Model predictive control of an industrial pyrolysis gasoline hydrogenation reactor

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Journal of Industrial and Engineering Chemistry 14 (2008) 175–181 www.elsevier.com/locate/jiec

Model predictive control of an industrial pyrolysis gasoline hydrogenation reactor Amornchai Arpornwichanop a,*, Paisan Kittisupakorn a, Yaneeporn Patcharavorachot a, Iqbal M. Mujtaba b a

Department of Chemical Engineering, Faculty of Engineering, Chulalongkorn University, Bangkok 10330, Thailand b School of Engineering, Design & Technology, University of Bradford, Bradford, West Yorkshire BD7 1DP, UK Received 11 January 2007; accepted 29 September 2007

Abstract This study focuses on the implementation of a nonlinear model predictive control (MPC) algorithm for controlling an industrial fixed-bed reactor where hydrogenations of raw pyrolysis gasoline occur. An orthogonal collocation method is employed to approximate the original reactor model consisting of a set of partial differential equations. The approximate model obtained is used in the synthesis of a MPC controller to control the temperature rising across a catalyst bed within the reactor. In the MPC algorithm, a sequential optimization approach is used to solve an openloop optimal control problem. Feedback information is incorporated in the MPC to compensate for modeling error and unmeasured disturbances. The control studies are demonstrated in cases of set point tracking and disturbance rejection. # 2007 The Korean Society of Industrial and Engineering Chemistry. Published by Elsevier B.V. All rights reserved. Keywords: Model predictive control; Fixed-bed reactor; Distributed parameter system; Reactor control; Simulation

1. Introduction The presence of the complexity in chemical processes due to higher product quality requirement and tighter environmental regulations posts challenging control problems that are difficult to handle with linear control techniques. With the limitation of linear controllers in achieving a satisfactory control performance, many advanced nonlinear control strategies have been devised over the past years. Among nonlinear control methodologies, a model predictive control (MPC) emerges as a powerful practical control technique. A key feature contributing to the success of MPC is its ability to cope with a multivariable system with constraints [1]. MPC has been widely applied in a wide range of applications, especially in the processes that their dynamic behavior is described by relatively simple models consisting of ordinary differential and/or algebraic equations. However, the implementation of MPC in complex systems like a distributed parameter system (DPS) which is naturally modeled by a set of nonlinear partial differential equations (PDEs) has been rarely

* Corresponding author. Tel.: +66 2 218 6878; fax: +66 2 218 6877. E-mail address: [email protected] (A. Arpornwichanop).

addressed [2,3] and therefore, control studies on this type of the systems are still the subject of interest. Controlling a DPS, i.e., a fixed-bed reactor and a tubular reactor has been accepted to be a difficult task. A control design may be considerably complicated due to inherent difficulties such as high nonlinearity and the presence of spatial variations [4]. Such difficulties prompt the need for an effective control algorithm. In general, the approaches to control a DPS are mainly based on various lumping techniques which can be classified into two different strategies: late lumping and early lumping methods. The late lumping method applies a distributed parameter control theory to full PDE models for designing a control system. After the controller design has been completed, the resulting control algorithms are then solved by lumping approximate techniques. Various approaches have been considered to directly use PDE models in controller designs. Examples include the control algorithm proposed by Dochain et al. [5], Renou et al. [6] and Christofides and Daoutidis [7]. However, this approach requires the greater knowledge of a distributed system control theory. The alternative early lumping approach, on the other hand, is a straightforward method and widely used in chemical engineering. The idea behind the approach is that a DPS is first converted into an approximate

1226-086X/$ – see front matter # 2007 The Korean Society of Industrial and Engineering Chemistry. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jiec.2007.09.009

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2. Catalytic gasoline hydrogenation reactor Nomenclature A ai Ci Cp Ea DH k0 Klai Q r T u z

reactor cross sectional area (m2) catalyst surface area/volume ratio (m2/m3) concentration of component i (kmol/m3) specific heat capacity (J/(kg K)) apparent activity energy (J/mol) heat of reaction (J/kmol) apparent pre-exponential factor liquid phase mass transfer coefficient (1/s) flow rate (m3/h) rate of reaction (kmol/(m3 h)) reactor temperature (K) superficial velocity (m/h) reactor length (m)

Greek symbols e void fraction r density (kg/m3) Subscripts Di diolefins f feed stream g gas phase H2 hydrogen l liquid phase Ole olefins Para paraffins q quench stream

model (lumping), with the use of discretization techniques, usually consisting of a set of ordinary differential equations (ODEs). Then, traditional control algorithms based on the resulting approximate model of the system are applied directly to design a control system. The advantage of this method is attributable to the fact that it can gain benefit of using welldeveloped and advanced control methodologies designed for a finite dimensional ODEs system. Thus, the method possibly conforms to the practical implementation [8]. Based on the early lumping technique, several approaches for the control of nonlinear PDE systems have been proposed as can be seen in Christofieds [4], Hanczyc and Palazoglu [9], Boskovic and Krstic [10] and Hua and Jutan [11]. The main objective of this study focuses on the implementation of a MPC algorithm via the early lumping approach to control a DPS. An industrial fixed-bed reactor where hydrogenations of raw pyrolysis gasoline are occurred, is considered to represent the DPS as a case study. The dynamic reactor models developed in our previous work [12] are used to design the MPC controller. Simulation studies are performed to investigate the performance of the MPC strategy in controlling the outlet temperature of a catalyst bed within the reactor. The control performance of the MPC is evaluated and compared with that of a conventional PID controller in cases of set point tracking and disturbance rejection under nominal and model mismatch conditions.

A hydrogenation reactor in which hydrogen and raw pyrolysis gasoline concurrently flow through a fixed bed of catalyst (Ni/Al2O3), is investigated here as an industrial case study for the design of a MPC strategy. This type of reactor is also known as a trickle bed reactor. Fig. 1 shows the schematic diagram of the fixed-bed gasoline hydrogenation reactor in which primarily chemical reactions involve the consecutive hydrogenation of diolefins to olefins and then to paraffins within the same carbon number group. The fixed-bed catalyst is divided into two sections (top and bottom catalyst beds) interconnected by an intermediate cooling section where a quench stream is added directly to prevent too large temperature increases. The mathematical model of the gasoline hydrogenation reactor is presented in Appendix A. It is noted that the space between successive catalyst beds is treated as a quench section which is assumed to be a perfectly stirred tank as shown in Fig. 1 and that the mixing process reaches steady state condition instantaneously. The developed models of a fixed-bed reactor result in a set of PDEs which are solved numerically using a method of lines approach. The spatial derivative term in Eqs. (A.1)–(A.6) is discretized by an orthogonal collocation method on finite elements; each fixed-bed of catalyst in the reactor is divided into 10 elements with an equal space and two internal collocation points are used for each finite element. An approximation of the spatial derivative term make the PDEs reduce to a system of differential and/or algebraic equations (DAEs). In this work, the resulting DAEs are solved using the backward difference method [13]. 3. MPC of a fixed-bed gasoline hydrogenation reactor In this work, the control of the outlet temperature of the top catalyst bed is the main focus. The temperature rising across the catalyst bed, which is caused by the heat released from hydrogenations of diolefins and olefins, is one of the significant operating variables that are necessary to regulate for smooth process operation and for meeting product specification. To ensure a hydrogenation reaction rate which completely eliminates the unstable compounds, high temperature is preferred. However, it should not be too high because higher temperature favors the production of polymers that are deposited on the catalyst bed. In addition, higher temperature reduces the flow of liquid through the catalyst bed, thus decreasing the washing effect. In industrial practice, the inlet temperature to the reactor is used to maintain the temperature rise within the reactor at a proper condition. Thus, in this work, the MPC is implemented to determine the inlet temperature (Tin as a manipulated variable) for controlling the outlet temperature of the top catalyst bed. 3.1. Formulation of MPC problem The formulation of the MPC controller based on solving an on-line optimal control problem is described as follows.

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177

Fig. 1. Schematic diagram of a fixed-bed gasoline hydrogenation reactor.

The optimal control problem can be given by the objective function (performance index): Z t0 þT P min ðT sp ðzB1 Þ  TðzB1 ; tÞÞ2 dt (1)

T in ðtÞ

t0

subject to the process models: Eqs: ðA:1ÞðA:13Þ

(2)

bound on the manipulated variable: 343  T in  403

(3)

and the end point constraint: TðzB1 ; T P Þ ¼ T sp ðzB1 Þ

(4)

where Tsp(zB1) is the set point value of the reactor temperature at outlet of the top catalyst bed. It should be noted that Eq. (4) is included in the MPC formulation to ensure the stability of the system; the optimal controls have to force the states of the system to a desired set point at the terminal time (t0 + TP). In this work, the prediction horizon (TP) is set to be 1 h which is large enough to guarantee that T(zB1, TP) ! Tsp(zB1). The frequency of updated control action is chosen to be 0.1 h. Therefore, the number of future controls in the MPC problem equals to 10 steps. The structure of the MPC employed in this study is shown in Fig. 2. It is assumed that all state variables of the system are measured. It can be seen that feedback information consists of (i) a measurement of both the reactant concentrations (C) and the reactor temperature (T), and (ii) an error signal (e) of the controlled variable (Tat the outlet of the top catalyst bed) which is

the difference observed between the measured value and the predicted value from the model. This error signal will go to zero if there are no unmeasured process disturbances to the system and the system model accurately represents the real plant. Otherwise, there is a disturbance feedback signal to the MPC controller. In this work, the effect of modeling error is treated as an additive step disturbance in the output and is determined at the kth sampling time as shown in Eq. (5). The disturbance term (dk) is added to the output prediction over the entire prediction horizon (TP) in the MPC objective function (Eq. (1)). dk ¼ ymeasured;k  ypredicted;k

(5)

After updating the feedback information from the system, the MPC computes a sequence of control inputs, Tin, by solving the optimal control problem (Eqs. (1)–(4)). However, only the initial value of this control profile is applied to the system and this procedure is repeated for the next sampling time. To find the solution of the open-loop optimal control of processes described by a system of PDEs, the orthogonal

Fig. 2. Structure of model predictive control strategy.

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collocation method on finite elements is applied to discretize the spatial derivative term. This converts the PDEs into ordinary differential equations. Then, the sequential model solution and optimization approach [14] is employed to solve the discretized optimal control problem. 4. Simulation results This section demonstrates the implementation of a MPC in controlling the outlet reactor temperature of the top catalyst bed within the fixed-bed reactor by adjusting the inlet feed temperature (a manipulated variable). The performance of the MPC controller is compared with that of a PID control algorithm in the velocity form [15]. In this study, the parameters of a PID controller were first tuned based on the Ziegler– Nichols technique. Then, fine tuning of the obtained controller parameters was performed by considering a closed-loop control performance and the resulting PID tuning parameters are Kc = 0.3, ti = 1, and td = 0.001. The control performance of both controllers is evaluated in set point tracking and disturbance rejection cases under nominal condition in which all model parameters are specified correctly. In all case studies, the set point change and disturbance are introduced to the system at time, t = 10 min. Since the MPC controller is based on the model of the system to be controlled, its robustness property under Table 1 Process parameters, kinetic constants and operating conditions Reactor length Reactor length Reactor diameter Inlet pressure Void fraction Diffusivity Liquid mass transfer Heat of reaction Specific heat capacity of hydrogen gas k0,1, k0,2, k0,3 Ea1 ; Ea2 ; Eaad

L1 = 7.9 m L2 = 15.75 m D = 1.8 m P = 29.73 atm ep = 0.4 D1;H2 ¼ 105 cm2 =s K 1;H2  ai ¼ 6:93 h1 DH = 125.52  106 J/kmol C p;H2 ¼ 14:63  103 J=ðkg KÞ 7345, 7839, 2.057 22,500, 35,370, 14,907 (J/mol)

Hydrogen gas feed Flow rate Temperature Hydrogen concentration

Q = 8350 kg/h T = 391.63 K C H2 ¼ 0:926 kmol=m3

Pyrolysis gasoline feed Flow rate Temperature Density Concentration Diolefins Olefins Paraffins Quench stream Flow rate Temperature Concentration Olefins Paraffins Hydrogen

Q = 68.69 m3/h T = 391.63 K r = 820 kg/m3 CDi = 1.48 kmol/m3 COle = 1.23 kmol/m3 CPara = 2.97 kmol/m3 Q = 15.43 m3/h T = 309.67 K COle = 2.20 kmol/m3 CPara = 3.11 kmol/m3 C H2 ¼ 0:25 kmol=m3

Fig. 3. Control response for set point tracking (Tsp = 455 K): (a) MPC control and (b) PID control.

model mismatch conditions, i.e., changes in k0,1 and DH is also investigated. In the simulation studies, we assumed that the fixed-bed reactor is started at a steady state condition. The values of nominal process parameters, kinetic parameters and initial inlet stream conditions which are based on industrial plant data collected from the gasoline hydrogenation unit of the olefin production plant are given in Table 1. The first set of simulations illustrates the nominal case; all process parameters of the reactor models used in the MPC algorithm are known exactly. In case of set point tracking of the outlet temperature at the top catalyst bed from its nominal value (449.6 K) to 455 K, it can be seen from Fig. 3(a) that the MPC controller can control the temperature at the desired set point with small overshoot. On the other hand, the PID controller gives more overshoot and some oscillation in the response as shown in Fig. 3(b). Similar results can be observed for the case where the temperature set point is decreased to 445 K. The MPC controller still gives a good control response as shown in Fig. 4(a). It is able to drive the temperature to its desired set point smoothly with short setting time whereas the PID provides a slower and oscillated response (Fig. 4(b)).

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Fig. 4. Control response for set point tracking (Tsp = 445 K): (a) MPC control and (b) PID control.

The MPC and PID controllers are then tested in cases of the disturbance rejection that consists of 20% change in feed flow rate of pyrolysis gasoline from its nominal value. With a 20% increase in the flow rate of gasoline feed, the simulation results show that the MPC controller (Fig. 5(a)) is able to eliminate this disturbance and give faster control response, compared with the PID controller (Fig. 5(b)). Table 2 summarizes the control results in terms of the ISE value of the MPC and PID controllers for all simulation studies in the nominal condition. It clearly indicates that the control performance of the MPC controller is better than that of the PID controller in all case studies. Next, the robustness property of the MPC controller is examined with respect to uncertainties in model parameters employed in the MPC algorithm. Fig. 6 shows the results of a mismatch in the heat of reaction, DH (20% decrease), in case of set point tracking of the reactor temperature (+5 K from its nominal set point). The MPC controller is able to drive the reactor temperature to a new desired set point; although, some error are observed in the first sampling time interval. This is because the MPC controller computes the control input based on the model with the incorrect parameter value. However, the

179

Fig. 5. Control response for disturbance rejection with 20% increase in feed flow rate: (a) MPC control and (b) PID control.

effect of modeling error is compensated through feedback information at the next sampling time and therefore the MPC controller can bring the reactor temperature to the desired value. Similarly, in case of disturbance rejection with the same model mismatch (20% decrease in DH), the MPC controller still provides reasonably a good control response; it can bring the system back to its original set point value as can be seen from Fig. 7. This shows the ability and robustness of the MPC Table 2 Comparison of the MPC and PID control performance in the nominal case Case studies

Set point tracking (1) Change Tsp to 455 K (2) Change Tsp to 445 K Disturbance rejection (3) 20% increase in feed flow rate (4) 20% decrease in feed flow rate R a Note: ISE ¼ ðT sp  TðtÞÞ2 dt:

ISE valuea MPC

PID

10.6028 7.0764

21.3657 19.3338

1.9850 3.3421

6.0848 3.3971

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Fig. 6. Control response of MPC controller for set point tracking (Tsp = 455 K) with a mismatch in DH (20% decrease).

Fig. 8. Control response of MPC controller for set point tracking (Tsp = 455 K) with a mismatch in k0,1 (20% decrease).

mismatch. The modeling error is reduced after the MPC receives new feedback information from the system. Table 3 summarizes the control performance in terms of the ISE value of the MPC controller for all case studies under model mismatch conditions. 5. Conclusions

Fig. 7. Control response of MPC controller for disturbance rejection (20% increase in feed rate) with a mismatch in DH (20% decrease).

controller to reject the disturbance in the presence of the model mismatch. With a 20% decrease in the reaction rate constant, k0,1, Fig. 8 illustrates the capability of the MPC controller to control the system at the desired temperature set point value; although, there are some error in the first sampling time due to the model

Appendix A. Model of a gasoline hydrogenation reactor

Table 3 The MPC performances under model mismatch conditions Case studies

The implementation of a MPC strategy to control the temperature of an industrial fixed-bed reactor where hydrogenations of pyrolysis gasoline take place is presented. The reactor model consisting of a set of PDEs is utilized to the formulation of a MPC problem via the early lumping approach. The orthogonal collocation method on finite elements is applied to discretize the PDEs and the on-line optimal control problem in the MPC algorithm is solved using an efficient sequential optimization approach. The simulations have shown that in a nominal condition where all model parameters are specified correctly, the MPC controller gives a better control performance compared to a conventional PID controller; it can control the reactor temperature at the desired set point value in both set point tracking and disturbance rejection cases. In the presence of a model mismatch, the MPC controller still provides reliable control performance and is robust with respect to errors in model parameters.

ISE value 20% decrease in DH

20% decrease in k0,1

Set point tracking (1) Change Tsp to 455 K (2) Change Tsp to 445 K

50.2645 43.1226

41.5614 44.1059

Disturbance rejection (3) 20% increase in feed flow rate (4) 20% decrease in feed flow rate

40.2710 40.8011

40.8956 39.0212

To develop a dynamic model of a gasoline hydrogenation reactor, the following assumptions have been made [16,17]; the reactor is operated under adiabatic and isobaric condition; axial dispersion in both gas and liquid phases is negligible; the mass transfer resistance at the liquid–solid interface and the resistance to pore diffusion can be ignored; the reactions are occurred in the liquid phase. Due to a large number of reactions and components which make the model complex, all hydrocarbon components in the system are refined into three hydrocarbon classes: diolefins, olefins and paraffins, which is

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based on the lumping criteria discussed by Somer et al. [18] and Cheng et al. [19]. The kinetic model scheme as shown below is considered to represent the hydrogenation of pyrolysis gasoline.

181

Quench section: ðQf þ Qq ÞCHB22 ;l ¼ Qf CHF 2 ;l þ Qq CHQ2 ;l

(A.9)

Q B2 F ðQf þ Qq ÞCDi;l ¼ Qf CDi;l þ Qq CDi;l

(A.10)

Diolefins þ H2 !Olefins þ H2 !Paraffins

Q B2 F ðQf þ Qq ÞCOle;l ¼ Qf COle;l þ Qq COle;l

(A.11)

The process models derived from mass and energy balances can be written as: Reactor model (the top and bottom catalyst bed):

Q B2 F ðQf þ Qq ÞCPara;l ¼ Qf CPara;l þ Qq CPara;l

(A.12)

r1

r2

dC H2 ;g ug dC H2 ;g K l;H2 ai ¼  ðC H2 ;g  C H2 ;l Þ dt eg dz eg

(A.1)

ð1  eÞ ðr 1 þ r 2 Þ el

(A.2)

dC Di;l ul dC Di;l ð1  eÞ ¼  ðr 1 Þ el dt el dz

(A.3)

dC Ole;l ul dC Ole;l ð1  eÞ ¼  ðr 1 þ r 2 Þ el dt el dz

(A.4)

dC Para;l ul dC Para;l ð1  eÞ ¼ þ ðr 2 Þ el dt el dz

(A.5)

(A.6) 

r2 ¼

k0;2 expðEa2 =RTÞCOle C H2 1 þ k0;ad expðEaad =RTÞC Di

[1] [2] [3] [4]

[5] [6]

dT ðug C p;g rg þ ul C p;l rl ÞdT=dz þ ðDHÞð1  eÞðr 1 þ r 2 Þ ¼ dt ðeg rg Cp;g þ el rl Cp;l Þ  Ea1 r 1 ¼ k0;1 exp C Di CH2 RT

ug Arg Cp;g T F þ rl Cp ðQf T F þ Qq T Q Þ ðug Arg Cp;g þ ðQf þ Qq Þrl C p;l Þ

(A.13)

References

dC H2 ;l ul dC H2 ;l K l;H2 ai ¼ þ ðC H2 ;g  C H2 ;l Þ dt el dz el 

T B2 ¼

(A.7)

[7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

(A.8)

where DH is the heat of hydrogenation which is approximated to 30 kcal/mole for each double bond reaction.

[17] [18] [19]

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