Control of Rotating Wave Trains in a Loop Reactor - Industrial

Article pubs.acs.org/IECR

Control of Rotating Wave Trains in a Loop Reactor Pietro Altimari† and Erasmo Mancusi*,‡,§ †

Dipartimento di Chimica, Università “Sapienza” di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italia Facoltà d’Ingegneria, Università del Sannio, Piazza Roma, 82100, Benevento, Italia § Departamento de Engenharia Química e de Alimentos, Universidade Federal de Santa Catarina, 88040-970, Florianópolis, SC, Brazil ‡

ABSTRACT: A control strategy is proposed to stabilize thermal wave trains in networks of catalytic reactors with periodically rotated inlet and outlet ports. The implemented approach entails updating the time between two successive port rotations based on the current values of reaction- and thermal-front velocities. These latter values are estimated online by an algorithm derived from geometric analysis of the spatiotemporal temperature pattern. Guidelines for the placement of temperature sensors and controller design that enforce robust stability at regime and during startup are illustrated. The illustrated strategy is validated on a simulated problem of controlling a loop reactor for the combustion of volatile organic compounds.

■

INTRODUCTION Spatiotemporally varying patterns, such as traveling fronts and periodic waves, can autonomously form in catalytic reactors as a result of the interplay between reaction and transport.1−6 The emergence of these solutions can lead to thermal runway and reaction extinction and has therefore been considered a critical concern for reactor operation.7,8 Numerous theoretical and experimental studies over the past decades have thus focused on the development of design and control methodologies preventing the formation of spatiotemporally varying patterns.9,10 Typically, stationary operation is targeted, and reactor parameter values far from the stability boundaries of such solutions are selected. In contrast with such an approach, theoretical and experimental evidence has been reported demonstrating the possibility of purposefully exploiting spatiotemporally varying patterns to optimize the performance of catalytic processes.11,12 The fundamental idea is to drive the motion of traveling fronts and waves to construct a prescribed spatiotemporal pattern.13 This methodology has been demonstrated to be successful in enhancing the yield and selectivity in reaction−diffusion systems.11,12,14 Scarce attention has been paid, however, to its application in convection−reaction−diffusion systems. This class of systems configures the most appropriate modeling framework to describe the dynamics of catalytic reactors and thus represents an obligatory benchmark for testing the reliability of the introduced control ideas. Particular interest has been attracted in this context to the application of a network of tubular catalytic reactors with periodically rotated inlet and outlet ports.15,16 A schematic description of the network, also referred to as a loop reactor or a simulated moving-bed reactor, is shown in Figure 1. The number of reactors rotated in the flow direction is defined as ns, and the time between two successive modifications of the feeding sequence is referred to as the switching time τ. The network is usually operated in T*-periodic regimes, where T* is the time needed to recover the initial feeding configuration, although multiperiodic, quasiperiodic, and chaotic solutions can arise.17−20 The emergence of highconversion solutions is governed by the interaction of traveling © 2013 American Chemical Society

temperature fronts with the periodic rotation of the inlet and outlet ports. This rotation enables, for example, a reaction front traveling in the flow direction to be trapped within the bed and can thus ensure autothermal operation at low adiabatic temperature rise. This has motivated significant interest in the application of such network technology to the combustion of gaseous streams with low contents of volatile organic compounds.18,21−28 The effect of radial heat losses in the system for a lean mixture of volatile compounds was analyzed by Barresi et al.29 This latter application covers the largest portion of the studies on such networks presented during the past two decades. Some studies have also provided indications about the network potential to enhance the yield and/or selectivity in reversible reactions or more complex reaction schemes. This has been confirmed for the synthesis of methanol,30−33 the selective reduction of nitrogen oxides,34−36 and the production of synthesis gas through the coupling of partial oxidation and steam reforming.37 Despite the satisfactory results obtained in these applications, the fundamental mechanisms governing the emergence of highconversion regimes have only recently been identified, shedding light on the actual network potential. Sheintuch and Nekhamkina38 elucidated the fundamental role played by the interaction between traveling fronts and periodic forcing. They demonstrated the existence of single thermal waves traveling in the flow direction for ns/Vth < τ < ns/Vfr, where Vfr and Vth are the velocities of a reaction front and of a purely thermal front (that is, a temperature front moving in the absence of reaction), respectively. Experimental demonstration of this latter solution regime was also reported in ref 39. Subsequent studies were focused on the formation of trains of traveling thermal waves with switching times of 0 < τ < ns/ Vth. These regimes allow the conversion−temperature pattern minimizing the amount of catalyst needed to attain a prescribed Received: Revised: Accepted: Published: 12134

April 4, 2013 July 10, 2013 August 1, 2013 August 1, 2013 dx.doi.org/10.1021/ie401067x | Ind. Eng. Chem. Res. 2013, 52, 12134−12145

Industrial & Engineering Chemistry Research

Article

Figure 1. Schematic representation of the network. Inlet and outlet ports are periodically rotated in a stepwise manner by ns reactors.

conversion in equilibrium-limited processes to be reproduced40 and can significantly reduce the maximum temperature experienced by the catalyst in combustion processes.41 Mancusi et al.42 demonstrated that increasing ns can enlarge the τ values and the τ range sustaining the emergence of thermal wave-train solutions. This is of practical importance, as the achievement of low switching times would require frequent modification of the feeding sequence and, thus, cause damage to the external valve system. Altimari et al.43 later demonstrated the possibility of controlling the structure of thermal wave trains. Infinitely many domains of thermal wave trains were demonstrated to exist for any value of ns and any number of reactors composing the network, and analytical approximations were derived for the stability boundaries of the predicted solutions based on geometric analysis of the spatiotemporal temperature pattern. The analysis was extended to reversible exothermic reactions.40 The coexistence of wave-train solutions with low-conversion multiperiodic and extinguished regimes was analyzed in ref 44. The main obstacles to network operation around T*-periodic wave-train regimes are the narrow switching time stability range of such solutions and their coexistence with low-conversion multiperiodic and extinguished regimes. Under these conditions, external disturbances and model uncertainties can likely cause network shutdown or transition to low-conversion regimes. For this reason, the switching time τ cannot be fixed a priori but must be tightly controlled. Several approaches have been developed for network control, all of them concerned with single-wave solutions. The proposed control strategies employ the switching time as a manipulated variable and can be grouped into two different classes: (1) strategies triggering switching when temperature values at one or two prescribed axial positions cross certain thresholds and (2) strategies updating the implemented switching time based on an explicit feedback control law. Among the former group, the strategy proposed by Barresi et al.45 entails switching when the catalyst temperature at the beginning of the fed reactor falls below a first set point and the temperature at the inlet of the subsequent reactor is above a second set point (significantly larger than the first). Zahn et al.46 investigated the performance of three different controllers, again falling in group 1, concluding that the most effective approach, requiring the minimum number of temperature sensors, is to install a thermocouple at the end of each reactor and to switch when the temperature at the end of the fed reactor becomes lower than a prescribed set point. The feasibility and robustness of this control strategy was also confirmed experimentally.

A proportional feedback control law was proposed by Smagina and Sheintuch,47 allowing the switching time to be computed in response to the difference between the catalyst temperature at a prescribed axial position and a set-point value. Stability analysis of the controlled network was also performed on a simplified model reported earlier by Sheintuch and Nekhamkina,38 providing indications for the selection of controller parameters. An analogous control structure was employed by Sheinman and Sheintuch48 for the robust stabilization of the single-wave solution of a network carrying out methanol synthesis. They implemented a proportional controller and a proportional− derivative controller to update the switching time in response to the difference between the axial position reached by the reaction front at the end of each cycle and a set-point value. The set point was updated during each cycle based on the online estimation of the reaction-front velocity. For this purpose, an algorithm, whose detailed description was not reported, was employed by the authors for the reconstruction of the temperature pattern. Other advanced approaches, such as the use of observers or soft sensors for forced unsteady-state reactors, have been proposed to control a single traveling temperature wave.49−51 Even though they ensure the stability of the single-wave solution, each of the aforementioned strategies is inadequate for operating the network in wave-train regimes.52 A study addressing the problem of stabilizing wave-train solutions was recently presented.52 In that study, the proportional control structure proposed by Sheinman and Sheintuch48 was employed, and a controller design procedure enforcing the stability of wave-train solutions was described. The implemented control was adaptive as the set point was updated during each cycle based on the estimated value of the reactionfront velocity. Analytical relationships previously determined for the stability limits of wave-train solutions43 were used for this purpose. The main limit of the approach presented in ref 52 is related to the application of an approximate form for the expression of the stability limits of wave-train solutions. These limits depend on both the reaction- and thermal-front velocities, Vfr and Vth, respectively. However, only Vfr was estimated online with the approach described in ref 52, and a simplified form for the expression of the stability limits was employed to update the set point based on the assumption Vfr ≈ Vth. However, this assumption is valid at zero adiabatic temperature rise, whereas Vfr becomes significantly lower than Vth as the adiabatic temperature rise is increased. Because the minimum adiabatic 12135

dx.doi.org/10.1021/ie401067x | Ind. Eng. Chem. Res. 2013, 52, 12134−12145


Article

temperature rise sustaining autothermal operation increases with the number of waves,41 difficulties can be found as solutions with increasing number of waves are considered. A further weak point of the control method described in ref 52 is related to the implemented startup strategy, which imposes that network operation be started with the catalyst uniformly heated at temperatures as high as possible. This approach is based on the observation that higher initial temperatures reduce the risk of transition of the controlled network to multiperiodic and extinguished solutions. However, there is no guarantee that increasing the initial temperature is sufficient to definitively rule out this risk. In this article, a control strategy overcoming the previous difficulties is presented. In accordance with the proposed approach, a switching time value ensuring stability of the prescribed solution is determined during each cycle based on estimated values of reaction- and thermal-front velocities. Again, analytical relationships previously determined for the stability limits of wave-train solutions are used for this purpose, but unlike in ref 52, no proportional control is employed. Because both the reaction- and thermal-front velocities are estimated during each cycle, the difficulties related to the application of an approximate form for the expression of the stability limits are overcome. Also, it is demonstrated that the proposed control strategy significantly enhances robustness during startup. The rest of this article is structured as follows: The mathematical model of the network is described next. Then, the stability characteristics and spatiotemporal temperature pattern of wave-train solutions of the uncontrolled network are reviewed. Subsequently, the proposed control strategy and its implementation are analyzed. Concluding remarks end the article.

=

( −ΔH )C0γ E u ,v= ,B= ,D (ρc p)f T0 RT0 u0

a=

(ρc p)f Lu0 (ρc p)eff AL , Peh = ,P exp( −γ ), Le = (ρc p)f ke u0

em =

L Lu0 ,L= 0 Df N

(2)

Danckwerts boundary conditions are applied at inlet and outlet sections of each reactor and are modified after each switching to take into account the permutation of the network feeding sequence. The boundary conditions can be written in dimensionless form as 1 ∂xi Pem ∂ξ

= −[1 − fi (t *)]x in − fi (t *) xi − 1(i − 1, t *) 0

+ xi(i − 1, t *) 1 ∂ϑi Peh ∂ξ

= −[1 − fi (t *)]ϑin − fi (t *) ϑi − 1(i − 1, t *) 0

+ ϑi(i − 1, t *) ∂xi ∂ξ

= 1

∂θi ∂ξ

= 0,

i = 1, ..., N

1

(3)

where f i(t) = h{t* − mod[(i − 1)ns, N]τ}, mod(*, ·) denotes the standard modulo function,53 and h(t*) is the following piecewise-constant periodic function

■

⎧ ⎛ t* ⎞ ⎪ 0 if 0 ≤ mod⎜ , N ⎟ < 1 ⎝τ ⎠ ⎪ h(t ) = ⎨ ⎛ t* ⎞ ⎪ ⎪1 if mod⎝⎜ τ , N ⎠⎟ > 1 ⎩

MATHEMATICAL MODEL We consider networks of catalytic reactors organized in a loop (see Figure 1 for a representative scheme). In the following discussion, the number of reactors composing the network is denoted as N. The inlet and outlet sections are periodically rotated in the flow direction so as to jump ns reactors each time. The number of shifted reactors, ns, is kept constant during operation. The switching velocity is defined as Vsw = ns/τ. N − 1 switching strategies can be identified as ns varies between 1 and N − 1.43 A first-order irreversible exothermic reaction is selected as representative of a large class of chemical processes. For the sake of simplicity, constant physical properties and negligible interphase gradients are assumed. Mass and energy balances for the ith reactor unit can be then written as

(4)

It is worth noting that the implemented operating strategy (that is, ns) enters the mathematical model through the forcing function h(t*) in the boundary conditions 3. The numerical analysis illustrated in the article was derived, unless otherwise specified, using the parameter values reported in Table 1. We investigated parameter values capable of sustaining the formation of a reaction front moving at a constant velocity in the flow direction, as these conditions are of practical interest. Indeed, under stationary operation, the Table 1. Dimensionless Parameter Valuesa

2 ⎛ ϑγ ⎞ ∂xi ∂x 1 ∂ xi +v i = + Da(1 − x) exp⎜ i ⎟ 2 ∂t * ∂ξ Pem ∂ξ ⎝ ϑi + γ ⎠

Le

tu T − T0 z C , t* = 0 , ϑ = γ ,x=1− ,γ L L T C0

ξ=

2 ∂ϑi ∂ϑ 1 ∂ ϑi +v i = + BDa(1 − x) ∂t * ∂ξ Peh ∂ξ 2 ⎛ ϑγ ⎞ exp⎜ i ⎟, i = 1, ..., N ⎝ ϑi + γ ⎠

(1)

parameter

value

Damkohler number, Da dimensionless activation energy, γ dimensionless adiabatic temperature rise, B dimensionless feed temperature,b ϑin dimensionless velocity, v Lewis number, Le Peclet number for heat conduction, Peh Peclet number for mass diffusion, Pem

0.0129/N 14.13 4 −4.82 1 800 312.5/N 500/N

a

with the following definitions for dimensionless variables and parameters (see also the Nomenclature section)

Reference dimensional parameters: u0 = 0.5 m/s, L0 = 0.5 m, C0 = 1 × 10−3 kmol/m3, T0 = 293 °C. bCorresponding to Tin = 100 °C.

12136



Article

emergence of a reaction front traveling in the flow direction causes extinction, and thus, periodic rotation of inlet and outlet ports is required to attain high conversions. The total length L0 of the network is maintained constant as N is varied. Therefore, the Peclet and Damköhler number values reported in Table 1 are scaled by N.

section. This determines the warming of the cold region ab generated during the previous cycle. The network energy balance is closed, as a new cold region ab is generated at the same time by displacement of the upstream reaction front. The warming of cold region ab generated by displacement of the reaction front is possible only if thermal fronts forming in a and b reach the network outlet section during the same cycle. For the pattern displayed in Figure 2, these two thermal fronts must reach the network outlet section within a time interval ranging between one and two cycles. This imposes the condition that the distance covered in two cycles by the front forming in a is larger than the axial position reached by the network outlet section within the same time interval, that is, 2Vthτ ≥ 2. Analogously, it can be derived that the distance covered by the front forming in b in a single cycle must be lower than the axial position reached by the network outlet section within the same time interval, that is, Vfrτ + Vthτ ≤ 2. The derived inequalities can be manipulated to determine the switching-time stability limits of the considered spatiotemporal pattern leading to 1/Vth ≤ τ ≤ 2/(Vfr + Vth). Infinitely many T*-periodic wave-train regimes can be generated as τ varies and the two fronts forming in a and b reach the network outlet section after the same number of cycles. In particular, a train of p thermal waves is found when the fronts forming in a and b spend between p − 1 and p cycles traveling through the network. Again, the switching-time values sustaining the emergence of such a solution can be determined by the geometric analysis illustrated earlier. This gives

■

SPATIOTEMPORAL STRUCTURE OF WAVE-TRAIN SOLUTIONS Thermal waves are formed in the network by the interactions of traveling temperature fronts with the rotating inlet and outlet sections. To elucidate this mechanism, we analyze in Figure 2

mod(nsp , N ) mod(nsp , N ) ≤τ≤ Vthp Vfr + Vth(p − 1)

(5)

The term mod(ns p, N) appearing in inequalities 5 defines the axial position reached by the network outlet section within p cycles. It is important to remark that τ values defined by inequalities 5 can also sustain the emergence of multiperiodic and extinguished solutions.44 There is therefore no guarantee that a T*-periodic wave-train solution will be reached even when parameter values fulfilling inequalities 5 are selected.

■

CONTROL OF ROTATING THERMAL WAVE TRAINS In the following subsection, a strategy for network control is illustrated. The objective is to bring the network to operate around a prescribed T*-periodic wave-train regime and to ensure stability in the presence of disturbance and model uncertainties. Control Strategy. In accordance with the analysis presented in ref 43 and reviewed in the preceding section, ensuring the stability of T*-periodic wave-train regimes requires that inequalities 5 be satisfied. Once ns, N, and p are fixed, this objective can be achieved by varying the velocities of the temperature fronts or the switching time τ. The velocities of the temperature fronts can be modified by varying the feeding conditions (e.g., feed temperature and composition). This approach is less efficient than varying the switching time, however. This latter variable can be instantaneously changed to a prescribed value without any additional cost for network operation, whereas modifications in the feed characteristics increase operation cost and take a finite time interval to be introduced. These considerations suggest that switching time should be employed as the manipulated variable. Obviously, improved control performance could be attained by simultaneously manipulating the switching time and the feed

Figure 2. T*-periodic two-wave solution (ns = 3 and N = 4 at τ = 830): (a) temperature profile predicted at one-half cycle and (b) spatiotemporal temperature pattern, where blue and red identify cold and hot regions, respectively. Each line separating such regions describes the evolution of the axial position of a temperature front.

the spatiotemporal temperature pattern of a T*-periodic regime corresponding to a train of two thermal waves. Vertical bold segments are used to denote the positions of the inlet and outlet sections during each cycle. It is apparent from Figure 2 that the fresh reactants are always fed after switching to a hot reactor section. This ensures the formation of a new reaction front moving in the flow direction with velocity Vfr and prevents transition to extinguished or multiperiodic regimes. Owing to the displacement of the reaction front, a cold region, ab, with a width of Vfrτ is generated during each cycle (Figure 2). Two thermal fronts then form after switching in a and b and move at constant velocity Vth = 1/Le until reaching the network outlet 12137



Article

characteristics. However, the design of such a multivariable control system goes beyond the scope of the present study. In the following development, therefore, the switching time is employed as the only manipulated variable. When reaction- and thermal-front velocities are known, inequalities 5 can be used to determine a switching-time value ensuring stability. Owing to model uncertainties and disturbances, variations in Vth and Vfr can occur, however, causing the selected switching time to fall outside the stability range of the prescribed solution. The approach we followed to prevent this possibility was to update the switching time during each cycle based on the current values of Vth and Vfr. Estimates for these velocities were derived online and substituted into inequalities 5 to determine the switching time. To enhance robustness, the middle point of the stability range defined by inequalities 5 was selected as the switching time τi to be implemented during the ith cycle. This gives the following control law τi =

⎤ mod(nsp , N ) ⎡ 1 1 ⎢ ⎥ + ⎢⎣ Vth, ip 2 Vfr, i + Vth, i(p − 1) ⎥⎦

(6) Figure 3. Geometry-based online estimation of reaction- and thermalfront velocities. The spatiotemporal temperature pattern displayed in Figure 2 is considered. The axial positions of the thermocouples used are indicated by vertical dashed gray lines, while vertical bold gray segments are used to denote the time intervals Δt*th and Δt*fr . The instant at which a temperature front reaches the thermocouple placed in the ith reactor is determined by thhe intersection between the front spatiotemporal path and the vertical dashed gray line at ξ = i + Δξ. This allows Δt*th and Δt*fr to be computed during each cycle.

where Vth,i and Vfr,i are the thermal- and reaction-front velocity values, respectively, estimated during the ith cycle. The problem to be solved is to determine Vth,i and Vfr,i. The algorithm that we implemented for this purpose was derived from the geometric analysis of the network spatiotemporal pattern illustrated in the previous section. Consider the wave-train solution displayed in Figure 2. Each temperature front separates a region characterized by temperatures close to the feed temperature from a region where significantly larger temperature values are attained. Owing to the attainment of large Peclet values (typically found in industrial practice), temperature fronts are very steep, and transitions between adjacent hot and cold regions take place within a very narrow spatial interval. Under these conditions, the instant at which any temperature front crosses a prescribed axial position ξ is marked by a sudden transition of in the temperature at that position, T(ξ), between a value close to the feed temperature Tin and a value considerably larger than Tin. A decreasing or rising temperature front is found when the transition is accompanied by an increase or decrease in temperature, respectively. In this framework, the reaction-front velocity Vfr can be estimated by placing a thermocouple at a distance Δξ from the inlet section of the fed reactor and recording the time interval Δtfr* needed for the measured temperature to reach Tin. Because the reaction front forms at switching near the inlet of the fed reactor, it must be the case that Δξ Vfr = Δtfr*

whereas the thermocouples placed in the fourth, third, and second reactors allow Δt*fr to be determined during the second, third, and fourth cycles, respectively. This approach to estimate Vfr is identical to the one employed in ref 52. We next demonstrate that Vth can be estimated without introducing any additional thermocouple. From geometric analysis of the spatiotemporal pattern, it is known that a thermal front always forms at a switching in a (Figure 3). If the kth cycle is considered, the point a is the inlet section of the reactor fed during the (k − 1)th cycle. To estimate Vth, it is sufficient to record the time interval Δtth * needed for the thermal front forming in a to cover the distance Δξ separating a from the subsequent thermocouple. This can be computed as the time needed for the temperature measured by the thermocouple at a distance Δξ from a to become larger than Tin. It can thus be found that Vth =

(7)

Δξ Δth*

(8)

In Figure 3, Δt*th is determined during the first cycle by monitoring the temperature measured by the thermocouple placed in the second reactor. The thermocouples placed in the first, fourth, and third reactors are then exploited to estimate Vth during the second, third, and fourth cycles, respectively. It should be noted that, because Vth > Vfr, a thermal front takes less time than the reaction front to cover the distance Δξ, implying that Δtfr* > Δtth *. This ensures that an estimate for Vth is available during each cycle before Vfr is computed. Influence of Controller Parameters. Three parameters need to be fixed to implement the formulated control algorithm: Δξ and two temperature limit values. Temperature

In network operation, the feed section is periodically modified, and an estimate for Vfr can therefore be derived during each cycle provided that a thermocouple is installed at a distance Δξ from the inlet of each reactor. A schematic representation illustrating the approach implemented to estimate Vfr is reported in Figure 3. The spatiotemporal temperature pattern displayed in Figure 2 is considered as a representative example, and the axial position of each thermocouple is indicated. The time interval Δt*fr is determined during the first cycle by monitoring the temperature measured by the thermocouple placed in the first reactor, 12138



Article

Figure 4. Impact of front deformation on the estimation of reaction- and thermal-front velocities: (a) snapshots of the reaction front crossing the axial position ξ* = 0.5, (b) snapshots of a decreasing thermal front crossing the axial position ξ* = 0.5. Owing to the deformation induced by thermal dispersion, the temperature T(ξ*) is different from Tin at the time instant when the front crosses ξ*.

Figure 5. Influence of the limit temperature values Ts,1 and Ts,2 on the estimated reaction- and thermal-front velocities (a) Vfr and (b) Vth, respectively. The displayed data were computed by numerical simulation of the controlled network. The reported Vfr and Vth values correspond to the achievement of the two-wave T*-periodic regime illustrated in Figure 2.

the front deformation caused by thermal dispersion, the use of a temperature T s,2 larger than Ts,1 is recommended, as demonstrated in Figure 4b. The evolution of the thermal front displayed in this latter figure indicates that Ts,2 values exceedingly close to Tin would lead to an underestimation the time needed to cross a prescribed axial position. Increasing the enthalpy Peclet number, on the other hand, makes each temperature front reducing the required differences between Ts,1, Ts,2, and Tin steeper. Because of the large Peclet values found in industrial practice, the stability of the controlled network is ensured within a wide range of variation of Ts,1 and Ts,2. To confirm this statement, we analyze in Figure 5 the effects of Ts,1 and Ts,2 on the Vfr and Vth values estimated with the proposed algorithm. The results displayed in Figure 5 were derived by numerical simulation of the controlled network operated around the two-wave T*-periodic regime arising with N = 4 and ns = 3. The curves displayed in Figure 5 can therefore be considered stable branches of this latter solution. The estimated Vfr value increases slightly with Ts,1 (Figure 5a). This can be explained by noting that, because of the front deformation caused by thermal dispersion (Figure 4a),

limit values are required to determine the instant at which a temperature front crosses the thermocouple axial position. The passage of a rising temperature front, for example, must be identified by recording the time instant at which the measured temperature becomes lower than a limit value Ts,1 > Tin and not when it reaches Tin. Because of the front deformation caused by thermal dispersion and the finitely large reaction rate, the tail of the front extends upstream in a wide spatial interval with temperatures higher than Tin.41 The time needed for the measured temperature to reach Tin is consequently longer than the time needed for the front to cross the thermocouple axial position. This is clearly shown in Figure 4a, where snapshots of the reaction front crossing a prescribed axial position ξ* are displayed. It is apparent that the front has entirely crossed and is far from ξ* when T(ξ*) = Tin, whereas it becomes progressively closer to ξ* as T(ξ*) increases above Tin. Ts,1 values slightly larger than Tin must therefore be selected to determine the instant at which the front crosses the thermocouple axial position. A second temperature limit value Ts,2 must be defined to identify the passage of a decreasing thermal front. Because of 12139



Article

Figure 6. Startup of the controlled network; (a,c) temperature profiles computed at one-half of each cycle under regime conditions and (b,d) spatiotemporal temperature patterns. (a,b) Implementation of a τ1 value fulfilling inequality 10 allows the T*-periodic five waves solution arising with N = 4 and ns = 3 to be reached. (c,d) Application of a τ1 value violating inequality 10 leads to the emergence of a four-wave 5T*-periodic solution coexisting with the prescribed T*-periodic one.

increasing Ts,1 determines a reduction in the time interval Δtfr* needed for the temperature at Δξ to reach Ts,1. Figure 5a also shows that the estimated Vfr value is barely affected by Ts,2. The reason for this latter behavior is that the reaction front forms near the network inlet section and travels in the flow direction regardless of the implemented Ts,2 value. Increasing Ts,2 causes an increase in the estimated Vth value (Figure 5b) This effect can, again, be explained by analyzing the front deformation induced by thermal dispersion. Vth is estimated by evaluating the time interval Δtth * needed for the thermal front forming in a to reach the thermocouple axial position. This time interval increases with Ts,2 because of front inflection (Figure 4b). Larger Ts,1 values, on the other hand, cause a reduction in the estimated Vfr, thereby decreasing the implemented switching time, in accordance with eq 6. This, in turn, causes a reduction in the maximum temperature attained by the front forming in a,41 leading to a reduction in Δtth *. It is important to remark that, in accordance with the results displayed in Figure 5, the implemented control law can ensure stabilization of wave-train solutions within a wide region of temperature limit values Ts,1 and Ts,2. The choice of Δξ must also be based on an analysis of the evolution of the temperature fronts. Δξ must be lower than the distance covered by the reaction front during a single cycle. The fulfillment of this requirement is essential to ensure that the reaction front can cross the thermocouple axial position during

each cycle. When implementing the illustrated control algorithm, the distance covered by the reaction front during the ith cycle is Vfr,iτi, where τi is defined by eq 6. The upper bound for Δξ is therefore given by Δξmax =

⎤ mod(nsp , N ) ⎡ 1 1 + ⎢ ⎥ 2 1 + α(p − 1) ⎦ ⎣ αp

(9)

where α = maxi(Vth,i/Vfr,i). Δξmax is therefore determined by the type of regime (the p value in eq 9) to be stabilized and by α. Because Vfr,i and Vth,i depend on the reactor parameters, the value of α to be used in inequality 9 should be selected based on the explored region of parameter space. At parameter values of practical relevance, the inequality Vth,i ≤ 1.5Vfr,i is typically fulfilled. Therefore, α = 1.5 is generally sufficient to determine Δξmax. Startup. A fundamental obstacle to network operation around T*-periodic wave-train solutions is multistability. Because of the coexistence of the target T*-periodic solution with extinguished and low-conversion multiperiodic regimes, temperature hot spots compromising catalyst activity and/or reductions in the average conversion are likely to occur owing to disturbances and/or model uncertainty. The most complex step is in this framework startup. Because large variations in temperature and concentration occur during this phase of 12140



Article

Figure 7. Evolution of the outlet temperature and switching time during startup, corresponding to the numerical simulations illustrated in (a,b) Figure 6a,b and (c,d) Figure 6c,d. (a,c) Temperature and (b,d) switching-time values computed at the end of each cycle are reported. Consequently, (a,b) a fixed point is attained when the prescribed five-wave T*-periodic solution is reached, whereas (c,d) a periodic evolution (with period = 5) is found as a result of the emergence of the four-wave 5T*-periodic solution.

operation, the risk of transitions to undesired solutions is significantly greater than for operation at regime. The approach typically followed for network startup is to preheat the catalytic bed with a hot inert gas flow and subsequently start to feed fresh reactants and to rotate inlet and outlet sections. It was observed in ref 52 that increasing the initial bed temperature reduced the risk of transitions to extinguished and multiperiodic solutions. However, neither any indication about the limit initial temperature to be imposed nor any guarantee to reach the target solution was provided. We next demonstrate that the control algorithm illustrated in the previous section can successfully cause the network to operate around any T*-periodic wave-train solution. The only prerequisite to be fulfilled for this purpose is that the initial bed temperature be large enough to ensure the formation of a reaction front close to the network inlet. To demonstrate the effectiveness of the formulated control strategy for startup, the problem of bringing the network to the five-wave solution arising with N = 4 and ns = 3 is considered. Adiabatic temperature rise values sustaining this latter solution generate Vfr values significantly larger than the corresponding Vth values, making the application of the control strategy proposed in ref 52 inadequate. The evolution of the temperature profile resulting from implementation of control law 6 during startup is displayed in Figure 6a,b. A reaction front is formed at t* = 0 at the network inlet section and moves in the flow direction during the first cycle. By monitoring the temperature at a distance Δξ < Δξmax from the inlet, an estimate for Vfr,1 can be obtained. In contrast, no estimate for Vth can be determined during the first cycle as no thermal front is present at this stage. This rules out the possibility of computing τ1 based on eq 6. It can be demonstrated that a transition to the prescribed solution is ensured provided that the following condition is fulfilled

τ1 ≤

⎞⎤ mod(nsp , N ) ⎡ p − 1⎛ 1 1 ⎢1 − ⎜ + ⎟⎥ ⎢⎣ Vfr,1 2 ⎝p p − 1 + 1/α ⎠⎥⎦ (10)

Derivation of inequality 10 is reported in the Appendix. As found for inequality 9, the value α = 1.5 can be used, with reactor parameter values of industrial relevance, to determine an upper bound for τ1. An estimate for such a bound, valid at any set of reactor parameter values, can be derived in the limit 1/α → 0. This limit corresponds to a thermal-front velocity that is infinitely larger than the reaction-front velocity and cannot therefore be realized. It can nonetheless be imposed to determine the theoretical minimum of the right-hand side of inequality 10. The pattern displayed in Figure 6a,b was generated by implementing an underestimated τ1 value computed based on eq 10. In particular, the employed τ1 value was selected to be large enough to generate an upstream cold region ab and significantly lower than the minimum switching time ensuring stability, that is, 1/Vth, computed with the actual Vth value. Once the region ab had been created by displacement of the reaction front, switching was performed, leading to the formation of a thermal front in a. An estimate for Vth,2 was derived during the second cycle by monitoring the temperature at a distance Δξ from the inlet of the first reactor. A new reaction front also formed at the beginning of the second cycle, and an estimate for Vfr,2 was determined by means of the thermocouple placed in the fourth reactor. The switching time τ2 was thus computed. Iterating this procedure enabled the prescribed solution to be reached. The evolutions of the switching time and of the network outlet temperature during startup are reported in Figure 7a,b. It is apparent that about 50 cycles were needed to reach the prescribed regime and 12141



Article

Figure 8. Response of the controlled network to disturbances in the (a−c) feed reactant concentration C0 and (d−f) gas velocity u. The illustrated results describe the startup to the three-wave T*-periodic solution arising with N = 4 and ns = 3.

This can, in turn, cause transitions to multiperiodic or extinguished solutions. The effects of step variations in C0 and u are examined in Figure 8. The problem of bringing the network to the T*periodic three-wave solution arising with N = 4 and ns = 3 is considered. To analyze the ability of the controller to reject disturbances both at regime and during startup, numerical simulations were performed starting from a uniform initial temperature profile with disturbances introduced before the target T*-periodic regime was reached. Moreover, the minimum C0 and u values considered in the numerical simulation were selected close to the stability boundaries of the target solution. It must be emphasized that the achievement of such parameter values would make the target solution unstable if an open-loop control strategy (i.e., an a priori fixed switching time) were implemented. The evolutions of the network outlet temperature and switching time reported in Figure 8 demonstrate the ability of the controlled network to reject disturbances. The controller modifies the switching time to prevent that the stability boundaries of the target solution are crossed. Large variations in the outlet temperature were initially found corresponding to a transition from the initial flat temperature profile to a threewave temperature pattern. Also, a sudden increase in the switching time was obtained following the first cycle. Because an underestimated τ1 value was implemented, a sudden increase in the switching time was observed as an estimate for Vth was derived during the second cycle. Following the illustrated initial transient, moderate variations in the outlet temperature and switching time were ensured by the controller in response to the introduced disturbances.

moderate deviations of the outlet temperature and switching time from the corresponding regime values are found. It is important to remark that, in accordance with the illustrated analysis, no lower limit was found for τ1. The implementation of a τ1 value even much lower than the minimum value actually required to ensure stability is sufficient to reach the prescribed solution. In contrast, extinction or transition to a coexisting multiperiodic solution was found when τ1 values not fulfilling inequality 10 were selected. This latter circumstance is illustrated in Figure 6c,d. In this latter case, the controller could not recover the error made during the first cycle, and a transition to a four-wave multiperiodic solution was found even though τi values enforcing stability were implemented for all i > 1. The evolutions of the switching time and the outlet temperature during startup are illustrated in Figure 7c,d. It is worth noting that a multiperiodic solution with a period of 5T* was attained, with the switching time oscillating at regime within the stability range of the T*-periodic solution displayed in Figure 6a,b. This was possible because of the coexistence of the two displayed solutions. The implementation of an underestimated τ1 value is recommended. It is nonetheless important that τ1 be selected large enough to generate a region ab at a temperature around Tin. This is essential to form a thermal front in a and thus estimate Vth during the second cycle. Controller Performance. The ability of the formulated control strategy to reject disturbances in the feed concentration C0 and in the gas velocity u was analyzed. Variations in these two parameters typically occur in industrial practice, causing significant changes in reaction- and thermal-front velocities. 12142



Article

Figure 9. Response of the controlled network to disturbances in the (a−c) feed reactant concentration C0 and (d−f) gas velocity u. The illustrated results describe the startup to the five-wave T*-periodic solution arising with N = 4 and ns = 3 and analyzed in Figure 6a,b.

analysis of the network spatiotemporal temperature pattern. The implementation of the illustrated control strategy for network startup was analyzed. A procedure was presented in this framework enabling the controlled network to be brought to any T*-periodic wave-train solution. Numerical simulations were used to demonstrate the effectiveness of the proposed controller in rejecting disturbances both at regime and during startup. We conclude by noting the need for further investigation of the impact of the implemented control strategy on the spatiotemporal structure and stability of the loop reactor. In particular, the effects of the values of controller parameters on the stability limits of multiperiodic solution regimes coexisting with the target T*-periodic regimes should be performed through a detailed bifurcation analysis. This study can allow for indications to be derived on how to design the proposed control structure to reduce the risk of transitions to undesired regimes.

The analysis of controller robustness was completed by extending this numerical experiment to a solution characterized by a larger number of temperature waves. This test was essential to demonstrate the effectiveness of the proposed control approach, as control difficulties increase significantly with increasing number of waves. Indeed, increasing the number of waves causes an increase in the number of coexisting multiperiodic solutions and reduces the network stability range. These two latter effects contribute to an increase in the risk of transitions to undesired solutions. The problem of bringing the network to the five-wave solution displayed in Figure 6a,b is considered. The evolutions of the outlet temperature and switching time in response to successive step variations in C0 and u are displayed in Figure 9. Again, disturbances were introduced before the regime was reached, and the C0 and u values imposed in the numerical simulation were selected very close to the stability boundaries of the target solution. Again, the controller effectively rejected the disturbances. Following the initial transient, corresponding to the transition from a flat profile to a five-wave temperature pattern, oscillations of moderate amplitude were observed for outlet temperature and switching time, and the target solution was eventually attained.

■

APPENDIX: DERIVATION OF INEQUALITY 10 Stability limits derived for the uncontrolled network in ref 41 and reviewed in the section Spatiotemporal Structure of WaveTrain Solutions of the present article apply almost unchanged to the controlled network. The only difference to be taken into account in the analysis of the controlled network is the possibility of variations in the reaction- and thermal-front velocities. Any T*-periodic wave-train solution can be sustained in the controlled network provided that the thermal fronts forming in a and b (Figure 2) reach the network outlet section during the same cycle. For a train of p thermal waves, this imposes the following two conditions:41

■

CONCLUSIONS A control strategy has been proposed to stabilize thermal wave trains in a network of catalytic reactors with periodically rotated inlet and outlet ports. The strategy relies on updating the switching time during each cycle based on the current values of the reaction- and thermal-front velocities. To estimate these two velocities online, an algorithm was derived from geometric 12143



Article

q = number of reactors covered by a purely thermal front = mod(nsp,N) r = reaction rate R = gas constant t = time T = temperature t* = dimensionless time T * = time needed to recover the initial feeding configuration (T*=Nτ) Tl = lowest maximum temperature of a temperature wave Tm = highest maximum temperature of a temperature wave u = gas rate v = dimensionless gas flow rate Vfr = dimensionless reaction-front velocity Vsw = dimensionless switching velocity Vth = dimensionless purely thermal-front velocity x = conversion z = axial coordinate

First, the distance covered in p cycles by the thermal front forming in a is greater than the distance (modulo the network length) covered during the same time interval by the network outlet section, that is p+1

∑ Vth,iτi ≥ mod(nsp , N )

(11)

i=2

Second, the distance covered in p − 1 cycles by the thermal front forming in b is less than the distance (modulo the network length) covered during the same time interval by the network outlet section, that is p

Vfr,1τ1 +

∑ Vth,iτi ≤ mod(nsp , N )

(12)

i=2

Inequalities 11 and 12 coincide with inequalities 5 in the case of constant reaction- and thermal-front velocities. Inequality 11 involves only τi values with i > 1. These latter values are computed based on eq 6, as estimates for Vth,i and Vfr,i can be derived during startup for all i > 1. This ensures the fulfillment of inequality 11. Inequality 12, in contrast, involves switching time τ1, which cannot be computed based on eq 6 because of the impossibility of estimating Vth,1 during startup. Inequality 12 can be exploited to obtain an upper bound for τ1. By replacing the implemented control law (eq 6) with τi for i > 1, inequality 12 can be recast in the form p

Vfr,1τ1 mod(nsp , N ) ≤1−

≤1−

∑ i=2

⎛ 1⎜1 1 ⎜ + 2⎜ p p−1+ ⎝

Greek Letters

α = maximum value assumed by the ratio of the thermalfront velocity to the reaction-front velocity = maxi (Vth,i/Vfr,i) ΔH = heat of reaction Δtfr* = time interval needed for the reaction front to cover the distance Δξ Δt*th = time interval needed for the thermal front to cover the distance Δξ Δξ = distance of each thermocouple from the upstream reactor inlet ε = reactor void fraction γ = dimensionless activation energy θ = dimensionless temperature ρ = density τ = dimensionless switch time ξ = dimensionless axial coordinate

⎞ ⎟ Vfr, i ⎟ ⎟ Vth, i ⎠

⎞ p − 1⎛ 1 1 ⎜ + ⎟ 2 ⎝p p − 1 + 1/α ⎠

(13)

where α = maxi(Vth,i/Vfr,i).

■

Subscripts and Superscripts

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (E.M.), pietro.altimari@ uniroma1.it (P.A.). Notes

■

The authors declare no competing financial interest.

■

NOMENCLATURE A = Arrhenius constant B = dimensionless adiabatic temperature rise C = concentration cp = specific heat capacity Df = mass axial dispersion coefficient Da = Damköhler number E = activation energy f(t) = forcing function gl(T) = function describing the dependence of Tl on Tm gm(T) = function describing the dependence of Tm on Tl h(t) = piecewise-constant function defined by eq 4 ke = solid-phase axial heat conductivity L = length of a single reactor unit L0 = network length N = number of reactors composing the network ns = number of reactors jumped by inlet and outlet sections p = number of temperature waves Pe = Peclet number

0 = reference conditions f = fluid h = enthalpy in = inlet m = mass max = maximum

REFERENCES

(1) Rhee, H. K.; Lewis, R. P.; Amundson, N. R. Creeping profiles in catalytic fixed bed reactors. Continuous models. Ind. Eng. Chem. Fundam. 1974, 13, 317. (2) Gatica, J. E.; Puszynski, J.; Hlavacek, V. Reaction front propagation in nonadiabatic exothermic reaction flow systems. AIChE J. 1987, 33, 819. (3) Sheintuch, M. Excitable waves in a fixed bed reactor: Observations and analysis. Chem. Eng. Sci. 1990, 45, 2125. (4) Yakhnin, V. Z.; Rovinsky, A. B.; Menzinger, M. Convective instability induced by differential transport in a tubular packed bed reactor. Chem. Eng. Sci. 1995, 50, 2853. (5) Sheintuch, M.; Nekhamkina, O. Pattern formation in models of fixed-bed reactors. Catal. Today 2001, 70, 369. (6) Yochelis, A.; Sheintuch, M. Principal bifurcations and symmetries in the emergence of reaction−diffusion−advection patterns on finite domains. Phys. Rev. E 2009, 80, 056201. (7) Pinjala, V.; Chen, Y. C.; Luss, D. Wrong-way behavior of packedbed reactors: II. Impact of thermal dispersion. AIChE J. 1988, 34, 1663−1672. (8) Luss, D. Temperature Fronts and Patterns in Catalytic Systems. Ind. Eng. Chem. Res. 1997, 36, 2931.

12144



Article

(9) Yakhnin, V. Z.; Menzinger, M. Convective instability and its suppression in packed-bed- and monolith reactors. Chem. Eng. Sci. 1999, 54, 4547. (10) Gudekar, K. G.; Riggs, J. B. Bifurcation and Stability Analysis of an Ethylene Oxide Reactor System. Ind. Eng. Chem. Res. 2003, 42, 3285. (11) Wolff, J.; Papathanasiou, A. G.; Kevrekidis, I. G.; Rotermund, H. H.; Ertl, G. Spatiotemporal addressing of surface activity. Science 2001, 294, 134. (12) Wolff, J.; Papathanasiou, A. G.; Rotermund, H. H.; Ertl, G.; Li, X.; Kevrekidis, I. G. Gentle dragging of reaction waves. Phys. Rev. Lett. 2003, 90, 018302. (13) Mikhailov, A. S.; Showalter, K. Control of waves, patterns and turbulence in chemical systems. Phys. Rep. 2006, 425, 79. (14) Papathanasiou, A. G.; Wolff, J.; Kevrekidis, I. G.; Rotermund, H. H.; Ertl, G. Some twists and turns in the path of improving surface activity. Chem. Phys. Lett. 2002, 358, 407. (15) Matros, Y. S., Ed. Catalytic Processes under Unsteady State Conditions; Elsevier: Amsterdam, 1989. (16) Haynes, T. N.; Caram, H. S. The simulated moving bed chemical reactor. Chem. Eng. Sci. 1994, 49, 5465. (17) Mancusi, E.; Russo, L.; Continillo, G.; Crescitelli, S. Computation of frequency locking regions for a discontinuous periodically forced reactor. Comput. Chem. Eng. 2004, 28, 187. (18) Russo, L.; Mancusi, E.; Maffettone, P. L.; Crescitelli, S. Nonlinear analysis of a network of three continuous stirred tank reactors with periodic feed switching: Symmetry and symmetrybreaking. Int. J. Bifurcation Chaos Appl. Sci. Eng. 2004, 14, 1325. (19) Altimari, P.; Maffettone, P. L.; Crescitelli, S.; Russo, L.; Mancusi, E. Nonlinear dynamics of VOC combustion loop reactor. AIChE J. 2006, 52, 2812. (20) Russo, L.; Altimari, P.; Mancusi, E.; Maffettone, P. L.; Crescitelli, S. Complex dynamics and spatiotemporal patterns in a network of three distributed chemical reactors with periodical feed switching. Chaos, Solitons Fractals 2006, 28, 682. (21) Brinkmann, M.; Barresi, A. A.; Vanni, M.; Baldi, G. Unsteady state treatment of a very lean waste gases in a network of catalytic burners. Catal. Today 1999, 47, 263. (22) Fissore, D.; Barresi, A. A. Comparison between the reverse-flow reactor and a network of reactors for the oxidation of lean VOC mixtures. Chem. Eng. Technol. 2002, 25 (4), 421−426. (23) Fissore, D.; Barresi, A. A.; Velardi, S.; Vanni, M. On the properties of the forced unsteady-state ring reactor network. Chin. J. Chem. Eng. 2004, 12 (3), 408. (24) Sheintuch, M.; Nekhamkina, O. Comparison of flow-reversal, internal-recirculation and loop reactors. Chem. Eng. Sci. 2004, 59 (19), 4065. (25) Barresi, A. A.; Baldi, G.; Fissore, D. Forced Unsteady-State Reactors as Efficient Devices for Integrated Processes: Case Histories and New Perspectives. Ind. Eng. Chem. Res. 2007, 46 (25), 8693. (26) Mancusi, E.; Russo, L.; Altimari, P.; Maffetone, P. L.; Crescitelli, S. Effect of the Switch Strategy on the Stability of Reactor Networks. Ind. Eng. Chem. Res. 2007, 46, 6510. (27) Zahn, V. M.; Mangold, M.; Krasnyk, M.; Seidel-Morgenstern, A. Theoretical Analysis of Heat Integration in a Periodically Operated Cascade of Catalytic Fixed-Bed Reactors. Chem. Eng. Technol. 2009, 32, 1326. (28) Zahn, V. M.; Mangold, M.; Seidel-Morgenstern, A. Autothermal operation of an adiabatic simulated counter current reactor. Chem. Eng. Sci. 2010, 65, 458. (29) Barresi, A. A.; Vanni, M.; Brinkmann, M.; Baldi, G. Influence of the wall heat losses on the performance of a network of nonstationary catalytic burners. Chem. Eng. Technol. 1999, 22, 237. (30) Vanden Bussche, K. M.; Froment, G. F. The STAR configuration for methanol synthesis in reversed flow reactors. Can. J. Chem. Eng. 1996, 74, 729. (31) Velardi, S. A.; Barresi, A. A. Methanol synthesis in a forced unsteady state reactor network. Chem. Eng. Sci. 2002, 57, 2995.

(32) Velardi, S.; Barresi, A. A.; Manca, D.; Fissore, D. Complex dynamic behaviour of methanol synthesis in the ring reactor network. Chem. Eng. J. 2004, 99, 117. (33) Sheinman, R.; Sheintuch, M. Loop Reactor Design and Control for Reversible Exothermic Reactions. Ind. Eng. Chem. Res. 2009, 48, 5185. (34) Fissore, D.; Garran Tejedor, D.; Barresi, A. A. Experimental investigation of the SCR of NOx in a Simulated Moving Bed Reactor. AIChE J. 2006, 52 (9), 3146. (35) Fissore, D.; Penciu, O. M.; Barresi, A. A. SCR of NOx in loop reactors: Asymptotic model and bifurcational analysis. Chem. Eng. J. 2006, 122, 175. (36) Fissore, D.; Barresi, A. A.; Botar-Jid, C. C. NOx removal in forced unsteady-state chromatographic reactors. Chem. Eng. Sci. 2006, 61, 3409. (37) Fissore, D.; Barresi, A. A.; Baldi, G. Synthesis Gas Production in a Forced Unsteady-State Reactor Network. Ind. Eng. Chem. Res. 2003, 42, 2489. (38) Sheintuch, M.; Nekhamkina, O. The asymptotes of loop reactors. AIChE J. 2005, 51, 224. (39) Madai, A. Y.; Sheintuch, M. Demonstration of loop reactor operation. AIChE J. 2008, 54, 2413. (40) Altimari, P.; Mancusi, E.; Crescitelli, S. Formation of Thermal Wave Trains in Loop Reactors: Stability Limits and Spatiotemporal Structure for Reversible Reactions. Ind. Eng. Chem. Res. 2012, 51, 9609. (41) Altimari, P.; Mancusi, E.; Brasiello, A.; Maffettone, P. L.; Crescitelli, S. Temperature wave trains of the loop reactor: The effect of thermal dispersion. Chem. Eng. Sci. 2012, 76, 108. (42) Mancusi, E.; Altimari, P.; Russo, L.; Crescitelli, S. Temperature and conversion patterns in a network of catalytic reactors for methanol synthesis for different switch strategies. Chem. Eng. Sci. 2010, 65, 4579. (43) Altimari, P.; Mancusi, E.; Russo, L.; Crescitelli, S. Temperature wave-trains of periodically forced networks of catalytic reactors. AIChE J. 2012, 58, 899. (44) Mancusi, E.; Russo, L.; Altimari, P.; Crescitelli, S. Multiplicities of temperature wave trains in periodically forced networks of catalytic reactors for reversible exothermic reactions. Chem. Eng. J. 2011, 171, 655. (45) Barresi, A. A.; Vanni, M.; Brinkmann, M.; Baldi, G. Control of an autothermal network of nonstationary catalytic reactors. AIChE J. 1999, 45, 1597. (46) Zahn, V. M.; Yi, C. U.; Morgenstern, A. S. Analysis and demonstration of a control concept for a heat integrated simulated moving bed reactor. Chem. Eng. Sci. 2011, 66, 4901. (47) Smagina, Y.; Sheintuch, M. Control of rotating pulses in a loop reactor. J. Process Control 2009, 19, 954. (48) Sheinman, R.; Sheintuch, M. Loop Reactor Design and Control for Reversible Exothermic Reactions. Ind. Eng. Chem. Res. 2009, 48, 5185. (49) Fissore, D.; Pisano, R.; Barresi, A. A. Observer design for the selective catalytic reduction of NOx in a loop reactor. Chem. Eng. J. 2007, 128, 181. (50) Fissore, D. Robust control in presence of parametric uncertainties: Observer-based feedback controller design. Chem. Eng. Sci. 2008, 63, 1890. (51) Fissore, D.; David, E.; Hammouri, H.; Barresi, A. A. Nonlinear soft-sensor design for unsteady-state VOC afterburners. AIChE J. 2006, 52 (1), 282. (52) Altimari, P.; Mancusi, E. Control of temperature wave trains in periodically forced networks of catalytic reactors for methanol synthesis. Chem. Eng. Process.: Process Intensif. 2013, 63, 25. (53) Koshy, T. Elementary Number Theory with Applications; Elsevier: Amsterdam, 2007.

12145


Control of Rotating Wave Trains in a Loop Reactor - Industrial

Recommend Documents