ACS Symposium Series; American Chemical Society: Washington, DC, 1990.


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WITKOWSKI ETAL.

Kinetic Parameter Estimation of Crystallization

Therefore, aflexiblemethod to evaluate physical and chemical system parameters is still needed (2, 3). The model identification technique presented in this study allows flexibility in model formulation and inclusion of the available experimental measure ments to identify the model. The parameter estimation schemefindsthe optimal set of parameters by minimizing the sum of the differences between model predictions and experimental observations. Since some experimental data are more reliable than others, it is advantageous to assign higher weights to the dependable data. A lack of measurable process information is the primary problem to be overcome in developing a reliable model identification and verification scheme. A measure of the solute concentration is commonly available. To provide another process measurement, a Malvern 3600Ec Particle Sizer, which is based on Fraunhofer light scattering theory, is used to determine the CSD. A primary drawback of this CSD measurement tech nique is that the inversion of the scattered light data is an ill-posed problem. Also, Fraunhofer diffraction theory is accurate only for a dilute collection of large spheri cal particles. Through the typical operating region of a seeded, batch crystallization experiment performed in this work, the particle concentration stays well within the manufacturer's specifications. However, this is not common for all crystallizers where the particle concentration can change from essentially no particles to a very dense slurry. These high particulate concentrations increase the probability of multiple scat tering which would corrupt the CSD approximation (4).

Dynamic Batch Crystallization Model The dynamic model used in predicting the transient behavior of isothermal batch crystallizers is well developed. Randolph and Larson (5) and Hulburt and Katz (6) offer a complete discussion of the theoretical development of the population balance approach. A summary of the set of equations used in this analysis is given below.

^

at

M^)

+ G

=

oL

0

(1

2

^ = -3pk G f°° n(L, t)L dL at Jo v

G = k (c-c.Y

(2) (3)

g

h

B = k (c-c ) h

)

(4)

s

in which n(L,t) is the CSD at time t. The population balance, Equation 1, describes the CSD dynamics. The solute concentration, c, on a per mass of solvent basis, is described by Equation 2. Constitutive relationships modelling particle growth rate, G, and new formation (nucleation) rate, B, are given by Equations 3 and 4, respectively. These processes are typically formulated as power law functions of supersaturation, c- c . The description is completed with the statement of the necessary initial and bound ary conditions, 5

n(L,t = 0) = n (X) n(X = 0,0 = B/G c(t = 0) = c 0

0


(5) (6) (7)

103

104

CRYSTALLIZATION AS A SEPARATIONS PROCESS

One of the most popular numerical methods for this class of problems is the method of weighted residuals (MWR) (7,8). For a complete discussion of these schemes several good numerical analysis texts are available (9,10,11). Orthogonal collocation on finite elements was used in this work to solve the model as detailed by Witkowski (12).


Parameter Estimation The parameter estimation approach is important in judging the reliability and accuracy of the model. If the confidence intervals for a set of estimated parameters are given and their magnitude is equal to that of the parameters, the reliability one would place in the model's prediction would be low. However, if the parameters are identified with high precision (i.e., small confidence intervals) one would tend to trust the model's predictions. The nonlinear optimization approach to parameter estimation allows the confidence interval for the estimated parameter to be approximated. It is thereby possible to evaluate if a parameter is identifiable from a particular set of measurements and with how much reliability. Several investigators have offered various techniques for estimating crystallization growth and nucleation parameters. Parameters such as g,k ,b, andfc&are the ones usually estimated. Often different results are presented for identical systems. These discrepancies are discussed by several authors (13, 14). One weakness of most of these schemes is that the validity of the parameter estimates, i.e., the confidence in the estimates, is not assessed. This section discusses two of the more popular routines to evaluate kinetic parameters and introduces a method that attempts to improve the parameter inference and provide a measure of the reliability of the estimates. A survey of crystallization literature shows that the most popular technique to estimate kinetic parameter information involves the steady-state operation of a mixed suspension, mixed product removal (MSMPR) crystallizer (15, 5). The steady-state CSD is assumed to produce a straight line when plotted as log n vs. L. The slope and intercept of this line determine the steady-state G and B values. The major disadvantages of this method are that the experiments can be time consuming and expensive in achieving steady state, especially when studying specialty chemicals, and that valuable dynamic information is ignored. Also, maintaining a true steady state is often difficult in practice. The simplicity of the method is the primary advantage. An alternative scheme, proposed by Garside et al. (16,17), uses the dynamic desu persaturation data from a batch crystallization experiment. After formulating a solute mass balance, where mass deposition due to nucleation was negligible, expressions are derived to calculate g and k in Equation 3 explicitly. Estimates of the first and sec ond derivatives of the transient desupersaturation curve at time zero are required. The disadvantages of this scheme are that numerical differentiation of experimental data is quite inaccurate due to measurement noise, the nucleation parameters are not esti mated, and the analysis is invalid if nucleation rates are significant. Other drawbacks of both methods are that they are limited to specific model formulations, i.e., growth and nucleation rate forms and crystallizer configurations. In the proposed technique, the user is free to pick the model formulation to suit his needs. The optimal set of parameters are found by solving the nonlinear optimization g

g


8.

105 Kinetic Parameter Estimation of CrystaUization

WITKOWSKI ETAL.

problem mm

*(y,y;$)

subject to Equations 1-4

(8)

T

in which 0 = \g,k b,k ]. The objective function, is typically formulated as the summed squared error between experimental measurements, y, and model predictions, y. The relationship should properly describe the experimental error present and best utilize available ex perimental data. In the common least squares estimation, the measurement error is assumed to be normally distributed and $ takes the form Downloaded by UNIV OF GUELPH on January 22, 2018 | http://pubs.acs.org Publication Date: September 21, 1990 | doi: 10.1021/bk-1990-0438.ch008

gj

g

n m

(9) where n represents the number of experimental time records and m is the number of independent measurements at each sample time. The weighting factors, LJJ, are used to scale variables of different magnitudes and to incorporate known information about measurement uncertainty. The details of the calculations involved to estimate the confidence intervals are given in Bard (18) and Bates and Watts (19). Briefly, the confidence interval is estimated by looking at the curvature of the objective function at the optimal set of parameters. The crystallizer model was solved using orthogonal collocation as previously dis cussed to provide the model predictions during the nonlinear optimization problem. Caracotsios and Stewart's (20, 21) nonlinear parameter estimation package, which solves the nonlinear programming (NLP) problem using sequential quadratic program ming, was used in this work to estimate the parameters and compute the approximate uncertainties. There are several sources for discussions of NLP (22, 23), and there are several established codes available for the solution of this problem. In 1980 Schittkowski (24) published an extensive evaluation of 26 different NLP codes available at that time.

Experimental Studies In this section, a brief description of the necessary experiments to identify the kinetic parameters of a seeded naphthalene-toluene batch crystallization system is presented. Details about the experimental apparatus and procedure are given by Witkowski (12). Operating conditions are selected so that the supersaturation level is kept within the metastable region to prevent homogeneous nucleation. To enhance the probability of secondary nucleation, sieved naphthalene seed particles are introduced into the system at time zero. Available process measurements include the dissolved solute concentration and CSD information. To estimate the concentration, a PAAR vibrating U-tube densito meter is used to measure frequency as a function of thefluid'sdensity and temperature. This information is used to interpolate a concentration value from an experimentally constructed database. The Malvern Particle Sizer 3600Ec uses light scattering measurements to infer the CSD. The device has a He-Ne laser that illuminates the particles suspended in a stream that is moving continuously through aflowcell. The diffracted light is then focused


106


through a lens onto a solid-state photodiode detector. The CSD inversion problem is performed using Fraunhofer diffraction theory. The primary assumption of the Fraun hofer theory is that the scattering pattern is produced by spherical particles that are large with respect to the wavelength of the light. Manufacturer's specifications show that the instrument is capable of measuring particles with a diameter of approximately 0.5 to 564 fim. The Malvern also measures the fraction of light that is obscured by the crystals in the flow cell. The obscuration is defined as


obscuration = 1 — where J is the intensity of undiffracted light after passing through the suspension of crystals, IQ is the intensity of unobstructed light. Note that the obscuration is mea sured directly and does not suffer from the ill-posed nature of the CSD measurement. The obscuration can be predicted by the model for comparison with experimental data using the following relations: (10) (11) where r is the turbidity, / is theflowcell width, A(L) and Q(L) are the projected crosssectional area and extinction efficiency factor, respectively, of crystals of characteristic length L. Fraunhofer diffraction theory gives Q = 2 for all L (25). From Equations 10 and 11 it can be seen that the obscuration provides a measure of the second moment of the CSD.

Model Identification Results Numerical Analysis. It is difficult to determine which measurements contain sufficient information to allow the independent determination of all model parameters. This issue can be studied by assessing the impact of the use of various measurements on the parameter estimation problem using pseudo-experimental data. Pseudo-experimental data can be generated by solving the model, Equations 1-4, for a chosen set of parameters and initial conditions, and then adding random noise to the model solution. For a given choice of measurement variables, the simulated data is then used in the parameter estimation problem. This procedure provides a means by which to evaluate the measurements that are required and the amount of measurement noise that is tolerable for parameter identification. First, for the case in which only solute concentration measurements are available, pseudo-experimental data are simulated and used in the parameter estimation scheme. Even with noise-free data, the recovered parameters differ greatly from the true pa rameters and the uncertainties of the nucleation parameters are large. This indicates that there exists a large set of quite different b and k\> pairs that would lead to very similar solute concentration profiles. The insensitivity of the objective function to the nucleation parameters can be attributed to the fact that the mass of a nucleated par ticle is almost negligible and the change in the solute concentration is primarily due to seed growth. The Malvern uses light scattering measurements to determine the weight percent of "spherical equivalent" crystals in each of 16 size classes and the mean size diameter.


8.

WITKOWSKI ETAL.



0.540

0

10

20

30

40

50

Time

Figure 1: Simulated concentration data. Using the analysis technique described above, it was determined that while the addition of the weight percent information narrowed the parameter confidence intervals, this additional measurement does not allow reliable estimation of all kinetic parameters. Using the concentration and obscuration measurements allow all of the kinetic parameters of interest to be identified. The simulated data is shown in Figures 1 - 2. The parameter estimation results corresponding to these measurements are given in Table 1. These results indicate that these measurements may provide enough process information to allow identification, even in the presence of measurement noise. This hypothesis is investigated experimentally in the next section.

Table 1: Parameter estimation results utilizing simulated process measurements

9 Mean Parameter Estimate 2-Sigma Confidence Interval True Parameter

Kinetic Parameters b In kt In kg

1.62

9.78

0.83

13.13

±0.015

±0.091

±0.50

±1.67

1.61

9.74

1.01

14.06


107


108


Figure 2: Simulated obscuration data. Experimental Analysis. The most reliable process measurement is the oscillator fre quency from the PAAR densitometer. Along with the frequency, the temperature is also measured (± 0.05 °C). These two states are used to interpolate the solute concen tration. CSD weight percent information and obscuration measurements were obtained from the Malvern Particle Sizer. Approximately 500 concentration data points and 200 CSD and obscuration measurements were recorded during a run of about 80 100 minutes. Therefore, the dynamics of the system were well monitored, i.e., the time constant of the crystallizer is much larger than the sampling time. We have per formed 25 experimental runs. This section summarizes the analysis of a single, typical experiment. The experimental operating conditions for the case study on which the parameter analysis was performed are given in Table 2. In most previous studies, the kinetic parameters of interest (g,k ,b, and &&) have been estimated using only desupersatu ration data or steady-state CSD information. To judge the parameter reliability using only the desupersaturation data, the nonlinear optimization routine was run using only the solute concentration information. Table 3 shows the resulting parameter es timates and confidence intervals. The predicted concentration profile is shown by the solid line in Figure 3, which fit the experimental data quite well. The growth rate parameters are evaluated easily and with reasonable confidence intervals. However, as predicted from the pseudo-experimental data, calculating the nucleation parameters with certainty using only the concentration information is not possible. Even though mean parameter values were calculated for b and fo, no confidence is placed in their values; the large confidence intervals indicate that the objective function surface was almost completely insensitive to these particular parameters. The mean parameter values are actually very dependent on their initial guesses in the optimization, again indicating the objective function insensitivity to these parameters. To investigate whether the jagged character of the data would cause an unnecessary g


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WITKOWSKI ETAL.

Kinetic Parameter Estimation of Crystallization Table 2: Case study operating conditions Value 24.0°C 0.5386 g Naph./g Tol. 0.5325 g Naph./g Tol. 180 - 212 /zm 1.3 g 500 rpm


Variable Temperature, T co c. Seed size (sieve tray range) Seed amount Stirrer speed

Table 3: Parameter estimation results utilizing only concentration data

9 Mean Parameter Estimate 2-Sigma Confidence Interval

Kinetic Parameters b In k In kb g

1.23

8.43

2.89

12.88

±0.028

±0.16

±5.02

±35.88

0.540

Time

Figure 3: Predicted concentration profile using concentration data only.


110


Table 4: Parameter estimation results utilizing both concentration and obscuration data

9 Mean Parameter Estimate

1.66


2- Sigma Confidence Interval ±0.011

Kinetic Parameters In kb b In kg 10.04

1.23

14.74

±0.063

±0.040

±0.21

burden on the optimization routine or be a source of uncertainty, the concentration measurements were smoothed with afirstorderfilterand the curve's jaggedness was removed. The re-evaluated parameters were the same as those using the raw data, indicating that the relative noise level in the concentration data was low and data smoothing offers no advantages. As shown in the previous section, the use of concentration and obscuration mea surements should contain enough information to identify all of the kinetic parameters of interest. Table 4 displays the parameters with corresponding confidence intervals determined using experimental concentration and obscuration data. Since the noiseto-signal ratio was much larger for the light scattering data than the concentration data, the latter was weighted more heavily (100:1) in the objective function. Details concerning the weight factor assignments are given by Witkowski (12). The model's fit to the experimental solute concentration data is shown in Figure 4. Thefitto the obscuration is also very good as displayed in Figure 5. While the CSD weight percent information was not used in the parameter es timation, Figures 6 and 7 demonstrate that the particle shape is important when comparing model predictions of CSD to those determined from light scattering meth ods. The circular points represent the weight percent information obtained from the light scattering device. The two curves represent the theoretical weight percents for crystals with two different shape factors. It appears that the results for spherical particles fit the experimental data better than for cubic particles. This is probably because the light scattering instrument determines the CSD by assuming spherical particles. It is clear that the "spherical equivalent" weight percent information from the light scattering instrument must be properly treated if it is used in the parameter estimation. Fortunately, the particle shape has little effect in the resolved parameter values when using solute concentration and obscuration measurements to identify the model. The parameters shown in Table 5 are estimated assuming spherical crystals. Compar ing these values to those for cubic particles (Table 4) shows little difference between the two sets of parameters.


8. WITKOWSKI ETAL.


0.540-

model experimental

0.538 -


I

0.536-

0.534 -

0.532 -

I 20

'

I

1

40

Time

I 60

1

I 80

^

100

Figure 4: Predicted concentration profile using concentration and obscuration data.


111


100-

o

80-

experimental cubic particles spherical particles

60-


i

40-

20-

"T" 20

n

1

40

r 60

80

100

Time

Figure 6: Crystal weight percent for the 160-262 y,m size class.

o

experimental cubic particles spherical particles

~T~ 80

100

Figure 7: Crystal weight percent for the 262-564 fim size class.


8. WITKOWSKI ETAL.


Table 5: Parameter estimation results utilizing both concentration and obscuration data assuming spherical particles

9 Mean Parameter Estimate


2-Sigma Confidence Interval

Kinetic Parameters In kb b In kg

1.63

10.46

1.37

14.08

±0.010

±0.058

±0.028

±0.15

Conclusions This study demonstrates that the nonlinear optimization approach to parameter es timation is a flexible and effective method. Although computationally intensive, this method lends itself to a wide variety of process model formulations and can provide an assessment of the uncertainty of the parameter estimates. Other factors, such as measurement error distributions and instrumentation reliability can also be integrated into the estimation procedure if they are known. The methods presented in the crys tallization literature do not have thisflexibilityin model formulation and typically do not address the parameter reliability issue. It is shown that while solute concentration data can be used to estimate the ki netic growth parameters, information about the CSD is necessary to evaluate the nucleation parameters. The fraction of light obscured by an illuminated sample of crystals provides a measure of the second moment of the CSD. Numerical and experi mental studies demonstrate that all of the kinetic parameters can be identified by using the obscuration measurement along with the concentration measurement. It is also shown that characterization of the crystal shape is very important when evaluating CSD information from light scattering instruments. Acknowledgments The authors would like to thank the Eastman Kodak Company for financial support of this work. The first author also gratefully acknowledges Shell Oil Company for a Shell Foundation Fellowship. Literature Cited 1. Witkowski, W. R. and Rawlings, J. B. Kinetic Parameter Estimation of Naphthalene-Toluene Crystallization. National AIChE Meeting, Houston, TX, 1989. 2. Halfon, A. and Kaliaguine, S. Can. J. Chem. Eng., 1976, 54:160-167. 3. Tavare, N. S. and Garside, J. Chem. Eng. Res. Des., 1986, 64:109-118. 4. Weiner, B. B. Particle and Droplet Sizing Using Fraunhofer Diffraction. In


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5.


6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

25.


Barth, H. G., editor, Modern Methods of Particle Size Analysis, pages 135-172. John Wiley&Sons, 1984. Randolph, A. D. and Larson, M. A. Theory of Particulate Processes. Academic Press, New York, 1988. Hulburt, H. and Katz, S. Chem. Eng. Sci., 1964, 19:555-574. Singh, P. N. and Ramkrishna, D. Comput. Chem. Eng., 1977, 1:23-31. Subramanian, G. and Ramkrishna, D. Math. Bio., 1971, 10:1-23. Becker, E. B.; Carey, G. F., and Oden, J. T. Finite Elements: An Introduction, Volume I. Prentice-Hall, Englewood Cliffs, NJ, 1981. Reddy, J. N. Applied Functional Analysis and Variational Methods in Engineer ing. McGraw-Hill, New York, 1986. Villadsen, J. and Michelson, M. L. Solution of Differential Equation Models by Polynomial Approximation. Prentice-Hall, 1978. Witkowski, W. R. PhD thesis, Univeristy of Texas at Austin, 1990. Garside, J. and Shah, M. B. Ind. Eng. Chem. Proc. Des. Dev., 1980, 19:509-514. Tavare, N. Chem. Eng. Commun., 1987, 61:259-318. Chen, M. R. and Larson, M. A. Chem. Eng.Sci.,1985, 40(7):1287-1294. Garside, J.; Gibilaro, L., and Tavare, N. Chem. Eng.Sci.,1982, 37(11):1625-1628. Tavare, N. AIChE J., 1985, 31(10):1733-1735. Bard, Y. Nonlinear Parameter Estimation. Academic Press, New York, 1974. Bates, D. M. and Watts, D. G. Nonlinear Regression Analysis and Its Applica tions. John Wiley&Sons, 1988. Caracotsios, M. and Stewart, W. E. Comput. Chem. Eng., 1985, 9(4):359-365. Caracotsios, M. PhD thesis, University of Wisconsin-Madison, 1986. Gill, P. E.; Murray, W., and Wright, M. H. Practical Optimization. Academic Press, 1981. Edgar, T. F. and Himmelblau, D. M. Optimization of Chemical Processes. McGraw-Hill, 1988. Schittkowski, K. Nonlinear Programming Codes: Information, Tests, Perfor mance. In Lecture Notes in Economic and Mathematical Systems. vol. 183, Springer-Verlag, 1980. van de Hulst, H. C. Light Scattering by Small Particles. Dover, 1981.

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Crystallization as a Separations Process - American Chemical Society

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