Control Structure Selection of an Ideal Reactive ... - ACS Publications

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Control Structure Selection of an Ideal Reactive Distillation Column Ioannis K. Kookos* Department of Chemical Engineering, University of Patras, 26504 Rio, Patras, Greece ABSTRACT: The selection of process control structures has attracted significant attention in the past 3 decades. Two approaches have emerged. The first approach uses heuristics or previous knowledge in order to develop plausible control structures, while the second is model-based and systematic. In this work a challenging case study is considered with the aim being to demonstrate that the systematic methodologies can outperform the empirical approaches.

1. INTRODUCTION The process control structure selection (CSS) problem is a structural and parametric optimization problem that has received considerable attention during the past 3 decades. Three are the constituting elements of the CSS problem: the selection of controlled variables, the selection of manipulated variables, and the (structural and parametric) design of their interconnection. It is important to emphasize that the CSS problem is a subproblem of the general synthesis problem. The CSS problem deals with the dynamic behavior of a process around a nominal, optimal operating point with the aim being to devise control structures that mitigate the effects of the main disturbances and/or process uncertainties. It is interesting to note that during operation of a real plant the identification and continuous adaptation of the optimal operating point relies within the responsibilities of the real time optimization system (RTO).1 The aim of the regulatory control structure system is to keep the plant operating point as close as possible to the optimal point and minimize economic losses due to the effect of high-frequency disturbances. In accordance with the more general problem of chemical process synthesis, two approaches have emerged for solving the CSS. In the first approach a set of rules of thumb and/or simple controllability indicators are used to develop plausible control structures.2,3 The main advantage of the methodologies that fall into this category is that no extensive quantitative information is required and as a results they can be applied easily even to largescale CSS problems (such as plantwide control problems). The main drawback stems from the fact that no indication is offered relative to the optimality of the control structure(s) proposed. The second approach is based on systematic methodologies that make use of classical mathematical programming techniques in order to represent CSS alternatives (superstructures) and to locate effectively the most promising solutions.4
8 Needless to say that methodologies that fall into this category are accurate, within the limits imposed from the mathematical models and mathematical programming techniques used, and this is their main advantage. However, they suffer from the limitations imposed by the inefficiencies of the currently available solution algorithms, such as convergence problems, and that only local optimality can be guaranteed. The aim of this paper is to apply a systematic methodology to an interesting and challenging case study, first proposed by AlArfaj and Luyben,9 where the development of a control structure r 2011 American Chemical Society

of an ideal reactive distillation column is examined. A large number of publications that deal with this case study have been presented in the literature during the past decade.9
12 In all previous works the control structure selection was made on the basis of existing knowledge and empirical rules. The application of a systematic methodology13 will be presented in order to show the advantages of the systematic approach to the CSS problem. The structure of the paper is as follows: in section 2 the case study under investigation is briefly presented followed by the solution of the steady-state optimization problem, in section 3 the problem of the optimal CSS selection is formulated, and in section 4 the results are summarized and closed loop simulations are presented.

2. PROCESS DESCRIPTION AND STEADY-STATE OPTIMIZATION The following reversible and exothermic reaction is taking place on the liquid phase of the reactive trays9 A þ BSC þ D

ð1Þ

The reaction rate r (kmol/h) is expressed in terms of the mole fractions in the liquid phase r ¼ Mcat: ðkf, 0 eð
Ef =ðRg TÞÞ xA xB
kb, 0 eð
Eb =ðRg TÞÞ xC xD Þ

ð2Þ

where Mcat. is the mass of catalyst used and kf,0 and kb,0 are the preexponential factors and Ef and Eb the activation energies of the forward and backward reactions, respectively. Product C is the most volatile component, while product D is the heaviest component. The relative volatilities of the reactants lie between the volatilities of the two products αC > αA > αB > αD

ð3Þ

In order for the reactive column to operate successfully, reactant B must be fed above reactant A and the reactive section of the column must be located between the rectifying section, where product C is the dominant component, and the stripping section, where product D is the dominant component. The physical Received: April 7, 2011 Accepted: August 11, 2011 Revised: August 8, 2011 Published: August 22, 2011 11193

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Table 1. Physical Properties symbol

numerical value

Units

Δhreacn heat of reaction


41 840

Δhvap

heat of vaporization

29 053.7

kJ/kmol

kf,0

preexponential factor

2.37  1019

kmol/(h 3 kmol)

kJ/kmol

Ef

activation energy

125 520

kJ/kmol

kb,0

preexponential factor

1.1106  1025

kmol/(h 3 kmol)

Eb

activation energy

167 360

kJ/kmol

Rg

ideal gas constant

8.314

kJ/(kmol 3 K)

Asat.

Antoine equation

A, 12.34; B, 11.65;

Figure 2. Representative tray of the reactive distillation column.

T in K, Psat. in bar

C, 13.04; D, 10.96

constant,

col can be reactive trays. The total number of trays is defined implicitly from the location of the tray where the reflux is fed to the column. The two reactants can be fed anywhere between tray 2 and tray NT
1, but the feed tray for component A must lie below the feed tray for component B. The following binary variables are defined:

ln Psat. j = Asat., j
(Bsat./T) Bsat.

Antoine equation

3862

K

constant

αi ¼ 1, if reactant A is fed on tray i;

0, otherwise

βi ¼ 1, if reactant B is fed on tray i;

0, otherwise

γi ¼ 1, if reflux is fed on tray i, δi ¼ 1, if tray i is a reactive tray,

ð5Þ

0, otherwise 0, otherwise

A generalized (or representative) tray i of the reactive distillation column is also shown in Figure 2. Apart from the usual incoming and outgoing liquid and vapor streams (Li+1, Li, Vi
1, and Vi), each tray may have a feed stream of reactant A (fA,i) and/ or reactant B (fB,i) and/or reflux (Ri). As a result the component material balances can be written as follows: Liþ1 xiþ1, j þ Vi
1 yi
1, j þ Ri xNT, j þ fA, i zA, j þ fB, i zB, j þ nj ri ¼ Li xi, j þ Vi yi, j ,

i ∈ col, j ∈ J

ð6Þ

where xi, j (yi, j) is the mole fraction of component j (j ∈ J = {A,B,C, D}) on tray i in the liquid (vapor) phase, Ri is the part of the column reflux that is fed to tray i, nj is the stoichiometric number of component j, and ri is the reaction rate on tray i. The component material balance for the reboiler and the condenser/reflux drum are Liþ1 xiþ1;j ¼ Li xi; j þ Vi yi; j ; Vi
1 yi
1; j ¼ ð Figure 1. Reactive distillation column superstructure.

∑

i0 ∈ col

i ∈ reb; j ∈ J

Ri þ DÞxi; j ;

i ∈ con; j ∈ J

Liþ1 þ Vi
1 þ Ri þ fA;i þ fB;i þ Nri ¼ Li þ Vi ;

¼ f1g ¼ f2, 3, :::, NT
1g ¼ fNTg

ð8Þ

The corresponding overall material balances are the following

constants that appear in the description of the case study are presented in Table 1. The steady-state model of the reactive column is based on the superstructure representation of Figure 1. NT potential trays (including the partial reboiler and total condenser) are postulated. Index i is used to denote the trays in the column (i ∈ I = {1, 2, ..., NT
1, NT}). We also define the following sets reb col con

ð7Þ

i ∈ col

ð9Þ Liþ1 ¼ Li þ Vi ; Vi
1 ¼

∑

i0 ∈ col

i ∈ reb

Ri þ D;

ð10Þ

i ∈ con

ð11Þ

The vapor
liquid equilibrium is described by the following equations that are based on the assumption of ideal behavior:

ð4Þ

∑ xi, j
j∑∈ J yi, j ¼ 0,

j∈J

which correspond to the reboiler, trays within the column shell, and the condenser (I = reb ∪ col ∪ con). All trays that belong to

yi, j P
xi, j Pjsat: ðTi Þ ¼ 0, 11194

i∈I

ð12Þ

i ∈ I, j ∈ J

ð13Þ

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and reflux

Table 2. Models Parameters symbol

numerical value

units

γi RL e Ri e γi R U ;

FA

fresh feed of reactant A

FB

fresh feed of reactant B

45.36

kmol/h

P

pressure

8.5

bar

Ureb

overall heat-transfer coeff.

0.568

kW 3 m
2 3 K
1

Ucond

overall heat-transfer coeff.

0.852

ΔTLM,reb

kW 3 m
2 3 K
1

45.36

kmol/h

log mean temperature difference

34.8

K

ΔTLM,cond log mean temperature difference

13.9

K

πHE

see eq 29

7296

$

nHE

see eq 29

0.65

TS

tray spacing

0.61

m

πcol

see eq 33

16640

$

nC,D

see eq 33

1.066

nC,H

see eq 33

0.802

πtray

see eq 33

229

nT,D

see eq 33

1.55

csteam

steam cost

5.155  10
6

$/kJ

8000

h/year

ty NT

no. of potential trays

∑

i ∈ col

iβi e

ð14Þ

i ∈ col

As can be easily seen, eqs 6, 9, and 12
15 consist of 12 equations per tray in the corresponding 12 unknowns (if all incoming streams are considered known), which are the vapor and liquid mole fractions, outgoing molar flow rates, temperature, and reaction rate. The condenser/reflux drum and the reboiler have 1 degree of freedom each (vapor boilup and reflux flow rate). To develop an optimal design of the column, the following linear constraints that involve integer variables are defined. Integer variable δi obtains the value of 1 if tray i is a reactive tray in which case the mass of catalyst (Mi,cat.) on tray i is a positive constant:

∑

ai ¼ 1

ð16Þ

ð17Þ ð18Þ

FA is the total molar flow rate of reactant A (see Table 2). The same applies for reactant B ð19Þ fB;i ¼ FB βi ; i ∈ col

∑ βi ¼ 1 i ∈ col

ð23Þ

∑

iγi

ð24Þ

i
1

i

∑ αi
i ∑¼ 2 βi i ¼2 0

ð25Þ

00

0

00

Qreb ¼ Areb Ureb ΔTLM;reb ¼ V1 Δhvap

ð26Þ

Creb ¼ πHE Areb nHE

ð27Þ

Qcon ¼ Acon Ucon ΔTLM;con ¼ VNT
1 Δhvap

ð28Þ

Ccon ¼ πHE Acon nHE

ð29Þ

The distillation column dimensions and TAC are calculated using the following equations pffiffiffiffiffiffi πDC 2 Þ g VNT
1 ð30Þ 1:22 FV ð 4 HC ¼ 1:2ð

is the maximum mass of the catalyst that can be loaded on trays. In the paper of Al-Arfaj and Luyben9 this constant has been given the value of 1 kmol. Because reactant A must be fed on one tray, we have the following constraints:

i ∈ col

iβi

i ∈ col

NT
1

MU cat.

i ∈ col

∑

i ∈ col

To develop an optimal steady-state design, the model presented above must be augmented with the equations that can be used to calculate an approximate total annual cost (TAC) for the ideal reactive column. The equations that follow are used to calculate the heat exchangers cost and utilities consumption

ð15Þ

fA;i ¼ FA ai ;

ð22Þ

The constraint that follows ensures that the reactive section is located between the trays into which the two reactants are fed

and the definition of reaction rate

i ∈ col

∑

i ∈ col

The model is completed by considering the following simplified energy balance

U , 0 e Mi, cat: e δi Mcat:

iαi e

δi e

ri ¼ Mi, cat: ðkf , 0 eð
Ef =ðRg TÞÞ xi, A xi, B
kb, 0 eð
Eb =ðRg TÞÞ xi, C xi, D Þ,

γi ¼ 1

∑

i ∈ col

$

i ∈ col

ð21Þ

where RL and RU are lower and upper bounds on the reflux flow rate. The contraints that follow ensure that reactant B will be feb above reactant A and below reflux (without excluding the possibility for all streams to be fed on the same tray, in which case the distillation column is simplified to a reactive flash):

30

ðVi
Vi
1 ÞΔhvap ¼ ri ð
Δhreacn Þ,

i ∈ col

∑

i¼1

iγi
2ÞTS

Ccol ¼ πcol DC nC;D HC nC;H þ πtray DC nT;D ð

ð31Þ NT
1

∑

i¼1

iγi
1Þ

ð32Þ

Ccol þ Creb þ Ccon Þ þ ðcsteam Qreb þ ccw Qcon Þty 3 ð33Þ The numerical values of all constants appearing in the model are given in Table 2. Finally the following product purity constraints are considered TAC ¼ ð

ð20Þ 11195

x1;D g 0:95

ð34Þ

xNT;C g 0:95

ð35Þ

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Figure 3. Optimal steady-state design of the reactive distillation column.

The model presented is a mixed integer nonlinear model of the following general form min JðxÞ ¼ c x T

x, y

s:t: hðxÞ ¼ 0

ð36Þ

Ax þ By e b x ∈ X, y ∈ f0, 1gny where J is the objective function, x is the vector of continuous variables, y is the vector of binary variables, h is the vector of equality constraints that are nonlinear in the continuous variables, A and B are constant matrices, and b is a constant vector. Observe that all binary variables appear in the linear inequality constraints only. The problem was solved using the SBB solver in GAMS, and the results are summarized in Figure 3. The TAC is $240,184/year of which $124,186/year is the steam cost. The column consists of five trays in the stripping section and seven reactive trays and five trays in the rectifying section. These results are in close agreement with the results of Kaymak and Luyben12 for the same case study which were found using exhaustive enumeration. It is important to note that the computational time for the systematic optimization applied in this work is of the order of 1 s in a 2.8 GHz, 2GB memory computer. The proposed mixed integer nonlinear formulation appears to be more efficient than previous models of ideal reactive distillation columns such as the one by Ciric and Gu.14

Figure 4. Open loop system considered (control valve shown on vapor boilup is used to denote the actual manipulator which is the steam flow rate).

3. CONTROL STRUCTURE SELECTION Following the steady-state optimization the dynamic plant operation is considered. Under dynamic conditions the most important disturbance that can seriously disrupt the reactive column operation is a potential step change in the production rate. One way to implement production rate changes is by changing the feed flow rate of one of the reactants. This disturbance has been considered by, among others, Al-Arfaj and Luyben9 and Kaymak and Luyben.10
12 They have proposed a number of control structures that are partially successful in controlling the plant. Composition measurements are expensive to install and maintain and can be particularly unreliable, and as a result, following the work by Kaymak and Luyben,9
12 they have been excluded from consideration in this work. Kaymak and Luyben have also excluded all structures that use ratio control of the two feed streams on the basis of its unreliability. In a more recent paper Kaymak and Luyben10 have increased the number of reactive trays (from 7 to 10) in an effort to improve controllability as they claim that it is the lack of enough reactive capacity in the reactive trays that is responsible for the poor dynamic performance. Furthermore, they have used ratio control of the two reactants feed streams that involves also a temperature measurement in order to calculate the set point. The problem considered here is to devise a promising control structure that is based solely on temperature measurements as controlled variables and uses all or some of the available manipulated variables. The open loop structure that is considered is the structure shown in Figure 4. Feed of reactant B is used in order to set the desired total production (total production manipulator or 11196

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TPM). Under dynamic conditions the process has 5 degrees of freedom or manipulated variables: feed flow rate of reactant A, heat input in the reboiler, bottoms product flow rate, reflux flow rate, and top product flow rate. Two of the manipulated variables are used to control the liquid levels in the column base and reflux drum. In the following it is assumed that apart from the use of feed flow rate of reactant B in order to control the total production, the bottoms flow rate is used to control the liquid level at the column base and the reflux flow rate is used to control the liquid level in the reflux drum (see Figure 4). As a result, under dynamic conditions, there are three potential manipulated variables. The same holds true under steady-state conditions. To complete the problem description, we assume that the molar flow rate of reactant B lies in the interval [30 kmol/h, 60 kmol/h]; i.e., the total production can vary by (33.33% of its nominal value. To solve the aforementioned control structure selection problem, a representative systematic methodology proposed by Kookos13 is used. The details of the methodology will not be presented here, and the reader is referenced to the original publication for further details. The method assumes that a number of scenarios are selected and each scenario corresponds to a particular realization of the disturbance(s). The aim is to select a subset of the potential controlled and manipulated variables, which when used in the control system result in near-optimal operation of the plant for all possible realizations of the disturbance(s). The potential controlled variables are the temperatures of the 17 trays (trays 2
18, see Figure 4.). When a temperature (Ti) is selected as a controlled variable, then its value is constant to the set point value, which also needs to be determined. An equal or greater number of manipulated variables needs to be selected among the potential manipulated variables so as to obtain control structures that can achieve perfect control under steady-state conditions. The mathematical formulation necessitates the introduction of the following binary variables: ψi ¼ 1; if measurement i is used as a controlled variable; 0; otherwise υj ¼ 1; if manipulated variable j is used in the control structure; 0; otherwise

ð37Þ Then, we have the following mathematical description for the controlled variables selection problem: sp

sp

Ti þ ð1
ψi ÞTL e Tis e Ti þ ð1
ψi ÞT U ;

i ∈ col ð38Þ

ψi TL e

sp Ti

e ψi T ; U

i ∈ col

ð39Þ

where Tsp i is the set point value of the temperature on tray i, TL and TU are the lower and upper bounds on tray temperatures, and Tsi is the value of the temperature on tray i at scenario s. In a similar way for the potential manipulated variables we have (the case of the vapor boilup is shown as an example and similar equations apply for the other two manipulated variables which are the top product flow rate and reactant A feed flow rate) opt opt ð40Þ V1 þ υ1 VL e V1s e V1 þ υ1 V U opt

ð1
υ1 ÞVL e V1

e ð1
υ1 ÞV U

ð41Þ

where Vs1 is the value of vapor boilup at scenario s, Vopt 1 is the constant value obtained by the vapor boilup at all scenarios when it is not used in the control structure (and as a results remains

constant). It is important to note that when the temperature on tray i is selected as a controlled variable (ψi = 1), then eqs 38 and 39 give the following conditions: sp

sp

ð42Þ

TL e Ti e T U

ð43Þ

Ti e Tis e Ti sp

Tsi

Tsp i

∑

ψi


and the temperature on tray i remains constant and i.e., = equal to the set point value at all scenarios. When the temperature on tray i is not selected as a controlled variable (ψi = 0), then U s from eq 39 we have that Tsp i = 0 and from eq 38 TL e Ti e T . In a similar way when V1 is selected as a manipulated variable (υ1 = s 1), then from eq 41 we have that Vopt 1 = 0 and from eq 40 VL e V1 U e V , while, when not selected (υ1 = 0), then from eq 41 we have U s opt that VL e Vopt 1 e V and from eq 40 V1 = V1 . The number of manipulated variables selected must always be equal to or greater than the number of controlled variables selected in order for the steady-state perfect control to be feasible: i ∈ col

∑

υj e 0

ð44Þ

j ∈ fV1 , D, FA g

The overall mathematical formulation for the selection of control structure consists of an appropriate objective function, eqs 6
15, 34, 35, 38
41, applied for all scenarios, and eq 44. This is again a mixed integer nonlinear programming problem of the special structure shown in eq 36. Because the operation of the plant at different realizations of the disturbance(s) is studied for fixed plant design, the objective function selected must be a function of the operating cost only. Vapor boilup ratio and reflux ratio are appropriate cost indices as they are directly related to utilities consumption. Reflux ratio (rr) is selected, and the average value is minimized (the assumption is that all scenarios have equal importance). The final mathematical programming problem is again solved using SBB in the GAMS15 environment. The solution time for seven periods defined by values of the reactant B feed flow rate uniformly distributed in the interval 30
60 kmol/h was again of the order of 1 s in a 2.8 GHz, 2GB memory computer. The results obtained are presented in the section that follows. It is important to note, before closing this section, that the proposed methodology takes into account the asymptotic (dynamic) behavior of the closed loop system but does not take into account transients. There are a number of important publications that discuss this issue.16
18 According to these authors the steady-state or asymptotic (dynamic) behavior that is of primary importance in taking structural decisions relative to the selection of process control structure. If the structural optimization problem has been solved and a set of promising control structures is available, then closed loop simulations can be used to rank these structures unambiguously according to prespecified performance criteria.19

4. RESULTS AND DISCUSSION The mathematical formulation for the selection of control structures for the ideal reactive distillation column presented in the previous section was applied and the first structure obtained is the one that uses all available manipulated variables, and the controlled variables are the temperatures on tray 2 (first tray on the stripping section), tray 7 (first reactive tray), and tray 18 (last tray in the rectifying section). Figures 5 and 6 show that under closed loop conditions the temperature profile as well as the 11197

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Figure 5. Temperature profile for different flow rates of reactant B.

Figure 7. Closed loop system proposed.

Table 3. Controller Parameters manipulated

Figure 6. Reactant composition profiles (mole fractions) for different flow rates of reactant B.

composition profiles remain fairly constant at steady state for the wide range of production levels considered. The only point where variations are significant is the last tray in the reactive section. The fact that the structure uses all available manipulating variables was expected as none of these variables can remain constant when the feed of reactant B changes (simple overall material balances show that all external and internal flow rates must change in order for the conversion to remain constant and satisfy purity constraints). However, the selected temperatures to control appear not to be optimal. This is based on previous classical SVD analysis by Luyben and co-workers,9
11 which has shown that it is preferable to control temperatures on trays that are located somewhere in the middle of the rectifying, reactive, and stripping sections. Despite that, it was decided to perform closed loop simulations and evaluate the dynamic performance of the structure obtained from the optimization problem. The problem that remains to be solved, in order to evaluate the structure, is that of controller design. In accordance with our initial objective to develop the simplest possible structure, it was decided to design a simple decentralized PI controller. To solve

loop

variable

1 2 3

controlled set point

proportional gain reset time

variable

(K)

(kmol/h)/K

(h)

V1

T2

431.933

20

0.1

D

T18

356.331

30

0.1

FA

T7

395.001


36

0.1

the pairing problem, a number of alternative empirical or semiempirical methods can be applied. However, it is easy to see that each controlled temperature can be paired with the manipulated variable that is actually fed to the corresponding tray on the basis of the fact that this manipulated variable has the most direct impact on the controlled temperature. On the basis of this reasoning the closed loop system shown in Figure 7 is proposed. As far as the dynamic model is concerned, this is based on the steady-state equations presented in section 2, which are easily transformed into dynamic equations (a simple model that is based on the Francis weir formulate is used to calculate the liquid outflow from each tray). The catalyst related holdup was set constant to 1 kmol. A 10 s time constant is also considered for the thermometers. Simple first-order process models are developed and initial tunings are selected followed by trial and error finetuning of the gains of the decentralized PI controllers. The final tunings are shown in Table 3. A step change in fresh feed of reactant B is considered at t = 1 h from 45.36 kmol/h down to 30 kmol/h (
33.3%) followed by a step increase to 60 kmol/h at t = 4 h (+100%) and finally a step decrease to 30 kmol/h (
50%) at t = 7 h (see Figure 9). This is clearly a sequence of aggressive changes in the total production manipulator selected to test the closed loop behavior under extreme conditions. In Figure 8 the deviation of the controlled temperatures from their set point is shown, and in Figure 9 the variation of the manipulated variables and disturbance are shown. 11198

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Figure 8. Controlled variables deviation from set point values. Figure 10. Product stream composition variation.

Figure 9. Manipulated variables and disturbance variation.

In Figure 10 the variation of the products compositions (mole fractions) are shown. The behavior of the closed loop system is more than satisfactory. The dynamics lasts for less than 1 h even for the case of 100% increase in FB. The responses are smooth and stable and without large extrusions in the errors or manipulating variables. It is important to note that the steady-state product compositions are within specifications, while the time periods for which the purity falls below specification under dynamic conditions is small and is compensated by the overpurification achieved under steady state. The performance can be improved further by using tight control, derivative action, or multivariable PI controllers (results not shown). It is important at this point that the reader compares the dynamic behavior of the proposed control structure with that of the control structures CS5, CS7, CS7-R, CS7-RR, CS8A, and CS8B proposed by Kaymak and Luyben.10
12 Structure CS5 is similar to the one proposed here but uses direct composition measurements on the first reactive tray in order to set the flow rate of reactant A, and the constant reflux ratio control is achieved by manipulating D. The only significant difference is that vapor boilup is used to control the temperature at the middle of the

Figure 11. Product stream composition variation for disturbance in the feed stream composition (5% C in A feed from 1 to 3 h, 5% D in A from 5 to 7 h).

stripping section (tray 4). This structure offers satisfactory closed loop control, but it is based on direct composition measurements and ratio control. The proposed structure offers comparable closed loop behavior despite the fact that it is based on temperature measurements only. Control structures CS7, CS7-R, and CS7-RR use vapor boilup as an indirect total production manipulator and appear to be particularly inefficient in handling production rate changes. On the basis of the inefficiency of the CS7 structures the same authors propose to increase the number of reactive trays in order to improve the dynamic performance, which was demonstrated to be true (by using 10 or 13 instead of seven reactive trays). However, the argument that it is the limited reactive capacity that causes the poor dynamic performance is questionable as the design of seven reactive trays can actually achieve 99.5% conversion which far exceeds the nominal 95% conversion. It is the opinion of the author that it is the selection of the vapor boilup as 11199

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Industrial & Engineering Chemistry Research a production rate handle that results in the observed unsatisfactory performance. The two structures CS8A and CS8B are similar to the structure proposed in this paper. These structures use, however, ratio control of the feed streams (where the ratio is set by a master loop that uses temperature measurements) as well as reflux ratio control. In CS8A reactant A sets the production level, while in CS8B reactant B sets the production level. Control structure CS8B appears to be superior to CS8A, but it is characterized by long transients and is clearly far less effective than the simpler structure proposed in this work. It is important to note that the proposed structure is also quite robust to disturbances in the composition of the reactants feed streams and was tested for mole fractions of either C or D up to 0.5. In Figure 11 a representative response of the closed loop system (product compositions) to consecutive disturbances in the feed stream compositions is shown (5% disturbance in the A feed stream composition followed by 5% disturbance in the B feed composition).

5. CONCLUSIONS The aim of this work is to apply a systematic methodology for solving the control structure selection problem of an ideal reactive distillation column and compare the results with the results presented by a number of researchers in the past decade. All previous researchers have used empirical methodologies for developing control structures for the system under investigation with little success apart from the cases in which direct composition measurements are available. In the present work no ratio or direct composition control is considered. Following the development of a superstructure-based mixed integer nonlinear model a steady-state optimization was performed and the optimal design and optimal nominal point of operation was discovered quite easily and effectively, avoiding the exhaustive enumeration used in previous works. A systematic methodology was then applied in order to develop an effective control structure that will satisfy purity constraint at steady state and minimize the economic penalty associated with product overpurification. The proposed novel control structure, apart from being superior to all previously proposed structures, achieves near-optimal operation for all disturbance realizations as the reflux ratio is implicitly minimized. These facts clearly demonstrate the superiority of the systematic methodologies over methodologies that are based on heuristics or previous knowledge.

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’ AUTHOR INFORMATION Corresponding Author

*Tel.: +30-2610-969567. Fax: +30-2610-969567. E-mail: i.kookos@ chemeng.upatras.gr.

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dx.doi.org/10.1021/ie201404j |Ind. Eng. Chem. Res. 2011, 50, 11193–11200