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Control of a reverse flow reactor for VOC combustion
Chemical Engineering Science 60 (2005) 1661 – 1672 www.elsevier.com/locate/ces
Control of a reverse flow reactor for VOC combustion D. Edouarda,∗ , H. Hammourib , X.G. Zhouc a LGPC, CPE Lyon, UMR CNRS 2214, 43, Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France b LAGEP, Université Claude Bernard Lyon 1, UMR CNRS 5007, 43, Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France c UNILAB, State Key Lab of Reaction Engineering, East China University of Science and Technology, Shanghai 200237, PR China
Received 4 February 2004; received in revised form 1 October 2004; accepted 4 October 2004
Abstract A flow reversal reactor for VOC combustion is controlled by the linear quadratic regulator (LQR), which uses dilution and internal electric heating as controls to confine the hot spot temperature within the two temperature limits, in order to ensure complete conversion of the VOC and to prevent overheating of the catalyst. Three phases of operation, i.e., dilution phase, heating phase and inactive phase, are identified. In dilution and heating phases, the cost functions of the LQR control are defined in quadratic forms. In the inactive phase, the controllers are inactivated. A linear model is derived by linearization of a countercurrent pseudo-homogeneous model at two nominal operating conditions in the dilution phase and the heating phase, respectively. The feed concentration and the temperature profile are estimated on-line by using a high-gain observer with three temperatures measurements and are used in the LQR feedback control. Experiments are carried out on a medium-scale reversed flow reactor to demonstrate the proposed LQR control strategy. Results show that the LQR controller is highly efficient in maintaining normal operation of the reactor. 䉷 2004 Elsevier Ltd. All rights reserved. Keywords: Reverse flow reactor; VOC combustion; LQR control; Nonlinear distributed parameter system; Linearized model; Real-time control
1. Introduction Reverse flow reactor (RFR) is a packed-bed reactor with the flow direction periodically reversed. It was conceived more than 20 years ago and ever since has evoked wide interests among academia and industry. Through periodic flow reversal, heat released by reaction is first trapped in the packing, which is then used to heat up the feed when the flow direction is reversed. In the RFR, heat exchange between the feed and the effluent is highly efficient. As a result, autothermal operation is possible even if the feed has an adiabatic temperature rise as low as 10–15 K. Moreover, owing to the large heat capacity of the packing, the hightemperature plateau established in the packed bed is not
∗ Corresponding author. Tel.: +33 04 72 43 17 62; fax: +33 04 72 43 16 99. E-mail address: [email protected] (D. Edouard).
0009-2509/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2004.10.020
sensitive to abrupt changes in the inlet concentration, making the reactor easy to operate and complete conversion of the reactant easily achievable. These features make RFR highly competitive for VOC combustion. For efficient temperature control, different reactor configurations of RFR have been proposed. For rich feed, one can use cold gas injection, hot gas withdrawing, or heat recovery through internal heat exchangers to suppress temperature run-away; while for lean feed, hot gas supply, or internal heating can be applied to prevent extinction (Cunill et al., 1997). Despite the large number of publications contributed to maintain safe and smooth operation of the RFRs, only a few papers have been found dealing with control of the RFR. In the pioneering work of Budman et al. (1996), a PID and a feed-forward controller were employed to control a reversed flow reactor for carbon monoxide combustion. Pikes were observed at times of flow reversal when a PID controller was used. The feed-forward controller provided
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more efficient control of the conversion. However, it required on-line measurement of inlet concentration. The other is the work of Xiao et al. (1996), who controlled an RFR for sulfur dioxide conversion and compared the performance of proportional, feed-forward proportional and bang-bang control. The feed-forward proportional control was found slightly better than the simple proportional control, but it also required on-line concentration measurement. The bang-bang control was easily applicable, however, at the cost of more cooling and resulting in lower sulfur dioxide conversion. Both research groups used a reversal flow reactor with long cycles of 1200 at 2500 s, and their investigation was made by simulation. Owing to the intrinsic unsteady-state nature of the RFRs, the state variables change periodically even if the reactor has reached a cyclic steady state. Therefore, the controller is needed only to confine the variation of state variable within a certain range. This is relatively easy if good control variables have been chosen. However, again because of the unsteadystate nature, the dynamic response of control, and the control cost in particular, may be disregarded. In this paper, the LQR control strategy with well-defined control costs and performance is proposed, which is then validated experimentally on an RFR for VOC combustion.
2. Process description and the countercurrent pseudo-homogeneous model 2.1. Process description A medium-scale RFR for VOC combustion, as schematically shown in Fig. 1 (Ramdani et al., 2001), is considered in this work. Liquid xylene is employed as the VOC, which is injected into the air through a capillary tube and is then vaporized before entering the reactor. Cordierite monoliths (Corning) of square cross-sections with channels of 1 × 1 mm2 are packed in the reactor. Monolith in the core region is catalytically active and is inert in both end regions. Nieken et al. (1994a,b) demonstrated that, by using this structured packing, more efficient temperature control can be achieved than by hot gas withdrawal. The reactor is encapsulated in a rectangular box, which has a square opening in the center. In the downstream of the RFR, a blower keeps aspiration at a constant flow rate. Between the two catalytic monoliths in the core region of the reactor, an electric heater is installed for ignition and also for heat supply in case of lean feed. Feeding rate in the upstream can be decreased by opening the lid of the square opening to lower the temperature of the reactor. Three thermocouples are installed in the reactor to measure the temperature profile, and a total carbon analyzer (F.I.D) is utilized to determine the inlet pollutant concentration. For this reactor, fast flow reversal with a period of 16 s is needed to keep the hot spot temperature within the catalytic region. The reactor is thermally well insulated and thus
Fig. 1. Left: main geometrical characteristics of the RFR; Right: the countercurrent model.
adiabatic operation can be assumed. However, in the core region, heat loss is unavoidable and should be taken into account, because in this region the temperature is high and the insulation is not perfect. The lid of the square opening is not completely airtight even if the lid is closed and, as a result of leakage, a small net amount (5 m3 h−1 ) of fresh air is aspirated into the core when the aspiration rate of the blower is 100 m3 h−1 . 2.2. Countercurrent pseudo-homogeneous model Taking advantage of the high-frequency of flow reversal, this reactor can be approximated by the countercurrent reactor model (Nieken et al., 1995), as illustrated in Fig. 1 (right). This countercurrent reactor model is a heterogeneous model described by a set of three nonlinear PDEs and two algebraic equations. In order to homogenize and simplify this model, the following model has been introduced by Edouard et al. (2004). It is based on the method described in Balakotaiah and Dommeti (1999), and it is assumed that the kinetic reaction can be neglected under strong mass transfer limitation (Ramdani et al., 2001). The following pseudo-homogeneous model, described by Eqs. (1)–(4) is obtained. It features one PDE, two algebraic equations to account for mass transfer limitation, and a periodic frequency correction. Normalizing some variables and assuming that the Nusselt and Sherwood numbers on one hand and the Schmidt and Prandtl numbers on the other hand, are equal gives h = kD cpmg and 2 1 1 + 2 * T s 1 − *T s + + 2 Pax 2P 2 *x *x *T s + P Tad (t)(x) = , *t *1 *2 + P 1 = 0, − + P 2 = 0 (1) *x *x with ha c H 1− , P = 20 uv0 cpmg 2 1 Pax
=
=
2sax
, (H 0 uv0 cpmg ) 1 − 2
(1 − )s cps H , 20 uv0 cpmg
= (x)
1 + 2 , 210 (t)
Tad (t) = x=
z , H /2
H 10 (t), Mcpmg (2)
D. Edouard et al. / Chemical Engineering Science 60 (2005) 1661 – 1672
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where x is the normalized space variable. (x) accounts for the type of monoliths: (x) = 0 in the inert monoliths (x < ) and (x) = 1 in the catalytic monoliths (x ). The boundary conditions are: at x = 0:
rived from the minimization of a linear quadratic cost that characterizes the control objective. The implementation of this controller requires a discrete space model together with an observer, which generates online estimation of the states used for feedback control.
1 = 10 (t), *Ts = T0 , Tg1 = Ts − P *x x
3.1. Space discretization
(3)
at x = 1; Qj (1 + N )(Tg2 − T0 ) = (Tg1 − T0 ) + , S 0 uv0 Cpmg *Ts , Tg1 = Ts − P *x x 1 *Ts , Tg2 = Ts − P *x x 1 = 2
(4)
and the initial condition is (t = 0): Tg1 = Tg2 = Ts = T0 . In the above equations, T0 is the ambient temperature and the feed temperature, Ts is the solid temperature and Tg1 and Tg2 are the upstream and down-stream gas temperatures, respectively. Heat loss, in terms of transfer units N , dilution rate (1 − ) (percentage of fresh air in downstream flow), and heating power Qj (in watts) are accounted for in the boundary condition at x = 1. The first term on the left-hand side of Eq. (1) involves an effective axial heat conductivity, which is given by 1 Pax
+
eff 1 + 2 = , 2P 0 uv0 cpmg (H /2)(1 − /2)
eff = sax
1 + 2 (0 uv0 cpmg )2 + . 2 ha c
Finite difference method is used for discretization. The normalized space domain of the partial differential Eqs. (1), (3) and (4) is [0,1]. The space variable is denoted by x ∈ [0,1] and the discretization points are denoted by xi . It is found by simulation that 200 subintervals are needed for discretization to obtain a satisfactory temperature profile. The discretization spaces for the inert monoliths and the catalytic ones are, respectively, denoted by 1 and 2 . It means that 1 = xi − xi−1 in the inert monoliths (1 i 101) and 2 = xi − xi−1 in the catalytic monoliths (102 i 201). (Ts (x0 , t), . . . , Ts (x100 , t)) and (Ts (x101 , t), . . . , Ts (x201 , t)) are the respective discretized temperature profile in the inert monoliths and the catalytic monoliths. Ts (x101 , t) and Ts (x201 , t) are given by the boundary conditions (Eqs. (3) and (4)). The finite difference in this article takes the following forms: i = 101 → *Ts (x, t) Ts (xi , t) − Ts (xi−1 , t) = , k *x i 2 * Ts (x, t) Ts (xi+1 , t) − 2Ts (xi , t) + T (xi−1 , t) = *x 2 2k i
with k = 1 for the inert monoliths and k = 2 for the catalytic monoliths.
*T (x , t) Ts (x101 , t) − Ts (x100 , t) s 101 = 1 *x x101 . For i = 101, x101 = → 2 * Ts (x101 , t) Ts (x102 , t) ( 1 + 2 )Ts (x101 , t) Ts (x100 , t) = − + 1 2 *x 2 22 22 1 x101
When = 1 (i.e., there is no dilution), eff reduces to the well-known estimate of Vortmeyer and Schäfer (1974) as used by Nieken et al. (1995). Finally, this model was shown to experimentally match the process behaviour (Edouard et al., 2004) during openloop control and identification experiments.
3. Controller design The controller design that we propose in this article is based on the LQR techniques, and feedback control is de-
In the following sections, we will use these notations: X 1 (t) Ts (x1 , t) 1 .. .. , X1 (t) = = . . 1 (t) Ts (x100 , t) X100 2 X1 (t) = Ts (x101 , t),
X 3 (t) Ts (x102 , t) 1 . = .. . .. X 3 (t) = .
Ts (x200 , t)
3 (t) X99
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Hence, the mathematical model of the system under control takes the following form:
with
X˙ 1 (t) = A1 ((t))X 1 (t) + G1 ((t))X 2 (t) + d 1 ((t))T0 (t), 1 (t) X˙ 2 (t) = A2 ((t))X 2 (t) + G21 ((t))X100 2 3 2 + G2 ((t))X1 (t) + d ((t))Tad (t), X˙ 3 (t) = A3 ((t))X 3 (t) + G3 ((t))X 2 (t) + B 3 ((t))Qj (t)
X 2 (t) = Tad (t) = X22 (t) ,
+ d13 ((t))Tad (t) + d23 ((t))T0 (t).
(5)
Ai , B i and Gi are the parameters of the nominal model that will be used to regulate the process, and d i ’s are the disturbances that must be compensated. In the appendix their detailed expressions are given. Practically, a model-based control or supervision strategy requires the use of the temperature profile inside the reactor. The temperature profile could be reconstructed by physical sensors. However, it would require a large number of sensors and therefore would be inconvenient for implementation. For this reason, on-line estimation of the state and/or physical parameters by soft sensor, which provides on-line estimation of the state and/or parameters by combination of a priori knowledge of the physical system (nominal model) and the experimental data (a few on-line measurements), is more practical. To control the RFR under investigation, only the temperature profile is required. However, this temperature profile is strongly disturbed by the unknown concentration of the pollutant (Tad ) at the inlet of the catalytic monolith. Therefore, for a satisfactory estimation of the temperature profile, this concentration needs to be estimated by the soft sensor. 3.2. High-gain observer The concentration of the VOC (Tad (t)) is considered here as an unknown disturbance and is to be estimated. As a physical signal it is assumed as the response of a secondorder system. This assumption is not a strong one since most physical signals can be approximated by a response of such a second-order filter. The model used to design the observer is the combination of the control model (5) with d Tad (t) = (t), dt d (t) = v(t), dt
(6)
where v(t) is an unknown and bounded signal. The global extended model for observer design takes the following canonical form: X˙ 1 (t) = A1 ((t))X 1 (t) + G1 ((t))X12 (t) + d 1 ((t))T0 (t), 2 ((t))X 2 (t) + G2 ((t))X 1 (t) X˙ 2 (t) = A 1 100 + G22 ((t))X13 (t) + E, X˙ 3 (t) = A3 ((t))X 3 (t) + G3 ((t))X12 (t) + B 3 ((t))Qj (t) + d13 ((t))X22 (t) + d23 ((t))T0 (t)
(7)
2 = A
Ts (x101 , t)
(t) 0 0 0
r(t) 0 0
0 1 0
X12 (t) X32 (t)
and
E=
0 . 0 v(t)
On-line measurements required by the observer are the temperatures at the inlet of the upstream inert monolith (Ts (x1 , t)), the temperature at the inlet of the upstream catalytic monolith (Ts (x101 , t)) and the temperature at the outlet of the upstream catalytic monolith (Ts (x200 , t)). There are many algorithms available from the literature for on-line state estimation. Among these, the Kalman filter is more frequently used in process control. However, it requires the solution of a non-linear dynamical Riccati equation. Besides the numerical difficulties that may be encountered in solving the dynamical R iccati equation, on-line estimation with Kalman filter will be very time consuming, because the dimension of the Riccati equation is very large, i.e., (n2 + n)/2, where n is the dimension of the state vector (for the reactor under investigation, n = 201). By using the high-gain techniques (see for instance, Bornard and Hammouri, 1991; Deza et al., 1992; Gauthier et al., 1992; Farza et al., 1998), an observer is derived, the gain of which can be easily calculated without solving any differential equation. The following notations are defined: 2 Xe1 (t) X1 (t) e1 .. , Xe2 (t) = X2 (t) and Xe1 (t) = . e2 1 (t) Xe100 2 (t) Xe3 X 3 (t) e1 .. . Xe3 (t) = . 3 Xe99 (t)
which are, respectively, the state estimates of the vector states. 2 X1 (t) 1 X (t), Tad (t) (t) and X 3 (t), where Tad (t) and (t) are given in Eq. (6). The physical measurements are denoted by y1 (t)=X11 (t)= 3 (t) = Ts (x1 , t), y2 (t) = X12 (t) = Ts (x101 , t) and y3 (t) = X99 Ts (x200 , t). With these notations, the high-gain observer takes the following form (Edouard et al. (2004)): X˙ e1 (t) = A1 ((t))Xe1 (t) + G1 ((t))y 2 (t) 1 + d 1 ((t))T0 (t) + L1 ((t))(Xe1 − y1 (t)),
D. Edouard et al. / Chemical Engineering Science 60 (2005) 1661 – 1672
Fig. 2. Comparison between the measured and estimated inlet VOC concentration (in adiabatic temperature rise).
2 ((t))Xe2 (t) + G2 ((t))X 1 (t) X˙ e2 (t) = A 1 e100 3 2 + G22 ((t))Xe1 (t) + L2 ((t))(Xe1 − y2 (t)),
X˙ e3 (t) = A3 ((t))Xe3 (t) + G3 ((t))y2 (t) 2 + B 3 ((t))Qj (t) + d13 ((t))Xe2 (t) 3 3 3 + d2 ((t))T0 (t) + L ((t))(Xe99 − y3 (t)).
(8)
2 and the observer gains L1 , L2 The matrices A1 , A3 and A 3 and L are given in Appendix A.1 and A.2. The convergence of the observer is guaranteed on the basis of the following facts: • The temperature measurements at the inlet of the inert monolith and the catalytic monolith Ts (x1 , t) = y1 (t), Ts (x101 , t) = y2 (t)) allows for the estimation of the temperature profile in the inert monolith. • The temperature measurement Ts (x101 , t) = y2 (t) is sensitive to the instantaneous reaction. Consequently, it can be used to estimate Tad . • The temperature profile in the catalytic monolith is sensitive to the disturbance Tad . Using the estimation of Tad , together with the temperature measurement at the outlet of the catalytic monolith, permits the estimation of the temperature profile in the catalytic monolith.
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Fig. 3. Filtered temperature measurements and their on-line estimation. (y2 (t), Ts (x101 , t))=[1], (y3 (t), Ts (x200 , t))=[2], (y1 (t), Ts (x1 , t))=[3]. For curves [1] and [3] experimental and estimated data are superimposed.
Because the lid of the RFR is not perfectly airtight, a small amount of air ( = 0.95) is aspirated into the core region of the reactor. At t = 6200 s, more fresh air (3 m3 h−1 STP or =0.92) is introduced through a valve connected to the core of the reactor, while the feed flow rate of the liquid pollutant and the outlet gas flow rate remain constant. With constant feed flow rate of the liquid pollutant, any fresh air addition will decrease the air flow rate at the inlet, and therefore result in an increase of the inlet concentration, i.e., Tad . This is readily observed on both the experimental and theoretical curves of Fig. 2 at t = 6200 s. In Fig. 3, comparison is made between the estimated temperatures at x = 0, and 1 and the experiments, indication of the good performance of the observer: • at the boundary between catalytic and inert monoliths [1], the difference is less than 1 K. • at the outlet of the catalytic monolith [2], the difference is less than 5 K. • at the inlet of the inert monolith [3], the difference is less than 3 K. These discrepancies between the experimental measurements and their estimates is mainly due to the parametric uncertainty and the physical noise.
3.3. Experimental validation of the observer 3.4. LQR controller In the experiments, the period of flow reversal is 16 s. The inlet VOC concentration can be varied by changing the liquid flow rate of xylene. As mentioned above, the inlet concentration is expressed as the adiabatic temperature rise (Tad ). In Fig. 2, the state estimates are compared with the experimental Tad . It can be seen that both the dynamic responses and the steady states are estimated with satisfaction by the observer.
In operating a RFR, the temperature has to be high enough for a complete conversion of the pollutant. However, too high a temperature is harmful to the catalyst and the reactor as well. On the other hand, the temperature is preferably to be low to depress sintering and evaporating of active metal so as to prolong catalyst life. But it should be kept high enough to ensure complete conversion of the VOC. Internal electrical heating and fresh air dilution provide two effective
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means to ensure normal operation and can be used as control variables. Because of the high dimension of the reactor model, the computation time, which typically takes 0.2 s for simulation of 20 s dynamic operation, is relatively long. Therefore a model based control strategy, e.g. predictive control, which solves the model repeatedly for a real-time solution would become ponderous. Bang-bang control is also possible to ensure normal operation, however, it is difficult to minimize with it the control cost. The ideal operation of the reactor is the operation without control (controller inactive phase). This is possible when the feed concentration is appropriate and the reaction temperature falls within an envelope outlined by the two temperature boundaries predicted by complete conversion of the pollutant and safety of the catalysts. In this case, neither internal electric heating nor fresh air dilution are active, and the temperature will be anywhere inside of the envelope. Indeed, it seems natural to avoid both heating and cooling together. If the feed of pollutant concentration is too lean to ensure ignition temperature, internal heating is required (heating phase). However, the heating power should be kept as small as possible since it is at the cost of electric energy. For rich feed (dilution phase), air dilution, cheap as it is, should also be as small as possible because it reduces the throughput of the feed. Therefore, the reactor works at the upper limit temperature in dilution phase when the feed of pollutant concentration is rich and at the lower limit temperature in heating phase when the feed is lean. Consequently, to control the RFR, two controllers are needed, one for rich feed which uses air dilution as the control variable, and the other for lean feed which uses electric heating as the control variable. However, for each controller, it is evoked only when the measured temperature deviates from the setpoint in one specified direction. According to the temperature measurement, both controllers may be deactivated, or a specific one is evoked. This task is fulfilled by a control supervisor. Obviously, the choice of the steady state around which we linearize the system is important to the control performance. Except for the autothermal case (controller inactive phase), the controller to design has to compensate the input disturbance Tad and preserve the stability of the system. On the other hand, most of the reaction takes place at the inlet of the catalytic monoliths and is instantaneous (Nieken et al., 1994a,b; Ramdani et al., 2001). Therefore, instead of the full temperature profile, only Ts (x101 , t) has to be maintained between 450 and 600 K. These values correspond, respectively, to the minimal temperature necessary for autothermal operation and the maximum temperature to avoid overheating of the catalytic monolith. In the industry, the input disturbance (Tad ), usually varies between 0 and 30 K. But in order to prevent any accident, worse cases have to be evaluated. The equivalent concentration Tad is therefore assumed to vary randomly between 0 and 115 K
Fig. 4. Nominal temperature profiles for high- and low-temperature operations.
(Fig. 5). Fig. 6 shows that if no control is applied to the RFR, the hot-spot temperature will go beyond the temperature limits (450 and 600 K). This clearly justifies the need for closedloop control. With this information, the nominal steady states for the two controllers can be calculated by solving the algebraic linear equation corresponding to Eq. (5) in which X˙ i = 0. • For the steady-state profile for electric heating control, 0 = 0K, 0 = 0.95, and T 0 (x assuming Tad 101 ) = 450 K, s 0 we have Qj = 500 W and the corresponding steady-state profile Z 0 (shown in Fig. 4), where 10 10 X Z 0 Z = Z 20 = X 20 . Z 30 X 30 • Similarly, for the steady-state profile for air dilution con1 =115 K, T 1 (x 1 trol, with Tad 101 )=600 K, and Qj =0 W, s 1 we have =0.75 and the corresponding steady-state profile Z 1 (also shown in Fig. 4), where 11 11 X Z 1 Z = Z 21 = X 21 . Z 31 X 31 Finally with the steady-state profiles, the system (Eq. (5)) is linearized in both cases in the following forms: *A1 () 1i 1 i 1 1 ˙ X (t) = A ( )X (t) + Z * i *G1 () 2i 1 i 2 + G ( )X1 (t) + Z * i *d 1 () + T0 (t), * i
D. Edouard et al. / Chemical Engineering Science 60 (2005) 1661 – 1672
*A2 () 2i 2 2 i 2 ˙ X (t) = A ( )X (t) + Z * i 1 + G21 (i )X100 (t) + G22 (i )X13 (t) *G21 () 1i *G22 () 3i + Z + Z * i 100 * i 1 *d 2 () + T i + d 2 (i )Tad (t) * i ad *A3 () 3i 3 3 i 3 ˙ X (t) = A ( )X (t) + Z * i *G3 () 2i 3 i 2 + G ( )X (t) + Z * i *B 3 () + Qi + B 3 (i )Qj * i j *d13 () + T i * i ad *d23 () + T0 (t) * i
with X = Xe (t) − Z i , where i = 0, 1 and Xe (t) is the estimated state given by the observer. The gain of the feedback matrix is obtained from the solution of the classical algebraic Riccati equation F 0 = (p20 )−1 (B20 )T S 0 , S 0 A(0 )+(A(0 ))T S 0 +C T C−S 0 B20 (p20 )−1 (B20 )T S 0 =0, F 1 = (p21 )−1 (B11 )T S 1 , S 1 A(1 ) + (A(1 ))T S 1 + C T C (11) − S 1 B11 (p21 )−1 (B11 )T S 1 = 0, where C = [C1 . . . C100 C101 . . . C200 ], [C1 . . . C100 ] = [0 . . . 0] and [C101 . . . C200 ] = [1 0 . . . 0]. 4. Results 4.1. Simulation
+ d13 (i )Tad (t),
(9)
where X j = X j − Z j i with j = 1, 2, 3 and i = 0 or 1. The above linearization is achieved around each steady state 1i 1i X Z Z i = Z 2i = X 2i , with i = 0 or 1. Z 3i X 3i Both systems are summarized as ˙ = A(i )X(t) + B1i + B2i Qj . X(t)
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(10)
On the basis of the linear system (10), the LQR controllers are designed as follows. • Heating phase: when the temperature at the inlet of the catalytic zone is below the lower limit (450 K), electric heating control is evoked and the amount of heating is determined by minimizing ∞ (p10 (X 2 (t))2 + p20 (Qj (t))2 ) dt. 0
• Dilution phase: when the temperature at the inlet of the catalytic zone is above the upper limit (600 K), air dilution control is evoked and the amount of dilution determined by minimizing ∞ (p11 (X 2 (t))2 + p21 ((t))2 ) dt.
Before the controller is implemented to the real reactor, its reliability needs to be validated by simulation. The disturbance in feed concentration used for validation is randomly chosen and is shown in Fig. 5 in terms of adiabatic temperature rise. From Fig. 6 one can see that if the reactor is not controlled, the hot-spot temperature will either be higher than the upper limit or lower than the lower limit. With the use of the high-gain observer, the full states of the system can be estimated in real time, which makes implementation of the LQR control strategy possible. Since the feedback gain can be determined off-line, the time used to compute control actions online is negligible. Control simulation is conducted and the results of continuous-time system are shown in Fig. 7. The weights are tuned as follows: P10 = P11 = 1, P20 = 5 × 10−4 and P21 = 500. Simulations results show that, in spite of the steep changes in feed concentration (Fig. 5), the temperature at the inlet of catalytic zone is tightly controlled between the two limits (Fig. 7). Both manipulated variables are correctly regulated. Between 500 and 1550 s, the lean feed tends to decrease the temperature of the reactor. With the compensation of the LQR, internal heating is activated, and the temperature is kept above the extinction temperature (Fig. 9). In this period of time the dilution control is inactivated. After 1550 s, the feed becomes rich and tends to induce too high a temperature. In this situation, the dilution control is activated (Fig. 8), and the heating control is deactivated, to maintain the temperature below the maximum temperature (Fig. 7). ¯ j is At the end of the run, the average of electrical power Q 83.4 W, the average of dilution rate ¯ is 0.894, whereas in this simulation the constraints are always satisfied.
0
In both cases, the control obtained from the minimization of these costs is a linear feedback of the form
(t) = −F 1 X,
or Qj (t) = −F 0 X
4.2. Experimental validation To further validate the reliability of LQR control, experiments are carried out on the medium scale RFR. Control
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Fig. 7. Controlled response of the temperature in simulation at the hot spot. Fig. 5. Random feed concentration used in simulation.
Fig. 8. Dilution control in simulation. Fig. 6. Evolution of the hot-spot temperature in simulation without control.
weightings and the nominal profile of temperatures remain the same as in simulation. The profile of temperature necessary for the LQR controller is obtained from the high-gain observer, with the three measurements filtered from a moving average over one period. Feed concentration is changed manually in a random manner. Fig. 10 shows the changes of inlet concentration (i.e., Tad (t)) measured by the total carbon analyzer (F.I.D). Fig. 11 shows the temperature at the inlet of catalytic layer as a result of LQR control. The temperature is well confined in the envelop except when the dilution valve is saturated (with the maximum dilution rate of 10 m3 h−1 ), as seen in Fig. 12. After 5000 s, the lean feed leads to a decrease in the temperature inside the reactor. LQR tunes the internal heating (Fig. 13) to a value needed to keep the temperature above the extinction temperature. In the meantime no dilution is taking place (Fig. 12).
Fig. 9. Internal heating control in simulation.
D. Edouard et al. / Chemical Engineering Science 60 (2005) 1661 – 1672
Fig. 10. Random feed concentration used in experiments.
Fig. 12. Control input (dilution).
Fig. 11. Controlled response of the hot-spot temperature.
Fig. 13. Control input (heating power).
5. Conclusion and discussion In this paper, on the basis of a high-gain observer, an LQR control strategy is proposed and investigated to control a reverse flow reactor for VOC combustion. The new key issue of the control is that the temperature is to be confined in an envelop of two limit temperatures, instead of maintaining the temperature at a set point. To achieve this, three phases of operation are identified and two LQR controllers are incorporated in a switching control structure. With the LQR strategy, the objective of control is well defined and the control cost minimized, an advantage over simple PID controller. The reliability of the proposed control strategy is validated by simulation and experiments. Results show that, in spite of the steep changes in inlet concentration, the hot spot temperature is well confined within the two limits,
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by dilution or internal heating in case of rich or lean feed, respectively. In all the control studies, the control weighting pji are fixed. However, it is the most sensitive design parameter that can be selected for better control performance. Smaller p will result in a tighter control, however, at the cost of more control actions. Since the LQR formulation does not directly allow achieving standard control system specifications, trial and error iteration over the values of the cost weighting is necessary to arrive at more satisfactory control. Notation ac cpmg
specific solid–fluid surface area, m−1 fluid heat capacity, J kg−1 K −1
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cps h H kD M N P Pax P Qj S t Tg1 , Tg2 T0 Tmax Ts or T uv0 x X Xe y z
D. Edouard et al. / Chemical Engineering Science 60 (2005) 1661 – 1672
solid heat capacity, J kg−1 K −1 solid–fluid heat transfer coefficient, W m−2 K −1 total length of monolith, m solid–fluid mass transfer coefficient, m s−1 VOC molecular weight, kg mol−1 number of transfer units for heat loss, dimensionless Peclet number for solid–fluid heat transfer, dimensionless axial Peclet number for heat conduction, dimensionless P corrected for the finite frequency, dimensionless external power supply, W total cross-section of the monolith, m2 time, s gas temperature in the upstream, downstream monolith, K inlet and external temperature, K maximum solid temperature in the RFR, K solid temperature, K superficial gas velocity in the reference state, m s−1 reduced abscissa, 2z/H , dimensionless state vector estimated state vector filtered temperature measurements, K abscissa, m
Greek letters
H Tad
0 s (x) (x) 1su , 2su 1 , 2 10
fraction of feed flow rate, dimensionless reaction enthalpy, J mol−1 adiabatic temperature rise, K fraction of open frontal area, dimensionless period of flow reversal, s reduced abscissa of the boundary between the inert and catalytic monoliths
corrected for the finite frequency, dimensionless fluid density, kg m−3 gas density in the reference state, kg m−3 solid density, kg m−3 heat storage time constant, s characteristic function of the catalytic monolith, dimensionless (x)(1 + 2 )/(210 ), dimensionless VOC mass fraction of solid phase in the upstream, downstream monoliths, dimensionless VOC mass fraction in the upstream, downstream monoliths, dimensionless VOC mass fraction in the feed, dimensionless calibration parameter of the observer, dimensionless
Appendix A A.1. Matrices expression
a4 (t)
a3 (t) 1 A (t) = 0 . ..
a1 (t)
0
···
a2 (t) .. .
a1 (t) .. . .. .
..
···
0
. a3 (t)
a1 (t)
0
···
a2 (t) .. .
a1 (t) .. . .. .
···
0
0 A2 (t) = a2c (t),
a2 (t)
a3 (t) 3 A (t) = 0 . .. 0
b1 (t) 0 d 1 (t) = ... ,
..
.
0 .. . .. .
, a1 (t) a2 (t)
0 .. . .. .
.. , . .. . a1 (t) a3 (t) a5 (t)
d 2 (t) = r(t)
with
0 P (x101 ) r(t) = , P (x )
102
0 P (x 103 ) 0 d13 (t) = , d23 (t) = ... , .. . b2 (t) P (x200 ) 0 . G1 (t) = .. , G21 (t) = a3c (t), a1 (t) 2 G2 (t) = a1 (t), a3 (t) 0 0 3 ... . G3 (t) = ... . B (t) = b3 (t) 0
The coefficients ai , aic and bi are given by
eff1 (t) eff1 (t) D(t) , a2 (t) = −2 + , 2 xk ( xk ) ( xk )2 eff1 (t) D(t) a3 (t) = − , xk ( xk )2 (t) 1 , a4 (t) = a2 (t) + a3 (t) (t) P x1 1+ P x1 a1 (t) =
D. Edouard et al. / Chemical Engineering Science 60 (2005) 1661 – 1672
1 + N + 2 (t) P x2 1 × 2 (t) 1 + (1 + N ) 1 + − (t) P x 2 P x 2
a5 (t) = a2 (t) + a1 (t)
and
eff1 (t) D(t) + , a2c (t) = ( x1 + x2 ) 2 x 1 x 1 ( x 2 ) eff1 (t) D(t) a3c (t) = , − ( x1 )( x2 ) x1 b1 (t) = a3 (t)
b2 (t) = a1 (t)
1 , (t) 1+ P x1
1 + N − (t) 1 2 (t) + (1 + N ) 1 + − (t) P x2 P x2
K1 1 1 K2 .. 1 A = .. . . 1 K99
K12
×
1 2 (t) 1 + (1 + N ) 1 + − (t) P x 2 P x 2
with 1 − (t) , D(t) = 2
1 1 + 2 (t) eff1 (t) = + Pax 2P
and xk = x1 = 8.3651 × 10−3 for k = 1 (inert monolith), xk = x2 = 1.5513 × 10−3 for k = 2 (catalytic monolith).
0
...
...
.
..
.
2 A = K22
0
1
K32
0
0
K3 1 3 K2 .. 3 A = .. . . 3 K99
... .. .
... 0
0
and
0
...
. . . 0 .. . . .. . .. 1 .. .. . . 0 . . . 0 1
0
...
...
1 0 .. . .. .
3 K100
1 0 uv0 cpmg S
...
0 .. . .. .
0 ..
1
and b3 (t) = a1
0
. . . 0 .. . .. .. . 1 . , .. .. . . 0 . . . 0 1
1
1 K100
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0 ..
.
..
.
... .. .
... 0
are stable (it means that the real part of the eigenvalues of i the A ’s are negative). i are given by
...
... .. . .. . .. . ...
0 0 2 0 0
0 0
3
0 . 1 = .. . . . 0
2 =
0
2 .. .
... .. ..
. .
0
0 .. . .. . 0
100
,
and A.2. Gain expression
L1 (t) = a1 (t)1 K 1 , L2 (t) = (t)2 K 2 and L (t) = a3 (t)3 K J 3
3
K3
K12
K 2 = K22
2 .. . ...
... .. . .. . .. . ...
... .. ..
. .
0
0 .. . .. . 0
99
where > 0 is the parameter of calibration of the observer, in our case = 3. Finally,
where, K1 1 . 1 , . K = . 1 K100
0 . 3 = .. . . . 0
0
and
K32
(t) =
1
K 3 = ... are such that the matrices 3 K99
2
A =
0 0 0
1 0 0
0 1/r(t) 0 r(t) 0 0
0 1 0
0 , 0 1/r(t)
and
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D. Edouard et al. / Chemical Engineering Science 60 (2005) 1661 – 1672
0 0 0 J = . ..
... ... ...
1
0
0 0 1
0 1 0 . . . . .. 0 0
1 0 0 . .. . 0
References Balakotaiah, V., Dommeti, S.M.S., 1999. Effective models for packed-bed catalytic reactors. Chemical Engineering Science 54, 1621. Bornard, G., Hammouri, H., 1991. A high gain observer for a class of uniformly observable systems. In: Proceedings of the 30th IEEE Conference on Decision and Control Brighton, England, p. 1494. Budman, H., Kzyonsek, M., Silveston, P., 1996. Control of a nonadiabatic packed bed reactor under periodic flow reversal. Canadian Journal of Chemical Engineering 74, 751–759. Cunill, F., van de Beld, L., Westertep, R.K., 1997. Catalytic combustion of very lean mixtures in reverse flow reactor using an internal heater. Industrial & Engineering Chemistry Research 36, 4198. Deza, F., Busvelle, E., Gauthier, J.P., 1992. High gain estimation for nonlinear systems. Systems and Control Letters 18, 295. Edouard, D., Hammouri, H., Schweich, D., 2004. Observer design for reverse flow reactor. A.I.Ch.E. Journal 50 (9), 2155–2166.
Farza, M., Hammouri, H., Busavon, K., 1998. A simple observer for a class of nonlinear systems. Applied Material Letters 11, 27. Gauthier, J.P., Hammouri, H., Othman, S., 1992. A simple observer for nonlinear systems, application to bioreactors. IEEE Transactions on Automatic Control 37, 875. Nieken, U., Kolios, G., Eigenberger, G., 1994a. Fixed-bed reactors with periodic flow reversal: experimental results for catalytic combustion. Catalysis Today 20, 335. Nieken, U., Kolios, G., Eigenberger, G., 1994b. Control of the ignited steady state in autothermal fixed bed reactors for catalytic combustion. Chemical Engineering Science 49 (24B), 5507. Nieken, U., Kolios, G., Eigenberger, G., 1995. Limiting cases and approximate solutions for fixed-bed reactors with periodic flow reversal. A.I.Ch.E. Journal 41 (8), 1915. Ramdani, K., Pontier, R., Schweich, D., 2001. Reverse flow reactor at short switching period for VOC combustion. Chemical Engineering Science 56, 1531. Vortmeyer, D., Schäfer, R.J., 1974. Equivalence of one- and twodimensional models for heat transfer processes in packed beds: one dimensional theory. Chemical Engineering Science 29, 485. Xiao, W.D., Wang, H., Yuan, W.K., 1996. The behavior of heat removal unsteady state SO2 converter (II) controlling plan for two-point heat removal converter. Journal of Chemical and Industrial Engineering (China) 49, 455–461.
Control of a reverse flow reactor for VOC combustion D. Edouarda,∗ , H. Hammourib , X.G. Zhouc a LGPC, CPE Lyon, UMR CNRS 2214, 43, Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France b LAGEP, Université Claude Bernard Lyon 1, UMR CNRS 5007, 43, Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France c UNILAB, State Key Lab of Reaction Engineering, East China University of Science and Technology, Shanghai 200237, PR China
Received 4 February 2004; received in revised form 1 October 2004; accepted 4 October 2004
Abstract A flow reversal reactor for VOC combustion is controlled by the linear quadratic regulator (LQR), which uses dilution and internal electric heating as controls to confine the hot spot temperature within the two temperature limits, in order to ensure complete conversion of the VOC and to prevent overheating of the catalyst. Three phases of operation, i.e., dilution phase, heating phase and inactive phase, are identified. In dilution and heating phases, the cost functions of the LQR control are defined in quadratic forms. In the inactive phase, the controllers are inactivated. A linear model is derived by linearization of a countercurrent pseudo-homogeneous model at two nominal operating conditions in the dilution phase and the heating phase, respectively. The feed concentration and the temperature profile are estimated on-line by using a high-gain observer with three temperatures measurements and are used in the LQR feedback control. Experiments are carried out on a medium-scale reversed flow reactor to demonstrate the proposed LQR control strategy. Results show that the LQR controller is highly efficient in maintaining normal operation of the reactor. 䉷 2004 Elsevier Ltd. All rights reserved. Keywords: Reverse flow reactor; VOC combustion; LQR control; Nonlinear distributed parameter system; Linearized model; Real-time control
1. Introduction Reverse flow reactor (RFR) is a packed-bed reactor with the flow direction periodically reversed. It was conceived more than 20 years ago and ever since has evoked wide interests among academia and industry. Through periodic flow reversal, heat released by reaction is first trapped in the packing, which is then used to heat up the feed when the flow direction is reversed. In the RFR, heat exchange between the feed and the effluent is highly efficient. As a result, autothermal operation is possible even if the feed has an adiabatic temperature rise as low as 10–15 K. Moreover, owing to the large heat capacity of the packing, the hightemperature plateau established in the packed bed is not
∗ Corresponding author. Tel.: +33 04 72 43 17 62; fax: +33 04 72 43 16 99. E-mail address: [email protected] (D. Edouard).
0009-2509/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2004.10.020
sensitive to abrupt changes in the inlet concentration, making the reactor easy to operate and complete conversion of the reactant easily achievable. These features make RFR highly competitive for VOC combustion. For efficient temperature control, different reactor configurations of RFR have been proposed. For rich feed, one can use cold gas injection, hot gas withdrawing, or heat recovery through internal heat exchangers to suppress temperature run-away; while for lean feed, hot gas supply, or internal heating can be applied to prevent extinction (Cunill et al., 1997). Despite the large number of publications contributed to maintain safe and smooth operation of the RFRs, only a few papers have been found dealing with control of the RFR. In the pioneering work of Budman et al. (1996), a PID and a feed-forward controller were employed to control a reversed flow reactor for carbon monoxide combustion. Pikes were observed at times of flow reversal when a PID controller was used. The feed-forward controller provided
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D. Edouard et al. / Chemical Engineering Science 60 (2005) 1661 – 1672
more efficient control of the conversion. However, it required on-line measurement of inlet concentration. The other is the work of Xiao et al. (1996), who controlled an RFR for sulfur dioxide conversion and compared the performance of proportional, feed-forward proportional and bang-bang control. The feed-forward proportional control was found slightly better than the simple proportional control, but it also required on-line concentration measurement. The bang-bang control was easily applicable, however, at the cost of more cooling and resulting in lower sulfur dioxide conversion. Both research groups used a reversal flow reactor with long cycles of 1200 at 2500 s, and their investigation was made by simulation. Owing to the intrinsic unsteady-state nature of the RFRs, the state variables change periodically even if the reactor has reached a cyclic steady state. Therefore, the controller is needed only to confine the variation of state variable within a certain range. This is relatively easy if good control variables have been chosen. However, again because of the unsteadystate nature, the dynamic response of control, and the control cost in particular, may be disregarded. In this paper, the LQR control strategy with well-defined control costs and performance is proposed, which is then validated experimentally on an RFR for VOC combustion.
2. Process description and the countercurrent pseudo-homogeneous model 2.1. Process description A medium-scale RFR for VOC combustion, as schematically shown in Fig. 1 (Ramdani et al., 2001), is considered in this work. Liquid xylene is employed as the VOC, which is injected into the air through a capillary tube and is then vaporized before entering the reactor. Cordierite monoliths (Corning) of square cross-sections with channels of 1 × 1 mm2 are packed in the reactor. Monolith in the core region is catalytically active and is inert in both end regions. Nieken et al. (1994a,b) demonstrated that, by using this structured packing, more efficient temperature control can be achieved than by hot gas withdrawal. The reactor is encapsulated in a rectangular box, which has a square opening in the center. In the downstream of the RFR, a blower keeps aspiration at a constant flow rate. Between the two catalytic monoliths in the core region of the reactor, an electric heater is installed for ignition and also for heat supply in case of lean feed. Feeding rate in the upstream can be decreased by opening the lid of the square opening to lower the temperature of the reactor. Three thermocouples are installed in the reactor to measure the temperature profile, and a total carbon analyzer (F.I.D) is utilized to determine the inlet pollutant concentration. For this reactor, fast flow reversal with a period of 16 s is needed to keep the hot spot temperature within the catalytic region. The reactor is thermally well insulated and thus
Fig. 1. Left: main geometrical characteristics of the RFR; Right: the countercurrent model.
adiabatic operation can be assumed. However, in the core region, heat loss is unavoidable and should be taken into account, because in this region the temperature is high and the insulation is not perfect. The lid of the square opening is not completely airtight even if the lid is closed and, as a result of leakage, a small net amount (5 m3 h−1 ) of fresh air is aspirated into the core when the aspiration rate of the blower is 100 m3 h−1 . 2.2. Countercurrent pseudo-homogeneous model Taking advantage of the high-frequency of flow reversal, this reactor can be approximated by the countercurrent reactor model (Nieken et al., 1995), as illustrated in Fig. 1 (right). This countercurrent reactor model is a heterogeneous model described by a set of three nonlinear PDEs and two algebraic equations. In order to homogenize and simplify this model, the following model has been introduced by Edouard et al. (2004). It is based on the method described in Balakotaiah and Dommeti (1999), and it is assumed that the kinetic reaction can be neglected under strong mass transfer limitation (Ramdani et al., 2001). The following pseudo-homogeneous model, described by Eqs. (1)–(4) is obtained. It features one PDE, two algebraic equations to account for mass transfer limitation, and a periodic frequency correction. Normalizing some variables and assuming that the Nusselt and Sherwood numbers on one hand and the Schmidt and Prandtl numbers on the other hand, are equal gives h = kD cpmg and 2 1 1 + 2 * T s 1 − *T s + + 2 Pax 2P 2 *x *x *T s + P Tad (t)(x) = , *t *1 *2 + P 1 = 0, − + P 2 = 0 (1) *x *x with ha c H 1− , P = 20 uv0 cpmg 2 1 Pax
=
=
2sax
, (H 0 uv0 cpmg ) 1 − 2
(1 − )s cps H , 20 uv0 cpmg
= (x)
1 + 2 , 210 (t)
Tad (t) = x=
z , H /2
H 10 (t), Mcpmg (2)
D. Edouard et al. / Chemical Engineering Science 60 (2005) 1661 – 1672
1663
where x is the normalized space variable. (x) accounts for the type of monoliths: (x) = 0 in the inert monoliths (x < ) and (x) = 1 in the catalytic monoliths (x ). The boundary conditions are: at x = 0:
rived from the minimization of a linear quadratic cost that characterizes the control objective. The implementation of this controller requires a discrete space model together with an observer, which generates online estimation of the states used for feedback control.
1 = 10 (t), *Ts = T0 , Tg1 = Ts − P *x x
3.1. Space discretization
(3)
at x = 1; Qj (1 + N )(Tg2 − T0 ) = (Tg1 − T0 ) + , S 0 uv0 Cpmg *Ts , Tg1 = Ts − P *x x 1 *Ts , Tg2 = Ts − P *x x 1 = 2
(4)
and the initial condition is (t = 0): Tg1 = Tg2 = Ts = T0 . In the above equations, T0 is the ambient temperature and the feed temperature, Ts is the solid temperature and Tg1 and Tg2 are the upstream and down-stream gas temperatures, respectively. Heat loss, in terms of transfer units N , dilution rate (1 − ) (percentage of fresh air in downstream flow), and heating power Qj (in watts) are accounted for in the boundary condition at x = 1. The first term on the left-hand side of Eq. (1) involves an effective axial heat conductivity, which is given by 1 Pax
+
eff 1 + 2 = , 2P 0 uv0 cpmg (H /2)(1 − /2)
eff = sax
1 + 2 (0 uv0 cpmg )2 + . 2 ha c
Finite difference method is used for discretization. The normalized space domain of the partial differential Eqs. (1), (3) and (4) is [0,1]. The space variable is denoted by x ∈ [0,1] and the discretization points are denoted by xi . It is found by simulation that 200 subintervals are needed for discretization to obtain a satisfactory temperature profile. The discretization spaces for the inert monoliths and the catalytic ones are, respectively, denoted by 1 and 2 . It means that 1 = xi − xi−1 in the inert monoliths (1 i 101) and 2 = xi − xi−1 in the catalytic monoliths (102 i 201). (Ts (x0 , t), . . . , Ts (x100 , t)) and (Ts (x101 , t), . . . , Ts (x201 , t)) are the respective discretized temperature profile in the inert monoliths and the catalytic monoliths. Ts (x101 , t) and Ts (x201 , t) are given by the boundary conditions (Eqs. (3) and (4)). The finite difference in this article takes the following forms: i = 101 → *Ts (x, t) Ts (xi , t) − Ts (xi−1 , t) = , k *x i 2 * Ts (x, t) Ts (xi+1 , t) − 2Ts (xi , t) + T (xi−1 , t) = *x 2 2k i
with k = 1 for the inert monoliths and k = 2 for the catalytic monoliths.
*T (x , t) Ts (x101 , t) − Ts (x100 , t) s 101 = 1 *x x101 . For i = 101, x101 = → 2 * Ts (x101 , t) Ts (x102 , t) ( 1 + 2 )Ts (x101 , t) Ts (x100 , t) = − + 1 2 *x 2 22 22 1 x101
When = 1 (i.e., there is no dilution), eff reduces to the well-known estimate of Vortmeyer and Schäfer (1974) as used by Nieken et al. (1995). Finally, this model was shown to experimentally match the process behaviour (Edouard et al., 2004) during openloop control and identification experiments.
3. Controller design The controller design that we propose in this article is based on the LQR techniques, and feedback control is de-
In the following sections, we will use these notations: X 1 (t) Ts (x1 , t) 1 .. .. , X1 (t) = = . . 1 (t) Ts (x100 , t) X100 2 X1 (t) = Ts (x101 , t),
X 3 (t) Ts (x102 , t) 1 . = .. . .. X 3 (t) = .
Ts (x200 , t)
3 (t) X99
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D. Edouard et al. / Chemical Engineering Science 60 (2005) 1661 – 1672
Hence, the mathematical model of the system under control takes the following form:
with
X˙ 1 (t) = A1 ((t))X 1 (t) + G1 ((t))X 2 (t) + d 1 ((t))T0 (t), 1 (t) X˙ 2 (t) = A2 ((t))X 2 (t) + G21 ((t))X100 2 3 2 + G2 ((t))X1 (t) + d ((t))Tad (t), X˙ 3 (t) = A3 ((t))X 3 (t) + G3 ((t))X 2 (t) + B 3 ((t))Qj (t)
X 2 (t) = Tad (t) = X22 (t) ,
+ d13 ((t))Tad (t) + d23 ((t))T0 (t).
(5)
Ai , B i and Gi are the parameters of the nominal model that will be used to regulate the process, and d i ’s are the disturbances that must be compensated. In the appendix their detailed expressions are given. Practically, a model-based control or supervision strategy requires the use of the temperature profile inside the reactor. The temperature profile could be reconstructed by physical sensors. However, it would require a large number of sensors and therefore would be inconvenient for implementation. For this reason, on-line estimation of the state and/or physical parameters by soft sensor, which provides on-line estimation of the state and/or parameters by combination of a priori knowledge of the physical system (nominal model) and the experimental data (a few on-line measurements), is more practical. To control the RFR under investigation, only the temperature profile is required. However, this temperature profile is strongly disturbed by the unknown concentration of the pollutant (Tad ) at the inlet of the catalytic monolith. Therefore, for a satisfactory estimation of the temperature profile, this concentration needs to be estimated by the soft sensor. 3.2. High-gain observer The concentration of the VOC (Tad (t)) is considered here as an unknown disturbance and is to be estimated. As a physical signal it is assumed as the response of a secondorder system. This assumption is not a strong one since most physical signals can be approximated by a response of such a second-order filter. The model used to design the observer is the combination of the control model (5) with d Tad (t) = (t), dt d (t) = v(t), dt
(6)
where v(t) is an unknown and bounded signal. The global extended model for observer design takes the following canonical form: X˙ 1 (t) = A1 ((t))X 1 (t) + G1 ((t))X12 (t) + d 1 ((t))T0 (t), 2 ((t))X 2 (t) + G2 ((t))X 1 (t) X˙ 2 (t) = A 1 100 + G22 ((t))X13 (t) + E, X˙ 3 (t) = A3 ((t))X 3 (t) + G3 ((t))X12 (t) + B 3 ((t))Qj (t) + d13 ((t))X22 (t) + d23 ((t))T0 (t)
(7)
2 = A
Ts (x101 , t)
(t) 0 0 0
r(t) 0 0
0 1 0
X12 (t) X32 (t)
and
E=
0 . 0 v(t)
On-line measurements required by the observer are the temperatures at the inlet of the upstream inert monolith (Ts (x1 , t)), the temperature at the inlet of the upstream catalytic monolith (Ts (x101 , t)) and the temperature at the outlet of the upstream catalytic monolith (Ts (x200 , t)). There are many algorithms available from the literature for on-line state estimation. Among these, the Kalman filter is more frequently used in process control. However, it requires the solution of a non-linear dynamical Riccati equation. Besides the numerical difficulties that may be encountered in solving the dynamical R iccati equation, on-line estimation with Kalman filter will be very time consuming, because the dimension of the Riccati equation is very large, i.e., (n2 + n)/2, where n is the dimension of the state vector (for the reactor under investigation, n = 201). By using the high-gain techniques (see for instance, Bornard and Hammouri, 1991; Deza et al., 1992; Gauthier et al., 1992; Farza et al., 1998), an observer is derived, the gain of which can be easily calculated without solving any differential equation. The following notations are defined: 2 Xe1 (t) X1 (t) e1 .. , Xe2 (t) = X2 (t) and Xe1 (t) = . e2 1 (t) Xe100 2 (t) Xe3 X 3 (t) e1 .. . Xe3 (t) = . 3 Xe99 (t)
which are, respectively, the state estimates of the vector states. 2 X1 (t) 1 X (t), Tad (t) (t) and X 3 (t), where Tad (t) and (t) are given in Eq. (6). The physical measurements are denoted by y1 (t)=X11 (t)= 3 (t) = Ts (x1 , t), y2 (t) = X12 (t) = Ts (x101 , t) and y3 (t) = X99 Ts (x200 , t). With these notations, the high-gain observer takes the following form (Edouard et al. (2004)): X˙ e1 (t) = A1 ((t))Xe1 (t) + G1 ((t))y 2 (t) 1 + d 1 ((t))T0 (t) + L1 ((t))(Xe1 − y1 (t)),
D. Edouard et al. / Chemical Engineering Science 60 (2005) 1661 – 1672
Fig. 2. Comparison between the measured and estimated inlet VOC concentration (in adiabatic temperature rise).
2 ((t))Xe2 (t) + G2 ((t))X 1 (t) X˙ e2 (t) = A 1 e100 3 2 + G22 ((t))Xe1 (t) + L2 ((t))(Xe1 − y2 (t)),
X˙ e3 (t) = A3 ((t))Xe3 (t) + G3 ((t))y2 (t) 2 + B 3 ((t))Qj (t) + d13 ((t))Xe2 (t) 3 3 3 + d2 ((t))T0 (t) + L ((t))(Xe99 − y3 (t)).
(8)
2 and the observer gains L1 , L2 The matrices A1 , A3 and A 3 and L are given in Appendix A.1 and A.2. The convergence of the observer is guaranteed on the basis of the following facts: • The temperature measurements at the inlet of the inert monolith and the catalytic monolith Ts (x1 , t) = y1 (t), Ts (x101 , t) = y2 (t)) allows for the estimation of the temperature profile in the inert monolith. • The temperature measurement Ts (x101 , t) = y2 (t) is sensitive to the instantaneous reaction. Consequently, it can be used to estimate Tad . • The temperature profile in the catalytic monolith is sensitive to the disturbance Tad . Using the estimation of Tad , together with the temperature measurement at the outlet of the catalytic monolith, permits the estimation of the temperature profile in the catalytic monolith.
1665
Fig. 3. Filtered temperature measurements and their on-line estimation. (y2 (t), Ts (x101 , t))=[1], (y3 (t), Ts (x200 , t))=[2], (y1 (t), Ts (x1 , t))=[3]. For curves [1] and [3] experimental and estimated data are superimposed.
Because the lid of the RFR is not perfectly airtight, a small amount of air ( = 0.95) is aspirated into the core region of the reactor. At t = 6200 s, more fresh air (3 m3 h−1 STP or =0.92) is introduced through a valve connected to the core of the reactor, while the feed flow rate of the liquid pollutant and the outlet gas flow rate remain constant. With constant feed flow rate of the liquid pollutant, any fresh air addition will decrease the air flow rate at the inlet, and therefore result in an increase of the inlet concentration, i.e., Tad . This is readily observed on both the experimental and theoretical curves of Fig. 2 at t = 6200 s. In Fig. 3, comparison is made between the estimated temperatures at x = 0, and 1 and the experiments, indication of the good performance of the observer: • at the boundary between catalytic and inert monoliths [1], the difference is less than 1 K. • at the outlet of the catalytic monolith [2], the difference is less than 5 K. • at the inlet of the inert monolith [3], the difference is less than 3 K. These discrepancies between the experimental measurements and their estimates is mainly due to the parametric uncertainty and the physical noise.
3.3. Experimental validation of the observer 3.4. LQR controller In the experiments, the period of flow reversal is 16 s. The inlet VOC concentration can be varied by changing the liquid flow rate of xylene. As mentioned above, the inlet concentration is expressed as the adiabatic temperature rise (Tad ). In Fig. 2, the state estimates are compared with the experimental Tad . It can be seen that both the dynamic responses and the steady states are estimated with satisfaction by the observer.
In operating a RFR, the temperature has to be high enough for a complete conversion of the pollutant. However, too high a temperature is harmful to the catalyst and the reactor as well. On the other hand, the temperature is preferably to be low to depress sintering and evaporating of active metal so as to prolong catalyst life. But it should be kept high enough to ensure complete conversion of the VOC. Internal electrical heating and fresh air dilution provide two effective
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means to ensure normal operation and can be used as control variables. Because of the high dimension of the reactor model, the computation time, which typically takes 0.2 s for simulation of 20 s dynamic operation, is relatively long. Therefore a model based control strategy, e.g. predictive control, which solves the model repeatedly for a real-time solution would become ponderous. Bang-bang control is also possible to ensure normal operation, however, it is difficult to minimize with it the control cost. The ideal operation of the reactor is the operation without control (controller inactive phase). This is possible when the feed concentration is appropriate and the reaction temperature falls within an envelope outlined by the two temperature boundaries predicted by complete conversion of the pollutant and safety of the catalysts. In this case, neither internal electric heating nor fresh air dilution are active, and the temperature will be anywhere inside of the envelope. Indeed, it seems natural to avoid both heating and cooling together. If the feed of pollutant concentration is too lean to ensure ignition temperature, internal heating is required (heating phase). However, the heating power should be kept as small as possible since it is at the cost of electric energy. For rich feed (dilution phase), air dilution, cheap as it is, should also be as small as possible because it reduces the throughput of the feed. Therefore, the reactor works at the upper limit temperature in dilution phase when the feed of pollutant concentration is rich and at the lower limit temperature in heating phase when the feed is lean. Consequently, to control the RFR, two controllers are needed, one for rich feed which uses air dilution as the control variable, and the other for lean feed which uses electric heating as the control variable. However, for each controller, it is evoked only when the measured temperature deviates from the setpoint in one specified direction. According to the temperature measurement, both controllers may be deactivated, or a specific one is evoked. This task is fulfilled by a control supervisor. Obviously, the choice of the steady state around which we linearize the system is important to the control performance. Except for the autothermal case (controller inactive phase), the controller to design has to compensate the input disturbance Tad and preserve the stability of the system. On the other hand, most of the reaction takes place at the inlet of the catalytic monoliths and is instantaneous (Nieken et al., 1994a,b; Ramdani et al., 2001). Therefore, instead of the full temperature profile, only Ts (x101 , t) has to be maintained between 450 and 600 K. These values correspond, respectively, to the minimal temperature necessary for autothermal operation and the maximum temperature to avoid overheating of the catalytic monolith. In the industry, the input disturbance (Tad ), usually varies between 0 and 30 K. But in order to prevent any accident, worse cases have to be evaluated. The equivalent concentration Tad is therefore assumed to vary randomly between 0 and 115 K
Fig. 4. Nominal temperature profiles for high- and low-temperature operations.
(Fig. 5). Fig. 6 shows that if no control is applied to the RFR, the hot-spot temperature will go beyond the temperature limits (450 and 600 K). This clearly justifies the need for closedloop control. With this information, the nominal steady states for the two controllers can be calculated by solving the algebraic linear equation corresponding to Eq. (5) in which X˙ i = 0. • For the steady-state profile for electric heating control, 0 = 0K, 0 = 0.95, and T 0 (x assuming Tad 101 ) = 450 K, s 0 we have Qj = 500 W and the corresponding steady-state profile Z 0 (shown in Fig. 4), where 10 10 X Z 0 Z = Z 20 = X 20 . Z 30 X 30 • Similarly, for the steady-state profile for air dilution con1 =115 K, T 1 (x 1 trol, with Tad 101 )=600 K, and Qj =0 W, s 1 we have =0.75 and the corresponding steady-state profile Z 1 (also shown in Fig. 4), where 11 11 X Z 1 Z = Z 21 = X 21 . Z 31 X 31 Finally with the steady-state profiles, the system (Eq. (5)) is linearized in both cases in the following forms: *A1 () 1i 1 i 1 1 ˙ X (t) = A ( )X (t) + Z * i *G1 () 2i 1 i 2 + G ( )X1 (t) + Z * i *d 1 () + T0 (t), * i
D. Edouard et al. / Chemical Engineering Science 60 (2005) 1661 – 1672
*A2 () 2i 2 2 i 2 ˙ X (t) = A ( )X (t) + Z * i 1 + G21 (i )X100 (t) + G22 (i )X13 (t) *G21 () 1i *G22 () 3i + Z + Z * i 100 * i 1 *d 2 () + T i + d 2 (i )Tad (t) * i ad *A3 () 3i 3 3 i 3 ˙ X (t) = A ( )X (t) + Z * i *G3 () 2i 3 i 2 + G ( )X (t) + Z * i *B 3 () + Qi + B 3 (i )Qj * i j *d13 () + T i * i ad *d23 () + T0 (t) * i
with X = Xe (t) − Z i , where i = 0, 1 and Xe (t) is the estimated state given by the observer. The gain of the feedback matrix is obtained from the solution of the classical algebraic Riccati equation F 0 = (p20 )−1 (B20 )T S 0 , S 0 A(0 )+(A(0 ))T S 0 +C T C−S 0 B20 (p20 )−1 (B20 )T S 0 =0, F 1 = (p21 )−1 (B11 )T S 1 , S 1 A(1 ) + (A(1 ))T S 1 + C T C (11) − S 1 B11 (p21 )−1 (B11 )T S 1 = 0, where C = [C1 . . . C100 C101 . . . C200 ], [C1 . . . C100 ] = [0 . . . 0] and [C101 . . . C200 ] = [1 0 . . . 0]. 4. Results 4.1. Simulation
+ d13 (i )Tad (t),
(9)
where X j = X j − Z j i with j = 1, 2, 3 and i = 0 or 1. The above linearization is achieved around each steady state 1i 1i X Z Z i = Z 2i = X 2i , with i = 0 or 1. Z 3i X 3i Both systems are summarized as ˙ = A(i )X(t) + B1i + B2i Qj . X(t)
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(10)
On the basis of the linear system (10), the LQR controllers are designed as follows. • Heating phase: when the temperature at the inlet of the catalytic zone is below the lower limit (450 K), electric heating control is evoked and the amount of heating is determined by minimizing ∞ (p10 (X 2 (t))2 + p20 (Qj (t))2 ) dt. 0
• Dilution phase: when the temperature at the inlet of the catalytic zone is above the upper limit (600 K), air dilution control is evoked and the amount of dilution determined by minimizing ∞ (p11 (X 2 (t))2 + p21 ((t))2 ) dt.
Before the controller is implemented to the real reactor, its reliability needs to be validated by simulation. The disturbance in feed concentration used for validation is randomly chosen and is shown in Fig. 5 in terms of adiabatic temperature rise. From Fig. 6 one can see that if the reactor is not controlled, the hot-spot temperature will either be higher than the upper limit or lower than the lower limit. With the use of the high-gain observer, the full states of the system can be estimated in real time, which makes implementation of the LQR control strategy possible. Since the feedback gain can be determined off-line, the time used to compute control actions online is negligible. Control simulation is conducted and the results of continuous-time system are shown in Fig. 7. The weights are tuned as follows: P10 = P11 = 1, P20 = 5 × 10−4 and P21 = 500. Simulations results show that, in spite of the steep changes in feed concentration (Fig. 5), the temperature at the inlet of catalytic zone is tightly controlled between the two limits (Fig. 7). Both manipulated variables are correctly regulated. Between 500 and 1550 s, the lean feed tends to decrease the temperature of the reactor. With the compensation of the LQR, internal heating is activated, and the temperature is kept above the extinction temperature (Fig. 9). In this period of time the dilution control is inactivated. After 1550 s, the feed becomes rich and tends to induce too high a temperature. In this situation, the dilution control is activated (Fig. 8), and the heating control is deactivated, to maintain the temperature below the maximum temperature (Fig. 7). ¯ j is At the end of the run, the average of electrical power Q 83.4 W, the average of dilution rate ¯ is 0.894, whereas in this simulation the constraints are always satisfied.
0
In both cases, the control obtained from the minimization of these costs is a linear feedback of the form
(t) = −F 1 X,
or Qj (t) = −F 0 X
4.2. Experimental validation To further validate the reliability of LQR control, experiments are carried out on the medium scale RFR. Control
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D. Edouard et al. / Chemical Engineering Science 60 (2005) 1661 – 1672
Fig. 7. Controlled response of the temperature in simulation at the hot spot. Fig. 5. Random feed concentration used in simulation.
Fig. 8. Dilution control in simulation. Fig. 6. Evolution of the hot-spot temperature in simulation without control.
weightings and the nominal profile of temperatures remain the same as in simulation. The profile of temperature necessary for the LQR controller is obtained from the high-gain observer, with the three measurements filtered from a moving average over one period. Feed concentration is changed manually in a random manner. Fig. 10 shows the changes of inlet concentration (i.e., Tad (t)) measured by the total carbon analyzer (F.I.D). Fig. 11 shows the temperature at the inlet of catalytic layer as a result of LQR control. The temperature is well confined in the envelop except when the dilution valve is saturated (with the maximum dilution rate of 10 m3 h−1 ), as seen in Fig. 12. After 5000 s, the lean feed leads to a decrease in the temperature inside the reactor. LQR tunes the internal heating (Fig. 13) to a value needed to keep the temperature above the extinction temperature. In the meantime no dilution is taking place (Fig. 12).
Fig. 9. Internal heating control in simulation.
D. Edouard et al. / Chemical Engineering Science 60 (2005) 1661 – 1672
Fig. 10. Random feed concentration used in experiments.
Fig. 12. Control input (dilution).
Fig. 11. Controlled response of the hot-spot temperature.
Fig. 13. Control input (heating power).
5. Conclusion and discussion In this paper, on the basis of a high-gain observer, an LQR control strategy is proposed and investigated to control a reverse flow reactor for VOC combustion. The new key issue of the control is that the temperature is to be confined in an envelop of two limit temperatures, instead of maintaining the temperature at a set point. To achieve this, three phases of operation are identified and two LQR controllers are incorporated in a switching control structure. With the LQR strategy, the objective of control is well defined and the control cost minimized, an advantage over simple PID controller. The reliability of the proposed control strategy is validated by simulation and experiments. Results show that, in spite of the steep changes in inlet concentration, the hot spot temperature is well confined within the two limits,
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by dilution or internal heating in case of rich or lean feed, respectively. In all the control studies, the control weighting pji are fixed. However, it is the most sensitive design parameter that can be selected for better control performance. Smaller p will result in a tighter control, however, at the cost of more control actions. Since the LQR formulation does not directly allow achieving standard control system specifications, trial and error iteration over the values of the cost weighting is necessary to arrive at more satisfactory control. Notation ac cpmg
specific solid–fluid surface area, m−1 fluid heat capacity, J kg−1 K −1
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cps h H kD M N P Pax P Qj S t Tg1 , Tg2 T0 Tmax Ts or T uv0 x X Xe y z
D. Edouard et al. / Chemical Engineering Science 60 (2005) 1661 – 1672
solid heat capacity, J kg−1 K −1 solid–fluid heat transfer coefficient, W m−2 K −1 total length of monolith, m solid–fluid mass transfer coefficient, m s−1 VOC molecular weight, kg mol−1 number of transfer units for heat loss, dimensionless Peclet number for solid–fluid heat transfer, dimensionless axial Peclet number for heat conduction, dimensionless P corrected for the finite frequency, dimensionless external power supply, W total cross-section of the monolith, m2 time, s gas temperature in the upstream, downstream monolith, K inlet and external temperature, K maximum solid temperature in the RFR, K solid temperature, K superficial gas velocity in the reference state, m s−1 reduced abscissa, 2z/H , dimensionless state vector estimated state vector filtered temperature measurements, K abscissa, m
Greek letters
H Tad
0 s (x) (x) 1su , 2su 1 , 2 10
fraction of feed flow rate, dimensionless reaction enthalpy, J mol−1 adiabatic temperature rise, K fraction of open frontal area, dimensionless period of flow reversal, s reduced abscissa of the boundary between the inert and catalytic monoliths
corrected for the finite frequency, dimensionless fluid density, kg m−3 gas density in the reference state, kg m−3 solid density, kg m−3 heat storage time constant, s characteristic function of the catalytic monolith, dimensionless (x)(1 + 2 )/(210 ), dimensionless VOC mass fraction of solid phase in the upstream, downstream monoliths, dimensionless VOC mass fraction in the upstream, downstream monoliths, dimensionless VOC mass fraction in the feed, dimensionless calibration parameter of the observer, dimensionless
Appendix A A.1. Matrices expression
a4 (t)
a3 (t) 1 A (t) = 0 . ..
a1 (t)
0
···
a2 (t) .. .
a1 (t) .. . .. .
..
···
0
. a3 (t)
a1 (t)
0
···
a2 (t) .. .
a1 (t) .. . .. .
···
0
0 A2 (t) = a2c (t),
a2 (t)
a3 (t) 3 A (t) = 0 . .. 0
b1 (t) 0 d 1 (t) = ... ,
..
.
0 .. . .. .
, a1 (t) a2 (t)
0 .. . .. .
.. , . .. . a1 (t) a3 (t) a5 (t)
d 2 (t) = r(t)
with
0 P (x101 ) r(t) = , P (x )
102
0 P (x 103 ) 0 d13 (t) = , d23 (t) = ... , .. . b2 (t) P (x200 ) 0 . G1 (t) = .. , G21 (t) = a3c (t), a1 (t) 2 G2 (t) = a1 (t), a3 (t) 0 0 3 ... . G3 (t) = ... . B (t) = b3 (t) 0
The coefficients ai , aic and bi are given by
eff1 (t) eff1 (t) D(t) , a2 (t) = −2 + , 2 xk ( xk ) ( xk )2 eff1 (t) D(t) a3 (t) = − , xk ( xk )2 (t) 1 , a4 (t) = a2 (t) + a3 (t) (t) P x1 1+ P x1 a1 (t) =
D. Edouard et al. / Chemical Engineering Science 60 (2005) 1661 – 1672
1 + N + 2 (t) P x2 1 × 2 (t) 1 + (1 + N ) 1 + − (t) P x 2 P x 2
a5 (t) = a2 (t) + a1 (t)
and
eff1 (t) D(t) + , a2c (t) = ( x1 + x2 ) 2 x 1 x 1 ( x 2 ) eff1 (t) D(t) a3c (t) = , − ( x1 )( x2 ) x1 b1 (t) = a3 (t)
b2 (t) = a1 (t)
1 , (t) 1+ P x1
1 + N − (t) 1 2 (t) + (1 + N ) 1 + − (t) P x2 P x2
K1 1 1 K2 .. 1 A = .. . . 1 K99
K12
×
1 2 (t) 1 + (1 + N ) 1 + − (t) P x 2 P x 2
with 1 − (t) , D(t) = 2
1 1 + 2 (t) eff1 (t) = + Pax 2P
and xk = x1 = 8.3651 × 10−3 for k = 1 (inert monolith), xk = x2 = 1.5513 × 10−3 for k = 2 (catalytic monolith).
0
...
...
.
..
.
2 A = K22
0
1
K32
0
0
K3 1 3 K2 .. 3 A = .. . . 3 K99
... .. .
... 0
0
and
0
...
. . . 0 .. . . .. . .. 1 .. .. . . 0 . . . 0 1
0
...
...
1 0 .. . .. .
3 K100
1 0 uv0 cpmg S
...
0 .. . .. .
0 ..
1
and b3 (t) = a1
0
. . . 0 .. . .. .. . 1 . , .. .. . . 0 . . . 0 1
1
1 K100
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0 ..
.
..
.
... .. .
... 0
are stable (it means that the real part of the eigenvalues of i the A ’s are negative). i are given by
...
... .. . .. . .. . ...
0 0 2 0 0
0 0
3
0 . 1 = .. . . . 0
2 =
0
2 .. .
... .. ..
. .
0
0 .. . .. . 0
100
,
and A.2. Gain expression
L1 (t) = a1 (t)1 K 1 , L2 (t) = (t)2 K 2 and L (t) = a3 (t)3 K J 3
3
K3
K12
K 2 = K22
2 .. . ...
... .. . .. . .. . ...
... .. ..
. .
0
0 .. . .. . 0
99
where > 0 is the parameter of calibration of the observer, in our case = 3. Finally,
where, K1 1 . 1 , . K = . 1 K100
0 . 3 = .. . . . 0
0
and
K32
(t) =
1
K 3 = ... are such that the matrices 3 K99
2
A =
0 0 0
1 0 0
0 1/r(t) 0 r(t) 0 0
0 1 0
0 , 0 1/r(t)
and
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D. Edouard et al. / Chemical Engineering Science 60 (2005) 1661 – 1672
0 0 0 J = . ..
... ... ...
1
0
0 0 1
0 1 0 . . . . .. 0 0
1 0 0 . .. . 0
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