Characteristics-based model predictive control of a catalytic flow reversal reactor.
Periodically forced catalytic reactors have attracted increasing attention in recent years. The most successful applications are based on a reverse-flow operation of catalytic reactors, where the dynamic operation improves the thermal efficiency of the process. Reverse-flow operation has been used in many applications (Matros and Bunimovich, 1996). Examples include the oxidation of S[O.sub.2], methanol synthesis, NOx reduction and C[H.sub.4] combustion.
Reverse-flow operation can be an attractive mode of operation for exothermic reactions. For exothermic reactions, reversing the flow direction periodically can create a heat trap effect. This effect can be used to achieve and maintain an enhanced reactor temperature compared to a single flow direction mode of operation. In reverse-flow operation of catalytic reactors, the thermal capacity of the solid material within the reactor acts as a regenerative heat exchanger, allowing auto-thermal operation without the use of heat exchangers.
The principle of the heat trap effect in a catalytic reactor with reverse-flow operation is illustrated in Figure 1. Figure 1a illustrates a reactor temperature profile that might be observed in a standard unidirectional flow operation for a combustion reaction. If a temperature pattern, shown in Figures 1a-b is established, the reverse-flow operation can then be used to take advantage of the high temperatures near the reactor exit to preheat the reactor feed. A quasi-steady state operation may be achieved in which the reactor temperature profile has a maximum value near the centre of the reactor, which slowly oscillates as the feed is switched between the two ends of the reactor, as shown in Figure 1c-e. The quasi-steady state operation will be referred to in this paper as stationary state.
The design and modelling of catalytic flow reversal reactors has been investigated; however, for a stable and optimal operation of the system a robust control strategy is required. Control of such systems is an area that has not received much attention until recently (Budman et al., 1996; Dufour et al., 2003; Dufour and Toure, 2004; Edouard et al., 2005).
[FIGURE 1 OMITTED]
When operating a catalytic flow reversal reactor, the minimum control requirement is to keep the reactor temperature in the solid catalyst material within a range that avoids overheating or deactivation of catalyst and/or extinction of the reaction. Nevertheless, controlling the whole temperature profile along the reactor may lead to an optimum operation of the reactor unit. Associated with the temperature is the concentration of reactants that are often required to be nearly fully depleted at the end of the reactor.
Because the temperature in the centre of the reactor (see Figure 1) can greatly exceed the adiabatic temperature rise, energy can be extracted from the centre both as a means of control and as a source of useful energy. For example, in Aube and Sapoundjiev (2000) a CFRR was used for oxidation of lean methane emissions and a heat exchanger was added in the centre of the reactor to remove heat. This ensured that the reactor did not overheat or deactivate the catalyst or damage the reactor. For the design of CFRR units, different control measures have been proposed to avoid catalyst overheating (Nieken et al., 1994) including: bypassing the flow in the midsection of the reactor; withdrawing of gas to a external cooler; and cooling of the entire midsection by using a heat exchanger.
For control of CFRR units, different control techniques including feedback PID and feed-forward (Budman et al., 1996), linear quadratic regulator (Edouard et al., 2005) and model predictive control (Dufour et al., 2003; Dufour and Toure, 2004) have been presented in the literature. In Dufour et al. (2003) and Dufour and Toure (2004) a model predictive controller (MPC) was designed to control the maximum and minimum temperature in a CFRR unit. Heat supply at the core of the reactor and the inlet dilution were used as control variables. In the MPC formulation, the state variables of a non-linear distributed parameter model used are lumped via the finite difference method. The resulting model is then linearized around an arbitrary state profile and then used in the MPC formulation. A fast flow reversal time was used in Dufour et al. (2003) for the operation of the CFRR unit.
Most conventional approaches to controller design for a distributed parameter system use lumping techniques that discretize the underlying partial differential equation (PDE) model into a finite number of ordinary equations (ODE's). Discretization techniques, such as finite difference, finite element and finite volume can be used to discretize the model into a set of ODE's. Based on the approximate difference equations obtained from discretization, controllers can be designed using control methods for ordinary differential equations (ODE's).
Recent approaches for control of distributed parameter systems have focused on the development of control methods that directly account for their spatially distributed nature. In this direction, the well-known classification of PDE systems into elliptic, parabolic and hyperbolic, according to the properties of the spatial differential operator, essentially determines the approach followed for the solution of the control problem (Christofides, 2000). Christofides and co-workers have provided a considerable amount of work on non-linear order reduction and control of non-linear parabolic PDEs for systems where diffusion dominates convection (Christofides, 2000). Various extensions of non-linear control techniques can be found in the literature for the control of hyperbolic PDE systems: sliding mode control (Hanczyc and Palazoglu, 1995); geometric control (Christofides, 2000); adaptive control (Hudon et al., 2005); and model predictive control (Shang et al., 2004; Dubljevic et al., 2005).
In Shang et al. (2004), it is shown that the underlying geometry of hyperbolic PDE systems can be exploited to produce high-performance MPC, termed characteristics-based model predictive control (CBMPC), which does not require substantial on-line computations and provides high accuracy in the predictions of the process output variables.
In this paper, we apply the characteristics-based model predictive control of Shang et al. (2004) to a catalytic chemical reactor with reverse flow operation. As the key element in the use of the MPC strategy, we study specifically the selection of the prediction horizon. We discuss the many issues related to the tuning of a model predictive control (MPC) for a reverse flow reactor and propose a formulation of an MPC controller that after each flow reversal the prediction horizon begins to shrink as time progress until the next flow reversal occurs. This type of application of MPC is similar to that used in batch applications. The MPC controller is used to keep the temperature in the reactor within bounds that guarantee safe operation. To have control of the temperature in the reactor, we use heat removal by mass extraction from the reactor midsection. The performance of the controllers is evaluated on a simulated CFRR unit for combustion of lean methane streams for reduction of greenhouse gases emissions.
The remainder of this paper is organized as follows. Firstly, the mathematical model for a catalytic reactor used in the model-based controller is described. Then, the numerical approach for the characteristics based model predictive control is presented to provide the background of the MPC approach used. This is followed by a discussion of the controller tuning for the MPC for a reverse flow reactor. The paper continues with the application of the MPC controller to a CFRR unit for the combustion of lean methane emissions. Computer simulations illustrate the performance of the MPC controller.
The dynamics of the catalytic flow reversal reactor can be described by partial differential equations (PDEs) derived from mass and energy balances. A wide range of models with different degrees of complexity have been proposed in the literature for catalytic flow reversal reactors. Mathematical models and their solution have been used to investigate a variety of operating parameters, either from a theoretical view or coupled with experiments. Models found in the literature are one-dimensional (1-D), either pseudo-homogeneous or heterogeneous (e.g. Matros and Bunimovich, 1996; Liu et al., 2001; Marin et al., 2005) or two-dimensional models (Aube and Sapoundjiev, 2000; Salomons et al., 2004).
One of the most complete models for a catalytic flow reversal reactor is given in Salomons et al. (2004). In Salomons et al. (2004) a two-dimensional dynamic heterogeneous model is used to simulate the behaviour of a reverse flow reactor for the combustion of lean methane emissions. Using such a complete model for the control calculation in a model-based controller would be computationally expensive. For the controller formulation, we use a simpler one-dimensional plug flow model where a single value for the state variables in the fluid and solid phase is used. This model is termed pseudo-homogeneous model.
The basic equations for the mass and energy balances for a catalytic reactor where the variables in the fluid and solid are considered to be the same (pseudo-homogeneous model) are:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
where the parameters [k.sub.O], [eta]and [rho]are given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [rho]= [[rho].sub.b] [C.sub.pf], and with the boundary conditions given, for t [greater than or equal to], by:
Y(0,t) = [Y.sub.in] (3) T(0,t) = [T.sub.in]
Y and T are the mole fraction of methane and temperature, respectively; [v.sub.s,in] is the inlet superficial velocity, [alpha]is the ratio of the actual superficial flow velocity and the inlet superficial flow velocity ([velcity]= [v.sub.s] /[v.sub.s,in]). Other model parameters are: [epsilon], bed porosity; E, activation energy; [[micro].sub.eff], catalyst efficiency; [C.sub.p], heat capacity, and [rho], density (subscript s is used to indicate solid material and f to indicate fluid properties). Values for the model parameters are given in Table 1. The initial conditions are given, for 0 [greater than or equal to]z [greater than or equal to]1 by:
Y(z,0) = [Y.sub.0] (z) (4) T(z,0) = [T.sub.0] (z)
In the model Equations (1) and (2), a reaction rate of first-order is used. Plug-flow is assumed. The fluid in the reactor is treated as an ideal gas: [rho]f = PM/[R.sub.g] and C = P/[r.sub.g]T We neglect any pressure drop in the reactor and we consider that the density of the fluid phase is constant and is evaluated at the inlet conditions.
MODEL PREDICTIVE CONTROL
To formulate an MPC controller, we use the characteristics-based model predictive control of Shang et al. (2004). The characteristics-based model predictive control consists of three basic steps:
1. The method of characteristics is used to transform the PDE model to an equivalent set of ordinary differential equations (i.e., transform the distributed parameter model to an equivalent lumped parameter model);
2. A non-linear predictive controller is designed based on the equivalent lumped parameter model. During each half cycle (full cycle = forward + reverse flow direction), the prediction horizon is divided in Hp intervals. The prediction horizon shrinks as time progress until a new half cycle (with a different flow direction) begins;
3. A constrained quadratic program is solved to obtain the optimal input sequence.
Method of Characteristics for Prediction of Output Variables
To predict the dynamic behaviour of the catalytic reactor, the model Equations (1) and (2) are transformed into a set of ODE's by using the method of characteristics. Equations (1) and (2) can be described by a system of ODEs along the characteristic curves [[xi].sub.1] and [[xi].sub.2]:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE TO ASCII] (5)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE TO ASCII] (6)
Along the characteristic curves, [[xi.sub.1] and [[xi.sub.2], the state variables Y and T are described by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE TO ASCII] (7)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE TO ASCII] (8)
To predict the future state variables at any spatial point, simultaneous integration of Equations (7) and (8) along the two nonparallel characteristic curves, Equations (5) and (6) are required. Predictions of future state values can be obtained by discretizing the initial state at a finite number of spatial points (N) and then projecting the characteristic curves from each of these points and by computing the values of the state variables at the intersection points (Shang et al., 2004). Figure 2 illustrates the calculation of the state variables at a point C from the values at point A and point B. As seen in Figure 2, the state variable at any future time is approximated by two points. Therefore, the discretization affects the accuracy of the predictions; however, the approximation of the state variable value within a segment AC by that at point A and B reflects the true solution of hyperbolic PDE systems more closely than other numerical methods (Shang et al., 2004). The method can accommodate the use of larger spatial grids and time steps with minimal loss of accuracy (Shang et al., 2004). By varying point C and repeating the procedure, the values of the state variables at different grid points and different future times can be calculated. These predictions of the state variables are then used to compute the optimal control input.
[FIGURE 2 OMITTED]
Model Predictive Control Formulation
Using the state prediction method described above, the value of the states within the prediction horizon is obtained for specific control actions. To use the predictions in the MPC scheme, an output vector is defined:
y = Cx
where C is an observation matrix, x is the vector of state variables [[Y.sub.1 x N],[[[T.sub.1 x N]].sup.T] at the discrete spatial points. It is assumed that the value of all state variables at all discrete spatial points are available; C = [I.sub.2N x 2N]. The future values of the controlled outputs ([[??].sub.c]) are expressed, at each control interval, in a locally linear form (Shang et al., 2004):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE TO ASCII] (7)
where C is an (m * [H.sub.p]) x ([H.sub.p] * [2.sub.N]) (m = number of controlled outputs) matrix that selects the outputs that are controlled, X is a matrix of dimensions (2N) X [H.sub.P] of the predicted states for a prediction horizon [H.sub.P], [X.sub.0] is a matrix of dimensions (2N) X [H.sub.P] of predicted states in the absence of further control actions ([u.sub.-1]), ?u is the vector of future control changes for a control horizon [H.sub.c]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and S is the rate of variation of the states about past control actions ([U.sub.1])
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where the elements of S are updated at each control interval and are computed via perturbation:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where [delta] is a numerical perturbation on past input [u.sub.-1,X|u-1 + [delta]and [x|.sub.u-1] are the predicted future states under the control actions [u.sub.-1+[delta] and [u.sub.-1, respectively.
The control actions are calculated at each control interval by solving an optimization problem with a quadratic objective:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE TO ASCII] (11)
subject to Equations (5) to (8) and input and output constraints
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE TO ASCII] (11)
where r is the vector of set points, [epsilon]is an m-dimensional vector of slack variables used to soften the output constraints and Q, R and T are symmetric positive definite weighting matrices.
To take into account possible plant/model mismatch and/or unknown disturbances, the locally linearized prediction is modified by adding a mismatch compensation term:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE TO ASCII] (12)
where e is a mismatch term
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE TO ASCII] (13)
where [e.sub.-1] is the mismatch term computed at the previous control interval, [y.sub.c],m is the measured controlled output at the current control interval and [y.sub.c-1] is the predicted controlled output at the previous control interval for the current control interval. e is taken to be zero initially and is updated iteratively at every control interval.
For the implementation of an MPC controller, the selection of the prediction horizon ([H.sub.p]) is one of the most important parameters to be chosen. In a typical MPC application, where linear lumped parameter models are used, a prediction horizon that approximates the time to steady-state of the process is usually chosen. In this way, most of the dynamic behaviour of the controlled process is known to the controller. This type of prediction horizon is not possible for a reverse flow reactor. Mathematical models are usually developed on the basis of a single flow direction, and then the boundary conditions and flow sign are changed to account for the flow reversal. On the basis of a single flow direction model, simulation of the process for a sequence of full cycles would require, for example, an analytical solution of the model equations that can then be assembled to generate the full solution. For complex distributed parameter systems, analytical solutions are generally not possible and thus we need to rely on a numerical approximation. Numerical approximations and their solutions applied to the dynamical models for reverse flow reactors require the solution of a sequence of PDE model problems.
In catalytic flow reversal reactors, the dynamic behaviour of the temperature is changed by the reversal of the flow direction and the dynamic behaviour is a function of the number of cycles. Thus, even if a sequential solution is used, the number of cycles required to capture the whole dynamic behaviour of a catalytic flow reversal reactor may be large.
A long prediction horizon in the MPC would capture the whole dynamic behaviour and would give a smooth regulation and tracking of temperature within the reactor; however, the computational time for a long prediction may be prohibitive. On the other hand, a short prediction horizon would be computationally feasible, but it would give an aggressive control of the temperature. For MPC control of a reverse flow reactor, the computation of the control input is more tractable using a prediction horizon that covers the reactor behaviour during either direct or reverse flow direction. By using such a short prediction horizon, we acknowledge that we may lose some control performance. As time progresses during each semi-cycle, the prediction horizon shrinks until next flow reversal occurs. This type of implementation of MPC is similar to that used in batch applications. While conventional MPC employs a receding horizon framework and is suited for continuous processes, batch processes require a shrinking horizon because the time available for control shrinks as the end of the batch approaches (Soni and Parker, 2004).
The selection of the control interval is another parameter for the MPC that needs to be selected. The control interval should be carefully chosen so that enough time is available for the control calculation. In general, for complex PDE-based reactor models, a long prediction horizon has a higher computational time. With a longer control interval, the control of fast processes may not be satisfactory, but for the control of the temperature in a catalytic flow reactor the control interval required to perform all the computations in an MPC, with a prediction horizon that shrinks until the next flow reversal, is usually much smaller than the settling time of the temperature dynamics.
APPLICATION TO A CFRR FOR COMBUSTION OF LEAN METHANE EMISSIONS
To show the effectiveness of the MPC controller, the controller is used for a CFRR unit for combustion of lean methane emissions. CFRR technology has been suggested for the combustion of lean methane streams for reduction of greenhouse gases emissions from natural gas transmission facilities, upstream oil and gas production facilities and coal beds (Hayes, 2004).
The main problem for combustion of lean methane mixtures is the necessity of a high reactor temperature. For ambient temperature feed, some pre-heating would be required to achieve ignition temperature, and to operate the reactor in an autothermal mode. To eliminate feed pre-heat, reverse flow operation is a suitable solution due to the heat trap effect.
The control of a CFRR unit needs, as the main goal, the control of the temperature of the catalytic material in the solid section. The temperature should be kept below a critical value to avoid overheating or deactivation of catalyst and above the extinction temperature of the reaction. Keeping the temperature along specified temperature profiles is also desired to achieve optimal operation of the reactor. Concentration in the gas phase is sometimes required to be kept below a high limit; however, concentration and temperature are usually related and the control of the concentration can be achieved indirectly by controlling the temperature.
For the combustion of lean methane emissions in a CFRR unit, the temperature in the centre of the reactor can greatly exceed the adiabatic temperature rise and a usual control objective involves the removal of useful energy from the midsection of the reactor. Energy can be used for many purposes such as heating and power generation. Other control objectives such as complete conversion are sometimes desired.
In the numerical simulation study presented in the next section, we focus on the control of the temperature within the reactor. To control the process, we use heat removal by mass extraction from the middle section of the reactor. Compared to the use of a heat exchanger, mass extraction from the midsection has the advantage that no sharp changes in the temperature distribution occur. Moreover, it is easier to implement in practice.
Slowly changing disturbances and operating conditions, are simulated. Due to large heat capacity of the solid material within the reverse flow reactor, the dynamics of the temperature in the reactor are slow and high frequency disturbances are substantially attenuated by the process.
To simulate the closed-loop operation of a CFRR unit for combustion of lean methane emissions, we perform two case studies. In the first one, we evaluate the MPC strategy and the proposed tuning by applying it to a CFRR plant that is simulated using the same model used in the MPC controller, i.e., the 1-dimensional pseudo-homogeneous model described in Equations (1) and (2). For the solution of the 1-dimensional pseudo-homogeneous model, we use the model parameters given in Table 1.
In the second case study, a CFRR plant that is more complex than the simple one used for the first case study and for the controller formulation is used. The CFRR plant, in the second case study, is modelled by a dynamical two(spatial)-dimensional heterogeneous model that incorporates the effect of a large insulation layer required to reduce heat loss from the reactor. The model equations solved to simulate the more realistic CFRR plant, as well as the parameters in the model, are taken from Salomons et al. (2004). A scheme of the reactor configuration used in simulations is given in Figure 3.
For the computation of the control input, u, the MPC is formulated so that the average temperature in the catalyst section is the controlled output variable [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE TO ASCII] A discussion that supports this choice of output variable is given in the Case Study 1 section. The objective function given in Equation (11) with
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE TO ASCII]
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
is minimized subject to input and output constraints. The spatial domain discretization used in the MPC in all numerical simulations is N = 21.
For the CFRR unit with mass extraction as the manipulated variable, the input u corresponds to [alpha]= [v.sub.s]/[v.sub.s,in]. An extra constraint is added to ensure that flow velocity after the extraction point is less or equal to that at the reactor inlet. The active set method was used to solve the optimization problem. The combined simulation of the CFRR and the MPC controller, including the solution of the optimization problem, was executed using MATLAB [R].
Case Study 1: Simplified Process Model
To perform the simulation of the closed-loop system, the following nominal operating conditions are used:
Y(0, t) = 0.1% T(0, t) = 298 K [v.sub.s,in] = 1 m . [s.sub.-1] Tcycle = 10 min
[FIGURE 5 OMITTED]
Using the nominal operating conditions the stationary state is computed (Figure 4). The stationary state is computed using a fraction of mass extraction (1 - [alpha]) equal to 5%. The simulation of the CFRR unit was performed using the finite element method in the COMSOL [R] Multi-physics. To tune the MPC controller, we need to know the dynamic behaviour of the system. To obtain an estimation of the dominant time constants of the simulated CFRR unit, we performed a step test in the inlet mole fraction ([Y.sub.f0]). Starting from the initial stationary state given in Figure 4, the maximum, minimum and average temperature along the catalyst section after a step change of [[DELTA]Y.sub..f0] = 0.1% in the inlet concentration follow the trajectory given in Figure 5. From Figure 5, it can be observed that the settling time of the process is, for the selected operating conditions, larger than 10 h. With a full-cycle time ([T.sub.cycle]) of 10 min, more than 60 full cycles are needed to reach the new stationary state.
A prediction horizon that covers the process dynamics up to the settling time is not computationally tractable. Despite the loss of information of the dynamics of the process, a prediction horizon of a one full semi-cycle is chosen so that the solution of the control problem within a control interval can be computed. The nominal control interval is chosen to be (1/6) [T.sub.cycle], which is fast enough for the control of the temperature. A control horizon ([H.sub.c]) of one control interval unit and weighting matrices Q = [I.sub.Hp] x [H.sub.p] and R = 1 were used.
Using the MPC controller formulated in the Model Predictive Control Formulation subsection, control of the temperature along the axis of the reactor can be achieved. Controlling the CFRR by means of mass extraction gives control over the maximum and minimum temperature along the catalyst section. The parameter [alpha] in Equations (1) and (2) is used as manipulated variable. Before the midsection of the reactor (z/L = 0.5), [alpha] = 1, and afterwards, [alpha] takes the value of the fraction of the inlet mass flow that remains after hot gas has been extracted.
Starting from the initial stationary state given in Figure 4 and for a step change in the mass extraction represented by [DELTA] [alpha]= -5%, the trajectories of the maximum, minimum and temperatures along the catalyst section and the temperature distribution along the axis of the reactor are given in Figures 6a-b, respectively. We observe that the maximum temperature increases initially and decreases afterwards. This behaviour is typical of the reverse flow reactor with mass extraction (Kushwaha et al., 2005). We note that for all temperature distributions in Figure 6, the same number of points was used. For an MPC controller with a short prediction horizon, the MPC will drive the system in the wrong direction; however, we can control the average temperature because this variable gives an indication of the total energy accumulated in the catalyst section. Therefore, instead of controlling the maximum temperature directly, we choose to control the average temperature. The MPC is designed to ensure that the average temperature in the catalyst section (see Figure 3) is within specified bounds. When the minimum bound is chosen so that significant reactant conversion is achieved, then we guarantee that the reaction does not extinguish.
[FIGURE 6 OMITTED]
If the average temperature in the catalyst section is kept along a desired set-point or within a small band, then the total energy in the catalyst section will remain constant or almost constants; however, as observed in Figure 6, an increase in the mass extraction increases, temporarily, the maximum temperature in the catalyst section.
If mass extraction is used to control the average temperature, then an increase in the maximum temperature in the catalyst section will come at the expense of a lower minimum temperature in the same section. The large temperature gradient increases the heat transfer by diffusion in the solid material, which will tend to reduce the temperature difference. Diffusion phenomena are not included in the simulated CFRR in Case Study 1, and thus we expect to observe a large difference between the maximum and minimum temperature within the catalyst section after changes in the mass extracted from the midsection of the reactor. Nevertheless, diffusion is considered in Case Study 2 and we expect a lower difference between the maximum and minimum temperature in the catalyst section after changes in the fraction of mass extracted.
When mass extraction is the only variable used to control the system and the average temperature is the only controlled variable, then some increase in the maximum temperature should be tolerated; but this can be included in the controller by choosing a conservative average temperature set point or upper/lower bound.
To evaluate the performance of the controller, closed-loop simulation are shown using typical operating conditions. Typical operating conditions for the CFRR unit for lean methane emissions are: 0.1 m/s [less than or equal to][v.sub.s,in] [less than or equal to]1.5 m/s and 0.1% [less than or equal to][Y.sub.f0] [less than or equal to]1%.
[FIGURE 7 OMITTED]
[FIGURE 8 OMITTED]
The performance of the controller to disturbances in the operating conditions is shown in Figure 7. The controller is set to keep the average temperature within arbitrary bounds, 650K [less than or equal to][T.sub.avg] [less than or equal to]750K. The predictions of the average temperature are obtained from the average of the locally linearized predictions of the temperature, Equation (12), at a discrete set of points along the catalyst section. For all the simulations performed, the mass extracted (1 - [alpha]) is kept at least at 5%. In this way, we guarantee a fixed rate of energy that can be used for other purposes. Nevertheless, this requirement can be omitted and the mass extraction can be used for the sole purpose of control. From Figure 7a, it can be observed that the controller can keep the average temperature within bounds. The temperature increases initially due to changes in the inlet flow velocity (Figure 7c). After the temperature reaches the upper bound, the mass extraction is increased (Figure 7b) to remove energy from within the system and reduce the total energy in the system.
[FIGURE 9 OMITTED]
The performance of the controller for disturbances in boundary condition is shown in Figure 8. From Figure 8a it can be observed that the controller can keep the average temperature within the specified bounds. As seen in Figure 8c, when the average temperature reaches the upper bound, more hot gas is extracted and when the lower bound is reached, less gas is extracted. The change in the boundary condition is shown in Figure 8d. For the same change in the boundary conditions, the behaviour of the maximum, minimum and average temperature without control is given in Figure 8b.
Case Study 2: Complex Process Model
To perform the simulation of the closed-loop system with the more complex, but more realistic two-dimensional dynamic heterogeneous model (see Numerical Simulations section), the following nominal operating conditions are used:
Y (0, t) = 0.5% T(0, t) = 298 K [v.sub.s,in] = 0.1 m . [s.sup.-1] [T.sub.cycle] = 10 min
The simulation of the CFRR unit was performed using the finite element method in the COMSOL [R] Multi-physics environment. The tuning parameters used in the MPC controller are the same as those used in the Case Study 1 subsection. Even with substantial plant/model structural mismatch, the observations about the maximum temperature changes after changes in the mass extraction still apply. Therefore, we used the average temperature in the catalyst section as the controlled variable.
The MPC is designed to have the average temperature in the catalyst section (see Figure 3) track a desired set-point trajectory. The set point should be chosen so that it is above the minimum temperature for the reaction to extinguish and below, by a conservative difference, the maximum temperature of catalyst deactivation.
[FIGURE 10 OMITTED]
The performance of the controller for disturbances in the operating conditions is shown in Figure 9. The simulation results indicate that the controller provides good tracking performance. As seen in Figure 9a, the average temperature tracks the specified set point well. As the inlet flow velocity changes (Figure 9c), more or less hot gas is extracted (Figure 9b) to keep the average temperature along the specified set point.
The performance of the controller for disturbances in boundary condition changes is shown in Figure 10. As seen in Figure 10a, the average temperature tracks specified set point well. As inlet mole fraction disturbances enter the system (Figure 10c) more or less hot gas is extracted (Figure 10b) to keep the average temperature along the specified set point.
SUMMARY AND CONCLUSIONS
In this paper we show the development of a model-based controller for a catalytic flow reversal reactor. The controller uses a combination of the method of characteristics and the model predictive control technique. The selection of the prediction horizon is addressed since its selection is essential for the application of a model predictive control to a reverse flow reactor.
We study the formulation of model predictive controller where during each semi-cycle, the prediction horizon begins to shrink as time progresses until the next flow reversal occurs. This formulation is proposed to overcome the need of predicting a large number of full cycles, which would cover the full transient behaviour of the process. We found by computer simulations that this tuning of the MPC gives good control despite the shortsightedness of the controller.
The controller is formulated for the control of the slow dynamics within the catalytic reactor, i.e., the temperature along the reactor. By computer simulations, we show that the use of the heat extraction by means of mass extraction in an MPC scheme can be used to control the temperature so that it is kept within safe operating conditions. We show that good control performance can be achieved for regulation and set-point tracking in the presence of inlet disturbances or changes in the operating conditions.
Manuscript received February 7, 2007; revised manuscript received May 1, 2007; accepted for publication May 1, 2007.
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Department of Chemical and Materials Engineering, University of Alberta, Edmonton, AB, Canada T6G 2G6
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Table 1. Model parameters Parameter Value Units [k.sub.[infinity]] 1.35E5 [s.sup.-1] [epsilon] 0.51 E 54,400 kg * [m.sup.-2] * [s.sup.-2] * [mol.sup.-1] [R.sub.g] 8.314 kg * [m.sup.-2] * [s.sup.-2] * [mol.sup.-1] * [K.sup.-1] [[rho].sub.s] 1240 kg * [m.sup.-3] [C.sub.pf] 1066 J * [kg.sup.-1] * [K.sup.-1] [C.sub.ps] 1020 J * [kg.sup.-1] * [K.sup.-1] [DELTA][H.sub.r] -802E3 J * [mol.sup.-1] P 101325 kg * [m.sup.-1] * [s.sup.-2] M 0.02896 kg * [mol.sup.-1] [[micro].sub.eff] 1 L 3 m
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|Author:||Fuxman, A.M.; Forbes, J.F.; Hayes, R.E.|
|Publication:||Canadian Journal of Chemical Engineering|
|Date:||Aug 1, 2007|
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