Dynamic Model Development: Methods, Theory and Applications, Volume 16

DYNAMIC MODEL DEVELOPMENT Methods, Theory and Applications

COMPUTER-AIDED CHEMICAL ENGINEERING Advisory Editor: R. GanI Volume 1: Volume2: Volume 3: Volume 4: Volume 5: Volume 6: Volume 7: Volumes: Volume 9: Volume 10: Volume 11: Volume 12: Volurne 13: Volume 14: Volume 15: Volume 16:

Distillation Design in Practice (LM. Rose) TheArtof Chemical Process Design {G.L Wells and L M . Rose) Computer Programming Examples for Chemical Engineers (G. Ross) Analysis and Synthesis of Chemical Process Systems (K. Hartmann and K. Kaplick) Studies in Computer-Aided Modelling. Design and Operation Part A: Unite Operations (I. Pallai and Z. Fonyo, Editors) Part B: Systems (I. Pallai and G.E. Veress, Editors) Neural Networks for Chemical Engineers (A.B. Bulsari, Editor) Material and Energy Balancing in the Process Industries - From Microscopic Balances to Large Plants (V.V. Veverka and R Madron) European Symposium on Computer Aided Process Engineering-10 (S. Pierucci, Editor) European Symposium on Computer Aided Process Engineering-11 (R. Gani and S.B. Jorgensen, Editors) European Symposium on Computer Aided Process Engineering-12 (J. Grievink and J. van Schijndel, Editors) Software Architectures and Tools for Computer Aided Process Engineering (B. Braunschweig and R. Gani, Editors) Computer Aided Molecular Design: Theory and Practice (L.E.K.Achenie, R. Gani and V Venkatasubramanian, Editors) Integrated Design and Simulation of Chemical Processes (A.G. Dimian) European Symposium on Computer Aided Process Engineering-13 (A. Kraslawski and I. Turunen, Editors) Process Systems Engineering 2003 (Bingzhen Chen and A.W. Westerberg, Editors) Dynamic Model Development: Methods, Theory and Applications (S.R Asprey and S. Macchietto, Editors)

COMPUTER-AIDED CHEMICAL ENGINEERING, 16

DYNAMIC MODEL DEVELOPMENT Methods, Theory and AppUcations Proceedings from a Workshop on The Life of a Process Modei - From Conception to Action October 25-26,2000, Imperial College, London, UK

Edited by

S.P. Asprey S.Macchietto Centre for Process Systems Engineering Department of Chemical Engineering Imperial College of Science, Technology and Medicine Prince Consort Road, London, SW72BY UK

2003 ELSEVIER Amsterdam - Boston - Heidelberg - London - New York - Oxford Paris - San Diego - San Francisco - Singapore - Sydney -Tokyo

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Foreword Detailed mathematical models are increasingly being used by companies to gain competitive advantage through such applications as model-based process design, control and optimisation. Thus, building high quality steady-state or dynamic, single or multiresponse empirical or mechanistic models of processing systems has become a key activity in Process Engineering. This activity involves the use of several methods and techniques including model solution techniques, nonlinear regression for parameter estimation, nonlinear systems identification, model verification and validation, and optimal design of experiments just to name a few. In turn, several issues and open-ended problems arise within these methods, including, for instance, use of higher-order information in establishing parameter estimates, establishing metrics for model credibility, closed-loop identification and parameter estimation issues, and extending experiment design to the dynamic situation. Papers included in the book cover such topics as mathematical model representation, implementation and solution, rapid model development (including advances in nonlinear regression; structural analysis; and, automated mechanistic model building / nonlinear systems identification), model quality (including validation and verification techniques; advanced statistical metrics; as well as nonlinearity issues), model selection and improvement (including optimal design of experiments, as well as recursive, on-line techniques), computeraided modelling tools, and finally, industrial applications / challenges. The material covered in the book conveys information to a wide audience including researchers and practitioners within the process industries and academia, allowing easier development and full use of detailed and high fidelity models with reliable and quantified characteristics. Potential applications of these techniques in all engineering disciplines are abundant, including applications in chemical kinetics and reaction mechanism elucidation, polymer reaction engineering, physical properties estimation, biochemical and tissue engineering, and crystallisation. These models, in turn, can increasingly be used in and become the source of competitive advantage for such applications as model-based process design, control and optimisation. As a result, we can expect to see substantial reduction in the costs and time of development of building mechanistic, first-principles models, as well as increase the precision and confidence in their subsequent use. On the academic side, the book will serve to generate research ideas to further develop the underlying methods for mechanistic model building. The book will also serve as an excellent reference for postgraduate and research students, and has excellent potential to be used as supplementary reading in a graduate course on process modelling.

S. Asprey and S. Macchietto, 2002

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Contents

Foreword

v

Methodological Aspects in the Modelling of Novel Unit Operations H. Haario and I. Turunen

1

Dynamic Modelling, Nonlinear Parameter Fitting and Sensitivity Analysis of a Living Free-Radical Polymerisation Reactor A. Flores-Tlacuahuac, E. Saldivar-Guerra, andR. Guerrero-Santos

21

An Investigation of Some Tools for Process Model Identification for Prediction N. R. Kristensen, H. Madsen, andS. Bay J0rgensen

41

Multivariate Weighted Least Squares as an Alternative to the Determinant Criterion for Multiresponse Parameter Estimation P. W. Oxby, T. A. Duever, and P. M. Reilly

63

Model Selection: An Overview of Practices in Chemical Engineering P. J. T. Verheijen

85

Statistical Dynamic Model Building: Applications of Semi-infinite Programming S. P. Asprey

105

Non-constant Variance and the Design of Experiments for Chemical Kinetic Models A. C. Atkinson

141

A Continuous-Time Hammerstein Approach Working with Statistical Experimental Design D. K. Rollins

159

Process Design Under Uncertainty: Robustness Criteria and Value of Information F. P. Bernardo, P. M. Saraiva, andE. N. Pistikopoulos

175

A Modelling Tool for Different Stages of the Process Life M Sales-Cruz and R. Gani

209

Appendix

238

Other Papers Presented at the Workshop

251

Author Index

253

Subject Index

255

Dynamic Model Development: Methods, Theory and Applications S.P. Asprey and S. Macchietto (editors) © 2003 Elsevier Science B.V. All rights reserved

Methodological Aspects in the Modelling of Novel Unit Operations H. Haario^ and I. Turunen ^Yliopistokatu 5, FIN-00014, University of Helsinki, Helsinki, Finland ^Lappeenranta University of Technology, P.O.Box 20, FIN-53851 Lappeenranta, Finland Modelling is discussed as a tool for development of novel unit operations. General aspects, including model selection, validation and integration, are considered in the first part. In the second part of the paper novel methods for model building are discussed. INTRODUCTION: R&D STRATEGIES The main tools in the development of novel unit processes are modelling and experimentation. These two activities support each other; therefore any up-to-date methodology of process development should be based on integrated utilization of models and experiments. The requirements and results of experimental work have to be taken into account when planning the modeling strategy, and vice versa. A good description of industrial R&D strategy, valid for the development of novel unit processes, has been given by Euzen, Trambouze and Wauquier [1]. An example of modelling in industrial R&D project has been described by Piironen, Haario and Turunen [2]. The following two roles of models are especially important from the viewpoint of process development: • The theory and the mechanisms of the process are presented by models, therefore models increase the understanding of the process. • The models decrease R&D costs, because they make it possible to decrease the number of experiments and to increase the scale-up ratios. The first item often calls for deep scientific study of the process. In practical process development, this might contradict the second item, which emphasizes limited efforts. More detailed models often require extensive theoretical and experimental work and thus increase R&D costs. Therefore a very challenging task in process development is to formulate modeling strategy where different models are combined with different types of experiments to obtain reliable information for all relevant purposes with minimum costs within acceptable

time limits. Optimal design of experiments can provide tools for this task. Another challenge is created by modem measurement instruments: how to calibrate rapidly and effectively utilize the information content of the often massive amount of data they produce. We will discuss here the general aspects of modelling as well as some more specific computational tools. 1. THE ROLE OF MODELLING IN R&D Different steps in the modeling of novel unit processes can be presented e.g. in the following way: • • •

• • • • • • • •

Identification of the main purpose of the model, i.e. clear statement of the industrial objective. Identificationof the different phenomena in the process. Identification of the most important phenomena and planning their experimental research. In this connection it is often necessary to divide the process into several subsystems to avoid excessive complexity and too large a number of parameters. An adequate combination of laboratory, mock-up and pilot scale experiments has to be chosen from the extremely large number of possible ones. Selection of the theoretical basis from possible several competitive theories. Formulation of equations Parameter estimation and model validation based on experimental data. Experimental design methodologies should be adopted. Presentation and interpretation of the results Documentation of the model Integration of the model in the total system, i.e. evaluation of the impact of the new technology on the whole industrial process. Further development of the model as the project proceeds.

This modeling sequence is iterative as the developer usually returns to earlier steps after checking the results. From the preceding list experimentation, model selection, validation and integration can be considered as the cornerstones of industrial modeling methodology. 1.1 Model selection Commercial simulation programs can be used only to a limited extent in the development of novel unit processes. Usually existing models are not detailed enough and are incapable of coping with the specific features of real development projects. Flexible tools which could reach the required specificity and detailed nature have been developed in the universities. For one reason or another, these have stayed at the level of research results and are not much utilized in practice. Therefore companies develop their own specific and detailed models for their own purposes. This is especially true in the case of more complicated process units. One can recognize several important sub-problems in model selection. Often there are competitive theories to describe the relevant phenomena. There is always a choice between a

mechanistic and an empirical approach. The goal should be a mechanistic model but usually a compromise is needed in practice. All these decisions in model selection should be based on the main purpose of the model. Therefore it is very important to identify this purpose in the beginning of modeling activities. The purpose of the model also determines the degree of detail in the model. Increasing the details and the theory in the models, while usually increasing the accuracy and reliability, also often brings more parameters to be estimated and therefore also more experimental activities. In practice the best way often is to use several levels of "granularity", so that detail can be provided in the critical areas but simplified or empirical approaches are used in non-critical areas. 1.2 Model validation Validation of the models on the basis of experimental results is extremely important. First of all, the complexity of the model has to be compatible with the quality of the experimental data available. Crude data with high noise level may only identify rather crude models. A proper interplay between experimental and modeling work is crucial. This is a topic still overlooked in many R&D projects. A good fit between the model and measurements is usually not sufficient, but in addition, the proper values of parameters have to be identified and possible mutual correlations between them revealed. This is especially important when the model is used for extrapolation, e.g. scale-up. 1.3 Integration Integration of the models can be considered from different viewpoints. First the new unit process, as well as its model, should be considered as a part of the whole process. Optimal design of a plant can be found only in that way. Models have to be integrated also in the whole process development project, i.e. the modelling activity should proceed concurrently with process development. Different models are now developed for different purposes and at different stages of the process life-cycle, e.g.: • • •

at development, i.e. to demonstrate or investigate an idea before applying it in the process at design, i.e. for equipment sizing and scale-up, selection of operating conditions etc. at plant operation, i.e. for operation support, control, optimization or trouble shooting.

Model integration can be implemented also using a single model with minor modifications through all those stages. In order to do this a model should be based on sound engineering concepts. It must be pointed out that both model and process are evolving things; as the model is modified once new information about the process becomes available, in the same way, the process can be modified as soon as the model highlights how to improve the performance.

Finally, another important reason why companies should integrate models in life cycle of a process, i.e. by developing models concurrently with technologies, is that nowadays customers require them. A model may be used, for instance, to optimize the plant operation thus greatly improving its profitability. Models represent, in fact, a modem version of the operational handbook of the plant - and more. 2. NEW COMPUTATIONAL TOOLS In the following, we discuss in more detail certain novel methods for two basic topics in process model building: estimation of kinetic parameters and optimal design of experiments. 2.1 Rapid estimation of kinetic parameters by implicit calibration Reaction kinetics is often the bottleneck of reliable simulation studies, especially for complex or fast reaction mechanisms. Online measurements of various spectra - UV, IR, NIR, etc, combined with chemometrical calculation tools are increasingly used to identify the kinetics. We present here a recent approach which combines nonlinear parameter estimation with implicit, online calibration between measured spectra and calculated concentrations. In the standard parameter estimation approach the kinetic parameters are fitted against measured concentration units. The kinetic is supposed to be known except for some unknown parameter values, the model would give the concentrations if all model parameters were known. The initial values for the unknowns are guessed and optimized (estimated) iteratively by nonlinear regression in order to give the best possible fit between measures and calculated concentrations. In the approach that uses chemometrics, a computational model predicts the concentrations on the basis of the measured spectra. This requires off-line calibration, 'training sets' of known mixtures of concentrations and the spectra measured of them. Various principal component type multivariate methods - PCR, the principal component regression, or PLS, the partial least squares, some of the most common of them - exist for creating the model that maps the spectra into the concentration units. Once the calibration model has been created, measurement of the spectra can be done on-line during the reaction, often with rather dense sampling intervals, and the spectra are transformed by the calibration model into concentration units. The parameter estimation is then carried out exactly as in the standard approach, using the calculated concentrations as the measurements. The benefit is not limited to larger quantities of data, also the quality of concentration data obtained in this way may clearly exceed that obtained by more traditional chromatographic methods, see [12] as an example. Difficulties in preparing known mixtures may arise, anyway. In cases with fast reactions or elusive intermediates it may be difficult or impossible to create the training set foiathe calibration model. Another pitfall may be the scope of the training set: a reliable performance of the calibration inodel requires that the training set has been designed well

enough to cover the situations that will be met in the reaction batches. A considerable laboratory work may be required because of these aspects. Here we discuss a novel approach where minimal or no off-line calibration is done. The calibration of spectra and concentration units is done computationally, together with the parameter estimation procedure. No preliminary concentration data are available, with the natural exception that often the initial values of the reaction batches are known, together with the measured spectra at the same moments. The idea is to produce the concentration values from the model and test how these values may be calibrated against the measured spectra. Hopeftilly, the correct values of the kinetic parameters can be calibrated better than the incorrect ones. Figure 1 gives a schematic picture of the various approaches: arrows 1 present the standard way of fitting parameters and concentrations. Arrows 2 represent the chemometrical calibration of spectra and concentration units. Arrows 3 exhibit the implicit calibration.

dt dCs dt

—kiCjiCs + k2CcCo

Figure 1. Schematic picture of the various approaches The idea of going the way indicated by arrow 3 is to avoid any (or many) explicit calibration work. So we have called it 'implicit calibration'. Calibrations are in fact done, but between computed values, as the algorithm iterates via the ways indicated by arrows 1 and 2. There are several ways to proceed, we have studied mostly the methods where the calibration step is a sub-task of the kinetic fitting problem. Other approaches are discussed briefly below. We will see how the approach succeeds in finding correct values for the kinetic parameters, also in a case where more traditional off-line calibration failed. Let us clarify the implicit calibration, the way of Arrow 3, with an example, esterification of methanol with formic acid. Consider the equilibrium reaction A + B C + D, where A is methanol, B formic acid, C methyl formiate and D water. Let us denote the forward and backward reaction constants as kl and k2. From the implicit calibration point of view, this is

an interesting case because the reaction is fast, traditional analyses are difficult to carry out because the reaction starts immediately already at room temperature. The spectroscopic IR data was produced at Lappeenranta University of Technology. It turned out that the standard chemometrical procedure, Arrow 2, was not successful: even the short time between getting the mixtures of the components done and getting them into the instrument producing the IR spectra measured of them, was too much. The reaction had advanced so that no 'known' concentrations existed anymore that would correspond to the spectra at the sampling time. Of course, the concentrations could be calculated, from the known initial values at mixing time, if the kinetic rate constants were known - but they are exactly the unknowns to be estimated. So this situation calls for a methodology that is able to combine the calibration and model parameter identification. We shall see below that a simultaneous calibration of the spectra and parameter estimation could indeed be successftilly done. Six reaction batches were run, all in the same (room) temperature, with different initial concentrations for the components A,B,C, D. The spectra were measured at about 20 sampling times during each batch. In implicit calibration, the calibration step is performed separately inside the kinetic parameter estimation loop, i.e. the calibration takes place between the measured spectra and the concentrations calculated by the model using the current values of the kinetic parameters. The improvement is based on the idea that with more correct kinetic parameter values the fit of the calibration should also be better. Several alternatives for the calibration are possible, they are discussed below. Here we give the results just for one choice. Inside the parameter estimation loop the spectra were calibrated with the PLS method using 4 PLS dimensions. Figure 2 below shows the contour lines of the R2 values of the concentration fit, the concentrations calculated by the kinetic model versus the 'data', the concentrations calibrated from the spectra. The R2 values are given in a grid, we can see a clear optimum around the point kl=0.1, k2 = 0.2.

Figure 2. Contour lines of the R2 values of the concentration fit The same reaction was earlier studied at Kemira Agro Oyj Research Centre where a reliable kinetic model was developed by traditional nonlinear parameter estimation, using a sophisticated on-line HPLC system especially designed for fast reaction. So we are able to

compare the results, obtained by quite different measurement approaches at different temperatures. The values obtained by implicit calibration turned out to practically coincide with those obtained by the traditional parameter estimation procedure, which, however, required much more difficult experimental work. In general, we have several alternatives to choose. The fit can be done either in the concentration units or in the absorbance units. We should also employ any a priori knowledge available to 'fix' the empirical calibration step: positivity, smoothness, stoichiometry, linear constraints and so on. Furthermore, various objective functions can be employed: ordinary least squares, the overall R2 value of the fit, separate R2 values for each chemical component, etc. This gives us a multitude of calibration alternatives. The basic choices are: • • • •

The 'direction' of calibration (fit in absorbance/concentration units) The calibration method (ordinary regression, principal component regression, PLS, Ridge regression) The constraints (positivity, smoothness, stoichiometry ....) The objective function

It is intuitively clear that a too flexible calibration method would fit too well to concentrations calculated even with wrong values for the kinetic parameters. The choice of a suitable 'rigidity' for the soft calibration step is crucial. In addition, our studies show that the most important factors in obtaining reliable results are proper design of the batch experiments, proper weighting of the initial measurements of each batch (in cases where both spectra and concentration data is available for initial values), and proper selection of the objective function. A method related to our approach is so-called curve resolution. There, one aims at factoring the measured spectral matrix into a product between pure component spectra and concentration matrixes. It should be noted that there is a major difference in the curve resolution and implicit calibration approaches. The former is a valuable tool when the kinetic mechanism is not known, helping us in finding the possible reaction schemes. The latter, in turn, provides a tool for finding a quantitative kinetic model when the kinetic mechanism is known. The pure curve resolution procedure may also be extended to take into account the kinetics that produces the data. Indeed, this is another way to do the computational calibration [3] and has been further studied by other authors in, e.g., [14-16]. Above, we have exclusively discussed absorbance spectra and concentration units. The approach is, however, more general. Instead of absorbance, we could have any analytical signal, and instead of concentrations, we could have any state variable of the kinetic system, mole fraction, mass fraction, temperature, etc. The proposed implicit calibration method seems to be able to overcome many difficulties: no calibration based on known mixtures or pure components is needed. The measurements correspond to the true state of the reaction at each time step. All intermediates effect the

measurements provided that the instrument is sensitive to them. Thus, reaction mechanisms that could not be determined by traditional methods can be handled, and even in cases where traditional methods v^ork, the implicit calibration approach can significantly speed up the kinetics estimation procedure. Nevertheless, a multitude of details that affect the results have to be further studied. These include preprocessing of spectra, the direction of the implicit calibration, the multivariate calibration method itself, the measure of goodness of fit, use of prior knowledge in constraining the solution, weighting of measurements, the dimension of the calibration model or the weights for the constraints and finally the design of the experiments. For more details and references see [3,4,14,15,16]. 2.2 Global criteria for design of experiments The traditional criteria - D, A, E, etc - for optimal design of experiments are all based on linear theory. For nonlinear models they are applied via linearization. In some cases this approach has, in addition to the inevitable distortion caused by linearization, a certain drawback. With correlated model parameters the Jacobian matrix becomes singular, and no good experimental points are found by numerical optimization. Thus methods are required that take into account the parameter effects more globally. In [6] we employed a method we called 'Parameter Discrimination^ as a global criterion in a case of enzyme kinetics. The traditional criteria did not produce reasonable results. So we used the idea of finding 'bad' parameter vector pairs from the confidence region: parameter combinations far away from each other that produce roughly equally good fits, i.e., that are not discriminated by the data available so far. The optimal design is then found by maximizing, in some norm, the difference of responses at those discrimination points with respect to the experimental variables. If a good experimental point exists, it will be one that separates the responses by an amount greater than the experimental error level. The method is, in fact, the familiar model discrimination principle [5],[11], but applied within one model: the model predictions at two (or more) parameter values are regarded as given by different models. Basically the same idea was applied already by D. Isaacson [8] under the title 'distinguishability' in the slightly different context of electrical impedance tomography and, more recently, by Asprey in process research [7]. Here we want to emphasize a novel aspect of the methodology: the role of MCMC methods in finding 'optimal' discrimination parameters. We shall illustrate the methodology with two examples. The first one is a simple case where the classical criteria also work well. We show how essentially the same design is obtained by the parameter discrimination principle. As the second example we utilize the same enzyme kinetics case as in [6]. However, now the search for the discriminative parameters is done more systematically by the MCMC chain of the parameters to be estimated.

Markov Chain Monte Carlo, MCMC methods Let us recall the standard nonlinear model fitting situation, y=f(x,e}^s,s

^N(%a^)

(1)

where, for simplicity, we assume the measurement error 8 to be independent Gaussian with variance a^. The least squares estimate 0 for the unknown parameter vector is obtained by minimizing the sum of residuals

i(e)=Yjy,-f(x„e))'

(2)

of the data y., and the model at the measurement points x.,/=l,2,...« . Classical formulae for the confidence intervals for the parameter 6 are taken from the theory of linear models, and applied here after linearizing the above sum. The result is naturally approximative and may be rather misleading. The Bayesian approach is consider the posterior distribution of the parameters in terms of probability densities. In the case of Gaussian measurement error, the likelihood function is obtained from the residual sum by the formula P(e\y)=Ce

-}-i(e)

2^

(3)

(we skip here more general cases, as well as the discussion about a possible prior distribution for the parameters ^ ) . The constant C in the above should be determined so that a true probability distribution, with total mass one, is created. In principle, the recipe is simple: integrate the exponential term in the above formula over ^ and take C as the inverse of the value obtained. The probability that 6 belongs to a certain region - the 95 % credibility region around the best fitting point 9, for instance - is then obtained by integrating P{6\y) over that region. There are, however, two obstacles that practically have blocked the use of this recipe: the numerical integration over 0 soon becomes prohibitive, if the dimension of the parameter space gets higher. And there is no direct way to find limits for integration over given confidence regions. The emergence of the MCMC methods has recently removed these difficulties, to a large extent.

10 MCMC can be considered as a Monte Carlo integration where random points are chosen from a suitable constructed Markov chain. The chain produces successive points 0.,i=l,2,... in such a way that the probability distribution of 0 is approximated. There is a great variety of MCMC algorithms; let us describe a simple version, the Metropolis algorithm. Suppose the process is at the point 0. at the i^^ step. For a new sample a candidate point 0 is first chosen from a suitable proposal distribution q(\d.), which can depend on the previous state. The candidate point is accepted as a new state in the chain with probability a(0^,6 )= min

,' P(0V,

(4)

If the candidate point ^ is accepted, the next state in the chain becomes 6 , -0 ^ otherwise 0 . =^.. It may seem that the calculation requires the proper, scaled probability fixnction P. However, in the above formula only the ratio of the P values appear, so the normalizing constant cancels out and the everything boils down to successive computations of the residual sums. It can be proved that with a proper, fixed proposal distribution q- e.g., a Gaussian distribution centered at 0^ - the chain converges towards the proper distribution of ^ .[17]. The choice of q is the only decision the user has to make. This may also be a source of difficulties: if the size or shape of the proposal distribution is not in accordance with the distribution of 0 (so that, e.g., many candidate points will give practically a zero value for PO and are thus not accepted), the convergence may be very slow, excessive many points 6^ are needed. In [9,10] an adaptive method to 'tune' the proposal q was introduced, which, according to our experience, practically removes this possible pitfall. As an example, consider a simple linear model y=X6 in dimension 2, with data generated at certain points x., j;.,/=l,2,...,10 (we will use this example again below, with more details). Figure 3 gives the least squares fit point and the points generated by the MCMC calculation. The 95 % credibility region can be constructed by the histogram of the MCMC points in a straightforward manner. Of course, in this linear case the same result is readily available by classical theory, too. The strength of the MCMC methods is that they apply equally well for nonlinear models, which are beyond the scope of the standard statistics. The credibility region tells us how well the parameters are identified by given data. The information may be utilized to design new optimal experiments for better identification. In the next section we show how the information provided by the MCMC chain may be employed to design experiments for nonlinear models.

11 1

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Figure 3. Credibility region of a linear model. The theoretical 95% Probability region (-) and the points of an MCMC run (.). Local criteria for experimental design The traditional criteria - D, A, E, etc ~ for optimal design are based on linear theory. For nonlinear models they are applied via linearization. For the model y=f{x,0) one computes the Jacobian matrix J consisting of the sensitivity derivatives Jij=

de,

(5)

where x.,/=!,...,« denote the experimental points and 6jJ=\,...,p the unknown parameters. The derivatives are evaluated at the point 0^, the best guess or last fitted value for the parameters. An approximate credibility region for the parameters is obtained as a p dimensional ellipsoid, by replacing the design matrix X in the linear theory by the matrix / . The traditional criteria aim at minimizing the size or optimizing the shape of the ellipsoid. Minimizing the volume of the credibility region, for instance, leads to the D-Optimality, where one maximizes the expression

det(J'J) with respect to the ne^ experimental points x.

(6)

12 For moderately nonlinear, reasonably well identifiable models the traditional methods often work perfectly well. Problems may be encountered, however, for several reasons: The result is sensitive with respect the assumed value for 9^ If the model is badly identifiable, due to too many or correlated parameters, the Jacobian J becomes nearly singular. In the latter case the objective function for the optimization of the design may have no welldefined optimum, optimizers may have difficulties in convergence, or the results may be misleading, see, e.g., [13] for an example. One could interpret that a common cause for the difficulties is the local character of the criteria used: the derivatives in the Jacobian matrix are computed by varying the unknown parameter values in a small neighborhood of 6^. For weakly identifiable parameters the differences in the model predictions might be negligible, the criteria only test a small 'flat' landscape around 9Q .

Global criteria for experimeiital design To see the effect of parameter variations for the behavior of the model, one should compare the model predictions at points that are at larger, 'global' distances in the parameter space. Bad parameter identification is due to distant parameter points 0, at which the model responses nevertheless are close to each other, at the points x, where the data so far has been measured. The design of experiments should find new points x where the responses at those 6 points are as different as possible. This is the idea of the global type designs we discuss next. The procedure, in principle, consists of the steps Find 'extreme' parameter combinations 6. not discriminated by data so far Maximize the difference of responses with respect to the experimental variables,

d(f(x,e,)j(x,ej)i i^j

(7)

Above, the distance d may be given in any norm (L\L^ , max). If the model predictions given by the different parameter vectors are interpreted as given by different models, this is indeed the familiar model discrimination procedure [5]. So we have called the approach 'Parameter Discrimination'. To arrive at a practical algorithm, one still has to specify how to select the discriminating 'extreme' parameter vectors 0., how to define the norm d to be maximized, and how to connect the procedure with proper statistical context, taking into account the size of the measurement error.

13 The credibility region gives the parameter values that, with certain data available, could statistically 'equally well' be the true values. So the extreme tips of the region show how well or badly the model parameters are determined by the data: they represent points whose model responses still are close, yet the distances between the points are largest. A good design for next experiments would bring such tips of the credibility region as close each other as possible. For linear models the selection of the discriminating points could be done by the known formulae that define the credibility region (we shall indicate below the connection to the classical criteria). For the general, nonlinear case the same idea can be realized by aid of the MCMC chain. If properly run, the MCMC algorithm produces points that cover the credibility region. Various criteria may now be created, depending on how the discriminating 6 points are selected from the MCMC chain matrix. The situation is analogous with the classical 'alphabetic' criteria, constructed by the semi axes of the credibility ellipsoid in various ways. In principle, the search for the 'extremal' parameter points 0 could be viewed as a constrained optimization problem: maximize ||^-^o|| under the constraint that ||/(;c,^o)-/(x,^)[| remains small. But no optimization is needed if an MCMC algorithm provides the set of 6 from which the extremal ones can be selected. In addition, for proper statistical analysis the credibility region should be created anyway. So we get the objective fimctions for global design criteria as byproducts of the parameter fitting followed by the construction of the MCMC chain. In fact, the use of an optimizer could be omitted even in least squares fitting, since the MCMC chain provides the means, first moments or the peak points of one dimensional marginal distributions of the individual parameters. In some ill—posed such estimates may be more robust than the least squares fit, see [9] for an example. Our standard approaches, however, to calculate first the least squares fit and then continue with the calculation of the MCMC chain by the adaptive methods, using the approximate covariance from the fit as the first guess for the proposal distribution. Example 1: a linear model Let us consider the situation using the L2 distance norm, the usual least squares objective function. For a fixed 6^, denote

d(e)=\f(x,9j-f(x,e)\l In the linear case, f{x,6)=x'0,

(8)

we have

d(9f=(e-ejx'x((e-e,),

(9)

14 where X denotes the design matrix of the x vectors. If 0. -0=v. is an eigenvector of X^ X with eigenvalue X^, we see that, for example,

Y\d(6^)-X^ ...X ^, D~optimality, i

^ d ( 0 . ) = /i^-h\,

A-optimality,

(10)

So various combinations of d(6.) have a connection with the classical designs. Of course, for linear models the classical criteria only depend on the X matrix, not on any selected parameter points 0. However, the following example shows how the same designs may be obtained by classical and global methods, if the criteria in both cases are suitably chosen. Consider again the simple two dimensional model y=x'0, with 0=(20,25), with 0 = (20,25) and the data given at 10 'experimental' points (especially badly designed points, for demonstration) as shown in Figure 5. Noise with cr = 0.1 was added to the y values, the resulting credibility region was already shown in Figure 3. We can now ask for one new optimal experimental point x, and do the optimization both with D-optimality and with the global criterion close to D-optimality as shown above. In Figures 4 and 5 we have plotted both design criteria as functions of x^,X2. The surfaces are naturally different, but qualitatively close to each other. The optimal result is in both cases the same: the point that lies in the comer ^ = (-1,1).

parameter no 1

Figure 4. The D-optimality surface

15

S

oK

Figures. The Parameter Discrimination surface The result is rather typical. For mildly nonlinear, relatively well-behaved systems the classical criteria give similar designs as the parameter discrimination, if the discriminating parameter values are chosen so that they correspond to the given classical criterion. Example 2: a nonlinear model Here we study an example where the D-optimality seems to fail to produce a proper design, while parameter discrimination works. The example comes from enzyme kinetics [6], mashing of beer. The complexity of the system is rather typical in chemical or biological kinetics, with respect to the number of ODE components, unknown parameters, control variables and observations. Below is a list of the states and the ODE system that models the situation. a=-HaMIVg{a^a) ag =HaMIv(a^

enzyme, -a^-ka^^a

enzyme,

P=-HbMIVg{P^-P)

enzyme,

P^=-HbMIV{l3^P)-kbp

enzyme,

x\ xi

=-a{x^-uT)c^Agmlt-\-Agdex) =a{x^-uT)Agdex-Px^(c^Bgl-\-

starch, C3 "^Bmal Km + Xi

+ Bldex

dextrins,

X3=Bgl Px^

glucose,

X4= Bmal P x^ I {Km -^x^)

maltose.

16 X5 =Agmlta(x^

maltotriose,

-uT)

limit - dextrins,

X6 =Bildex J3x2

The experiments had been done on an intuitive bases, not following any specific design methodology. Seven batches were run at different temperatures, with different initial values for X2,x^,x^,x^. In two of the batches the temperature, a control variable, was changed during the batch. The parameters 0=(Agdex,Agmlt,Bgl,Bmal,Bldex,Km) depend on the temperature via the Arrhenius law. They were to be estimated by the measured values of components x^,x^, x^ and x^-^x^. For simplicity, were assume here the activation energies, as well as other remaining parameters, to be known and estimate only the parameters in ^ at a reference temperature. We also skip here all further practical details, for our present purpose it is sufficient to deal with the example simply as an ODE system with certain parameters to be estimated. data set 3

data set 2 0--

^^•••••^^

O:-

o

o

Q

o

-

O

-Q-O—©—o-

© ^

O Q

_^__&— O"

100

Q-— -—O

"'•••'O

Q

50

Figure 6. Typical data and fits for the ODE system. Several test runs were performed, for these runs we replaced the original data with simulated values, with some Gaussian noise added. Figure 6 shows some of the data together with the fits. The MCMC chains were run without difficulties after the fits. Figure 7 represents one and two dimensional marginal posteriors of two of the parameters, with the 68%, 95% and 99% probability regions shown for the two dimensional distribution.

17

Figure 7. The one and two dimensional posterior distributions of the parameters Bldex and Km.

X10*

In this simulated situation the least squares fit is already close to the 'true' parameter value 6^ but new experiments for smaller error bounds are called for. Two variables T and M, the temperature and a concentration constant in the enzyme rates above, were selected as the experimental factors to be optimized for the next batch (again, we deal with a somewhat simplified situation for demonstration purposes). For the global criterion two discriminating points 6^,02 were selected, roughly at the tips of the main semi axes of the 6 dimensional MCMC sample of the credibility region. Figures 8 and 9 present the D-optimal and global objective functions as contour surfaces in dependence of T and M. Now we can see that the criteria offer rather different answers: the global criterion suggest the point T = 321, M = 0.07 while the point T = 330, M = 0.03 would be optimal according to the D-optimality. Figures 10 and 11 exhibit the model responses as calculated at the discriminating points 6^,6^ and the least squares fit point 9^. We see that the global criterion does find an experiment that will give separate trajectories, while the respective responses at the experiment suggested by Doptimality are closer than the selected measurement error level. Performing an experiment at this point would thus not bring us new information to shrink the credibility region.

18

0.03

0.035

0.065

Figure 8. The D-optimality surface

J^i

Figure 9. The Parameter Discrimination surface

o.o:

19 data set 1

60

r

50

"T

" "

i

i

i

80

100

/

40

m

g

1 s

1/

LU

Q O

20

10

i-****^"'*"

/

20

40

Figure 10. The responses at the D-optimal point.

Figure 11. The responses at the Parameter Discrimination point.

20 REFERENCES 1. 2. 3 4 5 6

7 8 9 10 11 12

13 14 15 16 17

Euzen, J.P., Trambouze, P., Wauquier, J.P., Scale-up Methodology for Chemical Processes. Editions Technip, Paris, 1993. Piironen, M., Haario, H., Turunen, L, Modelling of Katapak Reactor for Hydrogenationof Anthraquinones. Chem. Eng. Sci. 56 (2001), 859-864. Haario, H., Taavitsainen, V., Combining soft and hard modeling in chemical kinetic models, Chemometrics and Intell. Lab. Syst, 44 (1998), 77-98. Taavitsainen, V., Haario H., Rapid estimation of chemical kinetics by implicit calibration. I. J. of Chemometrics, 15 (2001), 215-239. Bard Y., Nonlinear Parameter Estimation, Academic Press, New York, 1974. Haario, H., Holmberg, A., Pokkinen, M., Oinas, R. Integrated information and quality management of biotechnical processes and its application to experimental design in process scale-up from R & D to production. Proc. of the 6th International Conf on Computer Applications in Biotechnology, May 14-17, 1995, Garmisch-Partenkirchen. Asprey, S.P. and Macchieto, S., Statistical tools for optimal dynamical model building. Comp.Chem.Eng., 24 (2000), 1261-1267. Isaacson, D., Distinguishability of conductivities by electric current computed tomography. IEEE Trans. Med. Imaging, 5 (1986), 91-95. Haario, H.,Saksman E., Tamminen J., Adaptive proposal distribution for random walk MetropoHs algorithm. Comput. Statistics, 14 (1999), No 3. Haario H., Saksman, E., and Tamminen, J., An adaptive Metropolis algorithm. J.of Bernoulli Soc, 7 (2001), 223-242. Atkinson,A.C, Fedorov, V.V., The design of experiments for discriminating between two rival models. Biometrika, 62, 1, 57-70, 1975. Helminen, J., Leppamaki M, Paatero E., Minkkinen P., Monitoring the kinetics of the ion exhange resin catalysed esterification of acetic acid with ethanol using near infrared spectroscopy with PLS model. Chemometrics and Intell. Lab.Syst. 44 (1998), 345-356. Oinas,P., A. Wild and N. Midoux, H. Haario: Identification of mass transfer parameters in cases of simultaneous gas absorption and chemical reaction. Chem. Eng. Process, 34, 503-513,1995. Bijlsma,S., Louwerse,D.J., Smilde,A.K., Rapid estimation of rate constants of batch processes using on-line SW-NIR. Aiche J., 44 (12), 2713-2723, 1998. De Juan,A, Maeder, M., M.Martinez, R. Tauler, Combining hard- and soft-modelling to solver kinetic problems. Chemometrics and Intell. Lab. Syst., 54, 123-141, 2000. Bezemer,E, Rutan, S.C,Multivariate curve resolution with non-linear fitting of kinetic profiles. Chemometrics and Intell. Lab. Syst., 59,19-31,2001 Gilks, W.R., Richardson S., Spiegelharter D.J (eds), Markov Chain Monte Carlo in Practice. London, Chapman & Hall, 1995.

Dynamic Model Development: Methods, Theory and Applications S.P. Asprey and S. Macchietto (editors) © 2003 Elsevier Science B.V. All rights reserved

21

Dynamic Modelling, Nonlinear Parameter Fitting and Sensitivity Analysis of a Living Free-Radical Polymerization Reactor Antonio Flores-Tlacuahuac^, Enrique Saldivar-Guerra, Ramiro Guerrero-Santos ^Departamento de Ciencias, Universidad Iberoamericana, Prolongacion Paseo de la Reforma 880, Mexico DF, 01210, MEXICO.

In this work modelling and nonlinear analysis tools were applied to an experimental living free-radical polymerization reaction system. The objectives were to build a dynamic mathematical model able to reproduce observed data, to fit experimental information and to get a clear idea about the influence of reaction rate constants on reactor response using sensitivity analysis techniques.

1

INTRODUCTION

Polymer production is one of the most important areas of applied chemistry due to its significant economic and social impact. Polymers as materials are present in almost every field of human activity. They range from commodity materials, such as polyethylene or PVC (poly vinyl chloride), up to highly specialized and expensive materials for drug release or space-craft related applications. Polymers are long molecules or "macromolecules" produced from simple small chemical components or monomers. The chemical reaction by which monomers are transformed into polymers is called polymerization and its control presents serious challenges for the chemical engineer, due to the fact that these reactions are usually highly exothermic and often proceed in very viscous media that render mass and heat transport difficult. Also, these reactions are famous for behaving in non-hnear fashion and several instances of multiplicities and sustained oscillations have been reported in the literature for even industrial scale

22 reactors [1],[2],[3]. In order to aid in the design, operation, control and optimization of this kind of reactors and reactions, mathematical modeling of the polymerization process is an invaluable tool. For many years free-radical polymerization has been industrially used for the large-scale production of different kinds of polymers. The kinetic mechanism of such polymerization method is well known and the process is relatively easy to carry out. However, this polymers manufacturing technique has some drawbacks: (1) well defined molecular weight distributions are not easy to obtain, (2) polymers with a desired molecular structure are diSicult to manufacture. The importance of controlling these factors has been recognized due to the rising need for speciality polymers. In traditional free-radical polymerization the life of each polymer chain is only some fractions of a second. On the other side, living polymerizations in which the polymer chains are active for minutes or hours allow the preparation of polymers such as: macromonomers, macroinitiators, functional polymers, block and graft copolymers, and star polymers [4],[5]. Usually such polymers are produced using anionic/cationic or group transfer polymerization. However, because this type of polymerization processes requires severe reaction conditions (i.e. a high level of purity) and the spectrum of suitable monomers is limited [6] it has little industrial impact. Therefore, it would be desirable to combine the industrial advantages of free-radical polymerization (tolerance to impurities, and unselectivity to monomers) with the use of living polymerization techniques, as an efficient way to manufacture new polymers. It is also worth mentioning that, depending on the thermodynamic nature of the different components, these new block and graft molecular architectures can bring about micro-segregation of domains which lead to nanometric structures and to selfensambling materials. These materials are provoking a revolution in Material Science since its unique properties can be used in a wide spectrum of applications [7]. These properties arise from the fact that physical laws manifest themselves in unique ways at the nanometric scale. Living, quasi-living or controlled radical polymerization (CRP) is an emergent technique to synthesize polymers with control on the molecular weight and low polydispersities. There are several versions of CRP [8]: (a) atom transfer radicalpolymerization (ATRP), (b) nitroxyl mediated radical polymerization (NMRP), (c) the use of initers and iniferters and (d) reversible addition fragmentation transfer (RAFT). In CRP a new compound called "regulator" or "controller" is used. Such species are capable of reversibly trapping the propagating radicals, thereby reducing the concentration of growing chains which brings about the minimization of the irreversible termination step. Under such circumstances growing chains can polymerize only to a certain extent (before being trapped). The adduct formed by the controller and growing chain regenerates -in the reverse reaction step- free radicals that start a new cycle of reactions (propagation, reversible termination and dissociation). In this way polymer chains grow at the same pace. Polymers with narrow molecular weight distributions can be achieved if the initiation step, i.e. the period of time employed to initiate all chains, is reduced. To date, the achievements of CRP are: (i) the control of molecular weight and poly-

23 dispersity for homopolymerizations and (ii) the production of block copolymers by free-radical reactions, without such stringent conditions as needed for anionic polymerization. The goal of many scientist world wide is the implementation of this new technology in the development of new materials or improvement of existing ones. Mathematical modelling and simulation can help to understand better the underlying kinetic mechanism governing these processes. Also, with the aid of parametric sensitivity tools, key kinetic steps and associated parameters can be identified and their estimability can be assessed. Once these parameters are identified, the synthesis of better regulating agents can be attempted. In principle, for achieving these purposes, several living radical techniques might be used. In a first trial the initer bis(trimethylsylyloxy) tetraphenylethane was used. Initers are compounds that can generate two radicals; one acts as initiator while the other reversibly scavenge growing radicals leading to temporarily "dormant" polymer chains. In this sense initers compounds act as both initiators and regulators. In this work a dynamic mathematical model for a styrene living free-radical polymerization reactor is developed. The model is cast in terms of the moments of the species so we can compute molecular averages of the molecular weight distribution. Experimental information is used to fit kinetic constant rate values. The fitting procedure uses traditional nonlinear optimization techniques. Open-loop dynamic simulations of a typical industrial living polymerization reactor are discussed. Sensitivity analysis of the above mentioned mathematical model is used to assess the way kinetic rate information impacts typical reactor behaviour in the form of monomer conversion, molecular weight and polydispersity. The aim was to gain insight into the kinetic mechanism and to identify key kinetic steps and its associated parameters. Because very few works have been published on the modelling, parameter fitting and sensitivity analysis of initer living polymerization reactors, this work represents a contribution to this new and challenging engineering field. In section 2 the dynamic mathematical model of the living free-radical polymerization reaction is derived from the basic reaction mechanism. In section 3 experimental data information is used for the nonlinear parameter fitting procedure. In section 3 some open-loop numerical dynamic runs are shown. In section 5 sensitivity coefiicients are computed. Finally section 6 contains discussion and conclusions of the results

2 2.1

DYNAMIC MATHEMATICAL MODEL Kinetic mechanism

The living polymerization kinetic scheme involves the famihar thermal decomposition, initiation, propagation and termination steps. The thermal decomposition reaction produces free-radicals through the following reversible equilibrium reaction (Table 1 contains the notation description of those species participating in the polymerization reaction system):

24 i?^25"

(1)

where R = TPSE and S — SDPM (sylyloxy diphenyl methyl) radicals. Free radicals undergo reaction with monomer "M" through the initiation step yielding Pi : (2)

S' + M -^Pi from which the chain polymer starts growing (propagation step):

(3)

Pn + M^Pn+X

However, living polymerization differs from the more traditional free-radical polymerization in the following reaction step:

>„ + 5-SP„5

(4)

i^rd

This reaction step temporarily stops the chain polymer growing process (P^) leading to a dormant polymer chain (PnS). However, under certain processing conditions, the dormant chain polymer can undergo "reactivation" and the growing chain polymer process continues. This reaction step is the main reason why living polymerization produces narrow molecular weight distributions. Finally, the irreversible termination steps (disproportionation and combination, respectively) take place, leading to dead polymer chains: Pn + Pm ^

(5)

D^ + D^

Pji + Pm —^ Dn+m

2.2

(6)

Model

Dynamic material balances were make for each one of the species in an isothermal batch reactor. The kinetic scheme results in a system of pure ordinary differential equations of infinite dimension. • Initer compound (7)

^ = -k^R + krS'' at • Radical dS'

^^ =2fkdR-KS*''-kiS'M-ktpS'J^Pn

+ krdY,PnS n=l

n=l

(8)

25

Species Agent Radical Monomer Growing radical Polymer chain Dormant polymer Dead polymer

Symbol R 5* M Pi Pn

PnS Dn

Table 1: Chemical species participating in the reaction mechanism. Monomer dM

- - =

-kiMS'-kpMj2Pn

(9)

n=l

• Growing polymer a)

n= 1 dPi

—i = kiMS' - kpMPi - ktpPiS' + KdPiS - ktdPi J2^--

^*-^i S ^-(^^)

n=l

a=l

b) n > 1 oo

ip

- ^

= kpM{Pn-i

- Pn) - hpPnS'

+ KdPnS

oo

^ ^ -Pm - hcPn

^

771=1

m=l

- hdPn

^^(11)

• Dormant polymer dPnS dt

— f^tp-tn^

f^rd-^n^

(12)

• Dead Polymer (n > 2) dD,

— f^tpPn / ^ Pm + ey '^tc / ^ PrnPn-n

dt

2.3

ri=l

(13)

m=l

Moments

Next the method of moments [9] was used in order to reduce the dimensionality of the problem. 1) Growing polymer.

•

26 • Zeroth moment

—^ = kiMS* - ktpS'Xo + krdCo - ktdXoXo - hcXoXo

(14)

• First moment dXi ^^ dt

kiMS* - kpMXo - ktpS^Xi + KdCi - hdXiXo - hcXiXo

(15)

• Second moment

- ^ - kiMS' - kpM{Xo + 2Ai) - ktpS'X2 + krdC2 - ktdX2Xo - fc,,A2Ao (16) 2) Dormant Polymer

•

• Zeroth moment "IT = ktpS^Xo - krdCo

(17)

^

(18)

First moment = hpS^X^ - krdCi

• Second moment ^=ktpS'X2-krdC2 3) Dead Polymer

(19)

•

• Zeroth moment

^-M^

+ ^«

(20)

• First moment djj.

, - = ktpXiXo + ktcXiXo

(21)

Second moment ^

= ktpX2Xo + hc{X2Xo + Xl)

(22)

27 by numerically integrating the system of o.d.e.'s represented by Eqs. 14-22, average molecular weights for the dead polymer are obtained: M„ =

M ™ ^ ^ )

M„

M^(^^)

=

Co + Mo/

(23) (24)

where Mm stands for monomer molecular weight. Monomer conversion XM was computed as: MQ-M

^M = - j ^

(25)

where MQ is the initial monomer concentration.

3

PARAMETER FITTING

The kinetic parameters to be fitted correspond to the bulk styrene polymerization process. In Table 2 experimental information in the form of conversion and molecular weight distribution is shown. For solving the dynamic parameter fitting issue the problem was cast in terms of the following unconstrained nonlinear optimization program:

«»|:(^)^

m

where k represents the set of kinetic rate parameters to be fitted, yi(t) is obtained by numerically integrating the set of equations representing the mathematical model and yi represents the set of experimental data. Fitted results for styrene at 1=0.016 M and 110 '^C are shown in Figure 1; the fitted kinetic rate constants are given in Table 3. The same set of fitted kinetic rate constants was used for predicting experimental information recorded at different conditions. For instance. Figure 2 depicts a comparison between experimental and predicted behaviour at 0.045 M initiator concentration and 110 ^C, while Figure 3 depicts experimental against predicted information for 0.016 M initiator concentration and 100 ^C. In order to predict the behaviour shown in Figure 3, the kp rate constant was corrected for temperature changes, an activation energy value of 30000 J/gmol-^K being used. Therefore the k^ value used in Figure 3 was 850.

4

O P E N - L O O P D Y N A M I C SIMULATION

Figure 4 shows the evolution of conversion, M^, M^, and polydispersity. There is a contrast with traditional free radical polymerization in which molecular weight averages remain nearly constant throughout the polymerization. In this case molecular

28 ] Styrene a=.016 M 1 Time (min) Conversion (%) 4.65 20 12.42 50 16.76 70 100 27 1 150 29.95 1 Styrene Ci=.016 M 1 Time (min) Conversion (%) 1.42 30 4.86 50 70 6.35 11.46 90 |_ 150 21.09 1 Styrene a-.045 M 1 Time (min) Conversion (%) .7182 20 40 10.7175 60 18.8606 100 25.4279 1 120 29.8043

T = 110 °C ]

M„

1

63480

1

4610 27290 41880 58530

T = 100 "C 1

M„

1

58360

1

1826 26930 32080 48080

T = 110 ° C l

M„

1

26030

1

1730 2180 13990 22880

Table 2: Experimental styrene conversion and molecular weight recorded at different initiator concentrations and temperatures.

kd rCy.

Ki Kp "kp

krd

he hd

.030482 28408.6 .0136 1095.8 227241797344 2.38 64903458 0

Table 3: Fitted kinetic rate constants for styrene 0.016M and 110 ^C.

29

7000

(a)

8000

9000

(b)

Figure 1: (a) Conversion and (b) Molecular weight fitted results for I=.016 M and llO^C. weight increases with conversion. This occurs because the polymeric chains remain living and adding monomer units during most of the reaction. However, at long reaction times, even in this case some irreversible termination takes place, increasing the polydispersity. Figures 5,6 and 7 depict the radical, dormant and dead polymer concentrations, respectively. In a first stage, TPSE decomposes readly (Reaction 1, kd=.0304:82)^ producing a large amount of free radicals S which participate in the formation of dormant polymers; reactions 2, 3 and 4. When the concentration of free radicals stabilizes, a maximum level of dormant species has been formed and a second stage begins. Here TPSE molecules have dissappeared and dormant species regenerate P and S radicals, producing a polymerization with living character. Nevertheless, the deactivation between chains, reactions 5 and 6, becomes evident as indicated by the decreasing concentration of dormant species in Figure 6. Each irreversible termination event leaves radicals A without partners. The temporary excess of free radicals S may accentuate the importance of reaction (6); however, its propensity to initiate chains, cuts down this effect. Dead polymer concentration behaves in the opposite manner with respect to dormant species (Figure 7).

5

PARAMETRIC SENSITIVITY

In this section output sensitivities (conversion, molecular weight distributions and polydispersity) with respect to the kinetic rate constants were computed. The aim of this study is to have a clear idea about which of the kinetic rate parameters have greater influence on the reactor response. The results of the sensitivity analysis help in knowing which parameters should be carefully evaluated and also which ones are estimable •'

30 4

x10

y ^

3.5

-

3

o

2.5 Q

-

S" 2

1.5

/

-

0

-

1

0.5

O O

0

1000

2000

3000

4000 5000 Time (s)

6000

7000

8000

9000

"o

(a)

100C

2000

3000

4000 5000 Time (s)

6000

Experiment Fitted 7000

8000

9000

(b)

Figure 2: (a) Conversion and (b) Molecular weight fitted results for 1==.045 M and llO^CAssume that a mathematical model of a given system is available: rfx

f(x,p)

(27)

where x G 5ft'^ stands for system states and p G W^ represents the system parameters. The equations that describe the way reactor behaviour depends on the parameters are given by [10]: 9x/

\9p

(28)

where, S-11 (29) Si\

besides, (30) One of the suggested numerical procedures to evaluate the parametric sensitivity coefficients consists in the simultaneous solution of the equations representing the system dynamic behaviour (27) and the set of equations representing the sensitivity coefficients (28). According to this the numerical procedure to compute Sij has the following steps.

31

X / /

o o 0

0

/

1000

°

2000

;

3000

4000 5000 Time(s)

6000

Experiment

Fitted 7000

1

8000

(a)

(b)

Figure 3: (a) Conversion and (b) Molecular weight fitted results for I=.016 M and lOO^C. • Integrate the mathematical model X = f(x,p),

x(to) = X o

• Evaluate the partial derivatives given by

af(x,p) 5f(x,p) .9p • Integrate the sensitivity equations ^gf(x,p)

s+

ax

gf(x,p)

ap

in order to compare output sensitivities on the same basis, scaled sensitivities were computed as (Caracotsios): Pj

(31)

Due to the fact that some of the initial states are zero it is more convenient to use the above semilogarithmic scaling instead of the complete logarithmic scahng procedure.

32

6

DISCUSSION AND CONCLUSIONS

Figure 8 shows the sensitivity of conversion to the kinetic parameters (the program Athena [11] was used to compute the numerical values of the sensitivity coefficients). The propagation rate constant turns out to be the most important factor followed by the initiation and the termination constant (negative sensitivity). Conversion is moderately influenced by the capping/decapping reaction constants, while it is almost not influenced by the initer decomposition parameters. The number average molecular weight exhibits sensitivity to parameters similar to that of conversion, except for the fact that it is more sensitive to the initer decomposition constants, perhaps due to the influence of these parameters in the number of active centers formed. For the weight average molecular weight it is noticeable the increase in importance of the capping/decapping reaction constants. This is also reflected in the fact that these parameters are practically the most important ones for polydispersity. A rapid rate of exchange is considered to be the determining factor for getting polymer chains growing at approximately the same rate. Figure 9 shows the sensitivity of conversion to the kinetic parameters. The propagation rate constant turns out to be the most important factor followed by the initiation and the termination constant (negative sensitivity). Conversion is moderately influenced by the capping/decapping reaction constants, while it is almost not influenced by the initer decomposition parameters. The number average molecular weight exhibits sensitivity to parameters similar to that of conversion, except for the fact that it is more sensitive to the initer decomposition constants, perhaps due to the influence of these parameters on the number of active centres formed. For the weight average molecular weight it is noticeable the increase in importance of the capping/decapping reaction constants. This is also reflected in the fact that these parameters are practically the most important ones for polydispersity. A rapid rate of exchange is considered to be the determining factor for getting polymer chains growing at approximately the same rate. An additional conclusion that can be drawn by examining Fig. 9 is that all the sensitivity curves for parameters ktp and krd are symmetrical, indicating that these parameters cannot be independently estimated from the outputs shown here. The ratio of them (equilibrium constant) can be estimated but not single values; independent experiments are required. Figure 10 is virtually identical to Fig. 9, indicating that sensitivities obtained at diff"erent reaction conditions are practically the same. This parametric sensitivity study confirms the importance of the dormant/living exchange parameters on the broadness (polydispersity) of the molecular weight distribution. An additional interesting conclusion is that this variable (as well as others) are very little affected by the initer decomposition parameters (as long as they have values in the order of the case studied). On the other hand, it will be difficult to accurately recover (estimate) initer decomposition parameters based on conversion or polydispersity data alone, although average molecular weights will be more useful for this task.

33 REFERENCES 1. N.A.Dotson, R.Galvan, R.Laurence and M.Tirrell, Polymerization Reactor Modeling, John Wiley, 1996. 2. C.M.Villa and W.H.Ray, Chemical Engineering Science, 55,2,275-290 (1999) 3. J.CVerazaluce-Garciay A. Flores-Tlacnahuac and E.Saldivar-Giierra, Industrial and Engineering Chemistry Research, 39,6,1972-1978 (2000) 4. M.K.Georges,G.K.Hamer and N.A.Listigovers, Macromolecules,31,(1998),9087-9089 5. H.Shinoda, RJ.Miller, K.Matyjaszewski, Macromolecules, 34(10),(2001),3186-3194 6. G.Odian, Principles of Polymerization, Wiley, 3rd Ed., (1991) 7. J.Texter and M.Tirrell, AICHE J,47,8,1706-1710 (2001) 8. K.Matyjaszewski, Overview: Fundamentals of Controlled/Living Radical Polymerization in Controlled Radical Polymerization, Edited by K.Matyjaszewski, ACS Symp.Series 685 (1997) 9. W.H.Ray J.Macromol.Sci.-Revs.Macromol.Chem., C8(l),(1972)1-56 10. H.K.Khalil, Nonlinear Systems, 2nd Ed. Prentice-Hall (1996) 11. http://www.athenavisual.com

34

(a)

(b)

(c)

(d)

Figure 4: Open-loop dynamic simulation of the batch styrene reactor.

35

Time (h)

Figure 5: Radical concentration.

4.5

x10

4

3.5

^ - ^ ^ ^

1 3 CO

"c

8

^

^

-

-

^

^

^

^

•

§2.5

>>E 1 2

^ ^ ^ ^ ^ - :

CO

E

H 1

\

0.5

\ Time (h)

Figure 6: Dormant polymer concentration.

36

Time (h)

Figure 7: Dead polymer concentration.

37

— ___

K k' Kn

CK tc

(a)

(b)

(c)

(d)

Figure 8: Sensitivity coefficients for Q =.016 M and 110 "C.

38

x'"'

I

—

-

- —

1 °" - 1/^' 1

1

0.05

k^ k' k'

\\.

0.15|

k,.

o

O

^

y /"

"""

^

0 is a constant. The derivative of this function with respect to z/|.

iMiJK

--JtS

(38)

is called the influence function and goes to zero for large values of ul. Missing observations. The algorithms within the present parameter estimation scheme make it easy to handle missing observations, i.e. to account for missing values in the output vector yl when calculating the term (39)

49 in (33) for some i and some k. The usual way to account for missing or non-informative values in an extended Kalman filter is to formally set the corresponding elements of the covariance matrix S in (12) to infinity, which in turn gives zeroes in the corresponding elements of {Rk\k-i)~^ and the Kalman gain matrix Kk, meaning t h a t no updating will take place in (15)-(16) based on the missing values. This approach cannot be used when calculating (39), however, because a solution is needed which modifies el and RW-i to reflect t h a t the effective dimension of yl is reduced due to the missing values. This is accomplished by replacing (2) with the alternative measurement equation yk = E{h{xk,Uk,tk,0)

+ ek)

(40)

where E is an appropriate permutation matrix, which can be constructed from a unit matrix by eliminating the rows corresponding to missing values in yj^. If, for example, y^ has three elements, and the one in the middle is missing, the appropriate permutation matrix is E=^'

(41) ^ ^

' ''^ 0 0 1

Equivalently, the equations of the extended Kalman filter are replaced with the alternative output prediction equations tk\k-i

= Eh{xk\k-uUk,

nk\k-i

= ECPk\k-iC^E^

tk, 0) + ESE^

(42) (43)

the alternative innovation equation (44)

'^k = y~k-^k\k-i the alternative Kalman gain equation Kk ^ Pk\k-iC

(45)

E Rk\k-i

and the alternative updating equations ^k\k = Xk\k~i + KkSk

(46)

Pk\k = Pk\k-i - 'KkRk\k-iKl

(47)

whereas the state prediction equations are the same. These replacements in turn provide the necessary modifications of (39) to .

expr_i(4f(fi;

)-iej;^ (48)

^det (a^uk-i^ ^k\k-

(V2^)'

except for the fact t h a t / must also be reduced with the number of missing values in y^.

50 2.3. Statistical tests and residual analysis The third step in the proposed modelhng cycle deals with investigation of the properties of the model once the unknown parameters have been estimated. This step involves applying statistical tests and performing residual analysis to assess the quality of the model. 2.3.1. Statistical tests An estimate of the uncertainty of the parameter estimates can be obtained by using the fact that the sampling distribution of the estimator in (35) is asymptotically Gaussian with mean 6 and covariance Sfl = H-'

(49)

where the matrix H is given by

and where an estimate of H can be obtained by equating the expectation with the observed value, i.e.

This equation can therefore be used to approximate the covariance matrix E^, which can then in turn be used for calculating the standard deviations of the estimates and their correlation matrix via Eg = a-gRcT^

(52)

where CTQ is a diagonal matrix of the standard deviations and R is the correlation matrix. The validity of the approximation depends on the amount of data used in the estimation, because of the Gaussianity assumption is only assymptotically correct. However, it is the experience of the authors that the amount of data needed to be sufficiently close to the limiting Gaussian distribution is often moderate, so this is not a serious limitation. The asymptotic Gaussianity of the estimator also allows t-tests to be performed to test the hypothesis, ifo, that any given parameter is marginally insignificant. The test quantity is the value of the parameter estimate, 6j^ divided by the standard deviation of the estimate, cr^., and under HQ this quantity is asymptotically ^-distributed with a number of degrees of freedom that equals the number of data points minus the number of estimated parameters, i.e. z{ej) = ^

e ^3

t

^Ni-p

(53)

1=1

To test the hypothesis, HQ, that some parameters are simultaneously insignificant, several tests can be applied, e.g. a likelihood ratio test, a Lagrange multiplier test or

51 a test based on Wald's W-statistic. Under iJo these test quantities all have the same asymptotic x^-distribution with a number of degrees of freedom that equals the number of parameters to be tested for insignificance (Hoist et a/., 1992), but in the context of the proposed modelling cycle the test based on Wald's W-statistic has the advantage that no re-estimation is required. The statistic is computed as follows Wid,)=f,j:7%

e

x'''dini(0,)'i

(54)

where ^* C ^ is the subset of the parameter estimates subjected to the test and S^^ is the corresponding covariance matrix, which can be computed from the full covariance matrix as follows E^^ = E^i:^E

(55)

where E is an appropriate permutation matrix, which can be constructed from a unit matrix by ehminating the rows corresponding to parameter estimates not subjected to the test. Tests for insignificance are important in terms of investigating if the structure of the model is correct. In principle, insignificant parameters are parameters that may be eliminated, and the presence of such parameters is therefore an indication that the model is incorrect or over parameterized. On the other hand, because of the particular nature of models of the type (l)-(2), where the diffusion term a{') in (1) is included to account for process noise due to approximation errors or unmodelled inputs, the presence of significant parameters in this term is an indication that the drift term /(•) in (1) is not correct. More details about tests for significance and other tests can be found in Hoist et al (1992). 2.3.2. Residual analysis Another important aspect in assessing the quality of the model is to investigate its predictive capabilities by performing cross-validation and examining the corresponding residuals. Depending on the intended application of the model this can be done in both a one-step-ahead prediction setting or in a pure simulation setting. In either case a number of different methods can be applied (Hoist et al, 1992). One of the most powerful of these methods is to compute and inspect the sample autocorrelation function (SACF) and the sample partial autocorrelation function (SPACF) of the residuals to detect if there are any significant lag dependencies, as this indicates that the predictive capabilities of the model are not perfect. Nielsen and Madsen (2001) recently presented extensions of these inherently linear tools to nonlinear systems in the form of the lag-dependence function (LDF) and the partial lag-dependence function (PLDF), which are based on the close relation between correlation coefficients and values of the coefficients of determination for regression models and extend to nonhnear systems by incorporating nonparametric regression. More specifically, to derive the LDF, which is a generalization of the SACF, the equivalence between the squared correlation coefficient between the stochastic variables Y and Xk, i.e. V{Y] - V{Y\Xk} ji Po{k) = V{V}~

^^^^

52 and the coefficient of determination of a linear regression of observations of Y on observations of Xfc, i.e. 2

_ SSQ - g5o(fc)

^«(^)-

5 ^ —

(^^)

is used. In (57) SSQ = Y^^iiVi — X^^ |f)^ and S'S'o(fc) is the sum of squares of the residuals from the regression, and the equivalence is due to i?Q/^N being the ML estimate of pQ/^^ when Gaussianity is assumed. For a time series of observations of a stationary stochastic process {Xt}, the squared SACF at lag k is equivalent to the squared correlation coefficient pg/^x between Xt and Xt-k^ and it can therefore be closely approximated by the corresponding value of R^rj^) obtained via a linear regression of observations of Xt on observations of Xt-k- Replacing the linear regression with a nonparametric fit of the conditional mean fk{x) = E{Xt\Xt-k = ^ } , e.g. by using a locally-weighted polynomial smoother, the LDP can be defined as a straightforward extension of the SACF, i.e. LDF(fc) = s i g n ( A ( 6 ) - A ( a ) ) ^ % J

(58)

where a and b are the minimum and maximum over the observations, and where Rhj^v is the corresponding value of the coefficient of determination. The sign is included to provide information about the average slope. To derive the PLDF, which is a generalization of the SPACF, the equivalence between the squared partial correlation coefficient between the stochastic variables (l^|-^i, •.., Xk-i) and(Xfc|Xi,...,XA;_i), i.e. 2 _V{Y\Xu--.,Xk-i}-V{Y\X,,...,Xk} P(o.)|(i,...,.-i) - — ]/{y|Xi,...,X,_i}

,_„. ^^^^

and the coefficient of determination

is used. In (60) 55O(I,...,A;-I) is the sum of squares of the residuals from a linear regression of observations of Y on observations of ( X i , . . . , Xk-i) and 550(1,...,A;) is the sum of squares of the residuals from a linear regression of observations of Y on observations of ( X i , . . . , X^), and the equivalence is due to Rfok)\(i,. .,k-i) being the ML estimate of P?o/e)|(i,...,A;-i)/^^^^ Gaussianity is assumed. For a time series of observations from a stationary stochastic process {X^}, the squared SPACF at lag k is equivalent to the squared partial correlation coefficient plok)\(i,...,k-i) between (Xt|Xt_i,... ,Xt_(/c-i)) and {Xt-k\Xt-i,...,Xt-(k-i)), and it can therefore be closely approximated by the corresponding value of ^?OA;)|(I,...,A;-I) obtained via linear regressions of observations of Xt on observations of (X^-i,..., Xt-(k-i)) and on observations of (Xt_i,..., Xt-k), i-e. via fits of the AR model Xt = (f)jQ + (l)jiXt-i +

h (j^jjXt-j -\-et

(61)

53 for j = k - l,k. Replacing the AR models with additive models (Hastie and Tibshirani, 1990), i.e. Xt = fjo + fn {Xt-i) + ••• + fn (Xt-j) + et

(62)

for j = A; — 1,fc,where each fji is fitted nonparametrically, e.g. by using a locally-weighted polynomial smoother, the PLDF can be defined as a straightforward extension of the SPACF, i.e. PLDF(fc) = sxgniMb)

- Ma)).

%^^^^^^

(63)

where a and b are again the minimum and maximum over the observations, and where Rfok)\(i k-i) ^^ ^^^ corresponding value of the coefficient of determination. Again, the sign is included to provide information about the average slope. Being an extension of the SACF, the LDF can be interpreted as being, for each A:, the part of the overall variation in the observations of X^, which can be explained by the observations of Xt-k- Likewise, being an extension of the SPACF, the PLDF can be interpreted as being, for each k, the relative decrease in one-step-ahead prediction variation when including Xt-k as an extra predictor. However, unlike the SACF and SPACF, the LDF and PLDF can also detect certain nonlinear dependencies and are therefore extremely useful for residual analysis. More details about the LDF and PLDF and other similar tools can be found in Nielsen and Madsen (2001). Details about locallyweighted polynomial smoothers and additive models can be found in Hastie and Tibshirani (1990) and Hastie et al (2001). 2.4. Model validation or invalidation The last step in the proposed modelling cycle deals with model validation or invalidation, or, more specifically, with whether, based on the information gathered in the previous step, the model is invalidated with respect to its intended application or not. If the model is invalidated, the modelling cycle is repeated by first changing the structure of the model in accordance with the information gathered in all steps of the previous cycle. 2.5. Software implementation The parameter estimation scheme described in Section 2.2 and some of the features for applying statistical tests and performing residual analysis described in Section 2.3 have been implemented in a software tool called CTSM (Kristensen et a/., 2001), which is an extension of the tool presented by Madsen and Melgaard (1991). 3. CASE STUDY: MODELLING A FED-BATCH BIOREACTOR To illustrate how the proposed modelling cycle can be used to develop or improve the quality of a first engineering principles model, a simple simulation example is given. The process considered is a fed-batch bioreactor described by a simple unstructured model of

54 biomass growth, i.e. (dX\ ( f^{S)X-^ \ ^ rf5 I = • - ^ + ( ^ £ ^ ,dt+

0

F

dV

hi\ y^

fan ,0

/X\ k

^

/eA

/e,\ k

^^

k

^^

0 (722 0

Ol 0 ,du;t ,tG

[0,3.8]

(64)

(733

/N(0,S„)\ k

^(O'^^aa)

where X is the biomass concentration, S is the substrate concentration, V is the volume of the fermenter, F is the feed flow rate, Sp (=10) is the feed concentration of substrate, Y (=0.5) is the yield coefficient of biomass and /i(S') is the growth rate, for which three different cases are considered, i.e. • A model structure with linear kinetics: M(S) = ^imaxS

(66)

• A model structure with Monod kinetics: S (67)

f^{S) = fimaxj^^ • A model structure with Monod kinetics and substrate inhibition: ^^^) = ^-K,S^

+ S + K,

^''^

In the following the model consisting of equations (64), (65) and (68) with K2 = 0.5, i.e. the model with Monod kinetics and substrate inhibition, is regarded as the true process to be modelled, and using the true parameter values in the top row of Table 3 the two data sets shown in Figure 3 are generated. Each data set consists of 101 equidistant samples of F , 2/1, 2/2 and ?/3 generated by stochastic simulation using a Milstein scheme with F being perturbed along an analytically determined optimal trajectory corresponding to maximum biomass productivity. The noise levels used are typical for real experiments of this type, if not slightly exaggerated. In the following, the data set on the left hand side of Figure 3 is used for estimation and the data set on the right hand side is used for vahdation. It is assumed that the intended purpose of the model to be developed is simulation or infinite-horizon prediction, e.g. for use in a model predictive controller (MFC). The essential performance criterion for the model to be developed is therefore that it has good prediction capabilities in pure simulation. 3.1. First cycle: Linear kinetics The model consisting of equations (64), (65) and (66), i.e. the model with linear kinetics, is regarded as an existing first engineering principles model to be investigated. In the context of the first step of the modelling cycle this is therefore the basic model.

55

.......O-il 1.5

2

ul>«i^ 2

2.5

2.5

(b) Validation data.

(a) Estimation data.

Figure 3. The data sets used for estimation y2, dash-dotted: ys).

and validation.

(Solid: F, dashed: yi, dotted:

Moving to the second step, the unknown parameters of the model are estimated with C T S M (ML on the estimation data set), and this gives the results shown in Table 1. Moving to the third step of the modelling cycle, Table 1 also includes the t-scores for performing tests for marginal insignificance. These tests show that ass and S22 are marginally insignificant, whereas all other parameters are marginally significant, including an and a22- Recalling t h a t the presence of significant parameters in the diffusion term o'(-) in (1) is an indication of approximation errors or unmodelled inputs in the drift term / ( • ) in (1), this in turn indicates that the drift terms of the equations for X and S in (64) are not correct in terms of describing the variations in the estimation data set. Otherwise an and (J22 would also have been insignificant. To investigate this further, residual analysis is performed. The left part of Figure 4 shows one-step-ahead prediction results on the validation data set and Figure 5 shows the SACF, SPACF, LDF and P L D F for the corresponding residuals. The one-step-ahead prediction results show discrepancies between the true and predicted values of yi and 2/2, which is confirmed by inspecting the corresponding residuals. Furthermore, the SACF,

Table 1 Estimation

results using linear

Parameter True value Estimate Std. Dev. t-score Significant

XQ 1 1.053 0.054 19.45 Yes

SQ 0.245 0.244 0.030 8.136 Yes

kinetics. VQ 1 1.000 0.011 93.63 Yes

fimax

(y\i

(^22

(^2,2,

0.804 0.052 15.33 Yes

0 0.442 0.061 7.245 Yes

0 0.451 0.035 13.06 Yes

0 0.000 0.000 0.000 No

-

Sii

0.01 0.004 0.001 3.067 Yes

0.001 0.000 0.000 0.000 No

0.01 0.011 0.001 7.221 Yes

56

(a) One-step-ahead prediction.

(b) Pure simulation.

Figure 4. Cross-validation (CV) results for the model structure with linear kinetics. (Solid: predicted values, dashed: true yi, dotted: true y2, dash-dotted: true ys).

SPACF, LDF and PLDF all reveal a significant lag dependence at lag 1 in the yi residuals. Especially the LDF and PLDF also reveal several significant lag dependencies in the y2 residuals, whereas in the ys residuals there are no significant lag dependencies. These results all provide additional evidence to suggest that the drift terms of the equations for X and S in (64) are not correct. A final piece of evidence that something is wrong is gathered from the pure simulation results in Figure 4. Moving to the last step of the modelling cycle, the information now available clearly invalidates the model, and the cycle must therefore be repeated by modifying the structure of the model. 3.2. Second cycle: Monod kinetics The information available suggests that it is the drift terms of the equations for X and S that need to be modified, i.e. precisely those parts of the model that depend on /x(5). Replacing (66) with (67) to yield a model with Monod kinetics and re-estimating the unknown parameters with CTSM, the results shown in Table 2 are obtained.

Table 2 Estimation results using Monod kinetics. Parameter True value Estimate Std. Dev. t-score Significant

XQ 1 1.042 0.014 72.93 Yes

SQ 0.245 0.250 0.010 24.94 Yes

VQ 1 0.993 0.001 689.3 Yes

fimax

-

0.737 0.008 96.02 Yes

Ki

-

0.003 0.001 2.396 Yes

(Jii

Cr22