Input–Output Linearizing Control Strategy for an Ethylene Dichloride

Article pubs.acs.org/IECR

Input−Output Linearizing Control Strategy for an Ethylene Dichloride Cracking Furnace Using a Coupled PDE-ODE Model Atthasit Tawai and Chanin Panjapornpon* Department of Chemical Engineering, Center of Excellence on Petrochemicals and Material Technology, Faculty of Engineering, Kasetsart University, Bangkok 10900, Thailand S Supporting Information *

ABSTRACT: An input−output (I/O) linearizing control scheme for a gas-fired thermal cracking furnace is developed for a tubular reactor coil, which is a type of tubular reactor surrounded by gas-fired burners in the furnace. Due to the simultaneous interaction between the spatially distributed dynamics of the reactor coil and the lumped radiating wall, the typical proportionalintegral-derivative control widely used in industry may have insufficient performance to handle the complexity. In this work, a feedback I/O linearizing controller is applied to control a cracking furnace system that is described by a coupled PDE-ODE model: ethylene dichloride cracking. The cracked gas temperature is manipulated through the fuel gas flow to achieve the desired trajectories. Control performances of the developed controller are illustrated through simulation results for servo and regulatory problems. The proposed method provides more robustness to handle control problems without offset.

1. INTRODUCTION Due to the fact that it offers better radiant heat transfer than many other methods, thermal cracking by a gas-fired furnace is an important process that is widely used in petrochemical industries for breaking up heavier hydrocarbon molecules into lighter ones.1 In the process, hydrocarbon vapors are fed into tubular reactor coils placed in the middle of a furnace and surrounded with gas-fired burners. The radiant heat from the furnace walls, heated by the burners, transfers to the reactor coils and leads to pyrolysis of hydrocarbon vapors. This furnace behavior is highly complex due to the interaction between process-side and fireside variables, and the mixed gas in the reactor coil is strongly affected by the radiative heat transferred from the furnace walls.2 Handling the gas temperature at the desired target by manipulating the fuel gas flow rate is quite challenging due to many factors involved: for example, the complex interaction of the furnace dynamics, with the process variables and the tube skin temperature are varied along with a spatial coordinate and time due to the ongoing cracking reaction, while the furnace wall temperature is considered as a lumped system.3 The robustness of the controller should be good enough to reject disturbances caused by the fluctuation of fuel gas composition, pressure, fuelgas heat value, or hydrocarbon feed rate. The poor control performance could lead to off-spec production, thermal runaway, or hazardous plant operation. To handle a thermal cracking furnace, control strategies combining a proportional-integral-derivative (PID) controller with the optimal steady-state set point obtained by solving the ordinary differential equation (ODE) or partial differential equations (PDE) reactor model have been proposed.4−6 However, these strategies may perform efficiently only in a limited operating region due to the complexity of spatial effects and interaction between the reactor coil and radiating wall dynamics. With the reactor coil being a type of tubular reactor governed by PDEs, several research works addressing the control problem of a PDE system may apply to the tubular reactor © 2015 American Chemical Society

system. A process model can be converted to a lumped model by using techniques such as the method of characteristics,7 Galerkin method,8 model reduction by Volterra series9 and finitedifference approximation10,11 before designing the controller. Alternatively, controller design by applying the PDE model directly has also been mentioned.12 However, in practice, the tube skin of the reactor coil is strongly affected by furnace radiation that results in complex interaction between process-side and fired-side dynamics.2 The reactor coil dynamics can be described by PDEs while the furnace wall dynamic can be modeled by an ODE. There are some control strategies that apply to the ODE-PDE system.13−20 Krstic and his colleagues have published a series of articles on backstepping boundary control for coupled ODE-PDE systems. For example, a backstepping boundary control has been developed for the system of Burgers’ equation (parabolic PDE), with the control inputs in boundary condition (ODEs) being used to control the PDE.13 The control study of the ODEPDE system was subsequently extended to a system in which the actuator is described by a diffusion PDE connected in cascade with an ODE plant. The input is fed to the PDE system that interacts in series with the ODE.14,15 Tang and Xie (2011) proposed an extended study for overcoming a problem of incomplete measurements for the PDE-ODE system by using the backstepping observer-based control.16 In addition, some proposed techniques such as a backstepping boundary control17 and H∞ fuzzy control18 have been studied for systems with bidirectional interconnections between PDE and ODE states. For a class of cascade-connected system for the hyperbolic PDEs and ODE, there are some proposed methods that are applicable for one-directional19,21 and bidirectional interconnection.22,23 Received: Revised: Accepted: Published: 683

October 7, 2015 December 9, 2015 December 29, 2015 December 30, 2015 DOI: 10.1021/acs.iecr.5b03759 Ind. Eng. Chem. Res. 2016, 55, 683−691

Article

Industrial & Engineering Chemistry Research Recently, there have been control methods proposed based on I/O linearization for a thermal ethylene dichloride (EDC) cracking furnace addressing the interactions of coupled PDEODE dynamics.24,25 Panjapornpon et al. (2012) proposed a control method based on I/O linearizing control and a PI controller for controlling of an EDC cracking furnace described by two hyperbolic PDEs coupled with two ODEs. The proposed method was formulated by using the I/O linearizing control for the closed-loop response of the lumped tube skin temperature and the fuel gas feed while the PI controller takes care of the VCM mass production-EDC feed velocity pair; four tuning parameters need to be adjusted in the control system.24 In subsequent work, the conduction effect of the tube skin temperature was taken into account in the furnace model which itself consists two hyperbolic PDEs describing EDC concentration and cracked gas temperature, a parabolic PDE describing the tube skin temperature and a lumped dynamic of furnace wall temperature. The I/O linearizing controller coupled with an integral controller was proposed to handle the cracked gas temperature by adjusting the fuel gas feed, and two tuning parameters were required.25 In this work, an extended study of the control system based on I/O linearizing control for an EDC cracking furnace, described by two hyperbolic PDEs, a parabolic-PDE and an ODE which addresses the bidirectional interconnection of the coupled PDEODE dynamics, is proposed. The objective is to control the outlet cracked-gas temperature by manipulating the fuel gas flow rate to track desired trajectories. A finite-element, open-loop state observer and a first-order error compensator are combined into the control system for estimating unmeasured variables and eliminating offset from the set points, respectively. An advantage of the proposed control system is being simplicity to achieve both set point tracking and robustness due to only a single tuning parameter being required. The first-order error dynamic compensates the output offset from the approximation of the input-output response and unmeasured disturbances by using the same desired closed-loop time constant of the I/O feedbackwhich make it quite easy to tune the control system compared with previous works. The paper is structured as follows. In section 2, preliminaries of the mathematical model and input-output linearization technique are explained. In section 3, the mathematical model of an EDC cracking furnace is described. section 4 presents the formulation of the control system, which is then applied to the process model. The simulated open- and closed-loop profiles of the process system are illustrated in section 5; the control performances of the proposed control scheme are demonstrated under the servo and regulatory tests by introducing set points and unmeasured disturbances.

∂xp(z , t ) ∂t

=A

∂xp(z , t )

+B

∂ 2xp(z , t )

∂z ∂z 2 + N (xo(t ), xp(z , t ))

+ M(xp(z , t ))

dxo(t ) = F(xo(t ), xp(z , t ), u(t )) dt y = H(x0(t ), xp(z , t )) (1)

with the following boundary and initial conditions: xp(0, t ) = xp, z = 0(t ) xp(L , t ) = xp, z = L(t ) xp(z , 0) = xp,0(z) xo(0) = xo,0

where xp(z,t) denotes the vector of state variables depending on the spatial coordinate and time, xo(t) denotes the vector of timedependent state variables, y denotes the vector of the output variables, z ∈ [0, L] is the spatial coordinate, t ∈ [0,∞) is the time, u(t) is the vector of manipulated variables, F and H are the vectors of nonlinear function, and A, B are constant matrices. 2.2. Input−Output Linearization for Coupled PDEsODEs. The dynamic behaviors of the states xp(z,t) and xo(t) are bidirectional interconnection and are considered in order to investigate the relationship between the controlled output y at the exit position (z = L) and the manipulated input u. The compact form of the system in eq 1 at the considered output can be written as x ̇ = f (x , xz , xzz , u) yL = h(x)|z = L

(2)

where x = [xp xo]T is the vector of state variables, xz = ∂x/∂z constitutes the first-order spatial derivatives of x, xzz = ∂2x/∂z2 constitutes the second-order spatial derivatives of x, yL is the output y at the exit position, and u is the manipulated input; f and

2. PRELIMINARIES

h are vectors of nonlinear functions. For the nonlinear system in

2.1. Problem Formulation. Consider the mathematical model of the system in eq 1, for which states are described by the coupling of parabolic partial differential equations and ordinary differential equations:

eq 2, the relative order of the controlled output yL is denoted by r, and it is finite; the notation in eq 3 is used: 684

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⎛a⎞ a! where ⎜ ⎟ = (a − b) ! b ! ⎝b ⎠ (r−1) (1) The time derivatives of state gradient (x(1) , xz̅ z , ..., z̅ , ..., xz̅ (r−1) xz̅ z ) are set to be zero. Therefore, eq 5 becomes

yL = h(x)|z = L dyL dt

⎡ ∂h ∂x ⎤ =⎢ ⎣ ∂x ∂t ⎥⎦ z = L

=h1(x , xz , xzz)|z = L ⎡ ∂h1 ∂x ∂h1 (1) ∂h1 (1)⎤ xz + xzz ⎥ =⎢ + ∂xzz ∂xz ⎣ ∂x ∂t ⎦z = L

2

d yL dt

h(x ̅ )|z = L +

2

(1r )h (x̅ , x̅ , x̅ ) 1

z

+ ... +

zz

z=L

(rr )h

r

(x ̅ , xz̅ , xzz ̅ , 0, ..., 0, u) z=L

(1) )|z = L =h2(x , xz , xzz , xz(1) , xzz

= ysp − δ

⋮

ϕ(x ̅ , xz̅ , xzz ̅ , u)|z = L = ysp − δ

dr − 1yL

= dt r − 1 ⎡ ∂hr − 2 ∂x ⎤ ∂hr − 2 (1) ∂hr − 2 (1) (r − 2) xz + xzz + ... + xzz + ⎢ ⎥ ∂xz ∂xzz ⎣ ∂x ∂t ⎦

The feedback controller can be achieved by solving eq 6 for manipulated input u, which the compact form can be denoted by u = ψ (x ̅ , xz̅ , xzz ̅ , ysp − δ)|z = L

z=L

(1) (r − 2) , ..., xzz )|z = L =hr − 1(x , xz , xzz , xz(1) , ..., xz(r − 2) , xzz

(7)

3. MATHEMATICAL MODEL OF ETHYLENE DICHLORIDE CRACKING FURNACE A cracking furnace of ethylene dichloride (EDC) is considered as a case study in this work: see Figure 1 for a schematic diagram of the furnace.

r

d yL

= dt r ⎡ ∂hr − 1 ∂x ⎤ ∂hr − 1 (1) ∂hr − 1 (1) (r − 1) xz + xzz + ... + xzz + ⎢ ⎥ ∂xz ∂xzz ⎣ ∂x ∂t ⎦

(6)

z=L

(1) (r − 1) , ..., xzz , u)|z = L =hr (x , xz , xzz , xz(1) , ..., xz(r − 1) , xzz

(3)

x(l) z

l

l

l

x(l) zz

l

where = d xz/dt and = d xzz/dt The measured (actual) value of the output at the exit position is given by yL̅ = h(x ̅ )|z = L + δ

where x̅ denotes the vector of the measured state variables and δ denotes an unmeasurable output disturbance. The predicted values of the output at the exit and states will be denoted by yL and x, respectively. The following assumptions are made (1) For every set point of the output at the exit position, ysp, and every δ, there exists an equilibrium pair that satisfies ysp − δ = h(xs̅ s)|z=L and f (xs̅ s, xz̅ ,ss, xz̅ z,ss, uss)|z=L = 0. (2) The nominal steady state pair of the process at the exit position, (xs̅ s, xz̅ ,ss, xz̅ z,ss, uss), is hyperbolically stable. All eigenvalues of the Jacobian of the open-loop process evaluated at (xs̅ s, xz̅ ,ss, xz̅ z,ss, uss) have negative real parts. To formulate a controller based on I/O linearization, we request a linear response of the closed-loop actual output at the exit of tube (yL̅ ) of the following form: (βD + 1)r yL̅ = ysp

Figure 1. Schematic of a typically EDC cracking furnace.

The tubular reactor coil is placed in the middle of the gas-fired furnace and surrounded with flames from burners, and the EDC vapor is fed into the reactor coil, which is heated to a desired temperature by gas-fired burners. EDC cracking is an endothermic gas-phase reaction which is considered to be only the reaction of C2H4Cl 2 → C2H3Cl + HCl

EDC is thermally cracked into smaller molecules, vinyl chloride monomer (VCM) and hydrogen chloride (HCl), in the absence of a catalyst. The process model takes into account the EDC concentration, the cracked-gas temperature in the reactor coil, the tube skin temperature, and the furnace wall temperature. The reaction rate of cracking is dependent on the cracked-gas temperature,26 the gas behavior of which is assumed to be in a turbulent flow regime and close to a plug flow pattern. In this example, the reactor coil length-diameter (L/D) ratio is large (L/D ≫ 50), meaning that variations in the radial direction can be neglected.27 The thermal dynamics can be divided into three parts: the cracked gas temperature, the tube skin temperature, and the furnace wall temperature. A nonlinear first-principles model for the cracking furnace is developed, with

(4)

where D is the differential operator (i.e., D = d/dt), ysp is the desired output set point, and β is the tuning parameter that adjust the speed of the output response. By substituting the definition in eq 3 into eq 4, one obtains: h(x ̅ )|z = L + r

h (x ̅ ,

(1r )βh (x̅ , x̅ , x̅ ) 1

(1) xz̅ , xzz ̅ , xz̅ ,

= ysp − δ

z

...,

+ ... +

zz

z=L

xz̅ (r − 1) ,

(rr )β

r

(1) (r − 1) xzz , u)|z = L ̅ , ..., xzz ̅

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ΔH denotes the heat of reaction, ρg denotes the average gas density, Cpg denotes the average heat capacity of the cracked gas, Vt denotes the volume of the reactor coil, Ro and Ri denote the constant external and constant internal coil radii, respectively, kt denotes the thermal conductivity of reactor coil, k denotes the thermal conductivity of cracked gas, Di denotes the internal tube diameter, and μ denotes the viscosity of cracked gas. 3.3. Energy Balance on the Tube Skin of the Reactor Coil. As noted in the previous section, the cracked gas temperature depends on the tube skin temperature Tt(z,t). A one-dimensional heat conduction equation for the tube skin is formulated based on the material properties of a high-nickel alloy that allow us to ignore the radial heat dispersion. The dynamic of the tube skin temperature, which is affected by radiant heat from the fire side and by the fluid temperature of the process side, can be described as follow:

additional simplified assumptions considered in the onedimensional EDC cracking furnace model: (1) The gas behaves as on ideal gas, and the flow in the coil is considered to be one-dimensional. (2) EDC is cracked to produce vinyl chloride monomer (VCM) and hydrogen chloride (HCl). Side reactions are neglected. (3) The cracked gas in the tube is in a turbulent flow regime, and the gas flow pattern is close to a plug-flow pattern. (4) The radial and axial dispersions of the cracked gas in the coil are negligible since the L/D ratio ≫50. (5) The properties of the cracked gas in the reactor coil are constant. 3.1. Mass Balance of the Ethylene Dichloride. The conservation of EDC is distributed along the cracking coil length. As the reaction proceeds, the dynamic of EDC concentration can be described as follows:

∂Tt(z , t ) k t ∂ 2Tt(z , t ) = ∂t Cpt ρt ∂z 2 A Fσ(TW(t )4 − Tt(z , t )4 ) + W mt Cpt

⎛ ⎞ ∂C(z , t ) ∂C(z , t ) E ⎟⎟ − C(z , t )k 0exp⎜⎜ − = −v ∂z ∂t ⎝ RTg(z , t ) ⎠ (8)

with the boundary and initial conditions

−

(1/mt Cpt )2πL

(Tt(z , t ) − Tg(z , t ))

C(0, t ) = C0

(ln(R o/R i)/k t) + (1/R ihg )

C(z , 0) = C0

with the boundary and initial conditions

where z ∈ [0,L] denotes spatial distance, C(z,t) denotes the EDC concentration, Tg(z,t) denotes the cracked gas temperature, v denotes EDC feed velocity, k0 denotes the kinetic constant, E denotes activation energy, and R denotes the gas constant. 3.2. Energy Balance on the Cracked Gas in the Reactor Coil. As noted in the previous section, the concentration of EDC depends on the cracked gas temperature Tg(z,t). The heat transfer of the cracked gas is mainly convective, but it is also coupled with the heat transferred from the tube skin and to the reaction. The cracked gas temperature dynamic can be described as ∂Tg(z , t ) ∂t

= −v

∂Tg(z , t ) ∂z

+

Tt(0, t ) = Tt,0 ∂Tt(z , t ) ∂t

⎞ ⎟ (Tt(z , t ) − Tg(z , t ))⎟ ⎟ ⎠

where Tw(t) denotes the furnace wall temperature, Cpt denotes the heat capacity of the tube, ρt denotes the density of tube, AW denotes the radiating area of the furnace wall, F denotes a shape factor, σ denotes the Stefan−Boltzmann constant, and mt denotes the mass of the tube. 3.4. Energy Balance on the Furnace Wall. As noted in the previous section, the tube skin temperature depends on the furnace wall temperature Tw(t). Heat is delivered to the cracker by combustion of a fuel gas consisting of methane and hydrogen. Due to the high temperatures in the cracking furnace, radiation is the dominant mechanism of heat transfer to the reactor coil. Given all of the above, the behavior of the furnace wall temperature can be written as

1 R ihg

mH σFAW (Tw(t )4 − Tt(z , t )4 ) dTw(t ) = f comb − dt m w Cpw m w Cpw

(9)

Tw(0) = Tw,0

Tg(0, t ) = Tg ,0

where mf is the fuel gas flow rate, Hcomb denotes the heat of combustion, mw denotes the mass of the furnace wall, and Cpw denotes the heat capacity of the furnace wall. Note that all process parameter descriptions and values are presented in Table S1 in the Supporting Information.

Tg(z , 0) = Tg ,0 where Tt(z,t) denotes the tube skin temperature. The function hg can be evaluated by the conventional Dittus−Boelter equation, which is recast as k ⎛ ρg Div ⎞ ⎜ ⎟ Di ⎝ μ ⎠

(12)

with the initial condition

with the boundary and initial conditions

hg = 0.023

=0 z=L

Tt(z , 0) = Tt,0

⎛ ⎞ ( −ΔH ) E ⎟⎟ k 0exp⎜⎜ − ρg Cpg ⎝ RTg(z , t ) ⎠

⎛ 1 ⎜ 2πL C(z , t ) + ⎜ ln(R o / R i) Vtρg Cpg ⎜ + kt ⎝

(11)

0.8

⎛ μCpg ⎞0.4 ⎜⎜ ⎟⎟ ⎝ k ⎠

4. FORMULATION OF THE CONTROLLER SYSTEM The cracked gas exit temperature is an important parameter affecting yields of VCM production in the cracking furnace.

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Figure 2. Schematic diagram of the developed control system for the EDC cracking furnace.

x∼ṗ = f1 (xp,̃ z , xp,̃ zz , xp̃ (z , t ), xõ (t ))

Thus, the control objective of this work is to handle the crackedgas exit temperature (y = Tg|z=L) at the desired set point by manipulating the fuel gas flow (u = mf). The optimal profile from a dynamic simulation of a finite-element model is used to select the set point value. To develop a control system that supports the complicated PDEs-ODEs dynamics, the controller system composed of a feedback input-output linearizing control, finiteelement-based state observer, and integrator is used, and a schematic diagram of the proposed control structure is shown in Figure 2. More details of the control system design are given in the following subsections. 4.1. Feedback Input−Output Linearizing Controller for the Cracking Furnace. Based on the definition of the relative order given in eq 3, for this furnace system, the relative order of the developed controller is chosen to be 3 (r = 3). The cracked gas temperature evaluated at the exit in the closed-loop system is arranged to a linear form as β 3D3Tg, L + 3β 2D2Tg, L + 3βDTg, L + Tg, L = Tg,sp

x∼ȯ = f2 (xõ (t ), xp,̃ L(z , t ), u(t )) y ̃ = h(xõ (t ), xp̃ (z , t ))

(15)

where the state observer vector is x̃p = [C̃ T̃ g T̃ t]T, x̃o = [T̃ w] and the controlled output observer is ỹ = [T̃ g]|z=L. C̃ denotes the estimated EDC concentration, T̃ g denotes the estimated gas temperature, T̃ t denotes the estimated tube skin temperature, and T̃ w denotes the estimated furnace wall temperature; x̃p,z and x̃p,zz denote the first- and second-order state gradients, respectively. The computed state observers are added into the control system to estimate unmeasurable states during the operations. To compensate the offset from the process/mathematical model mismatch and the error in the parameter estimation from the observer, the following error dynamics are introduced:

(13)

η̇ =

where Tg,L is the gas temperature at position z = L, Tg,sp is the set point, and β is a tuning parameter. By substitution of eqs 8−12 into eq 13, the control action equation can be solved with the combination of EDC concentration, cracked-gas temperature, and tube wall gradient fed into the controller. The local stability analysis and the formulation of the proposed method are shown in Appendices A and B in the Supporting Information. The compact form of the feedback controller can be written as

(ν − η ) β

Tg,̂ L = η ν = Tg,sp − (Tg, L − Tg,̂ L)

(16)

where η is the integral action and ν is a compensated set point that is integrated into the controller. 4.3. Controller System. In order to support various conditions, the feedback I/O controller in eq 14 is combined with the finite-based state observer in eq 15 and the integrator in eq 16; as a result, a schematic diagram of the developed control system can be formulated: see Figure 2. The developed control system can be written in a function of variables as

u(t ) = ψ (C(z , t ), Tg(z , t ), Tt(z , t ), Tw(t ), Tg, z , Tt, zz , Cz , Tg,sp)|z = L

(14)

The above equation requires values of distributed variables to compute the control actions. However, only the cracked gas temperature and tube skin temperature at the end of the reactor coil measurement are available in practical applications. Thus, a state estimation for distributed parameter system is proposed as a state observer in this work to evaluate the unmeasurable variables. 4.2. Finite-Element-Based State Observer Design and Compensator. The application of finite-element models to solve system equations numerically is an approach that has been used in many research works.28,29 In such works, the simulation of process dynamics can be implemented by computing software such MATLAB and COMSOL Multiphysics. In this work, the finite-based, open-loop state observer of the process as shown in eq 15 is used to estimate the unmeasurable states, x̃, and the state derivatives.

mf (t ) = ψ (ν , β , Tg, L , Tu. L , Tw , Tg,̃ z , Tt,̃ zz , C̃ , Cz̃ )|z = L (17)

The above control system is used to calculate the control action for each time step and then imparts the computed value to the control actuators. To avoid quenching of the fire surrounding the inside of the cracking furnace, a minimum value of the manipulated input needs to be specified.

5. SIMULATION RESULTS The dynamic behaviors of the thermal cracking furnace are simulated by using the coupled PDEs-ODEs system. Simulations are conducted for the open-loop and closed-loop systems, and the performances of the closed-loop system are tested through various servo and regulatory problems. 687

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Industrial & Engineering Chemistry Research 5.1. Open-Loop Profiles. The furnace variable profiles between the developed coupled PDEs-ODEs system in eqs 8−12 and the PDEs-ODE furnace model with lumped tube skin temperature24 at the outlet are compared using numerical simulation results. In both simulation, 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(z,0) = 713 K and C0 = 359.83 mol/m3, Tg,0 = 478 K, Tt,0 = 550 K. Figure 3 shows the

Figure 6. Evolution of open-loop profile of cracked gas temperature.

5.2. Closed-Loop Responses. The gas temperature at the outlet is controlled by manipulating the fuel gas flow rate of the closed-loop system. A PID controller is applied for comparing the control performance, with the tuning parameters tuned by a Ziegler-Nichols open-loop tuning method. The initial conditions of the cracking furnace are the same as those of the open-loop system, and the controlled set point is selected to be an optimal condition of the process. The PID controller is simulated with a set of tuning parameters, Kp = −0.001, KI = −0.00002 and Kd = −0.000001, and compared with the proposed I/O linearizing controller, which has a tuning parameter (β) of 4.9; the minimum fuel feed flow rate of both controllers is limited to 0.01 kg/s. The closed-loop response of the gas temperature, response of the tube skin temperature, response of EDC concentration and the profile of the fuel gas flow are shown in Figures 7−10,

Figure 3. Cracked gas and tube skin temperature profiles: lumped and coupled PDE-ODE model.

open-loop profiles of gas temperature and tube skin temperature while Figure 4 shows that of EDC concentration along the

Figure 4. EDC concentration profiles: lumped and coupled PDE-ODE model.

operating time. An increase in the tube skin temperature, itself resulting from changes in the furnace wall temperature, will be transferred to the gas for the pyrolysis reaction, which results in a decrease in the EDC concentration. However, the gas temperature of the furnace system without lumped tube skin temperature tends to be less than that of the system with a lumped tube skin temperature because of the effect of the heat transfer from both process and fire sides, as shown in Figure 5. The evolution of the open-loop profile of cracking gas temperature along the tube length with time is shown in Figure 6.

Figure 7. Cracked gas temperature response of the closed-loop system.

Figure 8. Tube skin temperature profile corresponding to the closedloop system of Figure 7.

respectively. The results indicate that the proposed controller can achieve better control performance with faster responses and less fluctuation compared with the PID controller. It is clear from the application that the proposed controller has fewer tuning parameters compared with the PID controller. Note that the PID controller with these tuning parameters can only stabilize the

Figure 5. Tube skin temperature profile: lumped and coupled PDEODE model. 688

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Figure 9. EDC concentration profile corresponding to the closed-loop system of Figure 7.

Figure 12. Profile of fuel gas flow rate (kg/s) corresponding to Figure 11.

decreasing the EDC feed velocity by 30% after the process system achieves the desired set point. The simulation results shown in Figures 13 and 14 demonstrate that the proposed

Figure 10. Fuel gas flow rate corresponding to the closed-loop system of Figure 7.

process within a narrow range due to the limitation of the PID controller under the highly nonlinear process. 5.3. Control Performances. The developed control system is evaluated for the servo and regulatory problems to demonstrate the control performance. To evaluate servo performance, the process is started at the same conditions as the open-loop simulation, and then the output is designed to track three desired set points: the set point is initially set at 677 K and then is changed to be 682 and consequently 692 K. This three step test is related to a situation which occur regularly: the cracked-gas temperature cannot be increased quickly since that may affect upstream or downstream process operations, for example, causing a coke formation problem. The responses of the servo problem test are shown in Figures 11 and 12, where it can

Figure 13. Closed-loop response of cracked-gas temperature (K) under regulatory tests.

Figure 14. Profile of fuel gas flow rate (kg/s) corresponding to Figure 13.

control system is successful in rejecting the effect of a disturbance within an hour and regulating the controlled output to maintain the desired set point. The controller requests an aggressive response at the beginning to compensate the reduced gas temperature affected by reduced convective term, and then the fuel gas flow rate is reduced to maintain a new corresponding flow rate. Meanwhile, the PID controller showed poor performance in this test since fluctuations remain after the system is disturbed, and it takes more than 9 h to control this problem completely. The system is further evaluated by introducing an EDC feed velocity change and a heat of reaction change as disturbances. The closed-loop simulation set point is increased to 690 K before an unmeasured disturbance, being a 30% decrease in EDC feed velocity, is introduced. Following this, two coupled disturbances are introduced: a 30% increase in EDC feed velocity and a 20% decrease in the heat of reaction. The simulation results of this test are shown in Figures 15 and 16. A rapid adjustment of the fuel gas

Figure 11. Closed-loop response of cracked-gas temperature (K) under servo tests.

be seen that the proposed controller successfully forces the outputs to the given conditions asymptotically while the PID controller cannot regulate the process so as to achieve the desired conditions. While the gas temperature set point is increased, the controller quickly adjusts more the fuel gas to supply heat for rising the gas temperature before asymptotically reduce amount. The regulatory test is performed by using the same initial conditions as the open-loop simulation with the set point set at 677 K. An unmeasured step disturbance is introduced by 689

DOI: 10.1021/acs.iecr.5b03759 Ind. Eng. Chem. Res. 2016, 55, 683−691

Article

Industrial & Engineering Chemistry Research

robustness of the cracking furnace. The I/O linearizing controller creates the trajectory path for set point tracking while the nonlinear state observer with error dynamic is provided to compensate the error, quickly reject disturbances, and stabilize the process responses.

6. CONCLUSION This paper has proposed a new control system based on feedback I/O linearizing control for an EDC thermal cracking furnace described by a coupled PDE-ODE model. The process model is formulated to explain the cracking furnace behavior which considers the coupling of a spatial coordinate system for the process-side variables and lumped system for the fire-side variable affected by thermal radiation. The control system, developed based on feedback I/O linearization technique, controls the cracked-gas temperature by manipulating the fuel gas flow rate of burners. The controller was integrated with an open-loop state observer and integral action to estimate the unmeasurable states and derivatives for handling process disturbances and compensate the model mismatch. Simulation results showed that the proposed control system successfully forces the controlled output to follow the desired set point asymptotically. The control performances of the control system were examined under various servo and regulatory problems that are typically present in practical operation and the capability to achieve the given set points and handle process disturbances efficiently.

Figure 15. Closed-loop response of cracked-gas temperature (K) under servo and regulatory tests.

Figure 16. Profile of fuel gas flow rate (kg/s) corresponding to Figure 15.

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flow rate is requested for rising the gas temperature when the set point is changed, and then the fuel rate is adapted to stabilize the process when EDC feed velocity is decreased. After the coupled disturbances are introduced, the fuel gas flow rate is decreased to reduce the gas temperature that rise rapidly due to the effects of the disturbances before it is adjusted to a new corresponding value. It can be seen that the developed controller is successful in rapidly handling the system without fluctuations under the proposed and can maintain the desired cracked-gas temperature, tracking the desired condition. The process responses under the PID controller, show fluctuations during the servo problem test and are nearly uncontrollable after coupling of the disturbances. Figure 17 shows the tube skin temperature and gas temperature

ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.5b03759. Appendices A, B and Table S1. (PDF)

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AUTHOR INFORMATION

Corresponding Author

*Tel: +66 2 797 0999/1230. Fax: +66 2 561 4621. E-mail: [email protected] (C.P.). 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, and the Faculty of Engineering, Kasetsart University. Support from these sources is gratefully acknowledged.

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Figure 17. Profile of tube skin temperature (K) corresponding to Figure 16 compared with the cracked gas temperature (K).

responses corresponding to the profile in Figure 16. These responses illustrate the importance of determining the cracked gas temperature as a controlled output since the tube skin temperature can give a different response when compared to the cracked gas temperature that directly affects the mass production for the same servo and regulatory problems. From the simulation results of the control performance tests, it is clear that the proposed control method can improve the performance and 690

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 f, f1, f 2, h, M, N = nonlinear functions DOI: 10.1021/acs.iecr.5b03759 Ind. Eng. Chem. Res. 2016, 55, 683−691

Article

Industrial & Engineering Chemistry Research 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) r = relative order R = gas constant (J mol−1 K−1) Ri = internal tubular reactor coil radius (m) Ro = external tubular reactor coil radius (m) t = time (hr) Tg, Tt, Tw = temperature of cracked gas, tube wall, and furnace wall, respectively (K) Tg,L = cracked gas temperature at position z = L (K) Tg,sp = temperature set point of cracked gas (K) u = manipulated variables 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 x̃ = estimated state variables xz, xzz = first- and second-order spatial derivatives of x y = output variables yL = output variables at position z = L ysp = output set points z = spatial coordinates ΔH = heat of reaction (J mol−1)

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Greek Letters

β = tunning parameter of I/O linearizing controller η = error dynamic parameter μ = viscosity of cracked gas (kg m−1 s−1) v = feed velocity (m s−1) ν = tunning parameter of error dynamics 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) ψ,ϕ = nonlinear functions

Superscripts

∼ = estimated variables

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DOI: 10.1021/acs.iecr.5b03759 Ind. Eng. Chem. Res. 2016, 55, 683−691