Model Identification and Control of Solution ... - ACS Publications

Ind. Eng. Chem.

Res. 1993, 32, 1275-1296

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Model Identification and Control of Solution Crystallization Processes: A Review James B. Rawlings,* Stephen M.

Miller, and Walter

R. Witkowskit

Department of Chemical Engineering, The University of Texas at Austin, Austin, Texas 78712

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The modeling of crystal quality (size, shape, and purity) is discussed. Numerical methods for the model solution are reviewed including collocation and Galerkin’s method on finite elements. Measurement technologies for on-line crystal size distribution are presented. The difficulties with laser light scattering methods are discussed in detail. Parameter estimation is reviewed and new methods based on nonlinear optimization are presented. Sample calculations of parameters and their uncertainty from experimental data are included. The research literature covering control of continuous and batch crystallization is reviewed, and sample calculations of optimal cooling profiles for batch crystallizers are presented. 1.

Introduction

review article provides an extensive discussion of population balance modeling, applications, and solution meth-

The objective of this article is to review the area of model identification and control of crystallization processes. The discussion is focused on the following topics: model development, model solution, measurement techniques,

ods.

2.1.1. Stochastic versus Deterministic Formulation. One of the first decisions that has to be made in

modeling a particulate system is whether to use a stochastic or deterministic framework. This decision is usually based on whether one considers the particle population fluctuations to be important. If fluctuations are large and need to be modeled for accurate predictions of observables of interest, they can be described in a stochastic framework. Ramkrishna [134] and Fan and coworkers [53, 80, 81] present stochastic population balances. Ramkrishna is concerned with modeling small populations of cells, for example in sterilization processes. At small population levels, the fluctuations about the mean number of cells become of the same order of magnitude as the mean itself, necessitating the more complex stochastic machinery. Fan and co-workers, however, recommend using the stochastic approach even for large populations of crystals. They attempt to model crystal size distribution (CSD) data displaying deviations from the decaying exponential size distribution expected of well-mixed crystallizers. It is not clear why a stochastic approach is required since the data can be fit by other deterministic models that include sizedependent growth or growth rate dispersion. It also seems plausible that, given the difficulty of measuring CSD, fluctuations in observed CSD are likely due to measurement noise rather than true crystal population fluctuations. In this case it would be reasonable to model the crystallizer with a deterministic population balance and simply append additive stochastic variables to account for the measurement noise. One could then design filters and regulators for CSD control that account for the expected CSD measurement fluctuations.

parameter estimation, and control. The final section draws conclusions and suggests possible future directions. In this review we attempt to assess the current state of understanding of this field. We identify the issues that are well understood and those that are not. Although effort has been expended to be reasonably thorough in the references, omissions undoubtedly have occurred. These are not intentional, and the authors would appreciate having them brought to their attention. 2.

Model Development

2.1. Population Balance. The objective of crystallization modeling is to provide a description of the population of the crystals dispersed in the continuous phase. The mathematical framework that has been

developed to describe general dispersed-phase systems is known as population balance modeling. The concept of a population balance arose as a generalization of the residence time distribution. In early work, Rudd [161] demonstrated that one cannot always formulate a model based on particle age, and it is necessary to consider a generalized describing variable (catalyst particle activity in his example). Other contributions during this time were made by Randolph [136,137,141] Curl [38], and Behnken et al. [8], These early efforts were followed by the classic paper by Hulburt and Katz [82] and book by Randolph and Larson [142]. These authors extended the concept of particle size distribution to include an arbitrary number of “internal” coordinates necessary to fully describe the state of a particle. Tsuchiya et al. [181], Eakman et al. [48], and Ramkrishna [132-135] apply this methodology to microbial populations. Hidy and Brock [77], Friedlander [56], and Hidy [ 76] review population balance methods in aerosol science applications. Rawlings and Ray [146-149] review the contributions in emulsion polymerization that employ population balance models, and describe the advantages of this description over other models. Ramkrishna’s [135]

Therefore in this article we focus exclusively on deterministic population balances. The key property of the particulate system is some general number distribution function. Let /(z,t) denote the crystal distribution function, which depends on the time, t, and other suitably chosen coordinates, z. The z coordinates include the usual three spatial (external) coordinates as well as the additional internal coordinates which are required to completely specify the state of the crystal. Let dV* denote an infinitesimal volume in the crystal phase space. The total number of crystals per volume of crystallizer in some

To whom correspondence should be addressed. Phone: (512) 471-4417; email: [email protected]. f Current address: Sandia National Laboratories, Division 1434, Albuquerque, NM 87185-5800. 0888-5885/93/2632-1275$04.00/0

©

1993 American Chemical Society

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subregion of crystal phase space,

,

7, 1993

speed films is vital for achieving their superior photographic properties. The aspect ratio of the particles in

is

J>. OdV, One derives a population balance by accounting for changes in the crystal number distribution due to changes in the coordinates z and any crystal formation and removal mechanisms. The mathematical arguments are analogous to those used in the development of conservation equations in transport phenomena [106]. The resulting population balance is

|+V.(|/)

=

B-D

(1)

where B and D are the density functions of formation of new crystals and removal of existing crystals, respectively. If one considers the ideally stirred reactor, which has no spatial dependence, it is convenient to split the z coordinates into the external, x, and internal, y, parts,

Z4y]

®

Since neither f, B, nor D have any x dependence, the microscopic population balance can be integrated to give —, d(fV) (dy ,3,

\

in which fk(y) is the crystal distribution function of the Ath stream, Qk is the volumetric flow rate of the Ath stream (positive sign denotes flow out), and V is the crystallizer slurry volume. With suitable choices of the Qk, eq 3 describes batch, semibatch, and continuous crystallizer operation. The crystallizer is clearly a distributed parameter system regardless of whether it is spatially distributed as well. To complete the description, one has

to choose the internal coordinates, y, specify the functional forms of dy/dt, fk, B, and D, and supply the necessary initial and boundary conditions. Specifying these functions and conditions requires a detailed understanding of the transport and kinetic events taking place, and is the major hurdle to overcome in modeling any particulate system. 2.1.2. Crystal Quality: Size, Shape, and Purity. Equation 3 is a general starting point, and has been used to address issues such as growth rate dispersion, coalescence and breakage of particles, and a variety of other mechanisms. The choice of internal coordinates in vector y is dictated by the intended use of the model. In industrial manufacture of crystals, the crystal quality is usually determined by the crystal size, shape (habit or morphology), and purity (composition). The vast majority of the engineering research literature is concerned with modeling the crystal size and size distribution. In this article we too shall review and formulate models intended to enable improved control of crystal size distribution. However, a brief review of the progress being made in crystal habit and purity modeling and control is worthwhile and points out several new opportunities for improved crystal quality control. The fact that crystal size is important in applications is rather obvious: crystals that are too small are not easily separated from the mother liquor and may cause handling and environmental problems due to dust formation. A poorly shaped CSD may promote agglomeration and caking and pose storage and processing problems. However, crystal morphology may play an equally important role in many applications. For example, precise control of the twinning grain morphology and size distribution in high-

powdered reinforcement influences the strength and toughness of composite materials [152]. If the final product is a suspension of crystals in a liquid, the fluid rheology is strongly affected by the particle morphology [111]. The relative growth rates of the different faces of a crystal determine its shape. The crystal habit is determined by the slowest growing faces. This phenomenon serves as the basis for attempts to control habit by developing tailor made additives to serve as habit modifiers [1, 2]. A tailor-made additive is similar in chemical structure to the crystallizing species, so it is incorporated into the crystal structure. The difference between the additive and the crystallizing species, however, inhibits the growth of the crystal face containing the additive. This can produce a dramatic effect on the crystal morphology. A comprehensive recent review of this area is provided by ref 40. Another recent development that may have a significant effect on industrial practice in the near future is the development of public domain software for morphology prediction [36,44,45]. The use of this software

for quantitative prediction of crystal morphology for certain ideal organic crystals is well established, and progress is being made for the more difficult ionic crystals as well [111]. Attempts have been made to calculate the effect of impurities and different solvent mixtures on final crystal habit [121]. This area has been reviewed by Myerson [119]. His conclusions are that although there are not as yet reliable methods to predict habit or select solvents and habit modifiers without extensive trial-and-error experimentation, recent progress may make habit and purity control a realistic goal for the near future. 2.1.3. CSD Modeling. Therefore, although states of a crystal such as purity and shape can be included as elements of y, we restrict ourselves to modeling the crystal size distribution (CSD), in which case the vector of internal coordinates is simply L, a characteristic crystal length. If one additionally makes the following assumptions: no crystal breakage or coalescence, and crystals are nucleated at negligibly small size, Lq, then eq 3 reduces to d(fV)

d(Gf)

~dT+V~dL=

—

VB°8iL

"

Lo>

"

W

in which G is the crystal growth rate, B° is the nucleation rate density, and 5(L- Lq) is the Dirac delta function acting at L = Lq. The functions fk(L) can be defined to include washout from a CSTR (mixed product removal) and sizedependent removal (classified product removal and fines destruction). If Lq 0, it can be shown that eq 4 is equivalent to -*

with the boundary condition

/(0,t)

(6)

=

G\lm0

2.1.4. Mass and Energy Balances. As illustrated by the constitutive relations for G and B° discussed in section 2.2, growth and nucleation rates are functions of supersaturation (a measure of the difference between concentration and saturation concentration) and temperature. Therefore, mass and energy balances are required to complete the description of the crystallization process.

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The solute mass balance in the continuous phase must account for flow of solute into and out of the system and mass transfer to the solid phase via formation and growth of crystals. The mass balance can be written as d(eCV)

^—,

—---^kQkCk-3Pckvy¡0fL2GdL

(7)

where C is the solute concentration in the continuous phase, and pc and kv are the crystal density and volumetric shape factor, respectively [114], The fraction of Vthat is solidsfree (continuous phase) is represented by 0, j

V2

mTBm

(20)

1,2,..., p

=

ETE + · , and c -2ETet. where B A comparison of Figures 4 and 5 demonstrates the benefit derived from the constrained, penalized inversion algorithm given by eq 20. The distribution is inferred with greater success using this inversion algorithm then when using simple least squares, even though the data contain 5% random noise while the data used in the least squares inversion contains no added noise. While Figure 5 illustrates that the constrained, penalized least squares approach can be effective, the choice of y is important. The tradeoff is that y must be large enough to avoid instabilities, but not so large that the solution is strongly biased. The approach used by many researchers is a trialand-error procedure for determining y [34], but other methods are available. Crump and Seinfeld [37], suggest a cross-validation method for determining an appropriate value for y. Shaw [166] demonstrates that y can be treated as a parameter of a low-pass filter that removes the oscillatory modes of the solution; the eigenvalues of EFE can then be used as a guide for the selection of y. Boxman [28] demonstrates the use of a weighted least squares technique to solve the inverse problem. The weighting matrix is the inverse of the covariance matrix of the light scattering measurements; it is obtained from observations of fluctuations of light intensity over time. This weighting and a nonnegativity constraint help to stabilize the solution. An iterative approach is used to minimize a measure of the goodness of fit to the data. Boxman also demonstrates the use of a statistical analysis =

1,....

=

=

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According to the Beer-Lambert law, I/IQ

=

exp (~rl)

(21)

(22) Jq7(L) Ap(L) Q(L) dL where is the slurry turbidity, l is the flow cell width, AP(L) is the projected area for a crystal of characteristic size L, and Q(L) is the extinction efficiency factor. Assuming Fraunhofer diffraction is the dominant scattering phenomena, Q 2 for all L. An extension to cases of nonspherical particles can be made by applying the theorem discussed above relating geometrical cross section to surface area, transforming eq 22 to the following: =

—

6. Malvern results for monodispersion of cylinders with 45-µ diameter and 195-µ length.

Figure

=

to assess the reliability of a solution and to guide the way in which the size domain is discretized. A further complication of inferring the CSD from scattered light intensity measurements is due to the assumption that the particles are spherical. Brown and Felton [29] suggest making particle shape corrections using a theorem proved by van de Hulst [285], that states that the average geometrical cross section of a convex particle with random orientation is one-fourth its surface area. The light scattering results are taken to be for equivalent spheres, and the theorem is used to shift the equivalentsphere diameter to a characteristic length of the actual particle. Jones et al. [95] demonstrate that agreement with sieve analysis is obtained when this method is used to correct particle size to correspond to the characteristic length of potassium sulfate crystals. Many crystals have needle or platelet morphological structures. Methods based on a shift of equivalent-sphere diameters could lead to large errors for such large aspect ratio particles. To illustrate, Figure 6 presents the results of a Malvern particle sizer for a monodispersion of cellulose acetate cylinders (45-µ diameter, 195-µ length). Note that instead of the description of a monodispersion, the distribution is bimodal with modes located near the primary dimensions of the cylinders. Obviously, no method that simply shifts the CSD would be adequate for crystals of large aspect ratios. Another assumption made in developing eq 17 is that there is no multiple scattering. For high-density slurries that are common in industrial crystallizers, this poses a problem. Some studies have been made of the effect of multiple scattering and on methods to extend the light scattering theory to account for this effect [46, 78], but most of these methods are empirical and require calibration. Jager et al. [85] present a different approach—the use of an automatic dilution unit. The challenge of this approach is maintaining a diluent of proper quality to assure crystal stability and a representative sample. 4.2.2. Transmittance. As discussed by Witkowski et al.[197], information about the CSD can be obtained from a measurement of the transmittance, the fraction of light that is transmitted by the crystal slurry:

transmittance

=

I/I0

where I is the intensity of undiffracted light that passes through the suspension of crystals and Iq is the intensity of the incident light. Most particle sizers based on light scattering allow measurement of transmittance, but a relatively inexpensive spectrophotometer could also be used to make this measurement.

where

Q

J“f(L)(l/4)(ka6L2) dL

ka is

=

3kaj¡f(L)L2 dL

(23)

the surface area shape factor (the ratio of surface

of a crystal of size L to a cube with sides of length L). Therefore, there is a one-to-one relationship between transmittance and the second moment of the CSD. The expressions given above assume that there is no multiple scattering. The rule-of-thumb given by van de Hulst [285] is that multiple scattering effects on scattering patterns can usually be ignored for rl< 0.3. The validity of the Beer-Lambert law for predicting transmittance for l ij and yy denote the ith measurement and model prediction, respectively, taken at the ;'th sampling instant during the experiment, , is the weighting assigned to the ith measured variable, Nm is the number of measured variables (i.e., concentration, transmittance, etc.), N¿ is the number of samples of each measured variable, and

where

0T= [kg, kb, b,g] represents the unknown parameter vector. While the weights of eq 32 can be chosen to incorporate the data error structure (e.g., letting aequal the inverse

of the variance of the ith measurement), it would require replicate experiments to assess this statistical information. Therefore, other than accounting for scaling of the measurements, the weights are usually set somewhat arbitrarily. The maximum likelihood method, on the other hand, enables the estimation of weights for the data along with the estimation of the parameters. If one assumes that the model structure is correct, then

the deviations between the data and the model predictions are due to random (and possibly biased) errors in the measurements. The maximum likelihood method can be thought of as determining the parameters that maximize the probability that a given set of data could be obtained, considering the error structure of the measurements. Consider the following additional assumptions about the error structure of the measurements: 1. Errors at the ;'th sampling instant are normally distributed with zero mean (no bias) and are homoscedastic (variance of errors is independent of the magnitude of the measured variable). 2. Errors at different sampling instants are uncorrelated. 3. Measured variables are independent (measurement covariance matrix is diagonal). With these assumptions, it can be shown [6] that the maximum likelihood method is equivalent to the minimization of

(30)

(0)

Cq, t= o

1285

Nd N*

Ni

_yiog(V 2

(33) ei;.2 (0)) >-i‘ where ey = 5>y yy. The method also yields the following estimate for the variance of the ith measured variable: =

1=1'

From section 2.2, reasonable choices for constitutive relations for operation in which the temperature variation is small are G = kgS* B° = kbSbp (31)

-

^ 6*2^

3

where As is the third moment of the distribution in units of cm3/g solvent.

where

0*

is the set of optimal parameters.

(34)

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Whether the objective function is chosen for weight least squares or for maximum likelihood, the optimal parameter values are found by solving the nonlinear optimization problem,

min (>( ),£( );0) 0

subject to eqs 27-31.

of Estimates. The estimation of as a nonlinear operator that transforms a data set into an estimate of the parameters. Due to the random errors of the measurements, the parameters are also random variables with a probability distribution that depends on the nature of the estimator and on the distribution of the measurement errors; Bard [6] refers to the probability distribution of the parameters as the sampling distribution. If many replicates of the experiment used to estimate the parameters were available, the sampling distribution 5.2. Interpretation process can be thought

could be rigorously characterized. Since replicate data from crystallization experiments are difficult to obtain, an alternative is to perform a Monte Carlo study in which simulated replicate data are derived by adding random errors to the original data set in accordance with the assumed measurement probability distribution. However, Monte Carlo studies are too time consuming for many purposes as well, because the estimation process can be computationally intensive. An approximation of the confidence region can be obtained by assuming that the model can be represented by linear functions in the vicinity of the estimate,

y,(0)«y;(0*)

+

(35)

B;(0-0*)

[yy(0).....yww(0)]T, and B, is the Nm X Np matrix (where Np is the number of parameters) given below where

yj(0)

=

These partial derivatives can be obtained by finite difference; however, since the model involves differential equations, they can be efficiently and accurately computed by integrating the sensitivity equations along with the model equations [6, 27]. With the linearized model and assumption that the measurement errors are normally distributed, the estimates are normally distributed and it can be shown that the quantity Ni

(0-0*)T(yB/V-1B;)(0-0*)

ft

is distributed as 2 with Np degrees of freedom [6], in which V is the diagonal measurement covariance matrix with elements V¡¡ = ,·2 = ;.2 Therefore, letting V^-1 = BjV_1B; (Va is the parameter covariance matrix for

the linearized problem), and letting Pr[E} denote the probability of some event E, Pr{(0

-

0*)TVÍ-1(0

-

0*) ^ Xivp2(«)}

=

1

-

a

(36)

In other words, the approximate 100(1 a) % confidence region from the linearization method is the ellipsoidal region defined by -

(0

-

0*)TV,"1(0

-

0*)

=

XjVp2(a)

(37)

This region can be characterized by the eigenvectors and eigenvalues of Vg"1. The eigenvectors give the direction of the ellipsoid’s axes, which have lengths that depend on

Figure 7. Confidence region demonstrating significant correlation

between parameters. Conservative confidence interval for 0; is defined

by

50¡.

the eigenvalues—the ith semiaxis lies along the ith eigenvector of Vr1 and has length [jo 2( )/ ;]1/2, where , is the ith eigenvalue. The shape and orientation of the ellipsoid indicate the degree of correlation between the parameters. As can be seen in Figure 7 from the elongated ellipse for a hypothetical two-parameter problem, 0i and 02 may be varied individually by only ±0.5; however, if they are adjusted simultaneously, they can be varied significantly and still be in the confidence region. Since it is difficult to report succinctly a confidence region for cases in which Np > 2, confidence intervals—Pr{0 G [0* ± 60(a)]} = (1 a)—are often desired. Conservative estimates of these can be obtained by locating the box in the parameter space that circumscribes the ellipsoid (cf. Figure 7). Bard [6] gives several alternative approximate techniques for obtaining approximate confidence regions and intervals, and Donaldson and Schnabel [47] give an extensive discussion of confidence regions for nonlinear least squares problems. As illustrated by several Monte Carlo studies [4, 47], approximate techniques can overestimate confidence regions, but there is no guarantee that the true confidence region will not be grossly underestimated. Nevertheless, the approximate technique outlined above does give a convenient method for assessing parameter uncertainty that at least gives qualitative information about how well-determined the parameters are. 5.3. Experimental Design. The goal of the experimental design can now be stated precisely. One wishes to choose operating conditions and measurements such that the correlation between parameters and the volume of the uncertainty region are minimized. As discussed, the fact that the crystallizer model is nonlinear in the parameters requires that the parameter estimates be obtained before the uncertainty can be assessed. A related complication exists for the experimental design problem—a model-based procedure requires the values of the parameters to minimize the parameter uncertainty. Whether the approach is trial-and-error or quantitative and model-based, experimental design is sequential in nature: collect data, estimate parameters and uncertainty region, design new experiments. 5.3.1. Measured Variables. Physical arguments provide some insight into the choice of measured variables. For example, intuition suggests that since the third -

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Time (minutes)

Figure 8. Temperature profile for batch crystallization of KNO3

from water.

Figure 10. Transmittance measurements from experiment with temperature profile given in Figure 8.

Time (minutes)

Figure 9. Concentration measurements from experiment with temperature profile given in Figure 8.

Figure

moment is the only solid-phase information that can be derived from the concentration measurement; this measurement alone is probably insufficient for the independent determination of all of the kinetic parameters. Nevertheless, it is not obvious which measurements or sets of measurements contain sufficient information for successful parameter estimation. In order to study this question, the data from benchscale, potassium nitrate/water crystallization experiments were analyzed to assess the impact on parameter estimation of different measurement choices. Concentration and transmittance measurements were made for a batch run with the temperature profile shown in Figure 8. Zero time corresponds to the time of the addition of seeds, which were added at approximately zero supersaturation (i.e., initial concentration is just above the saturation concentration). Details of the experimental setup are given by Miller [114]. The concentration and transmittance measurements are given in Figures 9 and 10, respectively. Since concentration is a common measurement choice in the literature previously reviewed, we first consider parameter estimation using this measurement alone. Applying the techniques outlined in sections 5.1 and 5.2 to the data shown in Figure 9 results in the optimal parameters and approximate uncertainties given in Table I. The fit to the data of the model with these parameters is shown in Figure 11. As can be seen, while the model fit to the data is good there is significant parameter uncertainty, especially in the nucleation parameters. Many combinations of parameters in the large region described in Table I have roughly the same ability to predict the observations. A physical explanation for this is that many different transient CSDs can take up the solute from the continuous phase to produce nearly identical C(t) curves.

Table I. Parameter Estimation Results Using Concentration Measurements

11. Fit to the data of the model with nominal parameters given in Table I.

estimate

interval

ln(ftg) 8.014 ±0.423

b

g

ln(feb)

1.10

14.710

0.85

±0.12

±6.268

±2.48

Table II. Parameter Estimation Results Using Concentration and Transmittance Measurements g

ln(feb)

b

8.167

1.09

15.779

1.39

±0.205

±0.06

±1.475

±0.406

ln(feg)

estimate

interval

The concentration measurement does not contain sufficient information to determine the parameters uniquely. This result is the central point that is overlooked in much of the crystallization literature and is a possible explanation for the disagreement between various investigators’ reported parameters. The practical significance is that crystallizer control based on the parameters identified using concentration measurements alone is not likely to achieve the product CSD that is desired. As discussed in section 4.2.2, there is a one-to-one relationship between transmittance and the second moment of the CSD. It is therefore reasonable to expect that the measurement of transmittance would provide independent information for the estimation process. A comparison of Tables I and II demonstrates that the transmittance measurement is in fact effective in reducing parameter uncertainty. The reduction in parameter uncertainty has important implications for controller design. Given the identified model in Table I, the controller design problem is difficult since one does not even know the sign of the nucleation order. In other words, the controller must take control action without knowing whether increasing supersatura-

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Time (minutes)

Figure 12. Temperature profile for batch crystallization of KNO3

from water.

13. 13: Fit to concentration data of model with nominal parameters given in Table III.

Figure

Table III. Parameter Estimation Results Using Concentration and Transmittance Measurements from Two Batch Runs

_ln(feg)_g_ln(feb)_b estimate

interval

8.918

1.32

17.207

1.78

±0.127

±0.03

±0.356

±0.09

tion will increase

or decrease the nucleation rate. This obviously extreme amount of model uncertainty is a consequence of the poor experimental design used for identification. The controller design problem using Table II is much simplified since the uncertainty is so small. It would be a much simpler matter to detune the nominal feedback controller to be robust to this small amount of uncertainty. 5.3.2. Operating Variables. The inspection of the kinetic expressions, eq 31, provides considerable insight for the choice of operating variables. For example, it is obvious that if one wishes to identify and b independently, then the variation in the supersaturation should be made as large as possible during the course of the experiment. The justification is simply that if the supersaturation is constant during the experiment, one can only estimate the product, fcb Sb. Similarly, if temperature dependence is being characterized, the temperature variation should be made as large as possible. The parameter estimation results obtained in the previous section were based on a single batch experiment using a temperature profile with a rapid initial drop and a slow approach to the final temperature. It is possible that the choice of a different temperature profile would lead to improved parameter estimation (a smaller parameter uncertainty region). As discussed previously, the experimental design procedure for a nonlinear problem is sequential. With the results obtained above, simulations could be used to determine a temperature profile that would be expected to reduce the parameter uncertainty. The process is actually often trial-and-error—analysis is performed using data from experiments with new operating conditions that were selected using intuition. For example, intuition suggests that additional information could be obtained by performing an experiment that is a replicate of the one described above, with the exception of the use of gradual cooling like that shown in Figure 12 instead of the rapid cooling shown in Figure 8. The results from parameter estimation using concentration and transmittance data from both experiments are given in Table III. As expected, the additional information contained in the data of the new experiment leads to a decrease in the parameter uncertainty. It should be noted that the nominal parameters of Table III do not fall within the previous confidence region (given in Table II); again, that

14. Fit to transmittance data of model with nominal parameters given in Table III.

Figure

Table IV. Comparison of Linear Statistics with Monte Carlo Analysis of 500 Replicate Data Sets

%

freq that param est from replicate data are within elliptical confidence region, %

50 70 80 90 95 98

49 67 76 87 93 97

linear statistics confidence value,

is the nature of experimental design for nonlinear problems. The fit of the model to the data is given in Figures

and 14. The seed load used in the experiments discussed above was quite light. This initial condition is another operating variable that could be used to affect the quality of parameter estimation. As pointed out in section 5.2, the validity of the approximate parameter confidence region obtained by linearizing the model and using linear statistics is problem specific but can be tested using a Monte Carlo study. The validity of the approximate confidence intervals for this model is confirmed by a Monte Carlo study in which replicates of the data used in the discussion above were produced. Table IV illustrates that the elliptical confidence regions from linear statistics adequately characterize the parameter uncertainty over a range of confidence values [115]. 13

6.

Control Issues

The control of batch and continuous industrial crystallizers recently has been reviewed by Rawlings et al. [150]. They discuss the following issues in detail: the

Ind. Eng. Chem. Res., Vol. 32, No. 7,1993

Figure 15. Oscillations in an industrial crystallizer, after Randolph [136],

interaction between crystallizer design and control, available measurements and manipulated variables, and guidelines for choice of sensors and final control elements. The state-of-the-art in practiced industrial crystallizer control is summarized for numerous types of common crystallization process designs. The focus of this review, therefore, is on the open research problems and areas in which research advances may lead to further improvement in industrial practice. 6.1. Continuous Crystallization. The need for largetonnage fertilizers created the demand for continuous crystallizers that could produce large and uniform crystals, and operate stably for long periods of time [9,113,154]. Although general estimates for the minimum production rate justifying continuous crystallization vary from 1 ton/ day [122] to 50 tons/day [10], the scale of the equipment unavoidably creates nonhomogeneous regions of temperature, product concentration, and slurry density within the crystallizer at locations such as the heat exchanger or boiling surface, feed entrance, and in areas of lower mixing intensity or within baffled regions. The design intent is achieved if the vessel operation is controlled in a manner such that the vessel locations of these regions are time invariant and the process variables within the regions do not violate desired bounds. Operation outside these bounds generally results in fouling, deviations in crystal size, and lowered product purity. Additionally, the rate of change of the variables within acceptable bounds must be moderated to the extent that a higher supersaturation, or driving force, is not created than can be relieved by crystal growth and secondary nucleation. Such rapid changes would trigger spontaneous nucleation, producing a cyclic swing of smaller crystals for the downstream product recovery equipment to handle [150]. Sustained oscillations in continuous crystallizers are an important industrial problem and can lead to off-specification products, overload of dewatering equipment, and increased equipment fouling. Figure 15 shows this oscillatory behavior for an industrial KC1 crystallizer. Similar results have been observed in academic investigations with bench-scale crystallizers [238,172]. 6.1.1. Stability Issues. The stability of the open-loop, continuous crystallizer has received considerable attention [5, 7, 52, 84, 90, 109, 138, 139, 142]. The analytical techniques employed are either Laplace transform of the linearized population balance, conversion to a set of ordinary differential equations (ODEs) by taking moments, or spectral analysis of the linear operator obtained from the population balance [167]. Recently analytical techniques have been developed to enable the calculation of the linearized integro-differential operator’s spectrum even

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for cases in which the moments do not close and the model cannot be converted into a set of ODEs [57, 298]. As summarized by Randolph [237], the two explanations that have been developed for CSD instability have been termed high-order cycling and low-order cycling. Highorder cycling corresponds to a high-order kinetic relationship between the nucleation rate and the supersaturation. Low-order cycling is due to nonrepresentative product removal (i.e., fk not equal to /). For an MSMPR crystallizer, the value of b required to cause cycling is much higher than has been estimated for most chemical systems. Therefore, low-order cycling is the more likely explanation for the commonly observed CSD instabilities. All of these stability analyses are local, and no one has determined Lyapunov functions to ascertain the global stability. There have not been any simulations of the nonlinear equations, however, that indicate the global stability is markedly different from the local stability. Numerous investigators have addressed the problem of stabilizing an oscillating crystallizer using feedback control. Many of the early studies proposed measuring some moment of the CSD on line and using a proportional feedback controller to adjust the flowrate of a fines destruction system [70, 72, 108]. After reviewing these studies, Rousseau and Howell [259] proposed using supersaturation as the measured variable, since accurate on-line measurements of properties of the CSD are difficult to make. They demonstrated through simulation that a proportional feedback controller could stabilize a cycling crystallizer, although the sensitivity to measurement error was more

severe.

Randolph and co-workers [7,11,137,140,143,145,160] proposed inferring the nuclei density from a measurement of the CSD in the fines stream in conjunction with a proportional feedback controller. Significantly improved disturbance rejection is demonstrated experimentally for the proposed control scheme. Recently, Rohani [255] reported preliminary tests on using a turbidimeter for an on-line measurement of slurry density in the fines loop. Again, stabilization and improved disturbance rejection were demonstrated by simulation using a proportional feedback controller to manipulate the fines removal rate. One of the primary motivations for feedback is to overcome model uncertainty. For the controller to work well in practice, it should be tuned to be robustly stabilizing. That is, the controller should not only stabilize the nominal model, but also all models within some uncertainty region that reflects how well the system has been identified. The areas of robust stability and robust performance are current topics of control research. To date, essentially none of this research has been applied to crystallization. There have been relatively few studies and no experimental implementations of multivariable controllers for continuous crystallizers. The multiple-input, multipleoutput control systems that have been suggested are based on linear state-space models. A variety of methods to obtain these models have been put forth. Tsuruoka and Randolph [182] use the method of lines to cast the population balance in state-space form, and de Wolf et al. [41] and Jager et al. [87] demonstrate that time-series analysis can be used to obtain a state-space model. Hashemi and Epstein [74] use the method of moments to transform the population balance into a set of ODEs and then linearize them. They apply singular value decomposition analysis to develop measures of the system observability and controllability. More recently Eek et al. [52] have studied the observability and controllability

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light scattering to measure the CSD and withdrawal rate of both fine and large crystals as manipulated variables. Myerson et al. [120] outline a scheme for multivariable, optimal control of an MSMPR crystallizer. The suggested strategy uses a quadratic optimal controller to manipulate heat exchanger duty and feed flow rate. The controller is based on the linearized moment equations, and mass and energy balance equations, which are relinearized at each sampling period, and uses state estimates determined by an extended Kalman filter. On-line densitometry was proposed as a means of measuring the third moment of the CSD. issues using forward

6.2. Batch Crystallization. Batch crystallizers are used extensively in the chemical industry, often for small volume, high value-added specialty chemicals. Batch operation offers a flexible and simple processing step for plants with frequently changing recipes and product lines.

The control objectives for batch operation are often quite different from those of continuous crystallization. In the production of specialty organic materials, such as pharmaceuticals and photomaterials, product purity and CSD are of prime importance. If acceptable CSD and purity standards are not met, a batch of crystals has to undergo further processing steps, such as milling or recrystallization. Most of the previous process control studies of batch crystallizers have dealt with finding the open-loop, supersaturation versus time trajectory that optimizes some performance criterion derived from the CSD. The “natural” cooling curve displays the greatest cooling in the initial period when the temperature difference between the hot crystallizer fluid and the surroundings is the greatest. This causes a large degree of supersaturation at early times that leads to excessive nucleation, smaller final crystals, and aggravated operability problems such as fouling. Mullin and N£vlt [118] show that adjusting the supersaturation to maintain a constant level of nucleation would increase crystal size and decrease fouling of heat exchanger surfaces. Jones and Mullin [97] calculate a cooling profile that maintains a constant supersaturation. They use simulations to conclude that this operating policy leads to a product CSD that is superior to that from natural or linear cooling policies. It was experimentally verified that the constant-supersaturation cooling policy improves both the mean size and the variance of the CSD compared with natural cooling. Ulrich [184] tests various cooling profiles on a system of an organic product in an acetone/water solution and finds that a temperature curve qualitatively similar to that suggested by Jones and Mullin gives products with larger crystals and less variation than that corresponding to natural cooling or linear cooling. Ajinkya and Ray [3] suggest that it would be desirable to determine the optimal operating policy that maximizes mean particle size and minimizes CSD variance in minimum time. Nevertheless, with the simplified model examined (assumptions included monodispersion of seeds and negligible nucleation), the only problem that could be addressed was that of manipulating cooling rate to maximize the size of the crystals. Morari [116] determines the analytical solution of this problem and tests the algorithm using a simulation study, but the model used is unrealistically simple and no experimental verification was performed. Jones [93] applies optimal control theory and control vector iteration to find the cooling curve that maximizes the final size of the seed crystals. The temperature of the size-optimal control profile decreases slowly at early times and then rapidly at the end of the batch time, in direct

16. Temperature profiles for batch crystallizer. Cooling policies illustrated include natural cooling, constant-rate or linear cooling, and cooling controlled to maximize final-time seed size.

Figure

contrast to the natural cooling curve. Chang and Epstein [28] also compute optimal temperature profiles for a variety of objective functions based on average size, volume, or variance of the final CSD. They use a gradient method to solve the two point boundary value problem for the optimal profile. Surprisingly, their results show an initial increase in supersaturation followed by a slow decrease, which is much closer to natural cooling than Jones’ result. Although these works employ slightly different objective functions and crystallizer models, the source of the disagreement is not immediately clear. This disagreement is discussed further by Jones [94], who provides a general review of attempts to control batch crystallizers. It should be noted that the open-loop optimal controllers of Jones and of Chang and Epstein are based on the ODEs that result from applying the method of moments to the population balance, so the objective function for the optimization must be a function of the moments of the CSD. The moment equations are a closed set only for the cases of no classified removal and for growth rates that are size-independent or linearly dependent on size; therefore, there are limitations of the applicability of these methods. As explained by Rawlings et al. [150], the open-loop optimal control problem can be posed as a nonlinear programming problem. This formulation is flexible with respect to the crystallizer configuration, the form of the kinetic expression, and the choices of objective function and manipulated variables. It also allows the incorporation of input, output, and final-time constraints (e.g., constraints on cooling rate, supersaturation level, and yield, respectively). To illustrate the method, the parameters given in Table III and the corresponding model were used to compute the temperature profile for maximizing the terminal seed size. The temperature profile was parameterized as piecewise linear, and the constraints imposed included a minimum temperature bound and a constraint that the yield must be greater than or equal to that resulting from a linear cooling profile. The optimal cooling profile is shown in Figure 16, and the corresponding supersaturation profile is given in Figure 17. Optimal cooling leads to a predicted 9% increase in the terminal seed size over that of linear cooling, and a 18% increase over that of natural cooling. Jones et al. [96] investigate the feasibility of using fines destruction in a batch crystallizer. While the energy costs of using fines destruction would have to be taken into account, this may provide an additional manipulated variable. Their experimental results indicate that fines destruction can remove the crystals formed by secondary

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17. Supersaturation profile corresponding to the cooling profiles in Figure 16.

Figure

nucleation, increase the average crystal size, and decrease the coefficient of variation of the CSD. Again, although we are not covering this area explicitly, one should note that the species feed flow rates can be used as manipulated variables in reactive crystallization, providing other handles for influencing supersaturation. Much insight is provided by computing the optimal open-loop cooling profiles for batch crystallizers. One can determine from such calculations general operating guidelines for producing CSDs with selected characteristics (cf. Figure 16). The optimal cooling profile by itself, however, is not sufficient to control the crystallizer. As demonstrated by Bohlin and Rasmuson [15] via simulation and experimentally by Chianese et al. [33], even the qualitative benefits expected from optimal cooling profiles may not be realized, at least not reproducibly. Poorly characterized kinetics and inaccurate process control are the probable causes of the shortcomings. Feedback is necessary to compensate for deviation from the nominal trajectory, to reject disturbances, and to account for the errors in the model on which the open-loop policy is based. Chang and Epstein [29] suggest a strategy for incorporating feedback into the optimal control scheme discussed above that is based on the assumption that the system state trajectory is in the vicinity of the nominal optimal trajectory. A one-step iteration in a gradient search technique is used to find a new trajectory with an improved performance index. Simulation studies were used to demonstrate the application of this control scheme, but it has not been experimentally verified, and it assumes that the moments of the CSD can be measured or estimated. A natural way to incorporate feedback into the controller is to periodically compute an optimal control trajectory after using available measurements to estimate the unmeasured states and possibly update the model’s parameters. Eaton et al. [49, 50] and Patwardhan et al. [126] have applied on-line nonlinear estimation and control strategies to several chemical engineering processes. A more conventional approach that could be used as a basis of comparison is to use state estimation with a linear quadratic (LQ) optimal controller based the linearized ODEs obtained after using some lumping technique on the crystallizer model (method of moments, method of lines, etc.). Kwakernaak and Sivan [104] and Bryson and Ho [21] are standard references for LQ design. A feedback control scheme that is not rooted in optimal control theory has been suggested by Rohani et al. [258]. The strategy is based on the hypothesis that the final CSD can be improved by maintaining the fines slurry density at some constant value over a batch run. They experi-

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mentally verify that by maintaining the fines slurry density at a set point using a conventional PI controller to manipulate the fines destruction rate, the final mean crystal size is increased and the coefficient of variation is reduced. A simulation study by Rohani and Bourne [256] shows a self-tuning regulator to be more effective in setpoint tracking and disturbance rejection than the PI controller. Another variable that can be used to affect the CSD is seeding. As shown by Bohlin and Rasmuson [25], and Chianese et al. [33], seeding increases average crystal size and enables more reproducible operation than unseeded crystallization. However, because of the inherent bimodal nature of seeded crystallizer products, seeding may actually increase the coefficient of variation, depending on the cooling policy, the seed loading, and the crystallization kinetics. Crystallizers are usually seeded at or very near the saturation temperature, but Karpinski et al. [100] demonstrates experimentally that seeding can take place at significantly higher supersaturation levels without leading to lower product quality. The timing of the seeding of a batch sucrose crystallizer is one of the steps that is automated in a control scheme discussed by Virtanen [288]. The goals of automating the existing batch equipment were to shorten the crystallization time, produce a more uniform CSD, and reduce the labor and energy costs. In the control strategy, a microcomputer adjusts the steam flow rate to maintain constant supersaturation based on on-line density and refractive index measurements. The computer also regulates the timing of the feed addition, concentration, seeding, and growing steps for each batch. The control scheme was implemented at the Finnish Sugar Co. Ltd. and the biggest gains were achieved in decreasing batch to batch variations, decreasing the labor costs, and improving power plant economy.

Conclusions and Suggested Future Directions 7.1. Model Development and Solution. The population balance methodology is well established and suc7.

cessful for macroscopic modeling of the dynamics of crystal

distribution. Quantitative microscopic prediction of crystal shape and purity is not as well developed, but as discussed in section 2.1.2, rapid and exciting progress is being made. A feasible and worthwhile goal for future research is to bridge these two levels of models so that the macroscopic design and control decisions can be made with models that contain the best available knowledge of the microscopic chemistry. For engineering design and control purposes, the crystal nucleation and growth expressions are empirical and will likely remain so in the near future due to the complexity of these phenomena. Perhaps what is required in the near term for developing models with an intended control purpose is a more flexible empiricism. The use of flexible nonlinear functions for nucleation and growth, such as neural networks, radial basis functions, or multivariate splines, within the fundamental population balance framework to describe the CSD may prove to be a useful approach. The use of fundamental models in conjunction with flexible empirical models is a developing area of research [91,103, 129], The solution of these models for prediction and control in real time is feasible with today’s computing capabilities. Although efficient numerical methods have been developed for particular crystallizer configurations, it would be reasonable to expect progress on the development of a unified approach to handle the entire class of problems, size

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a large class of ODEs are efficiently handled by implicit predictor/corrector methods with variable step size. The benefit would be that effort currently being

much as

expended on case-by-case solution methods could be spent on other issues, and the organization and dissemination of the software to researchers and industrial practitioners would be simplified and streamlined. 7.2.

Measurement and Parameter Estimation. The

single area in which improvements are in hand but not completely disseminated and used is parameter estimation. An effort should be made to use the latest and most powerful methods to perform parameter estimation because population balance models pose, in general, difficult parameter estimation problems. It is not obvious whether a set of data are sufficient to identify a crystallization model without carefully and systematically analyzing a nonlinear optimization problem. Investigators should report parameter uncertainty as well as best fit parameters. Without some measure of parameter uncertainty, it is essentially impossible for an investigator to deduce how informative someone else’s experimental data are. It is now possible to implement efficient experimental design. Models can be identified in some cases without time-consuming and expensive steady-state experiments. The choice of measurements is important in experimental design. It seems that laser light scattering does not as yet provide a reliable CSD measurement for the purposes of parameter estimation. The inherent assumptions of single scattering by optically smooth spheres seems to be the biggest problem with the technology, in spite of recent research efforts. The use of less informative but more reliable measurements, such as transmittance and turbidity, seems to be adequate for accurate parameter estimation. Direct digital imaging may prove to be the most effective technology for the future. Although the method is still problematic, progress is being made and the instrument cost is decreasing. 7.3. Control. Although considerable research effort has been devoted to CSD control, we are only now seeing the advances in measurement and computing technologies necessary for successful industrial implementation of the ideas. It is reasonable to expect closed-loop CSD control to become part of accepted industrial practice in the near

future. In most process control applications, one tries to develop a model with minimal effort that is suitable for control purposes. The main reason for feedback control is, after all, the mitigation of model error and unmeasured disturbances in the closed loop. Hence the popularity of linear, blackbox models in process control. Crystallizers, however, have complex dynamics such as open-loop instability and long time delays caused by the interaction between growth of the CSD and new crystal nucleation. Common control objectives include optimal regulation of a batch crystallizer over a large operating region. It seems likely that low-order, linear models will prove inadequate in some of these applications and the nonlinear population balance models will be necessary. Research is needed to address in which applications these models are necessary and how they best can be used in the on-line control of crystallizers.

Acknowledgment Financial support from Eastman Kodak Company and the National Science Foundation under Grant CTS8957123 is gratefully acknowledged. The cylinder samples for Figure 6 were made available by Chester W. Sink of Tennessee Eastman Co.

Notation coefficients in basis function expansion of f heat-transfer area for cooling crystallizer B = rate of crystal formation mechanisms B; = matrix for linearized model at jth sampling instant B° = rate of crystal nucleation at size L0 C = solute concentration (mass of solute/volume of slurry) C = solute concentration (mass of solute/mass of solvent) Cast = solute saturation concentration = heat capacity cp D = rate of crystal removal mechanisms = error between measurement and prediction, S'ij ya en et(s¡) = energy of scattered light measured by detector of inner radius s¡ Eb = nucleation rate activation energy = growth rate activation energy Eg f = crystal distribution function F = focal length of lens in particle sizer g = crystal growth rate order G = crystal growth rate h = conversion factor, volume of slurry per mass of solvent = smoothing matrix AH = heat of crystallization / = intensity of scattered light = intensity of incident light = j slurry density power law coefficient for nucleation rate expression J0 = Bessel function of the zeroth kind Ji = Bessel function of the first kind K = optical constant of particle sizer kt = crystal area shape factor kb = nucleation rate coefficient &bcf = Burton, Cabrera, and Frank growth rate parameter = growth rate coefficient in the BCF model k'g kv = crystal volume shape factor L = characteristic particle or crystal size m(L) = particle or crystal mass density distribution m = vector of mass density distribution values at quadrature locations Mi = slurry density, pjk^o N¿ = number of measurement samples during an experiment JVm = number of quantities measured = Np number of parameters estimated P = system pressure Q = volumetric flow rate r residual of the weighted residual method s = radius of annular scattered energy detector S = supersaturation, (C R = gas constant a¡

-

=

Ac

~

-

-

time temperature Tc = coolant temperature U = heat-transfer coefficient V = volume of the crystallizer’s contents V = measurement covariance matrix V$ = parameter covariance matrix for the linearized problem ith residual weighting function or quadrature weight at Wi the ith quadrature point x = vector of external coordinates y = vector of internal coordinates ith measurement taken at the jth sampling instant $/i¡ y¡j = model prediction for the ith measurement taken at the jth sampling instant z = vector of particle’s coordinates Greek Letters t

=

T

=

=

-

a

-

joint confidence limit

= chi-square statistic with Np degrees of freedom Dirac delta function 7= penalty weight in constrained linear inversion e = volume fraction of continuous phase

xn2 =

Ind. Eng. Chem. Res., Vol. 32, No. 7,1993 vector of parameters wavelength of light = ith eigenvalue X¡ Uk = fcth moment of the CSD p = density of the crystallizer contents pc = density of the crystal solid a¡2 = variance of ith measured variable r = turbidity or crystallizer residence time = t weight for the ith measurement in the parameter estimation objective function = parameter estimation objective function ; = ith basis function for the weighted residual method =

X

=

Subscripts k = feth flow stream 0 = initial conditions

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Received for review November 20, 1992 Accepted April 12, 1993