Simple Control Scheme for Batch Time Minimization of Exothermic

Article pubs.acs.org/IECR

Simple Control Scheme for Batch Time Minimization of Exothermic Semibatch Polymerizations Tiago F. Finkler,*,†,‡ Sergio Lucia,† Melik B. Dogru,† and Sebastian Engell† †

Process Dynamics and Operations Group, Technische Universität Dortmund, 44221 Dortmund, Germany Evonik Industries, 63457 Hanau, Germany

‡

ABSTRACT: Polymerization processes are typically characterized by strongly exothermic reactions with complicated nonlinear kinetics, and they usually require a very precise temperature control to maintain product uniformity. Polymerization reactors are usually operated with a constant monomer feed, while a cascade of PI controllers manipulates the heating and cooling power to keep the reaction temperature within the tolerance limits. Because a robust operation has to be guaranteed for different products under various disturbing influences, for example, fouling in the reactor, oscillations in the supply of the utilities and feed impurities, the monomers are usually slowly dosed into the reactor, leading to large batch times and limiting the process productivity. In this work, a simple and efficient control scheme for the time optimal operation of exothermic semibatch reactors that can be easily implemented in industry is introduced and evaluated by means of simulations using a well-known industrial benchmark problem. The results show that the new control scheme can shorten the reaction period by more than fifty percent without jeopardizing the quality of the temperature control and performs more robustly than a fully model-based controller that employs a fixed nominal process model.

1. INTRODUCTION Semibatch reactors are widely used for the production of many different types of products, for example, polymers, fine chemicals, pigments, and pharmaceuticals. As the reactions involved in such processes are often strongly exothermic and very precise control of the reaction temperature is required to ensure that the end product will have acceptable quality, for example, because of undesired side reactions, those reactors are usually equipped with a cooling jacket that has to remove the reaction heat to keep the reaction temperature within the tolerance bounds. In industrial reactors, this control problem is usually addressed by a cascade of PI controllers in which a slave controller tracks the set point for the jacket temperature that is provided by a master controller that controls the reactor temperature. This cascade control structure (CCS), which has been widely used in the last decades, has shown to be able to provide a robust operation under a variety of disturbing influences, for example, a changing environment temperature in winter and summer, presence of impurities in the raw materials, and oscillations in the utilities supply. However, to guarantee robustness under many uncertain scenarios, usually the monomers are fed slowly into the reactor and performance is sacrificed. A well-known industrial benchmark problem on reactor temperature control that exposes this challenge was introduced by Chylla and Haase.1,2 During the last years, a lot of work has been done to develop new schemes for improving the control of exothermic batch processes. In particular, several contributions addressed the case where the system is operated with constant monomer feed and the CCS is replaced by advanced structures so that the temperature control is improved. For example, in Gattu and Zafiriou,3 the nonlinear quadratic dynamic matrix algorithm was compared with the classical CCS solution for controlling the reactor temperature during a batch. In Helbig et al.,4 a control scheme was presented where an extended Kalman Filter is used to estimate © 2013 American Chemical Society

the amount of heat generated and the heat transfer coefficient along the batch and, based on those estimates, a nonlinear model predictive controller (NMPC) takes care of tracking the desired reaction temperature by manipulating the cooling power under the assumption that the heat generation will remain constant over a given prediction horizon. In Hinsberger et al.,5 the temperature control of semibatch reactors was analyzed from the point of view of optimal control theory, and a neural network based scheme was used in Bhat and Banavar.6 A control system based on a two cascade input-output linearizing controllers was proposed in Tyner et al.7 More recently, an adaptive feedforward control scheme and an exact linearizing controller were proposed in Graichen et al.8 and Beyer et al.,9 and a latent variable based model predictive control scheme was proposed in Shamek and Lennox.10 Motivated by the fact that the operation with constant monomer feed is a conservative approach that leads to long durations of the batch runs, a considerable amount of effort has also been made to identify alternative control schemes for semibatch polymerizations in which not only the ideal reaction temperature is tracked but the monomer dosage is optimized along the batch as well. For example, in Terwiesch et al.11 and Gentric et al.,12 different optimization schemes are proposed where optimal trajectories for the jacket temperature and optimal monomer inlet flow rates are computed offline and then incorporated into the operation recipe. In François et al.,13 a run-to-run optimization scheme for emulsion batch polymerizations was proposed. Several model-based schemes for online optimization of batch processes have been introduced as well. In Helbig et al.,14 Received: Revised: Accepted: Published: 5906

August 29, 2012 March 7, 2013 March 12, 2013 March 13, 2013 dx.doi.org/10.1021/ie302321k | Ind. Eng. Chem. Res. 2013, 52, 5906−5920

Industrial & Engineering Chemistry Research

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simulations where a model predictive control scheme was used to simultaneously optimize the monomer feed and the cooling usage in a semibatch reactor were reported. In Gesthuisen et al.15 and Vicente et al.,16 model-based control schemes with calorimetric state estimators that were used to calculate the monomer holdup and drive the reaction at the high reaction rates are presented, while in François et al.17 an optimization scheme where a model of the optimal solution is updated from batch to batch was proposed. In Mauntz,18 a state estimator based control scheme for the time-optimal operation of semibatch emulsion polymerizations was proposed and tested in a laboratory reactor. In Finkler et al.19 a NMPC (nonlinear model predictive control) scheme for the online optimization of an industrial semibatch polymerization was developed. Other online schemes where necessary conditions for optimality are tracked have also been reported.20−22 Several surveys that summarize the techniques for the optimization of batch and semibatch processes can be found in the literature.11,23 Despite all the effort that has already been made for developing high performance schemes for the temperature control of semibatch reactors, the operation strategy in which the monomers are inserted into the reactor with constant flow rate while the CCS takes care of keeping the reactor temperature at the desired set point is still by far the most popular solution that is applied in the industry nowadays. An important reason for this undoubtedly is the significant effort that is needed to develop and to implement model-based solutions in real production units. Therefore there is still a need for a suitable control solution that provides high operational performance and, at the same time, can be easily implemented in industry. In this work, instead of employing complex model-based solutions, a simple and efficient control scheme that aims at the time-optimal operation of exothermal semibatch reactors is introduced. In the proposed scheme, the conventional CCS is combined with an additional PI controller that manipulates the monomer feed in order to bring the cooling power as close as possible to the limit of the capacity of the cooling system, which is quantified by the position of the control valve that manipulates the flow of the cooling medium. This alternative control scheme is investigated here using the Chylla−Haase Benchmark Reactor (CHBR) to demonstrate its robustness. The results show that it is possible to reduce the duration of the reaction period by more than fifty percent without jeopardizing the quality of the control of the reactor temperature in the presence of significant uncertainties. This paper is organized as follows. First, in section 2 the original CHBR problem is presented, the process model is explained and the robustness issues that arise are introduced. A modified version of the CHBR problem in which variable monomer inlet flow rates can be employed is proposed. In section 3, the standard PI cascade solution is revisited, and a reference NMPC solution is presented in section 4. The proposed optimizing control scheme is discussed and evaluated in section 5. The paper is concluded in section 6.

Figure 1. Chylla−Haase benchmark reactor.

exchange system can be used either to cool the reactor down or to heat it up. If the system is operated in the heating mode, a steam valve that manipulates the injection of medium-pressure steam into the recirculation water is activated. If the system is in the cooling mode, a dumping valve that manipulates the amount of cold water that enters the recirculation loop is activated. The dumping valve regulates the amount of hot water that leaves the recirculation loop and, when hot water is discharged, the same amount of cold water is admitted at the jacket inlet through a pressure regulator. As it is usual in the polymer industry, this reactor is used to produce different products, that is, polymer grades, which are obtained through different recipes. Generally all the recipes can be described by the following steps that may or not be repeated: (1) charging, an initial charge of prepolymer, surfactants, and water is inserted into the reactor; (2) heating, the reactor is heated up until it reaches a specific temperature Tspec at which the reaction should take place; (3) feeding, monomer is added to the reactor with constant flow rate for a given period; and (4) holding, the reactant mixture is kept inside the reactor at the specified temperature during a hold period to achieve higher monomer conversions. The different polymer grades are mainly characterized by the polymer composition and the particle size distribution. As these properties are highly sensitive to the temperature at which the polymerization takes place, a very precise control of the inner reactor temperature is required in order to guarantee that the final product will have an acceptable quality. The challenge is to guarantee precise and robust temperature control under a variety of uncertainties and disturbing influences, for example, changing environmental temperatures in winter and summer, presence of impurities in the raw-materials and oscillations in the utilities supplied. In the original paper in which this benchmark problem was proposed,1 it was stated that the temperature must lie within a tight tolerance range of ±0.6 K around the specified reaction temperature. 2.2. Modified Chylla−Haase Benchmark Reactor. In the original version of the CHBR problem, there is no degree of freedom to shorten the batch duration, because the monomer is fed into the reactor with a constant flow rate. As the main purpose of this paper is to investigate the potential of a reduction of the batch time by an improved control scheme that manipulates the monomer dosage, a modified version of the benchmark problem in which the monomer feed can vary freely

2. DESCRIPTION OF THE BENCHMARK PROBLEM 2.1. Original Chylla−Haase Benchmark Reactor. The Chylla−Haase benchmark reactor has been widely used as a reference problem for the evaluation of alternative schemes for the temperature control of semibatch polymerization reactors. This polymerization process, which is illustrated in Figure 1, mainly consists of a stirred tank reactor where an emulsion polymerization reaction takes place and a heat exchange system that is composed of a jacket and a recirculation loop. The heat 5907

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from 0 to 22.7 × 10−3 kg/s during the feeding step is introduced here. If the Chylla−Haase reactor is considered as a semibatch reactor where a simple exothermic reaction takes place, the proposed modification should not affect the properties of the final product, as long as the temperature control is good enough to ensure that the reaction temperature does not violate the tolerance range during the feeding period, when almost all of the polymer is formed. For more complex emulsion polymerization processes with several monomers, the product properties might be affected if the system is driven along different monomer concentrations trajectories during the batch. One possibility to take care of the product properties in this case would be to adapt the product recipe to the new trajectories. This issue however is beyond the scope of this work. In contrast to the original version of the benchmark problem, where 32 kg of monomer has to be fed into the reactor during the feed period, an overall monomer load of 45 kg is considered here. This last modification is proposed to extend the feeding period into the region where the heat transfer coefficient reaches its lower limit (see section 2.3 for more details) and to investigate how the proposed control scheme performs in this region. 2.3. Process Modeling. A dynamic model that describes the monomer and polymer hold-ups, as well as the dynamics of the relevant temperatures, during the batch was presented by Chylla and Haase.1,2 In recent years, this model has been extensively investigated by the process control community and several mistakes in its formulation were identified and corrected. In Gattu and Zafiriou’ work,3 corrections in the energy balance for the reactor temperature and corrected values for the initial reactor charge were given. To have a more realistic description of the heat transfer phenomena, Graichen et al.8 and Beyer et al.9 recommended that a lower bound of 0.2 kW m2 K−1 should be imposed to the heat transfer coefficient. Moreover, the derivation of the energy balance equation for the recirculation loop was questioned in Hinsberger et al.5 However, as the model was experimentally validated, the original equation for describing the recirculation loop dynamics has been used in several recent works8,9 under the reasonable justification that it is an empirical relation that satisfactorily describes the system behavior. The model used in this work consists of a set of ODEs given by eqs 1−5.

ṁ M = FM − rP

(1)

ṁ P = rP

(2)

Ṫ =

make no attempt to describe the polymerization mechanism or the particle size distribution of the final product, but they can describe the conversion versus time behavior accurately. This allows one to compute the amount of heat that is produced by the reaction during the batch, which is sufficient for temperature control purposes. Detailed information about the values of the model parameters can be found in different sources in the literature1,2,8,9 and in the table given in Appendix B of this contribution as well. The heating/cooling function KP is a function of the valve position with the split-range characteristic given by eq 6. ⎧ 0.8 × 30−c /50(T − T in), c < 50% inlet J ⎪ ⎪ c = 50% KP(c) = ⎨ 0, ⎪ c /50 − 2 in ⎪ 0.15 × 30 (Tsteam − TJ ), c > 50% ⎩

2.4. Modeling of the Uncertainties. In Chylla and Haase,1 various uncertainties and disturbances were introduced in the model such that a variety of scenarios can be simulated to test the robustness of different control schemes. Moreover, information data for two different products, polymer A and polymer B, were also given. The investigations presented in this contribution are restricted to product A only and the following three scenario variables are considered: (1) The purity factor i varies from 0.8 to 1.2 so that the fluctuations in the reaction rate caused by impurities in the raw-materials can be described. It changes randomly from batch to batch, but it is constant during one batch. (2) The fouling factor 1/hf varies from zero to 0.704 m2 K kW−1 to simulate the decrease in U resulting from the formation of a polymer film on the reactor wall during successive batches. (3) The ambient temperature Tamb, which affects the monomer inlet feed and the initial conditions T(0), Tjin(0), and Tjout(0), can vary from 280 to 305 K (from winter to summer). By combining the upper and lower values of these scenario variables, one gets the eight extreme case scenarios that are presented in Table 1. The robustness of the new optimizing Table 1. Scenarios Considered for the Robustness Analysis

⎡ ⎢FMCp,M(Tamb − T ) + UA(Tj − T ) ∑i = M,P,W miCp, i ⎣ 1

+ UA loss(Tamb − T ) + out T ̇j =

rPΔHP ⎤ ⎥ MWM ⎦

(3)

(4) in out Tj̇ = Tj̇ (t − θ2) +

− θ2) − τP

T jin

+

KP(c) τP

scenario

i

1/hf (m2 K kW−1)

Tamb (K)

1 2 3 4 5 6 7 8

0.8 1.2 0.8 1.2 0.8 1.2 0.8 1.2

0 0 0.704 0.704 0 0 0.704 0.704

305.38 305.38 305.38 305.38 280.38 280.38 280.38 280.38

control scheme that is introduced in section 5 is evaluated by simulating these eight scenarios. In Chylla and Haase,1 it is suggested that dead times θ1 and θ2 for the cooling jacket and the recirculation loop may vary by ±25% around the nominal values of 22.5 and 15 s. However, it turned out during the simulations that these fluctuations on the transport delays do not affect the results much. Hence, for the sake of brevity, only the results corresponding to the nominal values of θ1 and θ2 are presented in this work.

1 [FCC p,C(T jin(t − θ1) − T jout) + UA(T − Tj)] mCC p,C

T jout(t

(6)

(5)

All the variables and parameters that appear in the model equations are listed at the end of this paper in the section entitled “List of Variables and Parameters of the CHBR”. The empirical relations that are used for the computation of rP and UA as a function of the system state along the batch are given by eqs 12−20 in Appendix A. According to Chylla and Haase,1 these simplified kinetic relations

3. ORIGINAL CASCADE CONTROL STRUCTURE In the original control solution for the Chylla−Haase benchmark problem, the system is operated using a constant monomer flow rate during the whole feeding period while a cascade of PI 5908



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Figure 2. Block diagram for the original cascade control structure.

cooling water will not evaporate or freeze inside the jacket, the set point for the jacket temperature is limited within the range of 275−370 K. The results of an investigation of higher monomer flows with the goal to reduce the feeding period using the CCS for temperature control are presented in Figure 3. In these simulations, the

controllers is used to track the trajectory for the reactor temperature given by eq 7. ⎧ 5 ⎛ t ⎞i ⎪ ⎟ , t ≤ theat ⎪Tamb + (Tspec − Tamb) ∑ ai⎜ t Tset = ⎨ i = 3 ⎝ heat ⎠ ⎪ ⎪Tspec , t > theat ⎩

(7)

The cascaded control structure (CCS), which is illustrated by the block diagram in Figure 2, is configured such that the master controller regulates the reactor temperature by manipulating the set point of the slave controller, which controls the recirculation water temperature by acting on the heating/cooling usage. As the investigations of this paper focus on the reduction of the batch time, the original cascade control structure is tested for cases where the value of ṁ inM is increased so that the feeding period can be shortened. Moreover, the smooth temperature set-point trajectory in eq 7 was changed to a simple step trajectory. In addition, as an attempt to improve the overall control performance, the slave and the master controller were retuned by applying the methodology proposed in Huang et al.24 After a fine-tuning procedure, the gain KpS and the integral time constant TIS of the slave controller were adjusted to −60 and 15 s, while the gain KpM and the integral time constant TIM of the master controller were tuned to −20 and 75 s, respectively. The master and the slave controllers are implemented in a discrete fashion using the velocity form of the PI algorithm. Every sampling period, a new set point for the jacket inlet temperature is computed by the master controller according to eq 8 and a new position for the cooling valve is computed by the slave controller according to eq 9. ⎛ eTi − eTi − 1 eTi ⎞ T J,in,set = T J,in,set + ⎟ i i − 1 + KpM ⎜ Δt TIM ⎠ ⎝

(8)

⎛ eT J,seti − eT J,seti − 1 eT J,seti ⎞ ⎟ + ci = ci − 1 + KpS⎜⎜ Δt TIS ⎟⎠ ⎝

(9)

Figure 3. Temperature control for different monomer inlet flow rates.

performance of the temperature control is compared for the cases where ṁ inM is set to 7.56 × 10−3, 15.1 × 10−3, and 22.7 × 10−3 kg/s. As 45 kg of monomer have to be fed into the reactor during the feeding period, the corresponding durations of the feeding period are 99.2, 49.6, and 33.1 min, respectively. In the simulations, a measurement noise with a standard deviation of 3.33 × 10−3 K is assumed in the temperature measurements. As it can be seen from Figure 3, during the whole feed period the cascade of PI controllers is capable to robustly operate the system as long as the monomer inlet flow is 7.56 × 10−3 or 15.1 × 10−3 kg/s. However, when FM is bigger than 15.1 × 10−3 kg/s, a temperature rise that largely violates the upper tolerance limit can be observed at the end of the feeding period. This temperature rise is a consequence of the continuous decrease in UA due to the polymer formation along the batch, which affects the closed-loop system as a ramp-like disturbance, as it can be seen in Figure 4. At large monomer inlet flow rates this disturbance becomes considerably faster and cannot be rejected well by the temperature controller. In the extreme case where FM is equal to 22.7 × 10−3 kg/s, this effect is amplified by the higher reaction rate and the cooling valve ends up being

The variable eTi denotes the control error of the master temperature control loop at the sampling period i, that is, the difference between the reactor temperature Ti and its set point Tspec while the variable eTinJ,i denotes the control error of the slave temperature control loop at the sampling period i, that is, the difference between the jacket inlet temperature TinJ,i and its set point Tin,set J,i . The variable Δt denotes the sampling period, which is set to two seconds. The position of the cooling valve is bounded within the range 0−100%, and to guarantee that the 5909



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saturated at the end of the feeding period, which drives the system out of control for more than 20 min. In all the three cases, the reaction temperature violates the lower tolerance limit at the beginning of the holding period. The occurrence of temperature undershoots after the monomer

feed suddenly stops is a well-known limitation of this classical PI cascade that has been already reported in the literature.8,9 However, as almost all the monomer (more than 99%) has already been converted into polymer at the point in time when the lower temperature constraint is violated, it is actually very unlikely that these relatively small temperatures undershoots will have a relevant effect on the product properties. Hence, although some advanced control schemes4,8−10 that can completely eliminate those undershoots have been discussed, no special attention is given to this issue in this work. The main challenge here is to find a control structure that can operate the system at large monomer inlet flow rates and reject disturbances resulting from the changes in UA and rP so that a fast and robust operation during the feeding period can be guaranteed.

Figure 4. Reaction rate, heat transfer coefficient and conversion for several monomer inlet flows.

4. REFERENCE NMPC SOLUTION To get a reference solution for evaluating the performance of the proposed control structure, an ideal NMPC scheme that computes the optimal monomer feed trajectory along the batch is introduced in this section. The NMPC scheme is implemented in such a way that it takes care of both keeping the reaction temperature within the desired tolerance range and bringing the monomer inlet flow rate as close as possible to its maximal value. This is realized by repeatedly solving an open-loop

Figure 5. Reference NMPC solution for the nominal case.

Figure 7. Comparison between OCS and NMPC solution (nominal scenario, cset = 30%).

Figure 6. Block diagram of the proposed control scheme. 5910



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optimal control problem where a set of discrete control movements that minimize the objective function J over a discrete prediction horizon are determined. After every sampling period, the first control movement is applied to the system, the control and prediction horizons are shifted, the controller is reinitialized and the open-loop optimal control problem is solved again. As only temperature measurements are available, in a real application it would be necessary to use a state estimator, for example,

an EKF, to reconstruct the state vector along the batch. A detailed discussion on state estimation for the CHBR can be found in the literature.8,9 Since the NMPC is here employed as reference solution to evaluate the performance of the proposed control structure, full state measurement is assumed. The mathematical formulation of the optimal control problem including temperature tracking and monomer feed maximization is given by eq 10

⎧ NP ⎪∑ wT (Ti − Tspec)2 + wc (ci − ci − 1)2 , t < theat or i i ⎪ ⎪ i=1 min J = ⎨ ci , FMi ⎪ NP w (T − Tspec)2 + wMi(F Mi − FMmax )2 + wci(ci − ci − 1)2 , otherwise ⎪ ⎪∑ Ti i ⎩ i=1

s.t. 0 ≤ ci ≤ 100, 0 ≤ F Mi ≤ FMmax ,

t

FM dt ≥ 45 kg

heat

(10)

optimizer, and to guarantee that the solution of the optimization problem respects the process model, the model equations are incorporated to the optimization problem as additional equality constraints, which are integrated by an implicit first order method with constant integration step. The sampling period was set to thirty seconds. The control horizon is set to one minute (NC = 2) and the prediction horizon is set to two and a half minutes (NP = 5). The optimization problem for computing the optimal control movements is solved by the solver SNOPT from the TOMLAB suite. It is beyond the scope of this paper to go deep into the numerical implementation of the NMPC scheme. For additional information on these issues, the reader is referred to Lucia et al.,27 where the numerical implementation of a NMPC scheme for the CHBR is discussed. The simulation results with the performance of the NMPC for the nominal scenario (purity factor equal to one, 1/hf = 0.352 m2 K kW−1, and summer season), that is, without any plant-model mismatch, are shown in Figure 5. As it can be seen from the plots, the NMPC scheme reduces the feeding period duration from 99.2 to 38.5 min when compared to the original CCS. The overall qualitative behavior of the reference solution can be described as follows. During the first part of the feeding period, the controller increases the monomer feed to its maximum value and adjusts the cooling usage such that the desired reactor

i = 1 , ..., NC i = 1 , ..., NC

Tspec − 0.6 < Ti < Tspec + 0.6,

∫t

i = 1 , ..., NP

where NP denotes the number of discretization points in the prediction horizon and NC denotes the number of discretization points in the control horizon (input moves). The variables wTi, Wci, and WMi are weighting factors which were set to 125, 75, and 125, respectively. By the choice of the cost function, the monomer feed is maximized as it is at the same time constrained to be less than the maximal value. The penalty term on the temperature error could be dropped and only the hard constraint could be implemented, but the quadratic term drives the temperature to the middle of the tolerance band and thus improves the robustness to model errors and disturbances. This NMPC controller was implemented in Matlab using a simultaneous approach.25,26 In this implementation, the dynamic optimization problem is formulated so that the system states and the control movements at each sampling period along the prediction horizon are degrees of freedom for the 5911



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Figure 10. Simulation results for Cset = 10% (blue dashed line, ideal NMPC; pink dotted line, standard NMPC; green solid line, OCS).

injects steam into the jacket in order to compensate the sharp decrease in the reaction rate that is caused when the monomer feed is abruptly stopped. As the ideal NMPC computes the optimal control action based on the exact same model that is used to simulate the plant and there is no uncertainty in the computation of the optimal control movements (it is assumed that the true values of i anf hf are known) it is reasonable to expect that it performs close to the best possible achievable

temperature is tracked. As it was discussed before, the heat transfer coefficient decreases continuously along the batch due to the polymer formation. Therefore, the cooling usage c does not stabilize but it approaches zero as the feeding period advances. When the cooling valve eventually saturates, the controller keeps the reactor temperature as close as possible to its set point by adjusting the monomer inlet flow accordingly. Just before the feeding period ends, the predictive controller stops cooling and 5912



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Figure 11. Simulation results for Cset = 20% (blue dashed line, ideal NMPC; pink dotted line, standard NMPC; solid green line, OCS).

robust performance. In fact, apart from some small oscillations at the turning points where the feed or the cooling constraints become active, the ideal NMPC drives the system almost exactly on its operation bounds. From the physics of the system, it is clear that the batch time is minimized when the monomer feed rate is maximized, and this value is determined by the available cooling power to remove the heat of reaction which is caused by the polymerization reaction. (More precisely, the cooling rate determines the maximum value of the monomer concen-

tration at each point in time, and this value is determined by the feed rate.) The NMPC scheme described above is taken as the reference solution to evaluate the performance of the optimizing control scheme that is proposed in the next section.

5. PROPOSED OPTIMIZING CONTROL STRUCTURE In this section, a simple and efficient control scheme for the time-optimal operation of semibatch reactors is proposed and 5913



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Figure 12. Simulation results for Cset = 30% (dashed blue line, ideal NMPC; dotted pink line, standard NMPC; solid green line, OCS).

and the monomer feed is set to zero. This control scheme can be considered as a feedback implementation of the necessary condition of optimality.20 As explained above, when the cooling power reaches its maximum value, the optimum monomer feed rate is the one that keeps the cooling power exactly at this bound without violating the temperature constraint. So ideally, the reference value cset would be the minimum possible value (0%). To improve the robustness, we use slightly higher

tested using the CHBR as a case study. The basic idea behind this new scheme, which is illustrated in Figure 6, is to combine the original CCS with an additional PI controller, such that the monomer inlet flow is continuously manipulated to keep the cooling usage c at a desired level cset. As the third PI controller manipulates the monomer feed, it is active during the feeding period only. When the desired amount of monomer has been fed into the reactor, the monomer feed controller is deactivated 5914



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Figure 13. Simulation results for the case where the standard NMPC is driven against Cset = 20% (blue dashed line, ideal NMPC; pink dotted line, standard NMPC; green solid line, OCS).

drive c to the given lower bound will approximately realize the necessary condition of optimality in the presence of uncertainties without any model-based prediction or optimization. Therefore, by tracking the optimal level cset, the additional PI controller automatically regulates the monomer feed so that the amount of heat produced by the reaction is balanced with the available cooling capacity. This results in an optimizing control structure (OCS) that can operate the system

values of the set point of the cooling usage. (Also in a real NMPC implementation, some safety margin of the cooling power would be respected). The main motivation for using this control structure is to take advantage of the fact that, for almost any external disturbances that may affect the system, for example, fluctuations in the cooling water or in the monomer temperatures, c will always provide a quantitative measure of how much cooling power is still available and to 5915



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Table 2. Reduction of the Reaction Period for the Different Operation Scenarios scenario

ideal NMPC (%)

standard NMPC (%)

standard NMPC with cset = 20% (%)

OCS cset = 30% (%)

OCS cset = 20% (%)

OCS cset = 10% (%)

nominal 1 2 3 4 5 6 7 8 average

61.9 62.0 62.3 60.7 61.6 65.9 66.0 65.6 66.0 63.6

61.9 60.4 62.3 60.6 61.5 63.0 66.0 65.1 65.6 62.9

61.1 60.2 61.8 59.7 60.6 64.1 65.6 64.7 65.2 62.6

53.6 55.2 55.3 52.2 51.9 59.3 61.7 60.3 60.8 56.7

58.6 58.9 59.5 56.7 57.1 60.5 63.0 61.4 63.2 59.9

60.2 57.2 59.6 58.7 59.6 64.0 64.6 63.7 64.6 61.4

Table 3. Overall Constraint Violation Period during a Batch for the Different Scenarios scenario

ideal NMPC (min)

standard NMPC (min)

standard NMPC with cset = 20% (min)

OCS cset = 30% (min)



nominal 1 2 3 4 5 6 7 8 average

0 0 0 0 0 0 0 0 0 0

0 21.5 0 0 16.8 20.2 0 0 17.0 8.39

0 21.7 0 0 16.9 20.7 0 0 17.3 8.51

0 0 0 0 1.2 0 0 0 0 0.13

0 0 0 0 0 3.0 0.4 1.6 0 0.56

0 2.2 0.8 0 0 2.3 1.7 0.8 0 0.87

close to the limit of its cooling system and significantly reduces the batch duration. Similar to the master and slave controllers, the monomer feed controller is also implemented in a discrete fashion using the velocity form of the PI algorithm. Every sampling period, the monomer inlet flow rate is updated according to eq 11 (the variable eci denotes the control error at the sampling period i, that is, the difference between the position of the heating/cooling valve ci and its set point cset). ⎛ ec − eci − 1 ci ⎞ F Mi = F Mi−1 + KpMF⎜ i + ⎟ Δt TIMF ⎠ ⎝

(11)

The monomer feed controller was tuned by simulations so that the batch time can be reduced as much as possible without harming the temperature control within the considered uncertainty range. After a fine-tuning procedure, the gain KpFM and the integral time constant TIFM were tuned to 2.5 × 10−5 and 30 s, respectively. The main idea is to make the variations in the monomer dosage enter the temperature control loop as a slow disturbance that can be easily rejected while the heating/cooling valve is being driven to the desired position cset. To make the closed loop system more robust to fast disturbances that may drive the system out of control, for example, the sharp decrease in the heat transfer coefficient, the monomer control is accelerated when the heating/cooling valve is close to saturation. This is realized by reducing the integral time constant of the monomer feed controller to 5 s when c ≤ 5%. A simulation study considering all the extreme case scenarios from Table 1 was conducted and the proposed control scheme was evaluated with respect to robustness and performance. In these simulations, the set point of the monomer feed controller was varied within the range from 10% to 30% and the performance of the OCS was compared to the ideal

Figure 14. Simulation with cooling failure between 45 and 55 min (nominal scenario, cset = 20%).

NMPC scheme for the different operation scenarios. The results for the nominal scenario (purity factor equal to one, 1/hf = 0.352 m2 K kW−1, and summer season) for the cases where Cset is equal to 10%, 20%, and 30% are presented in Figures 7−9. The outcome of the simulation shows that the proposed optimizing scheme performs quite close to the ideal NMPC solution if a small safety margin is used. As it can be seen from the plots in Figures 7−9, especially during the reaction period when the temperature control is crucial, the proposed scheme follows the reference solution very closely until the point at which the cooling valve position reaches the value of cset. After this point, instead of driving the system to the limit of the cooling power, the OCS regulates the monomer dosage 5916



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Figure 15. Simulation results for the case where a cooling failure events occurs during the time interval when t < 45 min and t < 55 min (blue dashed line, ideal NMPC; pink dotted line, standard NMPC; green solid line, OCS).

period is reduced by 60.2% and 58.6% in the first two cases and by 53.6% in the last case. The results of the full robustness analysis, considering the entire set of extreme cases scenarios, are presented in Figures 10−12. In these simulations, the proposed OCS is compared with the ideal NMPC that always uses a perfect process model and with a standard NMPC scheme that always uses the

such that the cooling valve position stays around cset. When the proposed OCS is compared to the original setup (mMin = 7.56 × 10−3 kg/s and CCS), a significant performance improvement is observed. In the nominal case, the time needed to feed 45 kg of monomer into the reactor could be reduced from 99.0 to 39.4, 40.9, and 45.9 min when cset is equal to 10, 20, and 30%. This means that the feeding 5917



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driving the temperature control out of the performance bounds. The investigations were based on the Chylla−Haase polymerization reactor problem, which is a widely accepted benchmark problem for the evaluation of alternative control strategies for semibatch processes in the presence of uncertainties. As in the original setup of the Chylla−Haase benchmark problem it is not possible to reduce the batch time because the monomer inlet flow is constant, a modified version of the benchmark problem where variable monomer dosages are allowed was considered. A simple and efficient optimizing control structure that maximizes the monomer feed by regulating the position of the heating/cooling valve was then introduced and evaluated using an ideal NMPC scheme as the reference solution. The simulation results show that the proposed scheme is capable to robustly operate the plant within the considered uncertainty range and provides almost the same performance as NMPC. In the presence of plant−model mismatch, the proposed scheme may even perform better than a full NMPC-based scheme that employs a fixed nominal model. When compared to the original recipe with a constant low monomer flow rate, the reaction period was shortened by more than fifty percent in most of the scenarios investigated. The tuning of the PI controllers in the proposed scheme is simple if the interaction of the controllers is kept small by making the feed control loop significantly slower than the temperature control loop. The conclusions of this investigation can be directly extrapolated to other processes where the temperature of exothermic reactors has to be controlled and the necessary condition of optimality is to maximize the feeding rate under the constraint of the limited cooling power.

nominal process model. The simulation of the standard NMPC scheme are introduced here to illustrate that the performance of the NMPC can deteriorate significantly in the presence of plant-model mismatch if the model uncertainties are not properly addressed, for example, by employing a robust NMPC approach27 or by compensating the model errors online during the batch.19 In Figure 13, an additional set of simulations where the standard NMPC is formulated such that a set point for the heating/cooling valve of cset = 20% is tracked, what is realized by adding an additional penalty term in the cost function. The reductions of the reaction period provided by the OCS and by the NMPC schemes for all the considered operation scenarios are presented in Table 2 and the overall period of time in which the temperature constraints are violated are given in Table 3. The results from Table 2 show that the OCS performs very close to the reference solution in terms of the resulting feeding times. Moreover, by analyzing the results from Table 3, it is clear that the proposed optimizing scheme provides a significantly more robust operation than the standard NMPC scheme under the uncertainty range considered here. By choosing cset as equal to 10% the temperature constraints are slightly violated for short periods, contrasting with the standard NMPC scheme which may lead to significant violations of the specification bounds for more than 20 min. As it can also be seen in Figures 10−12, if the value of value of cset is set to 20%, the violation of the temperature constraints becomes almost negligible. If a conservative value of cset is chosen, that is, 30% or less, the OCS can operate the system practically without any constraint violation (but the feeding time increases). Even for the case where the standard NMPC is realized with some safety margin by tracking the set point cset = 20%, the specification bounds are severely violated. This is because it is very difficult for a full model-based scheme to meet such tight temperature specification bounds (±0.6 K) if it is based on a wrong model. Finally, a set of simulations where a cooling failure takes place during the feeding period is reported. The cooling failure event is simulated by dividing the cooling term in eq 6 by two during the time interval when t > 45 min and t < 55 min. The results reported in Figures 14 and 15 show that the proposed control scheme is capable of rejecting this strong disturbance very well in all operation scenarios. The main disadvantage of the proposed optimizing control structure is that, due to its feedback nature, it is only capable to provide delayed reactions against disturbances, while in the NMPC schemes important disturbances as the continuous changes in rP and UA are explicitly included in the model. Although this may lead to a small loss of performance, if a proper value for cset is selected the OCS has been shown to be considerably more robust than the standard NMPC scheme in the presence of plant-model mismatch (note that in this case the reaction of the NMPC controller to the uncertainties in rP and UA is also delayed). Moreover, the proposed scheme only depends on the available measurements and does not require a state estimator nor online optimization, what makes its industrial implementation straightforward and promising.

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APPENDIX A: KINETIC AND HEAT TRANSFER RELATIONS OF THE CHBR rP = ikmM

(12)

k = koe−E / RT (k1μ)k 2

(13)

⎣

⎦

μ = coec1f 10c2⎡(a0 / T ) − c3⎤ f=

(14)

mP mM + mP + mW

(15)

⎛m m ⎞P m + B2 A = ⎜⎜ M + P + W ⎟⎟ ρP ρW ⎠ B1 ⎝ ρM

U=

−1

h

1 + h f −1

(17)

h = doed1μ W

(18) ⎣

⎦

μ W = coec1f 10c2⎡(u0 / TW ) − c3⎤

TW =

6. CONCLUSIONS In this work, the optimal operation of exothermic semibatch polymerization reactors was investigated. The main goal was to reduce the batch time as much as possible without

Tj̅ = 5918

(16)

(19)

T + Tj̅ (20)

2

T jout + T jin 2

(21) dx.doi.org/10.1021/ie302321k | Ind. Eng. Chem. Res. 2013, 52, 5906−5920


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APPENDIX B: PARAMETERS VALUES OF CHBR

Table 4. Parameters Values of CHBR parameter mM,0 mP,0 mW ρM ρP ρW Cp,M Cp,P Cp,W Cp,C MWM mC ṁ C k0 k1 k2 ΔHP E c0 c1 c2 c3 a0 d0 d1 Tset P B1 B2 R UAloss τP θ1 θ2 Tinlet summer winter Tsteam

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unit kg kg kg kg m−3 kg m−3 kg m−3 kJ kg−1 K−1 kJ kg−1 K−1 kJ kg−1 K−1 kJ kg−1 K−1 kg kmol−1 kg kg s−1 s−1 m s kg−1

value

K kW m−2 K−1 m s kg−1 K M m2 m2 kJ kmol−1 K−1 kW K−1 s s s

0.0 11.227 42.750 900 1040 1000 1.675 3.140 4.187 4.187 104.0 21.455 0.9412 55 1000 0.4 70152.16 29560.89 5.2 × 10−5 16.4 2.3 1.563 555.556 0.814 −5.13 355.386 1.594 0.193 0.167 8.314 0.00567567 40.2 22.8 15.0

K K K

294.26 277.26 449.82

kJ kmol−1 kJ kmol−1 kg m−1 s−1

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FC = water flow rate across the jacket (kg s−1) T = reactor temperature (K) Tout = jacket outlet temperature (K) j Tinj = jacket inlet temperature (K) T̅ j = jacket average temperature (K) T̅ w = wall average temperature (K) Tinlet = cooling water temperature (K) Tsteam = steam temperature (K) Tset = reaction temperature set point (K) Tamb = ambient temperature (K) rP = reaction rate (kg s−1) Qreac = reaction heat (kW) ΔHP = heat of reaction (kJ kmol−1) i = impurity factor A = heat transfer area [m2] U = overall heat transfer coefficient (kW m−2 K−1) h = heat transfer coefficient (kW m−2 K−1) hf = fouling factor (kW m−2 K−1) UAloss = heat loss coefficient (kW K−1) ρi = specific mass of the different substances (kg m3, i = M, P, W, C) Cp,i = specific heat of the different substances (kJ kg−1 K−1, i = M, P, W, C) MWi = molar mass of the different substances (kg mol−1, i = M, P, W, C) θ1, θ2 = time delays in jacket and recirculation loop (s) c = heating/cooling usage (%) Kp(c) = heating/cooling function (K) τp = heating/cooling time constant (s)

REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*E-mail: tiago.fi[email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge the support provided by the Ziel2.NRW Program funded by the State of North RhineWestphalia and the European Union via the European Fund for Regional Development.

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LIST OF VARIABLES AND PARAMETERS OF THE CHBR mM, mP, mW = mass amount monomer, polymer, and water (kg) mC = mass of water in the jacket (kg) FM = monomer feed rate (kg s−1) 5919



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(13) Franço is, B.; Srinivasan, G.; Bonvin, D. Run-to-run Optimization of Batch Emulsion Polymerization. Proc. 15th IFAC World Congr. 2002, 1258−1263. (14) Helbig, A.; Abel, O.; Marquardt, W. Model Predictive Control for Online Optimization of Semi-batch Reactors. Proc. Am. Control Conf. 1998, 1695−1699. (15) Gesthuisen, R.; Krämer, S.; Engell, S. Hierarchical Control Scheme for Time-Optimal Operation of Semi-batch Emulsion Polymerizations. Ind. Eng. Chem. Res. 2004, 43, 7410−7427. (16) Vicente, M.; Leiza, J. R.; Asua, J. M. Maximizing Production and Polymer Quality (MWD and Composition) in Emulsion Polymerization Reactors with Limited Capacity of Heat Removal. Chem. Eng. Sci. 2003, 58, 215−222. (17) François, G.; Srinivasan, B.; Bonvin, D.; Barajas, J. H.; Hunkeler, D. Run-to-Run Adaptation of a Semiadiabatic Policy for the Optimization of an Industrial Batch Polymerization Process. Ind. Eng. Chem. Res. 2004, 43, 7238−7242. (18) Mauntz, W. A Contribution to Observation and Time-Optimal Operation of Semi-batch Emulsion Polymerizations, Ph.D. Thesis, Technical University of Dortmund, Schaker Verlag, Aachen, Germany, 2010. (19) Finkler, T. F.; Kawohl, M.; Piechottka, U.; Engell, S. Realization of Online Optimizing Control in an Industrial Polymerization Reactor. In Proceedings of the International Symposium on Advanced Control of Chemical Processes, 2012. Available via the Internet at http://www.nt. ntnu.no/users/skoge/prost/proceedings/adchem-2012/files/0025. pdf. (20) François, G.; Srinivasan, B.; Bonvin, D. Using Measurements for Enforcing the Necessary Conditions for Optimality in the Presence of Constraints and Uncertainty. J. Process Control 2005, 15, 701−712. (21) Bonvin, D.; Srinivasan, B.; Ruppen, D. Dynamic Optimization in the Batch Chemical Industry. Proc. CPC-VI Conf., Am. Inst. Chem. Eng. 2001, 326 (98), 255−273. (22) Bonvin, D.; Bodizs, L.; Srinivasan, B. Optimal Grade Transition for Polyethylene Reactors via NCO Tracking. Trans. Inst. Chem. Eng., Part A 2005, 83 (A6), 692−697. (23) Bonvin, D.; Srinivasan, B.; Hunkeler, D. Control and Optimization of Batch Processes. IEEE Control Syst. Mag. 2006, 26 (6), 34−45. (24) Huang, H. P.; Chien, I. L.; Lee, Y. C.; Wang, G. B. A Simple Method for Tuning Cascade Control Systems. Comput. Chem. Eng. 1998, 165, 89−121. (25) Diehl, M.; Findeisen, R.; Schwarzkopf, S.; Uslu, I.; Allgöwer, F.; Bock, H. G.; Gilles, E. D.; Schlöder, J. P. An Efficient Algorithm for Nonlinear Model Predictive Control of Large-Scale Systems Part I: Description of the Method. Automatisierungstechnik 2002, 50, 557− 567. (26) Kirches, C.; Wirsching, L.; Bock, H. G.; Schlöder, J. P. Efficient direct multiple shooting for nonlinear model predictive control on long horizons. J. Process Control 2012, 22, 540−550. (27) Lucia, S.; Finkler, T. F.; Basak, X.; Engell, S. A New Robust NMPC Scheme and Its Application to a Semi-Batch Reactor Example. In Proceedings of the International Symposium on Advanced Control of Chemical Processes, 2012. Available via the Internet at http://www.nt. ntnu.no/users/skoge/prost/proceedings/adchem-2012/files/0035. pdf.

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Simple Control Scheme for Batch Time Minimization of Exothermic

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