Control of Ethylene Dichloride Cracking Furnace ... - ACS Publications

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Control of Ethylene Dichloride Cracking Furnace Using an Analytical Model Predictive Control Strategy for a Coupled Partial Differential Equation/Ordinary Differential Equation System Atthasit Tawai,†,‡ Chanin Panjapornpon,*,†,‡ and Peerapan Dittanet†,‡ †

Department of Chemical Engineering, Center of Excellence on Petrochemicals and Material Technology, Faculty of Engineering, Kasetsart University, Bangkok 10900, Thailand ‡ The Center for Advanced Studies in Industrial Technology, Faculty of Engineering, Kasetsart University, Bangkok 10900, Thailand S Supporting Information *

ABSTRACT: A nonlinear optimization-based control system with analytical model predictive control (AMPC) structure is formulated in cascade with an off-line pseudo-steady-state calculator for an ethylene dichloride (EDC) cracking furnace process described by a coupled partial differential equation/ordinary differential equation model. The objective of the proposed control system is to control the EDC cracking rate at the desired set points by manipulating the fuel gas flow rate with constraints to avoid extensive coke formation. To handle the complex behaviors that are affected by radiating walls interacting with spatial dynamics of the reactor coil, the set point calculator is employed to provide an optimal target for the constrained optimization-based controller in calculating the control actions. Simulation results show that the proposed control system is successful to regulate the controlled output at the desired set points. Control performance tests with servo and regulatory problems demonstrate that the developed control system is capable of providing excellent responses to achieve the desired set point and reject process disturbance.

1. INTRODUCTION An ethylene dichloride (EDC) thermal cracking furnace is an important unit of vinyl chloride monomer (VCM) production. EDC vapor is fed to a tubular reactor coil and thermally decomposed to yield VCM and hydrogen chloride (HCl) by heat energy from gas-fired burners in the radiating walls of the furnace. Because spatial effects in the tubular reactor coil cause the process variables to vary along the coil length and over time, researchers have proposed control techniques based on a partial differential equation (PDE) model for regulating the cracking furnace. Such PDE control systems have been published in the literature for similar chemical processes. For instance, an unsteady-state optimal control based on hyperbolic PDEs has been applied to an ethane cracking reactor.1 Also, a propane thermal cracking reactor, which is divided into several zones where each zone is regulated by a proportional− integral−derivative (PID) controller to follow an optimal gas temperature trajectory, has been proposed and obtained from PDE models.2 Due to the interaction between the process-side and fire-side variables of the thermal cracking furnace, there are some techniques to control an EDC cracking furnace governed by the PDEs coupled with an ordinary differential equation (ODE) with bidirectional interconnection. For example, a combination of the PI controller and input−output (I/O) linearizing controller was proposed to control the VCM mass © XXXX American Chemical Society

production and the tube skin temperature of an EDC cracking furnace where the state dynamics are described by two hyperbolic PDEs (EDC concentration and gas temperature) coupled with two ODEs (tube skin temperature and furnace wall temperature).3 Furthermore, recent work by Tawai and Panjapornpon (2016)4 proposed an I/O linearizing control system for controlling the cracked gas temperature of an EDC cracking furnace, for which the tube skin temperature distribution was taken into account, and only a single tuning parameter was needed to be adjusted. There are some control strategies that have been developed for a process system which can be described by the coupled PDE−ODE system.5−10 A backstepping observer-based control was applied for overcoming a problem of incomplete measurements for a PDE− ODE system in Tang and Xie (2011).5 The optimal boundary (LQ) control systems are developed for a coupled hyperbolic PDE−ODE system6 and a coupled parabolic PDE−ODE system8 to reduce the amount of online calculations by applying the off-line state feedback gain. In addition, some proposed techniques such as a backstepping boundary control9 Received: March 7, 2016 Revised: August 22, 2016 Accepted: September 1, 2016

A

DOI: 10.1021/acs.iecr.6b00916 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research and H∞ fuzzy control10 have been studied for systems with bidirectional interconnections between PDE and ODE states. In brief, the furnace wall temperature, a state of the ODE subsystem of the coupled PDE−ODE furnace system, has a strong effect on the system behavior because the heat energy is delivered to the reactor coil by the combustion of a fuel gas through the radiation emitted by the furnace wall. There are also some literature works proposing partial control strategies for the dominant states of the process with a vector of large state variables.11−18 Arbel and colleagues have published a series of articles on partial control for a fluidized catalytic cracker (FCC) described by a set of ODEs.11−14 The process model of the FCC was developed5 and analyzed for the dominant variables,12 which was then used for a control system design based on the partial control method.13,14 A reduced partial control structure was proposed for the Tennessee Eastman process described by ODE models to maximize production rate.15−18 In this work, an alternative control strategy formulated by an I/O linearization-based analytical model predictive control (AMPC) in cascade with an off-line pseudo-steady-state calculator for an EDC cracking furnace is investigated. Generally, the gas temperature is considered as the controlled output since it can be measured in real time during the operation and relates directly to the cracking rate. However, the changing EDC feed velocity from upstream processes may lead to inconsistency of the EDC cracking rate despite a constant gas temperature set point. In practice, the EDC cracking rate should be conducted at about 50−60% to avoid an increase in excessive byproduct, and the furnace wall temperature also needs to be constrained to reduce the coke formation.19 To address the concerns about the optimal manipulated inputs under process constraints and lessen the complex computation of the control action, a partial control with nonlinear optimizing control strategy is considered. Thus, the objective of this work is to control the EDC cracking rate at the desired set point by adjusting the fuel gas flow. The optimization procedure minimizes the squared error between a requesting response of the furnace wall temperature and desired target, obtained by the pseudo-steady-state calculator of the furnace, at each time instant by manipulating the fuel gas flow rate with consideration of the input and target constraints. To evaluate the control performance, the cracking furnace addresses the bidirectional interconnection of a coupled PDE−ODE model by using the finite element method for the process simulation. Integrators are applied to compensate unmeasured variables and output offset. Although the proposed strategy and the control method by Tawai and Panjapornporn (2016)4 are both based on a concept of I/O linearization, there are several differences in the controller formulation. The I/O feedback controller4 is directly formulated by using a relative order developed for coupled PDEs−ODEs, where both PDEs and ODEs are involved in the controller synthesis. The proposed method is formulated by I/O linearization with the concept of a partition system. An ODE subsystem is applied in the developed feedback while the steady-state solution obtained from solving the PDE system is used as the desired target. The advantages of the proposed scheme are that the controller has less complexity in formulation compared with the I/O feedback controller4 and it does not require the PDE state observer. By taking into account the constraints of the input and desired target, the proposed controller provides gradual manipulation to prevent an aggressive control action that can cause

excessively high furnace wall temperature and extensive coke formation. This paper lists the preliminaries of the mathematical model and nonlinear optimization-based controller with AMPC structure in Section 2, followed by the governing equations of an EDC cracking furnace in Section 3. The formulation of the control system is presented in Section 4, followed by the controller to the process model. The effect of EDC feed velocity on the cracking rate and closed-loop responses of the developed control system are illustrated in Section 5, along with the servo and regulatory tests to demonstrate the controller’s performance.

2. PRELIMINARIES 2.1. Problem Formulation. Consider a nonlinear system, for which the states are modeled by the coupling of parabolic partial differential equations and ordinary differential equation: ∂x p(z , t ) ∂x p(z , t ) ∂ 2x p(z , t ) =A +B + M(x p(z , t ), x o(t )) ∂t ∂z ∂z 2 dx o(t ) = fo (x o(t ), x ̃p(t ), u(t )) dt y = h(x p)|z = L (1)

where x ̃ p(t ) =

∫0

L

x p(z , t ) dz

with the following boundary and initial conditions:

x p(0, t ) = xp, z = 0(t ) x p(L , t ) = xp, z = L(t ) x p(z , 0) = xp,0(z) x o(0) = xo,0 where xp(z, t) denotes the vector of state variables depending on the spatial coordinate and time, xo(t) denotes a timedependent state variable, x̃p(t) denotes the lumped state of xp, y denotes an output variable at the exit position, z ∈ [0, L] is the spatial coordinate, t ∈ [0, ∞) is the time, u(t) is a manipulated variable, A and B are constant matrices, M is the vector of nonlinear function, and fo, h are nonlinear functions. 2.2. Analytical Model Predictive Control for the Coupled PDE/ODE System. In the system of eq 1, the dynamics of the states xp(z, t) and xo(t) are bidirectional interconnection, for which the manipulated input is fed into the ODE subsystem while the desired output is in the PDE subsystem at the exit position (z = L). A schematic of the bidirectional interconnection between the PDE and the ODE subsystem is illustrated in Figure 1.

Figure 1. Schematic of the coupled PDE−ODE interconnection. B


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(A1) For every set point of the process output at the exit position, ysp, and every δ, there exists an equilibrium set that satisfies [xos̅ s, xpL̅ ,ss, uss] = F(ysp − δ) and is asymptotically stable. (A2) Although the parabolic PDEs are an infinite dimensional system, the dominant dynamics behavior can be approximately characterized by finite numbers. With the assumption that the coupled PDE−ODE system is stable, we can design the controller to stabilize the system. To formulate the set point tracking controller based on I/O linearization, consider the nonlinear ODE subsystem:

Considering the EDC cracking furnace system studied in this work, the state xo(t) of the ODE subsystem (the furnace wall temperature) is a coupled variable and the heat energy is delivered to the reactor coil by the combustion of a fuel gas through the radiation emitted by the furnace wall. A proposed control strategy is to apply a set point tracking controller of the state xo(t), for which the set point is obtained from the steadystate pair of eq 1 corresponding to the desired output at z = L. To obtain the steady state pair, the following set of equations needs to be solved. ⎞ ⎛ ∂x p ∂ 2x p 0 = ⎜A + B 2 + M(x p , x o)⎟ ⎠ ⎝ ∂z ∂z

x ȯ = fo (x o , x ̅ p , u) yo = xo

z=L

where yo is a controlled output of the ODE subsystem and xp̅ is the lumped PDE state that can be calculated from available measurements along the spatial distance. A nonlinear optimization is formulated by modifying an I/O linearization-based AMPC by Panjapornpon et al. (2006).21 A concept of AMPC based on I/O linearization has been considered as a special case of the model predictive control framework,22,24,25 of which the prediction horizon is equal to a relative order number and the control horizon is equal to 1. The nonlinear state feedback is derived by minimizing a function norm of the deviations of the ODE output (yo) from its requesting linear reference trajectory; the relative order of the output is equal to 1. The following constraint optimization problem is solved at each time instant (t ∈ [t0, tf]):

0 = fo (x o , x ̃p , u) ysp = h(x p)|z = L

(2)

A compact form of the corresponding nominal steady-state pair at the exit position obtained by solving eq 2 is denoted as p [xsso , x L,ss , uss] = F(ysp )

(3)

where xoss, xpL,ss, and uss denote the steady-state pair of the coupled system at the exit location. The steady-state information on the coupled system is then used as a steadystate profile. Numerical methods such as a finite element method (FEM), finite difference method (FDM), and model reduction-based optimization (e.g., Bonis and Theodoropoulos, 2012)20 can be applied to solve the steady-state pair of the system. Note that there is no guarantee that a unique or multiple steady-state solution may exist in the system. For the exothermic system, multiple steady states are often the result of crossover between the heat generation and heat removal. However, the multiple steady states are unlikely to occur in the endothermic reaction. Therefore, such behavior is not expected for this endothermic EDC cracking reaction, in general, as studied here. A change of the wall temperature has the direct effect on the process-side dynamics. The radiated energy from the furnace wall, the coupled variable of the ODE subsystem, raises up the reactor coil temperature and subsequently the gas temperature. The dynamics of cracked gas are relatively fast due to a short residence time of the reaction (less than 70 s) compared with a time constant of the furnace wall (more than 10 000 s). Thus, the pseudo-steady state of the PDE subsystem is assumed and valid. It should be noted that the coupled variable in the ODE subsystem may not always justify the pseudo-steady state of the PDE subsystem. To avoid aggressive responses of the control actions obtained by solving the optimization problem and to prevent the states exceeding the bounds, the development of steady-state profiles with the feasible bounds of the states (xp, xo) and input (u) have been taken into account while solving the steady-state pairs. The measured (actual) value of the output at the exit position is given by y ̅ = h(x ̅ p)|z = L + δ

(5)

⎡ (βD + 1)y o − v o ⎤2 min⎢ ⎥ u ⎣ ⎦ β

(6)

subject to x ȯ = fo (x o , x ̅ p , u),

x o(t0) = x ̅ o

ulb ≤ u ≤ u ub o v lbo ≤ v o ≤ vub

where D is the differential operator (i.e., D = d/dt), vo is the reference ODE output set point obtained from the pseudosteady-state profile, and β is the tuning parameter that adjusts the speed of the ODE output response. For the system in eq 5, the optimization problem in eq 6 can be rewritten as min J(x o , x ̅ p , u , y o , v o)

(7)

u

⎡ y o + βf (x o , x ̅ p , u) − v o ⎤2 o ⎥ J(x o , x ̅ p , u , y o , v o) = ⎢ β ⎦ ⎣

subject to x ȯ = fo (x o , x ̅ p , u) ulb ≤ u ≤ u ub o v lbo ≤ v o ≤ vub

(4)

where ulb and uub denote the lower bound and upper bound of the manipulated input u, respectively. volb and voub denote the lower bound and upper bound of the reference output set point vo, respectively. Figure 2 illustrates a flow diagram of the proposed AMPC calculation.

where xp̅ denotes the vector of the measured PDE state variables and δ denotes an unmeasurable output disturbance. The predicted value of the output at the exit for the ODE and PDE states will be denoted by y,̅ xo̅ , and xp̅ , respectively. The following assumptions are made: C


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The governing equations of the EDC cracking furnace composed of dynamic behaviors of the EDC concentration, the cracked gas temperature in the reactor coil, and the tube skin temperature, which are described by PDE models as follows: ⎛ ⎞ ∂C(z , t ) ∂C(z , t ) E ⎟⎟ − C(z , t )k 0 exp⎜⎜− = −v ∂z ∂t ⎝ RTg(z , t ) ⎠ ⎛ ⎞ (−ΔH ) E ⎟⎟C(z , t ) k 0 exp⎜⎜− ∂t ∂z ρg Cpg ⎝ RTg(z , t ) ⎠ ⎞ ⎛ ⎟ 1 ⎜ 2πL + − ( ( , ) ( , )) T z t T z t ⎟ ⎜ t g Vtρg Cpg ⎜ ln(R o / R i) + 1 ⎟ kt R ihg ⎠ ⎝

∂Tg(z , t )

= −v

∂Tg(z , t )

+

∂Tt(z , t ) k t ∂ 2Tt(z , t ) A Fσ(TW(t )4 − Tt(z , t )4 ) = + W 2 ∂t Cpt ρt mt Cpt ∂z (1/mt Cpt )2πL (Tt(z , t ) − Tg(z , t )) − (ln(R o/R i)/k t) + (1/R ihg )

Figure 2. Flow diagram of the calculating sequences of the proposed analytical model predictive control.

(8)

with boundary and initial conditions C(0, t ) = C0 ,

3. MATHEMATICAL MODEL OF ETHYLENE DICHLORIDE CRACKING FURNACE A simplified schematic diagram of a VCM process is shown in Figure 3. An EDC cracking furnace studied in Tawai and

C(z , 0) = C0

Tg(0, t ) = Tg,0 ,

Tg(z , 0) = Tg,0

Tt(0, t ) = Tt,0 ,

Tt(z , 0) = Tt,0

∂Tt(z , t ) ∂z

=0 z=L

and the furnace wall temperature described by an ODE model: mf Hcomb dTw(t ) σFAW (Tw(t )4 − Tt̃ (t )4 ) = − dt mwCpw mwCpw

(9)

with the initial condition Tw(0) = Tw,0

where z ∈ [0, L] denotes spatial distance, C(z, t) denotes the EDC concentration, Tg(z, t) denotes the cracked gas temperature, Tt(z, t) denotes the tube skin temperature, T̃ t(t) denotes the integrated tube skin temperature over the entire coil length, Tw(t) denotes the furnace wall temperature, v denotes the EDC feed velocity, mf is the fuel gas flow rate, and hg is the convective heat-transfer coefficient of the cracked gas. The cracking rate of EDC and the VCM mass production rates can be defined as eq 10 and eq 11, respectively.

Figure 3. Schematic of a simplified VCM process.

Panjapornpon (2016) is considered as a case study in this work,4 in which extensive information regarding the development of a mathematical model and process parameters of the furnace can be found. EDC vapor is fed into the tubular reactor coil placed in the middle of the gas-fired furnace. The thermal energy from radiating walls and burners surrounding the tubular reactor coil is transferred to the feed vapor for breaking up the EDC into smaller molecules, vinyl chloride monomer (VCM) ,and hydrogen chloride (HCl) as the following reaction.

REDC =

MVCM =

C2H4Cl 2 → C2H3Cl + HCl

C0 − C(L , t ) × 100 C0

(10)

MwVCMvπDi 2(C0 − C(L , t )) 4

(11)

Note that all process parameter descriptions and values are presented in Table S1 in the Supporting Information.

In this example, the process-side variables change in the axial direction while the gradients in the radial direction can be neglected due to the high reactor coil length-diameter (L/D) aspect ratio (L/D ≫ 50).23 The properties of the cracked gas are assumed to be constant, and the gas velocity inside the reactor coil is assumed to follow plug flow behavior. The outlet stream from the cracking furnace is subsequently fed to the quench and VCM purification sections to separate the VCM product from other components, and the VCM mass production rate can be measured.

4. FORMULATION OF THE CONTROL SYSTEM In the studied cracking furnace, the EDC concentration, cracked-gas temperature, and tube skin temperature are classified as a PDE subsystem while the furnace wall is considered as an ODE subsystem. The vectors of the PDE and ODE subsystem states are xP = [C, Tg, Tt]T and xo = [Tw], respectively. Figure 4 illustrates the interconnection relations between the state variables. D


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profiles were proposed to reduce the burdened on the computational time of the control system and to provide the corresponding set points for the optimization problem. To calculate the optimal profiles, the process model in eqs 8 and 9 are considered at the steady state as the following equation. 0 = fp,1 (C , Tg)ss, z = L 0 = fp,2 (C , Tg)ss, z = L 0 = fp,3 (Tt , Tg , Tw )ss, z = L 0 = fo (Tw , Tt̃ , mf )ss

(13)

The steady-state information is obtained by using a numerical method in varying the fuel gas flow rate and the EDC feed velocity via a computing software such COMSOL Multiphysics. Figures 6 and 7 demonstrate the relationships of process variables under steady-state conditions with a nominal EDC feed velocity (v = 4.855 m/s).

Figure 4. Schematic of the process variable interconnection.

The dynamics of furnace wall temperature has a strong impact on the tube skin temperature that directly affects all reaction rates. The objective of this study is to control the EDC cracking rate (y = REDC) by manipulating the fuel gas flow rate (u = mf). The EDC cracking rate (REDC) can be expressed as a relation of measured VCM mass production rate (M̅ VCM) as in eq 12. 4 REDC = M̅ VCM × × 100 MwVCMC0vπDi 2 (12) The effects of variation in the feed composition, the delay time of the purification units, and the time lag in transport and measurement are not considered in this work. The following assumptions are used in a development of a feedback control system: (B1) The variables T̅ g,L,T̅ w, T̅ t,L, v,̅ and M̅ VCM are measurable for each time instant. (B2) The VCM produced by the cracking furnace is entirely purified and is considered as the VCM outlet stream of the purification units (M̅ VCM). (B3) The initial and boundary conditions of the EDC cracking process state variables are assumed to be constants. The relation of process parameters in eq 12 is used to apply the measured M̅ VCM for calculation of REDC to update the feedback responses for the control system. Figure 5 shows a schematic diagram of the proposed control structure; the integrators are added into the proposed control system to compensate the unmeasured disturbance and model mismatch. More details of the proposed control structure design are given as follows. 4.1. Off-Line Set Point Calculator. Due to the complexity of the coupled PDE/ODE system, a direct formulation of an I/O linearization controller can lead to a significant increase in the computational load. Thus, the off-line optimal steady-state

Figure 6. Steady-state profile of the fuel gas flow rate and the furnace wall temperature at 4.855 m/s of EDC feed velocity.

Figure 7. Steady-state profile of the furnace wall temperature and the EDC cracking furnace at 4.855 m/s of EDC feed velocity.

Figure 5. Schematic diagram of the developed control system for the EDC cracking furnace. E


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Figure 8. Steady-state profile for the cracked gas temperature.

Figure 9. Steady-state profile for the furnace wall temperature.

To support the operating ranges of the EDC cracking rate (30−80%) and the EDC feed velocity (3.5−6.0 m/s), a relationship of the cracking rate and feed velocity for the optimal cracked gas temperature is created as shown in Figure 8. Figure 9 shows a relationship of the cracked gas temperature and feed velocity for the furnace wall temperature used as the optimal set points for the nonlinear optimization problem. 4.2. Nonlinear Optimization Problem for the Cracking Furnace. The objective of the formulated optimization problem is to minimize the squared error between the requesting response of the furnace wall temperature and the desired target. By applying the process models of eq 9 into the optimization problem of eq 7, the following constrained optimization problem is solved at each time instant: min J(Tw , Tt,L , u , Tw,ss) u

J(Tw , Tt,L , u , Tw,ss) ⎡ ⎛ uHcomb ⎢ β ⎜⎝ mw Cp − w =⎢ ⎢ ⎢⎣

σFAW (Tw 4 − Tt̃ ) ⎞ ⎟ m w Cpw ⎠ 4

β

⎤2 + Tw − Tw,ss ⎥ ⎥ ⎥ ⎥⎦

subject to Tẇ = ϕ(Tw , Tt̃ , u) ulb ≤ u ≤ u ub Tw,ss,lb ≤ Tw,ss ≤ Tw,ss,ub

where T̃ t is the integrated tube skin temperature, Tw,ss is the wall

(14)

temperature set point, u is the manipulated input, and β is a

where

tuning parameter. F


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Industrial & Engineering Chemistry Research 4.3. Control System. To compensate the offset from the process−model mismatch, first-order integral actions are proposed as the following equations. εṘ = −λ1(REDC,sp − R̅EDC) ε ̇ = −λ 2(Tw,ss − Tw̅ ) νR = ysp − εR ν o = Tw,ss − ε

(15)

where εR is the integral of error of the controlled output REDC, ε is the integral of error of the output Tw, λ1 and λ2 are positive constants, vR is the compensated set point of the outer loop, and vo is the compensated set point of the inner loop. The terms ε̇R and ε̇ are integrated over the time to obtain εR and ε for calculation of vR and vo, respectively. The integral action in eq 15 is introduced into the control system to compensate the offsets of the set point tracking. The structure of the proposed control system is the integral controller incorporating with the AMPC controller, with the stability of the closed-loop system depending on the AMPC stability. As mentioned in Soroush and Kravaris (1996),22 the optimization problem formulated with the I/O linearization strategy does not have an analytical solution with active constraints. If the state feedback makes the performance index zero, the linear closed-loop response is inducedwhich implies a closed-loop stability. Implementation of the proposed control system requires the feedback measurements from the exit of the reactor coil and VCM mass production rate. As shown in the flowchart of Figure 10, the set point REDC,sp is compensated and sent as vR to calculate the Tw,ss by the off-line optimal profiles. The compensated target Tw,ss is then fed as vo to the optimization problem with the defined process constraints to calculate the manipulated input u. The input and output constraints are applied to the optimization problem to limit the adjusting of the control actions adjustments and avoid excessive furnace wall temperature. To conduct closed-loop studies, a cosimulation environment between MATLAB and COMSOL Multiphysics is deployed. The model of the EDC cracking furnace in eqs 8 and 9 developed in COMSOL Multiphysics is considered as a plant while the algorithm of the proposed control strategy is implemented in MATLAB. The control action is obtained by using f mincon function to solve the proposed optimization.

Figure 10. Flowchart of the proposed control system for the EDC cracking furnace.

output is presented.4 The typical disturbance, feed velocity change, is introduced into the process by decreasing the EDC feed velocity by 20% at 4 h, and the control system can regulate the gas temperature at the desired set point. However, the comparison of responses in Figure 11 shows that the EDC cracking rate cannot be maintained at the same set point as the feed velocity directly affects the EDC concentration.

5. SIMULATION RESULTS In such works, the simulation of process dynamics and control system can be implemented by computing software such MATLAB and COMSOL Multiphysics. The proposed control system is applied to the EDC cracking furnace by using the coupled PDE/ODE model, and the measured states are fed back to the controller for updating the current states. In this section, the output variables are compared to illustrate the responses under a given set of conditions to determine the controlled output for the proposed control system. The off-line optimal profiles are used to provide the set point of the nonlinear optimizing controller. Simulations are conducted for closed-loop systems, and the control performances are tested through servo and regulatory problems. 5.1. Effect of EDC Feed Velocity on Cracking Rate. To illustrate the importance of determining the EDC cracking rate as a control output, a simulation of the EDC cracking furnace with a controller that selects the gas temperature as the control

Figure 11. Cracked gas temperature and EDC cracking rate responses with a change in feed velocity.

5.2. Closed-Loop Responses. The closed-loop simulation results of the proposed control system are compared with a traditional PID control and an I/O linearization control system. In all simulations, the same initial and boundary conditions are applied: C(z, 0) = 359.83 mol/m3, Tg(z, 0) = 478 K, Tt(z, 0) = 550 K, Tw(0) = 713 K, C0 = 359.83 mol/m3, Tg,0 = 478 K, and Tt,0 = 550 K. The set point of the closed-loop system is 52% of the EDC cracking rate, REDC = 52%, with the tuning parameters G


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Industrial & Engineering Chemistry Research β = 20 s, λ1 = 0.05 s−1, and λ2 = 0.05 s−1; the fuel feed flow rate is limited between 0.01 and 0.3 kg/s (ul = 0.01, uh = 0.3), and the furnace wall temperature is operated between 700 and 1000 K (700 ≤ Tw,ss ≤ 1000). For the PID controller, the cracking rate of EDC, REDC, is also used as the control output by applying the mass production rate of VCM, MVCM, which proposed in this work to obtain the feedback of REDC. To obtain tuning parameters, the internal model control (IMC) tuning method is applied to tune the PID controller for testing the control performance. To compare with the I/O linearizing controller4 for which the cracked gas temperature is selected as a control output, optimal set point Tg,sp = 685 K is provided to obtain REDC = 52%. The tuning parameters of the I/O linearizing controller and the proposed controller are selected by the IMC tuning method. The I/O linearizing control system is simulated with a tuning parameter, β = 4.85 s, while the PID controller is applied by using a set of tuning parameters, KP = 0.005, KI = 0.000072, and Kd = 0.000001. The closed-loop responses of the EDC cracking rate, the response of EDC concentration, and the response of the tube skin temperature are shown in Figures 12−15, respectively. The performance

Figure 14. EDC concentration corresponding to the closed-loop system.

Figure 15. Tube skin temperature of the closed-loop system.

Figure 12. Profile of EDC cracking rate of the closed-loop system.

Figure 16. Furnace wall temperature response of the closed-loop system.

Figure 13. Cracked gas temperature response of the closed-loop system.

index of the controllers is shown in Table S2 in the Supporting Information. The results show that the proposed controller can achieve more optimal performance with faster responses when compare with the PID controller and less overshoot than both PID controller and I/O linearization controller. As mentioned in Schirmeister et al. (2009),19 the operation with a furnace wall temperature of more than 1053.15 K can increase the coke layer significantly. Although the I/O controller has a better performance of the integral of the squared error (ISE) index than the proposed controller, the adaptation of furnace wall temperature and fuel gas feed rate in Figures 16 and 17 show

Figure 17. Fuel flow rate corresponding to the closed-loop system.

that the proposed controller can provide more gradual control actions compared with the PID and I/O controller that violate H


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Industrial & Engineering Chemistry Research the process constraintswhich can extend the lifespan of the furnace. 5.3. Control Performance. To evaluate the control performance, servo and regulatory problems are introduced to the developed control system. For servo performance test, the control output is designed to track a new set point after the process achieved the first set point of the closed-loop simulation: the set point is initially set at 52% and then changed to 58%. The gas temperatures corresponding to the desired EDC cracking rate are 685 and 692.4 K, respectively, where these numbers must be provided as set points of the I/O linearization controller. It can be seen from the process responses to the servo problem test that are shown in Figures 18−21 that the proposed controller successfully forces the

Figure 21. Fuel gas flow rate corresponding to the servo test.

furnace wall temperature aggressively while the PID controller takes more than 10 h to control this change completely. The regulatory test is performed by decreasing the EDC feed velocity by 20% as a step disturbance after achieving the set point at the same condition of the closed-loop system. For the I/O linearization controller, the regulatory performance cannot be directly applied in this case because maintaining the cracked gas temperature will not stabilize the EDC cracking rate. Thus, a servo test of the I/O linearization to a new Tg = 677.8 K which corresponding to REDC,sp = 52% is applied. The simulation results shown in Figures 22−25 demonstrate that Figure 18. Closed-loop response of EDC cracking rate under the servo test.

Figure 22. Closed-loop response of EDC cracking rate under the regulatory test.

the proposed controller is successful in controlling the system and rejecting the disturbance within 3 h. It can track the process response to the optimal trajectory with a suitable adaptation of control actuator while the I/O linearizing control

Figure 19. Cracked gas temperature corresponding to the servo test.

Figure 20. Furnace wall temperature response under the servo test.

outputs to the desired set points asymptotically. To achieve the new set point, the I/O linearizing controller needs to adapt the

Figure 23. Cracked gas temperature corresponding to the regulatory test. I



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AUTHOR INFORMATION

Corresponding Author

*Tel.: +66 2 797 0999/1230. Fax: +66 2 561 4621. E-mail: [email protected] (C. Panjapornpon). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research is financially supported by the Graduate School of Kasetsart University, Center of Excellence on Petrochemical and Materials Technology, Center for Advanced Studies in Industrial Technology, and the Faculty of Engineering, Kasetsart University. Support from these sources is gratefully acknowledged.

Figure 24. Furnace wall temperature response under the regulatory test.

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Figure 25. Fuel gas flow rate corresponding to the regulatory test.

needs to be adjusted aggressively to achieve a fast response. For the PID controller, it encounters fluctuations in the response of PID controller after the system is disturbed, and it takes more than 10 h to reject the disturbance completely.

6. CONCLUSION A control system based on AMPC strategy in cascade with offline optimal profiles has been proposed to control the cracking rate of an EDC cracking furnace described by a coupled PDE/ ODE model. The off-line profiles provide the optimal pseudosteady-state pair to be used as the set points for the constrained optimization problem which calculate the fuel gas feed rate of burners in the cracking furnace. The controller is combined with integrators to compensate the offset and handle the process disturbances. Application to the process models demonstrates that the proposed control system is successful to regulate the controlled output at the desired set point optimally. Investigation of control performance by introducing the control problems showed that the control system has the capability to stabilize the process when the set point is changed or some disturbances are presented in the system. Also, the proposed control system is successful to regulate the furnace wall temperature under the constraints to avoid the extensive coke formation and byproducts with careful adjustment of the manipulated input.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.6b00916. Table S1. Appendix A, and Table S2 (PDF) J

NOMENCLATURE Aw = radiating area of furnace wall (m2) A, B = constant matrices C = EDC concentration (mol m−3) Cpt = heat capacity of tubular reactor coil (J kg−1 K−1) Cpg = average heat capacity of cracked gas (J kg−1 K−1) Cpw = heat capacity of furnace wall (J kg−1 K−1) D = differential operator Di = internal tube diameter (m) E = activation energy (J mol−1) Di = internal tube diameter (m) F = shape factor fo, f p, h, M = nonlinear functions Hcomb = heat of combustion (J mol−1) hg = convective heat-transfer coefficient of cracked gas (J s−1 m−2 K−1) k0 = kinetic constant k = thermal conductivity of cracked gas (W m−1 K−1) kt = thermal conductivity of tubular reactor coil (W m−1 K−1) L = tube length (m) mf = fuel gas flow rate (kg s−1) mt = mass of tubular reactor coil (kg) mw = mass of furnace wall (kg) MVCM = mass production rate of VCM (g s−1) r = relative order R = gas constant (J mol−1 K−1) REDC = EDC cracking rate (%) REDC,sp = set point of EDC cracking rate (%) Ri = internal tubular reactor coil radius (m) Ro = external tubular reactor coil radius (m) t = time (h) Tg, Tt, Tw = temperature of cracked gas, tube wall, and furnace wall, respectively (K) Tg,sp = temperature set point of cracked gas (K) Tt,L = cracked gas temperature at position z = L (K) u = manipulated variables v = feed velocity (m s−1) Vt = volume of tubular reactor coil (m3) Vw = volume of cracking furnace (m3) x = state variables xp = state variables depending on spatial coordinate and time xo = state variables depending time x̃p = lumped variable of PDE subsystem xpL = state variable at the exit position y = output variables yo = output of ODE subsystem DOI: 10.1021/acs.iecr.6b00916 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article


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ysp = process set point z = spatial coordinates ΔH = heat of reaction (J mol−1) Greek Letters

β = tuning parameter of I/O linearizing controller (s) δ = unmeasured variable ε = integral of error of furnace wall temperature (K) εR = integral of error of cracking rate (%) λ1, λ2 = tuning parameters (s−1) μ = viscosity of cracked gas (kg m−1 s−1) vo = refference output set point vR = compensated set point ρg = average density of cracked gas (kg m−3) ρt = density of tubular reactor coil (kg m−3) ρw = density of furnace wall (kg m−3) σ = Stefan−Boltzmann constant (W m−2 K−4) ϕ = nonlinear function Subscripts

ss = steady-state variable lb = lower bound ub = upper bound Superscript

− = measured variable ∼ = lumped variable

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