Separator Processes with

Ind. Eng. Chem. Res. 2004, 43, 1853-1862

1853

Balanced Control Scheme for Reactor/Separator Processes with Material Recycle Rosendo Monroy-Loperena,† Rocio Solar,‡ and Jose Alvarez-Ramirez*,†,‡ Departamento de Ingenieria de Procesos e Hidraulica, Universidad Autonoma Metropolitana-Iztapalapa, Apartado Postal 55-534, Mexico D.F., 09340 Mexico, and Programa de Investigacio´ n en Ingenierı´a Molecular, Instituto Mexicano del Petroleo

The simple reactor/separator process with recycle has recently been used as a benchmark case to gain insights into the problem of plantwide control structure selection. This paper explores the use of a balanced control scheme aimed at improving the disturbance rejection capabilities of the controlled process. The idea is to change the operating conditions in both the reactor and the separator so that the composition control effort in the presence of production rate changes is distributed. To this end, a parallel control structure is proposed where the product composition is regulated by means of simultaneous feedback manipulations of the vapor boilup rate and the reactor temperature. In this way, the use of the reactor temperature as a secondary control input reduces the snowball effects reflected in large vapor boilup rate changes in response to relatively small production rate changes. Nonlinear simulations show that effective composition control can be obtained with moderate vapor boilup control efforts. 1. Introduction The simplest reactor/separator with material recycle (RSR) process shown, in Figure 1, has been considered recently as a benchmark against which insights into the dynamics and control of process with recycles can be gained. Earlier works in this area focused on the dynamics induced by material recycling.1-6 Luyben4 showed that the recycle loop can induce the so-called “snowball effect” (SBE) as feed conditions change. This effect is reflected, for instance, in the fact that a small change in the fresh feed flow rate can lead to significant increases in the flow rates of recycle streams. In practice, high recycle rates are undesirable because they can lead to increased operating costs because of, e.g., high vapor boilup rates required in the distillation separator. To reduce the drawbacks of the SBE, Wu and Yu7 proposed the use of balanced control schemes to handle disturbance effects by changing operating conditions in several units of the process, not just one. Following these ideas, Wu et al.8 and Larsson et al.9 have explored several balanced control structures for the RSR process. Wu et al.8 used steady-state disturbance sensitivity analyses to simplify balanced control structures by eliminating some of the composition loops. It was shown that, for the simplest RSR process in Figure 1, only one composition loop is sufficient to keep all of the compositions near set points. On the other hand, Larsson et al.9 used the concept of self-optimizing control10 to address the problem of control structure selection. They found that the reactor level should be kept at its maximum to optimize the economic performance. Indeed, this result rules out many of the control structures proposed in the literature from being economically attractive, such as those that propose varying the reactor holdup.4 * To whom correspondence should be addressed. E-mail: [email protected]. Fax: +52-55-58044900. Tel.: +52-5558044650 † Instituto Mexicano del Petroleo. ‡ Universidad Autonoma Metropolitana-Iztapalapa.

In the control structures proposed so far, extensive variables (e.g., flow rates) are used to balance the work evenly among process units as the production rate changes. This is similar to Georgakis’s approach,11 in which intensive variables are kept constant at different operating conditions. Distribution of the control work is achieved by identifying suitable controlled output and manipulated input one-to-one pairings. On the other hand, in all of the balanced control schemes reported in the literature, the reactor temperature is kept constant by manipulating the jacket coolant flow rate. However, the reactor temperature is an intensive variable that can be changed to induce significant composition changes with relatively small steady-state control efforts. From a cascade control scheme, the reactor temperature set point can be seen as an additional degree of freedom that can be exploited, within a safe operating range, to enhance the performance of the controlled process. An interesting problem is determining how to manipulate the reactor temperature to alleviate the control effort addressed by balanced feedback control structures on the basis of extensive variables only. This paper focuses on this problem by exploring an alternative balanced control structure for the control of an RSR process. The main features of the proposed balanced scheme are as: (i) The reactor temperature set point is used as a manipulated input for feedback control purposes. In this way, the balanced control scheme uses both extensive and intensive manipulated variables. (ii) A parallel, also called habituating,12 control structure is used to distribute the control effort between the reactor and the separator. The idea is to manipulate simultaneously the reactor temperature and the vapor boilup rate to regulate the process product composition. As a result, the adverse consequences of the SBE (e.g., high recycle flow rates) are reduced by means of relatively small changes in the reactor temperature. Rigorous nonlinear simulations show that effective composition control can be achieved with moderate

10.1021/ie0305770 CCC: $27.50 © 2004 American Chemical Society Published on Web 03/05/2004

1854 Ind. Eng. Chem. Res., Vol. 43, No. 8, 2004

Figure 1. Schematic diagram of the RSR process.

vapor boilup control efforts in the presence of fresh feed flow rates changes. 2. Reactor and Separator with Recycle Process The RSR process benchmark and design data are taken from the work of Wu and Yu.7 The model assumes a binary feed with nominal fresh feed conditions z0 ) 0.9 mol/mol, F0 ) 460 kmol/h, and T0 ) 70 °F. The reactor has a maximum holdup of 2800 kmol, and a first-order reaction A f B is taking place. The distillation column has 22 stages including reboiler and condenser, liquid feed at stage 13, constant relative volatility RAB ) 2, and constant molar flows. The purity requirement for the product is xB ) 0.0105 mol/mol. The nominal reactor operating temperature T ) 156 °F, corresponding to the nominal reactor concentration xr ) 0.43 mol/mol. Figure 1 describes the RSR process and its nominal operating conditions. Let us discuss briefly the SBE introduced by Luyben.1-3 The idea is to describe what can happen to the recycle process in response to a change in the fresh feed conditions. After steady-state material balances in the reactor and the separator have been achieved, one concludes that the recycle flow rate D can be expressed as

D)

F0 - βxB βxD/F0 - 1

(1)

kVR z0 - xB

(2)

where

β)

VR is the reactor volume, and k is the reaction constant at the reactor operating temperature. Because the RSR process is commonly designed to operate under conditions of high product purity, it is useful to consider the limiting case in which xD f 1 and xB f 0. Under these conditions, one obtains

D)

z0F02 kVR - z0F0

(3)

It is noted that, because kVR - z0F0 > 0, an increment in the fresh feed parameters F0 and z0 increases the numerator and decreases the denominator. This effect can be explained as follows: As F0 and z0 increase at constant reactor temperature and holdup, the reactor composition increases. In turn, the distillation separator feed composition increases correspondingly. To meet product composition specifications, the vapor boilup rate must be increased, which induces subsequent increments in the reflux and recycling flow rates. 3. Conventional Control Configuration The aim of this section is to describe a conventional control configuration for the RSR process. By a conventional control configuration, we mean an input/output pairing based on traditional practical approaches for the control of distillation columns and chemical reactors. As in previous papers,8,9 it is shown that the resulting conventional control scheme does not overcome the adverse SBEs. In a subsequent section, the conventional control structure is modified to obtain a balanced control effort by distributing the work in the distillation column and the chemical reactor. In terms of possible manipulated inputs, the RSR process has several degrees of freedom that should be specified or linked to a controlled output to guarantee stable operation. The lower control hierarchy corresponds to level (i.e., holdup) control, which must be implemented to ensure safe equipment operation. In the case of the process shown in Figure 1, there are three levels to be regulated: two correspond to reboiler the and condenser drums in the distillation column, and the third corresponds to the reactor holdup. Following earlier results on the control of the RSR process (see, for instance, Wu et al.8 and Larsson et al.9), the input/ output pairing for level control is made as follows: (i) For distillation column holdups, use the distillate flow rate D to regulate the condenser drum level. On the other hand, use the bottoms flow rate B to regulate the reboiler drum holdup. (ii) Varying the reactor holdup has been proposed as an alternative for obtaining balanced operation of the RSR process.7 However, because of defined operating constraints (e.g., heat transfer area), practitioners have

Ind. Eng. Chem. Res., Vol. 43, No. 8, 2004 1855

certain opposition toward varying the reactor holdup during normal operation. On the other hand, aside from economic considerations, Larsson et al.9 have shown that maximizing the reactor holdup minimizes the SBE and maximizes economical process revenues. On the basis of these considerations, control structures with varying reactor holdups are discarded in this work. As suggested by Larsson et al.,9 the reactor is constrained to operate at a maximum and constant holdup value. The reactor effluent flow rate F is then used to regulate the reactor holdup. In the following, perfect level control in both the distillation column and the reactor are assumed. In this way, for instance, perfect reactor level control implies that F(t) ) F0(t) + D(t), for all t g 0. Given that D and B are used for inventory control and that the column recirculation flow rate R is given by the perfect condenser drum level balance R ) V - D, the vapor boilup rate V degree of freedom must be specified to fulfill the distillation column material balances. Recall that the task of the process is to convert reactant A into product B. In this way, the aim of the RSR process is to obtain the largest amount of product B in the process effluent, namely, in the bottoms flow. Consequently, a control objective is to regulate the bottoms composition xB. In classical studies on the control of distillation columns, the vapor flow rate V has been used to regulate the bottoms composition xB (see, for instance, Morari and Zafiriou13 and references therein). Among other reasons to take this input/output pairing is that the vapor flow rate has an almost-immediate (i.e., low transport delays) effect on the reboiler drum dynamics. Thus, the following input/output pairing is proposed: (iii) Use the vapor boilup rate V to regulate the bottoms composition xB. The purity requirement for the product is xB e 0.0105 mol/mol. Nextly, let us consider the operation of the chemical reactor. In general, industrial reactors are operated in a limited temperature range IT ) [Tmin, Tmax] around a nominal operating point T h . There are several reasons to impose this restriction. For instance, the chemical reaction A f B can be carried out by means of catalyst whose optimal temperature range is around T h . Large deviations from the nominal operating temperature T h can induce a loss of catalyst efficiency or catalyst deactivation. Therefore, the reactor temperature set point Tsp should be contained in the range IT. The following conventional input/output pairing is used to regulate the reactor temperature: (iv) Use the jacket temperature Tj to regulate the reactor temperature at a set point Tsp ∈ IT. Classical PI compensators tuned with internal model control (IMC) tuning guidelines were implemented in the loops for both the bottoms flow composition and the reactor temperature. Measurements delays for temperature and composition were taken as 1 and 6 min, respectively. The parameters of the reactor temperature controller are the same as in Wu and Yu7 (see also Wu et al.8), namely, Kc,T ) -1.81 × 10-2 and τI,T ) 5.4 h. On the other hand, the input/output step response for the path V f xB around the nominal operating point (described in Figure 1) can be approximated with a stable first-order (i.e., one-time-constant) model as follows

KVxB ∆xB ) exp(-θxBs) ∆V τVs + 1

(4)

Figure 2. Response of the conventional control configuration to a step change in the set point of the bottoms composition.

Figure 3. Response of the conventional control configuration to a +25% step change in the fresh feed flow rate.

where the steady-state gain is KVxB ) -8.25 × 10-5 mol/ mol h/lbmol and the time constant is τVxB ) 3.14 h. The time delay, θxB ) 0.1 h, accounts for composition measurement delays. The IMC tuning guidelines13 with assigned closed-loop time constant τc,V ) 0.75τV give the control gain Kc,xB ) 1.61 × 104 lbmol/(h mol/mol) and the integral reset time τI,xB ) 5.4 h. Figures 2 and 3 show the response of the control system under a set-point change in the bottoms composition xB and a +25%


2 lbmol has to be vapored in the bottom of the column and pumped into the reactor. In addition to the negative economic impacts, such excessive vaporization requirements might not be possible because of equipment limitations (i.e., the boiling vapor flow rate might attain its maximum value Vmax). This situation shows that conventional control structures that work quite well when used in a single piece of equipment might display poor performance when used in an interconnected equipment process, such as the quite simple RSR process in Figure 1. 4. Balanced Control Scheme

Figure 4. Steady-state separator flows as a function of the fresh feed flow rate for the conventional control configuration.

disturbance in the fresh feed flow rate F0. In these simulations, the reactor temperature is controlled at its nominal value T h ) 156 °F, such that the control system adjusts only the column operating conditions to maintain the product specification. In the set-point-change case, it is noted that the control system has an initial rapid response and then converges sluggishly, mainly because of changes in the recycling streams required to maintain the column and reactor holdups at their reference values. This sluggish behavior is clearly observed in the response of the control system in rejecting the disturbance in the fresh feed flow rate (see Figure 3). Excessive adjustment of the PI controller parameters can lead to oscillatory behavior mainly as a result of the excitation of recycling high-frequency dynamics.1 Although the control system is able to carry out setpoint changes and disturbance rejection in a stable manner, SBEs in the operation of the conventional control system are displayed. In fact, Figure 3 shows that, for +25% change in the fresh feed flow rate F0, the control system induces a +70% change in the vapor boilup rate V. Similar effects are observed in the distillate (D) and the reflux (R) flow rates. Figure 4 shows the steady-state values of the flow rates D, R, and V as functions of the fresh feed flow rate F0 for the same operating conditions as in Figures 2 and 3. It is noted that the recycled distillate flow rate D changes linearly with F0, with a slope on the order of 2.7. This excessive usage of recycled and vaporized material in the process can lead to unacceptable increments in the operating costs and, consequently, to a significant decrement in the process revenues. For instance, for each additional 1 lbmol of fresh feed material, more than

The excessive burden on the vaporization equipment in the presence of a sustained disturbance in the fresh feed flow rate (see Figure 4) is induced by an imbalance in the conventional control scheme. In fact, in the presence of a disturbance in the fresh feed flow rate, the conventional control scheme adjusts the operating conditions only in the distillation column by increasing the vaporization rate. Given the structure of the conventional control scheme, such adjustment is required to compensate for composition increments in the column feed, which are induced by decrements in the reactor conversion that are due mainly to a reduction in the reactor residence time (which is, in turn, induced by increments in the reactor feed). Several unconventional control schemes to overcome the adverse consequences of the SBE have been proposed. The basic idea is to balance the control effort by distributing the work in both pieces of equipment: the distillation column and the chemical reactor. Luyben4 and Wu and Yu7 proposed that the reactor holdup be varied to compensate for increments in the reactor feed. The rationale is that reactor feed flow rate increments induced by increments in the fresh feed flow rate can be compensated (at least partially) by increments in the reactor residence time, so that the reactor conversion is maintained around the nominal value. In this way, the processing work performed by the distillation column can be reduced with a consequent reduction in the vaporization rate. Although control strategies based on varying reactor holdups are attractive, several practical implementation problems can arise, including the fact that operators can be resistant to varying reactor levels during normal operations. Moreover, Larsson et al.9 pointed out that a liquid-phase reactor should normally be operated at maximum holdup (liquid level) to optimize steady-state economics. Following this idea, Larsson et al.9 proposed a control scheme in which the reactor holdup is maintained at its maximum level and the control effort is directed at controlling the ratio R/F. In this way, the reflux ratio R/F plays the role of a self-optimizing variable in the sense that constant set points give an acceptable economic loss. In this form, by fixing the value of R/F, additional fresh flows can be redistributed between the separation process (via recomputation of the reflux R) and the reactor. In a different form that will be developed below, this idea of Larson et al., i.e., of distributing the processing work in the presence of fresh feed disturbances, will be exploited to design a balanced control scheme based on manipulations of the reactor temperature. Specifically, whereas Larsson et al.’s approach is based on redistributing the excessive flows entering the process (via fresh feed disturbances), our approach is based on redistributing the processing work between the reacting and separating units.


It is very likely that the conventional control scheme described in the above section is installed in existing equipment. On the other hand, the implementation of new control schemes might be difficult, for instance, because of reduced budgets (as in developing countries). Next, a relatively slight modification of the conventional control scheme aimed at dealing with SBEs, is presented. Following Luyben’s ideas,4 the objective is to obtain a balanced control scheme by distributing the work in both pieces of equipment. Contrary to the balanced control schemes reported so far, we propose that the reactor operating temperature be varied to compensate, at least partially, for decrements in the reactor conversion. In this way, the distillation column operation is distressed, yielding no excessive increments in the vaporization rate. The rationale behind this control idea is that, because of the exponential (i.e., Arrhenius) dependence of the reaction on temperature, small changes in the operating temperature can induce large changes in the reactor conversion. Changing the reactor operating temperature in a limited range has a steady-state effect on the process through its effect on the conversion. In this way, if an excessive amount of work is detected in the separation column, reflected by significant vaporization rates, an adjustment in the reactor temperature should be made, which, in turn, should produce a decrement in the column feed composition. This control action should alleviate the operation effort devoted by the separation process. At this point, a natural question should be addressed: How can the adjustments in the reactor operating temperature be implemented in an automatic and feedback framework? In other words, how does one implement a feedback control scheme to manipulate simultaneously the reactor operating temperature Tsp and the vapor boilup rate V so that the control effort in the presence of fresh feed disturbances is distributed? To provide an answer to this question, we resort to parallel (e.g., habituating) control design methodologies.12 To this end, the following considerations are made: (i) For the sake of bottoms composition regulation, the reactor temperature set point Tsp is manipulated in a secure operating range IT around the value T h (i.e., T h ∈ IT). Once the corrective behavior of Tsp has been specified via the feedback function Tr ) φM(xB), the reactor temperature is changed by means of a series cascade control architecture where the function φM(xB) plays the role of the master controller and the standard PI-IMC controller acts as the slave controller.14 (ii) To design a feedback controller for the simultaneous manipulation of the reactor temperature set point Tsp and the vapor boilup rate V, a nonsquare 2 × 1 input/output model is formulated as follows

y(s) ) G1(s) u1(s) + G2(s) u2(s)

(5)

where, for convenience, the following notation has been used: y(s) ) ∆xB(s) is the regulated output, and u1(s) ) ∆V(s) and u2(s) ) ∆Tsp(s) are the manipulated inputs. In this way, the problem is to design a parallel feedback controller u(s) ) C(s)ey, where u(s) ) [u1(s), u2(s)], C(s) ) [C1(s), C2(s)], and ey(s) ) ysp(s) - y(s) is the regulation error, such that ey(t) f 0 asymptotically. Note that the resulting feedback controller manipulates in parallel form both control inputs u1(s) and u2(s) to regulate the single output y(s).

Figure 5. Schematic diagram of the parallel control architecture.

(iii) From step responses around the nominal operating point, one finds that the input/output dynamics can be approximately modeled as stable first-order process described by

G1(s) )

KVxB τVs + 1

exp(-θxBs)

(6)

exp(-θxBs)

(7)

and

G2(s) )

KTxB τTs + 1

That is, the input/output dynamics V f xB and Tsp f xB are modeled as stable first-order processes. The timedelay operator exp(-θxBs) accounts for delays in composition measurements. In the case of the RSR process (see Figure 1), one obtains the following set of parameters: KVxB ) -8.25 × 10-5 mol/mol h/lbmol, τV ) 3.14 h, KTxB ) -1.81 × 10-2 mol/mol/oR, and τT ) 5.4 h. Motivated by the habituating control system responsible for mammalian blood pressure regulation, Henson et al.12 proposed a systematic methodology for the synthesis of parallel controllers. The underlying idea is to exploit specific characteristics and operating objectives of a process with two different types of manipulated variables: (a) a slow, inexpensive type and (b) a fast, expensive type. In the case of the RSR system, the vapor flow rate V plays the role of the fast, expensive manipulated variable, whereas the reactor temperature Tsp becomes the slow, inexpensive variable. That is, because the reactor operation involves only heating/ cooling processes and the separation equipment requires vapor boilup, it is expected that manipulations of the vapor flow rate are more expensive than manipulations of the reactor temperature Tsp. This is because, for typical chemicals, the ratio Cp/λv is around 0.01, where λv is the vaporization heat: In general, it is more expensive to vaporize than to heat (within certain practical temperature range) a liquid of the typical chemical industry. Thus, it would be expected that small changes in the reaction operating temperature will have a significative impact on the vaporization rate burdens of separation processes. In this form, the control effort devoted to regulation of the overall process product composition is distributed between manipulations of the inexpensive but slow-response reactor temperature and manipulations of the expensive but fast-response vapor flow rate. In the spirit of Henson et al.’s approach,12 in the following discussion, a simple controller synthesis procedure for the input/output dynamics given by eqs 5-7 is proposed. The controller synthesis algorithm is strongly based on the factorization approach.15 Specifically, the procedure for obtaining a parallel controller consists of four steps (see Figure 5): (i) factorization of the plant transfer function, (ii) computation of a master controller, (iii) use of an optimization problem to obtain an input


divisor, and (iv) inversion of certain lead-lag filters to obtain the actual control inputs. The control design steps are described below. (i) Factorization. A factorization of process eqs 5-7 can be obtained as follows

y(s) ) H(s)[KVxBu1(s) + KTxBFLL(s) u2(s)]

(8)

where

H(s) )

exp(-θxBs) τVs + 1

and FLL(s) is a lead-lag filter given by

FLL(s) )

τVs + 1 τTs + 1

(9)

Notice that FLL(0) ) 1. (ii) Computation of the Master Controller. The second step consists of computing a master controller, CM(s), for the plant H(s). The idea is to obtain a single controller to regulate the output y(s) from a standard single-input-single-output (SISO) control problem. To this end, we introduce the intermediate control input w(s), defined as follows

w(s) ) KVxBu1(s) + KTxBFLL(s) u2(s)

(10)

so that one obtains the SISO plant

y(s) ) H(s) w(s)

(11)

In this way, once a control input w(s) for the plant described by eq 11 has been computed, the following step will be to distribute the control effort between the two actual control inputs u1(s) and u2(s). This corresponds to a rectangular (with one degree of freedom) inversion problem, which, after a suitable regularization procedure described in steps iii and iv below, leads to control functionalities for the control inputs u1(s) and u2(s) as functions of the intermediate control input w(s) (see Figure 1). In this way, rather than defining w as a function of u1 and u2, eq 10 provides a functional relationship between the signals u1, u2, and w. It is noted that H(s) is a stable, unitary, steady-state-gain [i.e., H(0) ) 1] process with input delay θxB. For this type of plant, classical PI controllers tuned with IMC tuning guidelines13 can be used. In fact, if

(

CM(s) ) Kc,M 1 +

the control input w(s) is composed of the contributions of two independent control inputs, namely, KVxBu1(s) and KTxBFLL(s) u2(s). At this point, a procedure for distributing the control input w(s) into such contributions is required. Several strategies can be followed, ranging from heuristic rules to rather formal rules based on some (e.g., H∞) optimal control problem (see, for instance, Morari and Zafiriou13). As in the approach proposed by McLain et al.16 for nonlinear habituating control, the regularization of the rectangular control problem is achieved by minimizing the cost associated with affecting control. Let v2(s) ) FLL(s) u2(s). By a manipulation of notation to denote the control input u1(s) in the time domain as u2(t) [and similarly for v2(s) and w(s)], the following least-squares optimization problem is posed

)

1 τI,Ms

1 min {R[KVxBu1(t)]2 + (1 - R)[KTxBv2(t)]2} (13) 2 (u1,v2) subject to the constraint

KVxBu1(t) + KTxBv2(t) ) w(t)

(14)

where 0 e R e 1 is a constant that weights the role of the inputs KVxBu1(s) and KTxBv2(s). In this way, one can see that eqs 13 and 14 correspond to a timewise optimization problem where the computed control inputs u1(t) and v2(t) minimize the least-squares index in eq 13 at time instant t. Of course, more sophisticated indices could be used, such as in an infinite-time-horizon version of eq 13. However, because our main objective here is to show how parallel control can be used to improve the operation of recycle processes, we consider only the timewise case of eq 13 to maintain simplicity in presentation. By using Lagrange operator methods,17 the optimization problem in eqs 13 and 14 leads to the conclusion that u1(s) and v2(s) must satisfy the relationships

u1(s) ) KVxB-1(1 - R)w(s)

(15)

v2(s) ) KTxB-1Rw(s) (iv) Recovery of u2(s) via Inversion. The last step consists of recovering the actual control input u2(s) from the eqs 15. By recalling that v2(s) ) FLL(s) u2(s), this can be easily accomplished by noticing that FLL(s) is a stable and minimum-phase filter, so that

u2(s) ) RFLL(s)-1 w(s)

then

Kc,M )

τV τc,M + θxB

(12)

τI,M ) τV where τc,M is a prescribed closed-loop time-constant, which, as a heuristic rule, can be taken to be on the order of 0.5-0.75 times τV. (iii) Computation of the Control Input Divider. We have computed the master control input w(s) to ensure the stability of the closed-loop system and the asymptotic tracking of the desired output, ysp. However,

That is, the actual control input can be recovered simply by inverting the lead-lag filter FLL(s). In this way, the controller that manipulates the vapor boilup rate V is simply a detuned PI compensator with detuning factor 1-R

(

u1(s) ) (1 - R)Kc,M 1 +

)

1 e (s) τI,Ms y

On the other hand, the controller that manipulates the reactor temperature set point Tsp is also a detuned PI


Figure 6. Response of the balanced control configuration to a step change in the set point of the bottoms composition.

compensator in series with a lead-lag compensator

u2(s) ) RKc,M

(

)(

)

τTs + 1 1 1+ e (s) τVs + 1 τI,Ms y

where e(s) is the regulation error. It is noted that the role of the lead-lag compensator (τTs + 1)/(τVs + 1) is to synchronize the phase of controller u2(s) with respect to the phase of controller u1(s). The following comments are in order: (i) Notice that u1(s) ) 0 [u2(s) ) 0] when R ) 1 (R ) 0). That is, when R ) 1, the vapor boilup rate V is set at its nominal value V h , and all dynamical and stationary control efforts are performed by the reactor temperature control. Conversely, when R ) 0, the reactor temperature set point is maintained at its nominal value T h , and all dynamical and stationary control efforts are executed by the column distillation controller. This corresponds to the conventional control scheme described in the preceding section. When 0 < R < 1, both controllers make nontrivial contributions to the regulation of the bottoms flow composition xB. In this way, the proposed parallel control scheme has the structure of a balanced controller for 0 < R < 1. As R is increased, more processing work is transferred into the reactor. (ii) Parallel control has an additional advantage over the conventional control scheme, namely, it can deal better with input saturation. In the event that the vapor boilup rate V achieves its limiting value Vmax, the reactor temperature controller is able to move the process dynamics into an operating region where either the vapor boilup rate is not more saturated or the

Figure 7. Response of the balanced control configuration to a +25% step change in the fresh feed flow rate.

bottoms flow composition is regulated at its set point by the reactor controller alone. (iii) A classical PI compensator has been proposed as the master controller CM(s). As shown in Figures 2 and 3 (see also Figures 6 and 7 below), one of the weaknesses of PI compensation is its inability to speed the convergence of the regulation error. In fact, in the presence of step set-point changes and external disturbances, the controller has an initial fast response; however, the convergence rate decreases as the regulation error approaches zero. This sluggish behavior is induced by the material recycling loop of the process. That is, step disturbances entering the process are partially recycled and integrated by the capacitance dynamics of the reactor and the column. In this way, after an initial stage where the disturbance is directly transmitted as a step change, the disturbance is reinjected into the process loop as a ramp-type signal. Belanger and Luyben18 studied the inability of PI control to reject ramp-type disturbances efficiently and proposed the use of low-frequency compensators to deal with ramp-type disturbances. A type of compensator proposed is the double integrator (called a PII controller)

(

CM(s) ) Kc,M 1 +

)

1 1 + τI,Ms τ 2s2 II,M

(16)

where τII,M is the reset time of the double integrator. The addition of this type of compensator forces the error in tracking a ramp disturbance to eventually go to zero.13 Following the tuning guidelines described by Belanger and Luyben,18 the following controller settings


Figure 8. Steady-state separator flows and reactor temperature as a function of the fresh feed flow rate for the balanced control configuration.

are proposed: Choose Kc,M and τI,M as in eqs 12, and set

τII,M ) 2τV Some adjustment of the controller parameters might be required to achieve the required performance; however, these settings provide a good initial estimate. In addition to this PII control strategy, for the case of fresh feed disturbances, one could use a feedforward/feedback scheme in the following form: Determine the ratio of the fresh feed to the set point of the reactor temperature controller, and use the output of the product composition controller to adjust the ratio. In this way, the feedback controller would be equipped with a predictive-type mechanism to prevent excessive departures from the composition set point induced by fresh feed disturbances, hence leading to better convergence rate properties. We now illustrate the performance of the balanced control scheme. To ensure secure reactor operation, the

reactor temperature set point was constrained to take values in the interval IT ) [152, 158]. On the other hand, the vapor boilup rate was subjected to the saturation limits Vmin ) 1200 lbmol/h and Vmax ) 2200 lbmol/h. The nominal control input values for the reactor set point and vapor flow rate were chosen as the flowsheet design values (see Figure 1): T h sp ) 156 °F and V h ) 1600 lbmol/h, so that the control inputs become these constant values when the corresponding controllers are turned off. Figures 6 and 7 show the response of the proposed balanced control scheme for a step change in the bottoms flow composition set point xB,sp and a +25% step change in the fresh feed flow rate F0, respectively. It is interesting to note that, in the extreme case with R ) 1 (i.e., only the reactor controller compensating for changes in the fresh feed flow rate), the regulation objective is achieved without excessive changes in the reactor temperature (no larger than about 5 °F). Also notice that, as more processing work is done by the reactor, the response becomes more sluggish, mainly


have confronted some problems in obtaining an acceptable tuning. However, the PII compensator tuning guidelines discussed before are easy to use because they are straigthforward extensions of well-known PI control tuning approaches. The simulation results show that the balanced control scheme is able to distribute automatically the processing work in the reactor and the separation column. A positive impact of using the balanced control scheme is that the vapor boilup rate is significantly reduced, which, in turn, has a positive impact on the economics of the process. In fact, for R ) 0.5, a vapor boilup rate loss of about 150 lbmol/h is obtained with only a 1 °F reduction in the reactor operating temperature (see Figure 6). However, one negative effect is that the rate of convergence to the desired composition set point is reduced, which can lead to a reduction of the robustness margin of the controlled process. Thus, at least for this class of processes, it seems that, similarly to the tradeoff between robustness and performance, a tradeoff exists between the economy and the operability of the process. That is, the less expensive the operation of the processes (via a larger control effort devoted to the reactor operating temperature), the lower the capacity of the control loop to reduce the adverse effects of exogenous disturbances. 5. Conclusions Figure 9. Response of the balanced control configuration for a classical PI compensator and an IMC-tuned PII compensator.

because of the indirect effect of the reactor temperature on the bottoms flow composition. In fact, the dynamical effect introduced by a change in the reactor temperature is received by the bottoms flow composition after being “filtered” by the lower section of the separation column. The balanced control scheme was conceived to alleviate the negative results of SBEs. Figure 8 shows the steady-state values of the flow rates D, R, and V as a function of the fresh feed flow rate for five different values of R. As expected, as more processing work is done by the reactor, the sensitivity of the column flows to changes in F0 is significantly reduced. In fact, for the conventional control scheme (i.e., R ) 0), the slope of the V-F0 relationship is around 2.7. However, for a balanced control scheme with R ) 0.5, the slope is about 1.43. When R is further increased, the sensitivity of the separation column flow rates is reduced significantly; however, the dynamical response of the overall control system becomes more sluggish. Thus, in addition to economical indices, the selection of the parameter R to balance the equipment processing work relies on a tradeoff between steady-state processing work balance and dynamical convergence rate. This tradeoff can be partially addressed by using a PII controller. For R ) 0.5, Figure 9 shows that the PII compensator is able to adjust the bottoms composition at its set point faster than a traditional PI compensator. Similar results are obtained for other values of R and for step disturbances in the fresh feed flow rate. We have also tested a feedforward/feedback control strategy in which the controller accounts for fresh feed disturbances and acts in advance to reduce their adverse effects. Although the closed-loop response is similar to the response obtained with the PII control scheme (as shown in Figure 9), we

Summarizing, we have presented a balanced control scheme for a liquid-phase reactor with recycle plant. The basic idea is to manipulate simultaneously the vapor boilup rate in the separator and the reactor temperature set point, expecting that the processing work be distributed in both pieces of equipment. The balanced control scheme is designed by means of parallel control methodologies to design, in a systematic way, the control effort required of each process unit. Numerical simulations demonstrate the ability of the resulting feedback control scheme to handle, under both dynamic and steady-state conditions, disturbances in the fresh feed conditions (i.e., composition and flow rate). In an overall sense, the main conclusion of this paper is that the problem of selecting a control structure and designing the corresponding feedback compensators for plantwide control should involve novel control configurations and more structured compensators (e.g., parallel control, low-frequency compensators, etc.) that exploit the specific structure of the process. By doing so, it is expected that more efficient plantwide operation and more robust control performance can be obtained.10 Acknowledgment This work was supported by Universidad Autonoma Metropolitana-Iztapalapa, Instituto Mexicano del Petroleo, and CONACyT. Literature Cited (1) Luyben, W. L. Dynamics and control of recycle systems. 1. Simple open-loop and closed-loop systems. Ind. Eng. Chem. Res. 1993, 32, 466. (2) Luyben, W. L. Dynamics and control of recycle systems. 2. Comparison of alternative process designs. Ind. Eng. Chem. Res. 1993, 32, 476.

1862 Ind. Eng. Chem. Res., Vol. 43, No. 8, 2004 (3) Luyben, W. L. Dynamics and control of recycle systems. 3. Alternative process designs in a ternary system. Ind. Eng. Chem. Res. 1993, 32, 1142. (4) Luyben, W. L. Snowball effects in reactor/separator processes with recycle. Ind. Eng. Chem. Res. 1994, 33, 299. (5) Luyben, M. L.; Luyben, W. L. Design and control of a complex process involving two reaction steps, three distillation columns, and two recycle streams. Ind. Eng. Chem. Res. 1995, 34, 3885. (6) Tyreus, B. D.; Luyben, W. L. Dynamics and control of recycle systems. 4. Ternary systems with one or two recycle streams. Ind. Eng. Chem. Res. 1993, 32, 1154. (7) Wu, K. L.; Yu, C. C. Reactor/separator processes with recycle 1. Candidate control structure for operatibility. Comput. Chem. Eng. 1996, 20, 1291. (8) Wu, K. L.; Yu, C. C.; Luyben, W. L.; Skogestad, S. Reactor/ separator processes with recycles 2. Design for composition control. Comput. Chem. Eng. 2002, 27, 401. (9) Larsson T.; Govatsmark M. S.; Skogestad S.; Yu, C. C. Control structure selection for reactor, separator, and recycle processes. Ind. Eng. Chem. Res. 2003, 42, 1225. (10) Skogestad, S. Plantwide control: the search for selfoptimizing control structure. J. Process Control 2000, 10, 487. (11) Georgakis, C. On the use of extensive variables in process dynamics and control. Chem. Eng. Sci. 1986, 41, 1471.

(12) Henson, M. A.; Ogunnaike, B. A.; Schwaber, J. S. Habituating control, strategies for process control. AIChE J. 1995, 41, 604. (13) Morari, M.; Zafiriou, E. Robust Process Control; Prentice Hall: Englewood Cliffs, NJ, 1989. (14) Luyben, W. L. Process Modeling, Simulation, and Control for Chemical Engineers; McGraw-Hill: New York, 1990. (15) Vidyasagar, M. Control System Synthesis: A Factorization Approach; MIT Press: Cambridge, MA, 1985. (16) McLain, R. B.; Kurtz, M. J.; Henson, M. A.; Doyle, F. J., III. Habituating control for nonsquare nonlinear processes. Ind. Eng. Chem. Res. 1996, 35, 4067. (17) Luenberger, D. G. Optimization by Vector Space Methods; John Wiley and Sons: New York, 1969. (18) Belanger, P. W.; Luyben, W. L. Design of low-frequency compensators for improvement of plantwide regulatory performance. Ind. Eng. Chem. Res. 1997, 36, 5339.

Received for review July 10, 2003 Revised manuscript received October 13, 2003 Accepted February 3, 2004 IE0305770

Separator Processes with

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