influences of pressure on the operation of reactive distillation

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Influences of Pressure on the Operation of Reactive DistillationColumns Involving Kinetically Controlled Exothermic ReactionsChao Wang, Liang Zhang, Kejin Huang,*, Haisheng Chen, Shaofeng Wang, Wei Liu,

and Zhigang Lei

College of Information Science and Technology, and State Key Laboratory of Chemical Resource Engineering, Beijing University ofChemical Technology, Beijing 100029, Peoples Republic of China

ABSTRACT: For the reactive distillation column involving a kinetically controlled exothermic reaction, operating pressure canpresent nonmonotonic influences on process dynamics and operation. While the enhancement of operating pressure benefitsprocess dynamics and controllability in the lower part of its feasible region (which is confined by the temperature levels of theavailable cold and hot utilities), it turns to deteriorate process dynamics and controllability in the higher part of its feasible region.Three reactive distillation systems, including an ideal reactive distillation column performing a hypothetical exothermic reaction,

A + B k

k

b

fC + D, and two real ones producing, respectively, methyl acetate from acetic acid and methanol and methyl tertiary

butyl ether (MTBE) from isobutylene and methanol, are thoroughly studied in this work, and the results confirm the existence ofthis unique phenomenon. The intricate behavior of such kind of reactive distillation columns is essentially governed by theconflicting effects of operating pressure on reaction rate and chemical equilibrium constant and determines actually a favorableregion of operating pressure for process dynamics and operation. Because the region may or may not coincide with the one interms of process synthesis and design, operating pressure can therefore serve as an important decision variable to trade-offprocess design and operation, rendering the resultant process design with balanced steady-state performance and processdynamics and controllability.

1. INTRODUCTION

Operating pressure is one of the most important decisionvariables in the synthesis and design of reactive distillation

columns because it can present a strong impact to not only thereaction operation and the separation operation involved butalso their combination (i.e., the so-called process intensifica-tion).1,2 Recently, we systematically studied the complicatedrelationship between operating pressure and steady-stateperformance in terms of a variety of reactive distillationcolumns involving either equilibrium-limited or kineticallycontrolled reactions with endothermic or exothermic effect.3

It was found that for the reactive distillation column involving akinetically controlled exothermic reaction, there existed acomplicated relationship between operating pressure andsteady-state performance. Whereas the enhancement ofoperating pressure reduced the heat duties of condenser andreboiler in the lower part of its feasible region (which is usually

confined by the temperature levels of the available cold andhot utilities), it turned to increase the heat duties of condenserand reboiler in the higher part of its feasible region. This uniquephenomenon was essentially dominated by the conflictingeffects of operating pressure on reaction rate and chemicalequilibrium constant and determined actually a favorableregion of operating pressure for process synthesis and de-sign. Because the conflicting effects of operating pressure onreaction rate and chemical equilibrium constant might alsoaffect considerably process dynamics and controllability, it istherefore necessary to ascertain the detailed mechanism forsuch kind of reactive distillation columns. The obtainedresults could not only reveal the inherent characteristics of

their complicated interplay but also offer a potential way tobalance process design and operation with operating pres-sure as an effective decision variable at the early stage of pro-cess development.

Despite that operating pressure is an important decisionvariable for the developments of reactive distillation columns,only a few studies have been conducted so far about its detailedimpacts on process dynamics and controllability. Al-Arfaj andLuyben once addressed the design and control of an olefinmetathesis reactive distillation column to be operated,respectively, at low and high operating pressures.4 Theynoticed the strong effect of operating pressure on the steady-state and dynamic performance but gave no detailedexplanations about its impacts on process dynamics andcontrollability. Kaymak addressed the design and operation ofan ideal reactive distillation column performing a hypothetical

exothermic reaction, A + B k

k

b

fC + D.5 He demonstrated the

existence of a nonmonotonic relationship between operatingpressure and steady-state performance, but did not examinefurther the effect of operating pressure on process dynamicsand controllability according to the characteristics of therelationship already obtained. Despite the fact that a certaindegree of degradation in closed-loop responses had been

Received: June 16, 2011Revised: January 10, 2012Accepted: February 8, 2012Published: February 8, 2012

Article

pubs.acs.org/IECR

2012 American Chemical Society 3692 dx.doi.org/10.1021/ie202241p | Ind. Eng. Chem. Res. 2012, 51, 36923708
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observed with the enhancement of operating pressure from8.5 to 10 bar, he did not further discuss the phenomenon andsimply drew a conclusion that operating pressure did not leadto any improvement in process dynamics and controllability.Because of the complicated interaction between the reactionoperation and the separation operation involved, the steady

states adopted to explore the impact of operating pressure onprocess dynamics and controllability must be carefully chosenfor the reactive distillation column involving a kineticallycontrolled exothermic reaction; otherwise, an incomplete andeven wrong interpretation is quite likely to be generated. This,on the other hand, reflects again the special feature of this kindof reactive distillation columns.

The current work attempts to examine the influences ofoperating pressure on the dynamics and controllability of thereactive distillation columns involving kinetically controlledexothermic reactions. Three reactive distillation systems,including an ideal reactive distillation column performing a

hypothetical exothermic reaction, A + B

k

k

b

fC + D, and two real

ones producing, respectively, methyl acetate from acetic acidand methanol and methyl tertiary butyl ether (MTBE) fromisobutylene and methanol, are selected as representativeexamples for this kind of reactive distillation columns andthoroughly studied in this work. The unique influences ofoperating pressure on process dynamics and controllability aredemonstrated, and their implications on process design andoperation are then highlighted. Some important conclusions aresummarized in the last section of this Article.

2. INFLUENCES OF OPERATING PRESSURE ON THEDYNAMICS AND OPERATION OF REACTIVEDISTILLATION COLUMNS INVOLVING KINETICALLY

CONTROLLED EXOTHERMIC REACTIONSFor any kind of reactive distillation columns involving eitherequilibrium-limited or kinetically controlled reactions, operat-ing pressure is one of the most important design variables thatcan affect simultaneously the reaction operation and theseparation operation involved. Regarding its influences on theseparation operation involved, they lie primarily on the relative

volatilities between the reacting components, affecting, ingeneral, the number of stages in the rectifying and/or strippingsections during process synthesis and design. Therefore, the

variation of operating pressure usually cannot affect signifi-cantly the dynamics and controllability of the separationoperation involved. However, the situation is quite different

with regard to its influences on the reaction operation involved.They lie mainly on the reaction rate and/or the constant ofchemical equilibrium, affecting eventually the reaction con-

version during process synthesis and design. Therefore, thevariation of operating pressure can affect significantly thedynamics and controllability of the reaction operation involved.The reaction operation can pose further a strong impact to thecombination between the reaction operation and the separationoperation involved, which determines eventually the dynamicsand controllability of the resultant reactive distillation columns.

Figure 1 shows a typical relationship between operatingpressure and steady-state performance for a reactive distillationcolumn involving a kinetically controlled exothermic reaction.It is generated based on our previous work.3 In the left side of

the dashed line (which corresponds to the minimum of the heatduty of condenser/reboiler), whereas the enhancement ofoperating pressure increases the reaction rate, it also decreasesthe constant of chemical equilibrium, both of which are caused

by the resultant temperature elevation. Because the favorableeffect of the former outweighs the unfavorable effect of thelatter, the net result is the improvement of the steady-state per-formance. In the right side of the dotted line, because thefavorable effect of the former is not as competitive as theunfavorable effect of the latter, the net result is the degradationof the steady-state performance. The nonmonotonic relation-ship between operating pressure and steady-state performanceis further affected by the interaction between the reaction

Figure 1. Optimal points for steady-state performance and process dynamics and controllability versus operating pressure for the reactive distillationcolumns involving kinetically controlled exothermic reactions.

Figure 2. An ideal reactive distillation column performing a

hypothetical exothermic reaction: A + B k

k

b

fC + D (HR < 0)

(example I).

Industrial & Engineering Chemistry Research Article

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operation and the separation operation involved and helps todetermine a favorable region of operating pressure (i.e., thegray region in this case) in terms of process synthesis anddesign for the given reactive distillation column.

Because the combination between the reaction operation and

the separation operation involved can give a strong impact toprocess dynamics and controllability, it is quite likely thatoperating pressure may also present a nonmonotonic influenceon process dynamics and controllability. More specifically,

whereas the enhancement of operating pressure gives a favorableinfluence to process dynamics and controllability in the lowerpart of its feasible region, the reverse is true in the higher part ofits feasible region. Confirming this unique behavior can certainlyfacilitate the simultaneous consideration of process design andoperation with operating pressure as an effective decision variableat the early stage of process development.

In what follows, three reactive distillation systems, includingan ideal reactive distillation column performing a hypothetical

exothermic reaction, A + B k

k

b

fC + D, and two real ones

producing, respectively, methyl acetate from acetic acid andmethanol and MTBE from isobutylene and methanol, areemployed as representative examples to explore the influencesof operating pressure on the dynamics and controllability of thereactive distillation columns involving kinetically controlledexothermic reactions.

3. EXAMPLE I: AN IDEAL REACTIVE DISTILLATIONCOLUMN PERFORMING A HYPOTHETICAL

EXOTHERMIC REACTION A + B k

k

b

fC + D

3.1. Process Description. The hypothetical ideal reactivedistillation column was originally proposed by Luyben and hisco-workers and has been studied intensively by a lot ofresearchers so far.611 As shown in Figure 2, a typical processdesign is chosen here, accommodating a three sectionalstructure, 7/6/7, that is, a rectifying section, a stripping section,and a reactive section between. It is equipped with a totalcondenser at the top and a partial reboiler at the bottom. Twopure reactant feeds, FA and FB, are fed onto the bottom and the

Table 1. Physicochemical Properties and OperatingConditions of Example I

parameter value

number of stages rectifying section 7

reactive section 6

stripping section 7

stage holdup (kmol) 1

activation energy (kJ kmol1) forward 125 520backward 167 360

specific reaction rate at 366 K(kmol s1 kmol1)

forward 0.008

backward 0.004

feed flow rate of reactant A (kmol s1) 0.0126

feed flow rate of reactant B (kmol s1) 0.0126

feed location of reactant A 14

feed location of reactant B 9

thermal condition of FA 1.0

thermal condition of FB 1.0

relative volatility A:B:C:D 4:2:8:1

heat of reaction (kJ kmol1) 41 840

latent heat of vaporization (kJ kmol1) 29 053.7

overhead product composition (C, mol %) 95

bottom product composition (D, mol %) 95vapor pressure constants A(Avp/Bvp) 12.3463/3862

B(Avp/Bvp) 11.6531/3862

C(Avp/Bvp) 13.0394/3862

D(Avp/Bvp) 10.96/3862

Figure 3. Effect of operating pressure on the steady-state performanceof example I.

Table 2. Controller Parameters for Example I

operating pressure

(bar) control loop KC TI (min)6 top composition 3.25 103 84

bottom composition 5.34 103 60

reactant stoichiometry 7.25 103

6.7 top composition 6.25 103 44



9 top composition 6.65 103 44







bottom composition 1.27 10

2 38reactant stoichiometry 8.46 103

Figure 4. A decentralized control scheme for example I.




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top of reactive section, respectively, and the specification of the

top and bottom products is set to be 95 mol %.

The hypothetical reversible reaction occurring on reactive

stages is

+ + = HA B C D 41 840 kJ kmol

k

k

R1

b

f

(1)

where the symbol k

k

b

findicates that the reaction is a kinetically

controlled one.

The volatilities are such that the products C and D are the

lightest and heaviest, respectively, in the reaction system. The

net reaction rate for component i on stage j in the reactive

section is given by

= v H k x x k x x( )i j i j j j j j j j, f, A, B, b, C, D, (2)

Figure 5. Regulatory responses of example I for a 5% step change in the feed flow rate of reactant B, respectively (6/6.7/9 bar): (a) composition ofcomponent C in the top product, (b) reflux flow rate, (c) composition of component A on stage 14, (d) feed flow rate of reactant A, (e) compositionof component D in the bottom product, and (f) bottom vapor flow rate. Gray curves, negative responses; black curves, positive responses.

Table 3. IAE of Example I

scenario

operatingpressure

(bar) distillatebottom

withdrawal

Iincrease by 5% in the feedflow rate of reactant B

6 7.132 102 2.901 102

6.7 1.554 102 1.497 102

9 9.945 103 3.721 103

decrease by 5% in the feedflow rate of reactant B

6 7.277 102 2.955 102

6.7 1.456 102 1.378 102

9 1.012 102 3.692 103

increase by 50% in the feedflow rate of reactant B

9 1.411 101 3.578 102

12 1.579 101 3.722 102

15 2.133 101 4.494 102

decrease by 15% in the feedflow rate of reactant B

9 2.958 102 1.056 102

12 2.974 102 1.207 102

15 3.273 102 1.225 102




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where kf, j and kb, j are the forward and backward specific

reaction rates and are given by

= k k( ) ej

E R Tf, f 366

( / )(1 / 1/366)jf(3)

= k k( ) ej

E R Tb, b 366

( / )(1/ 1 /366)jb(4)

Here, the liquid holdup Hj is an important design parameterthat can reflect the amount of catalyst installed on a reactivestage. A large value represents an operating condition that alarge amount of catalyst has been installed on a reactive stage,and vice versa.

Figure 6. Regulatory responses of example I for a +50% and 15% step change in the feed flow rate of reactant B, respectively (9/12/15 bar): (a)composition of component C in the top product, (b) reflux flow rate, (c) composition of component A on stage 14, (d) feed flow rate of reactant A,(e) composition of component D in the bottom product, and (f) bottom vapor flow rate. Gray curves, negative responses; black curves, positiveresponses.

Figure 7. A reactive distillation column producing methyl acetate fromacetate acid and methanol (example II).




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Ideal vapor and liquid phase behavior is assumed for thereaction system, and the vaporliquid equilibrium relationshipcan be expressed as

= + + +P x P x P x P x Pj j j j jA, As

B, Bs

C, Cs

D, Ds

(5)

=y x P P/i j i j i j j, , ,s

(6)

The vapor saturation pressure is calculated as

= P A B Tln /i j i i j,s vp, vp, (7)

Equimolar overflow is assumed, so the liquid and vapor flowrates are constant in the nonreactive rectifying and strippingsections. Because the reaction is exothermic, HR < 0, the

vapor and liquid flow rates change from stage to stage in thereactive section because the thermal heat of reaction vaporizes acertain amount of liquid on each reactive stage.

= +V V r H H /j j j1 ,C R V (8)

= + L L r H H/j j j1 ,C R V (9)

The physicochemical properties and nominal steady-stateoperating conditions for the hypothetical ideal reactivedistillation column are summarized in Table 1, and otherrelevant information can be found in the correspondingreferences. Because constant relative volatilities are assumed

between the reacting components, the variations of operatingpressure will not affect the separation operation involved in thiscase. The steady-state and dynamic behaviors of the hypo-thetical ideal reactive distillation column can be well predicted

with the first-principle models shown in Appendixes A and B,respectively.

3.2. Effect of Operating Pressure on Steady-StatePerformance. In Figure 3, the relationship between operating

pressure and the steady-state performance of the hypotheticalideal reactive distillation column is illustrated. It is noted that ittakes a shape somewhat similar to the one shown in Figure 1.The heat duties of condenser and reboiler reach simultaneouslytheir minimum values at the operating pressure of 9 bar. Awayfrom this value, they turn to increase monotonically in bothdirections, portraying a nonmonotonic relationship betweenoperating pressure and steady-state performance.

3.3. Effect of Operating Pressure on ProcessDynamics and Controllability. In the following subsections,the closed-loop control of the hypothetical ideal reactivedistillation column is used to ascertain the effect of operatingpressure on process dynamics and controllability. A decentralized

Table 4. Physicochemical Properties and OperatingConditions of Example II

parameter value


reactive section 25

stripping section 9

liquid holdup (m3) 0.0379584

activation energy (kJ kmol1) forward 49 190backward 69 230

pre-exponential factor(kmol s1 kgcat

1)forward 29 610

backward 1 348 000

feed flow rate of reactant HAc (kmol h1) 50

feed flow rate of reactant MeOH (kmol h1) 50

feed location of reactant HAc 4

feed location of reactant MeOH 28

thermal condition of FHAc 1.0

thermal condition of FMeOH 1.0

heat of reaction (kJ kmol1, 100 kPa, 330 K) 33 566.80

latent heat of H2O (kJ kmol1, 100 kPa, 330 K) 42 657.62

overhead product composition (MeAc, mol %) 95

bottom product composition (H2O, mol %) 95

Figure 8. Effect of operating pressure on the steady-state performanceof example II.

Figure 9. A decentralized control scheme for example II.

Table 5. Controller Parameters for Example II

operating pressure (atm) control loop KC TI (min)

0.9 top composition 1.95 83.45

bottom composition 1.49 78.32reactant stoichiometry 0.92


bottom composition 1.68 64.45

reactant stoichiometry 1.05











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control structure shown in Figure 4 is adopted here, where thepurities of the top and the bottom products are measured andcontrolled. Operating pressure is regulated with the heatremoval of condenser and is assumed to be perfectly controlled.In the top product, the composition of component C is

controlled by manipulating the reflux flow rate. In the bottomproduct, the composition of component D is controlled bymanipulating the heat duty of reboiler. The concentration ofreactant A on the lower feed stage (i.e., stage 14) is measuredand controlled by manipulating the feed flow rate of reactant A(i.e., the so-called reactant stoichiometry control loop). Theemployment of direct composition control here avoids thedrawbacks of the feedforward ratio control scheme, forexample, the severe noises in flow rate measurements andfailures to deal with feed composition changes. The feed flowrate of reactant B is the production rate handle and is flowcontrolled. The levels of the reflux-drum and bottom reboilerare controlled by the distillate and the bottom product flow

rates, respectively. The dynamics of concentration measure-ments is assumed to be two first-order lags of 30 s in series.

While proportional-only (P) controllers are used for all levelcontrol loops, proportional plus integral (PI) controllers areadopted for the top and the bottom composition control loops.

For the reactant stoichiometry control loop, a P controller(instead of a PI one) is chosen, and this cannot affect the stableoperation of the hypothetical ideal reactive distillationcolumn.7,10 The gains of the level controllers are assigned to

be 2. All composition controllers are tuned according to theTyreusLuyben rule (i.e., KC = KCU/3 and TI = 2PCU), andslight adjustments are still necessary to get satisfactoryresponses.12 The obtained results are listed in Table 2.

With reference to the optimum performance of thehypothetical ideal reactive distillation column shown in Figure 3,it is reasonable to divide the feasible region of operating pressureinto two parts, the left (below 9 bar) and right (above 9 bar)subregions. To assess the influence of operating pressure on

Figure 10. Regulatory responses of example II for a 5% step change in the feed flow rate of acetate acid, respectively (0.9/1.5/2.1 atm):(a) composition of methyl acetate in the top product, (b) reflux flow rate, (c) composition of methanol on stage 28, (d) feed flow rate of methanol,(e) composition of water in the bottom product, and (f) heat duty of reboiler. Gray curves, negative responses; black curves, positive responses.




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process dynamics and controllability, we choose, respectively,two values of operating pressure in these two subregions. Whileoperating pressures of 6 and 6.7 bar are selected in the leftsubregion, they are chosen to be 12 and 15 bar in the rightsubregion. Together with the boundary point of 9 bar, these

values form two groups of operating pressures: (i) 6, 6.7, and9 bar, and (ii) 9, 12, and 15 bar, based on which the closed-loopresponses of the hypothetical ideal reactive distillation columnsare to be compared.

3.3.1. Comparison of Process Dynamics and Controlla-bility between Operating Pressures, 6, 6.7, and 9 bar. InFigure 5, the regulatory responses of the hypothetical idealreactive distillation column operated under different operatingpressures (i.e., 6, 6.7, and 9 bar, respectively) are depicted whenthe feed flow rate of reactant B has been upset by 5%,

respectively. In case the operating pressure is fixed at 6 bar, theprocess exhibits rather sluggish responses and needs a long timeto reach the new steady states. In particular, initial oscillationsare found in the top and bottom product control loops. Withthe increase of operating pressure to 6.7 or 9 bar, the settlingtimes are sharply reduced irrespective of the negative orpositive changes in the feed flow rate of reactant B. As for thecomparison of dynamic performance between those operatedunder 6.7 and 9 bar, the former appears inferior to the latter in

both deviations of product qualities and settling times. Integralabsolute error (IAE) is calculated and tabulated in Table 3. It isnoted that a good accordance has been achieved with the aboveobservations.

The closed-loop simulation results demonstrate that the

enhancement of operating pressure benefits process dynamicsand controllability in the operation range below 9 bar.Moreover, the hypothetical ideal reactive distillation columnoperated under 9 bar can maintain its higher thermodynamicefficiency than those operated under 6 and 6.7 bar at the newlyreached steady states.

3.3.2. Comparison of Process Dynamics and Controll-ability between Operating Pressures, 9, 12, and 15 bar. Theregulatory responses of the hypothetical ideal reactive distillationcolumn operated under different operating pressure (i.e., 9, 12, and15 bar, respectively) are presented in Figure 6 when the feed flowrate of reactant B has been upset by +50% and 15%, respectively.

With the increase of operating pressure, the process tends to display

increasingly great maximum deviations in the top and bottomproduct qualities regardless of the negative or positive changes in thefeed flow rate of reactant B. The IAE for these circumstances is alsocalculated and listed in Table 3, confirming again the degradation ofprocess dynamics and controllability incurred by the increase ofoperating pressure from 9 bar.

The closed-loop simulation results show that the enhance-ment of operating pressure deteriorates process dynamics andcontrollability in the operation range above 9 bar. Moreover, atthe newly reached steady states, the process operated at 9 barmaintains its higher thermodynamic efficiency than thoseoperated at 12 and 15 bar.

4. EXAMPLE II: A REACTIVE DISTILLATION COLUMNPRODUCING METHYL ACETATE FROM ACETICACID AND METHANOL

4.1. Process Description. A methyl acetate reactivedistillation column by Tang et al. is adopted here.13 The processincorporates a three sectional structure, 2/25/9, with thespecification on the top and bottom products as 95 mol %,respectively (cf., Figure 7). The relevant physicochemical

properties and nominal steady-state operating conditions aresummarized in Table 4.The esterification reaction occurring on the reactive stages is

+

+

=

H

methanol(MeOH) acetic acid(HAc)

methyl acetate(MeAc) water(H O)

33 566.80 kJ kmol

k

k

2 R,330

1b

f

(10)

where the symbol k

k

b


controlled one.The net reaction rate is expressed by the pseudohomoge-

neous model with components represented in terms of activityand catalyst weight-based kinetics.

= m k a a k a a( )MeAc cat f HAc MeOH b MeAc H O2 (11)

= k (2.961 10 )e RTf4 49 190/( )

(12)

= k (1.348 10 )e RTb

6 69 230/( )(13)

The steady-state and dynamic simulation of the methylacetate reactive distillation column is performed using thecommercial software ASPEN PLUS and ASPEN DYNAMICS,respectively, and the UNIQUAC model is used to calculate theactivity coefficients accounting for the nonideal vaporliquid

equilibrium. The dimerization of acetic acid is described by theHaydenOConell second virial coefficient model, and the

ASPEN PLUS built-in association parameters are used tocompute its fugacity coefficients.

4.2. Effect of Operating Pressure on Steady-StatePerformance. The effect of operating pressure on the steady-state performance of the methyl acetate reactive distillationcolumn is illustrated in Figure 8 (it is slightly different from theone reported in ref 3 because of the minor adjustments ofsteady-state operating condition necessary for dynamicsimulation with ASPEN DYNAMICS). It is readily noted thatit again takes a shape somewhat similar to the one shown inFigure 1. The heat duties of condenser and reboiler first

Table 6. IAE of Example II

scenario

operatingpressure

(atm) distill atebottom

withdrawal

increase by 5% in the feedflow rate of acetate acid

0.9 9.875 103 1.643 102

1.5 6.314 103 1.186 102

2.1 7.250 103 1.273 102

2.4 9.958

10

3

1.614

10

2

2.6 4.890 103 1.131 102

decrease by 5% in the feedflow rate of acetate acid

0.9 9.932 103 1.539 102

1.5 5.214 103 9.394 103

2.1 3.979 103 1.052 102

2.4 4.807 103 1.039 102

2.6 3.752 103 6.062 103

increase by 0.01 in the topand bottom productpurities simultaneously

2.1 1.755 102 1.838 102

2.4 4.459 102 4.672 102

2.6 7.757 102 8.324 102

decrease by 0.01 in the topand bottom productpurities simultaneously

2.1 1.369 102 1.378 102

2.4 1.997 102 1.889 102

2.6 4.103 102 4.060 102




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decrease with the enhancement of operating pressure and reachtheir minima when the operating pressure is 2.2 and 2.1 atm,respectively. Beyond the optimum points, the heat duties ofcondenser and reboiler turn to increase, delineating anonmonotonic relationship between operating pressure and

steady-state performance.4.3. Effect of Operating Pressure on ProcessDynamics and Controllability. As shown in Figure 9, adecentralized control system is adopted here. Operatingpressure is again assumed to be perfectly controlled with theheat removal of condenser. The levels of the reflux-drum andreboiler are controlled, respectively, with the distillate and the

bottom product flow rates, and two P controllers with gains of2 are adopted. In the top product, the composition of methylacetate is controlled by manipulating the reflux flow rate, and aPI controller is employed. In the bottom product, thecomposition of water is controlled by manipulating the heatduty of reboiler, and a PI controller is employed. The

concentration of methanol on the lower feed stage (i.e., stage 28)is measured and controlled by manipulating the feed flow rate ofmethanol, aiming to keep the stoichiometric balance betweenthe reactants acetic acid and methanol, and a P controller isused. The feed flow rate of acetic acid is the production rate

handle and is flow controlled. The dynamics of concentrationmeasurements is assumed to be a pure time-delay of 3 min. Allcomposition controllers are first tuned in terms of the relay-feedback tests and ZieglerNichols rule and then detunedsomehow to take account of the interaction between differentcontrol loops. The obtained controller parameters are listed inTable 5.

Analogous to the case of the hypothetical ideal reactivedistillation column outlined in the preceding section, we dividethe feasible region of operating pressure into two parts basedon the optimum design of the methyl acetate reactivedistillation column, the left (below 2.1 atm) and right (above2.1 atm) subregions. In each of the subregions, three operating

Figure 11. Regulatory responses of example II for a 5% step change in the feed flow rate of acetate acid, respectively (2.1/2.4/2.6 atm):(a) composition of methyl acetate in the top product, (b) reflux flow rate, (c) composition of methanol on stage 28, (d) feed flow rate of methanol,(e) composition of water in the bottom product, (f) heat duty of reboiler. Gray curves, negative responses; black curves, positive responses.




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pressures are selected, and they are 0.9, 1.5, and 2.1 atm for theleft one and 2.1, 2.4, and 2.6 atm for the right one. In terms ofthese operating pressures, the regulatory responses of themethyl acetate reactive distillation column are to be comparedin the following subsections.

4.3.1. Comparison of Process Dynamics and Controlla-bility between Operating Pressures, 0.9, 1.5, and 2.1 atm.Figure 10 displays the regulatory responses of the methylacetate reactive distillation column operated under differentoperating pressures (i.e., 0.9, 1.5, and 2.1 atm, respectively)

when the feed flow rate of acetic acid has been upset by5% atthe instant of 0.5 h, respectively. As can readily be seen, theprocess displays increasingly small maximum deviations in thetop and bottom product qualities with the enhancement ofoperating pressure, no matter whether negative or positivechanges have been encountered in the feed flow rate of acetic acid.

Figure 12. Servo responses of example II for a 0.01 step change in the top and bottom product purities, respectively (2.1/2.4/2.6 atm):(a) composition of methyl acetate in the top product, (b) reflux flow rate, (c) composition of methanol on stage 28, (d) feed flow rate ofmethanol, (e) composition of water in the bottom product, and (f) heat duty of reboiler. Gray curves, negative responses; black curves, positiveresponses.

Figure 13. A reactive distillation column producing MTBE fromisobutylene and methanol (example III).




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In the aspect of settling time, however, the opposite trendsare found when the operating pressure increases from 1.5 to2.1 atm. With the simultaneous consideration of these two per-formance indexes, it is reasonable to conclude that the methylacetate reactive distillation column operated under 1.5 atm should

be superior to those operated under 0.9 and 2.1 atm. The IAEis calculated and listed in Table 6, which justifies the aboveinterpretations.

The closed-loop simulation results obtained indicate a

nonmonotonic effect of operating pressure on the dynamicsand controllability of the methyl acetate reactive distillationcolumn in the left subregion.

4.3.2. Comparison of Process Dynamics and Controlla-bility between Operating Pressures, 2.1, 2.4, and 2.6 atm.The regulatory responses of the methyl acetate reactive distillationcolumn operated under different operating pressures (i.e., 2.1, 2.4,and 2.6 atm, respectively) are showed in Figure 11 when the feedflow rate of acetic acid has been upset by5% at the instant of0.5 h, respectively. Although the process displaysincreasingly smallmaximum deviations in the top and bottom product qualities withthe enhancement of operating pressure, the settling times areextended considerably. In case the operating pressure has beenfixed at 2.6 atm, the methyl acetate reactive distillation column

exhibits initial inverse responses in the top product quality inaddition to the strong oscillations in the bottom product quality,

both phenomena signifying sharply deteriorated process dynamicsand controllability in this situation. In Table 6, the IAE is alsocalculated and listed for the three circumstances. Despite the factthat the value of the IAE at the operating pressure of 2.6 atmappears to be the smallest one for both positive and negativeresponses, the operating condition should evidently be consideredas the worst one in terms of process dynamics and controllability.Through the comparison of the other two circumstances, theyshow a good agreement with the observations made above; theenhancement of operating pressure deteriorates process dynamicsand controllability in the operation range above 2.1 atm.

Table 7. Physicochemical Properties and OperatingConditions of Example III

parameter value


reactive section 8

stripping section 5

stage holdup (kg) 1800

heat of reaction (kJ kmol1, 1100 kPa, 298 K) 37 700latent heat of MTBE (kJ kmol1, 1100 kPa, 298 K) 29 829

mole fraction of C4 IC4 0.357

NC4 0.643

feed flow rate of reactant MeOH (mol s1) 198

feed flow rate of reactant C4 (mol s1) 547

feed location of reactant MeOH 4

feed location of reactant C4 11

thermal condition of FC4 1.0

thermal condition of FMeOH 0.0

overhead product composition (NBUT, mol %) 84

bottom product composition (MTBE, mol %) 90

Figure 14. Effect of operating pressure on the steady-stateperformance of example III.

Figure 15. A decentralized control scheme for example III.

Table 8. Controller Parameters for Example III

operating pressure (atm) control loop KC TI (min)






reactant stoichiometry 0.527.1 top composition 9.32 22.00









Table 9. IAE of Example III

scenariooperatingpressure

(atm) distillatebottom

withdrawal

increase by 5% in the feedflow rate of methanol

6.5 2.770 102 3.061 102

6.8 3.344 103 7.802 103

7.1 1.839 103 5.889 103

8.5 5.645 104 1.508 103

10.1 5.673 104 6.764 103

decrease by 5% in the feedflow rate of methanol

6.5 1.099 102 9.627 103

6.8 1.922 103 3.869 103

7.1 1.201 103 2.957 103

8.5 5.340 104 1.562 103

10.1 2.992 103 2.284 102




12/17

To gain further insights into the intricate relationship betweenoperating pressure and process dynamics and controllability, wealso examined the servo responses of the methyl acetate reactivedistillation columns operated under different operating pressures(i.e., 2.1, 2.4, and 2.6 atm, respectively). In Figure 12, the obtainedresults are illustrated when the process has been upset by a 0.01step change in the set-points of the top and bottom products at

the instant of 0.5 h, respectively. With the enhancement ofoperating pressure from 2.1 to 2.4 atm, the settling time is greatlyincreased, implying a considerable reduction of tracking capability.

When operating pressure is further increased to 2.6 atm, thesettling time is further magnified in case that the set-points of thetop and bottom products are reduced simultaneously to 0.94. Incase that the set-points of the top and bottom products areincreased simultaneously to 0.96, the process cannot move to thedesired steady state, leaving a divergent oscillation in the bottomproduct. The IAE is also calculated and tabulated in Table 6,displaying a good agreement with the above observations.

These closed-loop simulation results evidence definitely thatthe enhancement of operating pressure deteriorates the

dynamics and controllability of the methyl acetate reactivedistillation column in the right subregion.

5. EXAMPLE III: A REACTIVE DISTILLATION COLUMNPRODUCING MTBE FROM ISOBUTYLENE ANDMETHANOL

5.1. Process Description. MTBE is produced by areversible exothermic reaction of methanol and isobutylene inthe presence of either a heterogeneous catalyst, for example, astrong acidic ion-exchange resin, or a homogeneous catalyst, forexample, sulfuric acid. In this work, the reaction catalyzed bysulfuric acid is to be investigated. Instead of being assumed to

be an equilibrium-limited reaction as shown in our previouswork,3,14 it is now taken as a kinetically controlled one. Thereactive distillation column shown in Figure 13 is employedhere. A C4 vapor at 350 K (qC4 = 0), comprised of 36 mol %isobutylene and 64 mol % inert n-butene (NBUT), is employedto react with a pure methanol liquid at 320 K (qMEOH = 1). Theprocess possesses a three sectional configuration: 2/8/5, that is,a rectifying section, a stripping section, and a reactive section

between, in addition to a total condenser at the top and apartial reboiler at the bottom. The top and bottom products arespecified to be 84 and 90 mol % for NBUT and MTBE,respectively. Other relevant physicochemical properties andnominal steady-state operating conditions are summarized inTable 7.

The reaction occurring on the reactive stages is

+

=

H

methanol(MeOH) isobutylene(IBUT)

methyl tertiary butyl ether(MTBE)

37 700 kJ kmol

k

k

R,298

1b

f

(14)

where the symbol k

k

b


controlled one.

The kinetic model proposed by Rehfinger and Hoffmann isused in this study.15

=

m q k

a

a

a

K aMTBE cat acid f

IB

MeOH

MTBE

eq2

MeOH (15)

=

k (3.67 10 )e

T

f

12 11 110 /

(16)

=K 284ef Teq( )

(17)

= + +

+ +

+

f T A

T TA

T

TA T T

A T T A T T

A T T

( )1 1

ln ( )

( ) ( )

( )

10

20

3 0

42

02

53

03

64

04

(18)

where T0 = 298.15 K, A1 = 1.49277 103 K, A2 = 77.4002,

A3 = 0.507563 K1, A4 = 9.12739 10

4 K2, A5 = 1.10649 106 K3, and A6 = 6.27996 10

10 K4.

The steady-state and dynamic simulation of the MTBEreactive distillation column is carried out using the commercialsoftware ASPEN PLUS and ASPEN DYNAMICS, respectively,and the liquid phase activities are calculated using theUNIQUAC model with the binary interaction parametersreported by Rehfinger and Hoffmann.15

5.2. Effect of Operating Pressure on Steady-StatePerformance. Figure 14 displays the effect of operatingpressure on the steady-state performance of the MTBE reactivedistillation column. It is noted that it again takes a shapesomewhat similar to the one shown in Figure 1. The increase ofoperating pressure leads to reductions in the heat duties ofcondenser and reboiler, and the minima locate at the operatingpressure of 7.2 and 7.1 atm, respectively. Beyond the optimumpoints, the heat duties of condenser and reboiler begin toincrease with the enhancement of operating pressure, therebydepicting a nonmonotonic relationship between operatingpressure and steady-state performance.

5.3. Effect of Operating Pressure on ProcessDynamics and Controllability. The decentralized controlscheme for the MTBE reactive distillation column is sketchedin Figure 15. Operating pressure is again assumed to beperfectly controlled with the heat removal of condenser. Thelevels of condenser and reboiler are controlled, respectively,

with the distillate and the bottom product flow rates, and two Pcontrollers with gains of 2 are adopted here. The topcomposition of NBUT is controlled with the reflux flow rate,and a PI controller is employed. The bottom composition ofMTBE is controlled with the heat duty of reboiler, and a PI

controller is employed. The isobutylene composition on thelower feed stage (i.e., stage 11) is controlled with the feed flowrate of C4, serving to keep the stoichiometric balance betweenthe reactants, and a P controller is used. The feed flow rate ofmethanol is the production rate handle and is flow controlled.The dynamics of concentration measurements is assumed to bea pure time-delay of 3 min. The composition controllers aretuned sequentially in terms of relay-feedback tests and ZieglerNichols tuning method.16 Table 8 lists the obtained controllerparameters for the MTBE reactive distillation column operatedunder different operating pressures.

In light of the optimum operating pressure of the MTBEreactive distillation column shown in Figure 14, the feasible




13/17

region of operating pressure is divided into two parts, the left(below 7.1 atm) and right (above 7.1 atm) subregions. In eachof the subregions, three operating pressures are selected, andthey are 6.5, 6.8, and 7.1 atm for the left one and 7.1, 8.5, and10.1 atm for the right one. In terms of these operating

pressures, the regulatory responses of the MTBE reactivedistillation column are to be examined in the followingsubsections.

5.3.1. Comparison of Process Dynamics and Controlla-bility between Operating Pressures, 6.5, 6.8, and 7.1 atm. InFigure 16, the regulatory responses of the MTBE reactivedistillation column operated under different operating pressures(i.e., 6.5, 6.8, and 7.1 atm, respectively) are shown, when thefeed flow rate of methanol has been upset by5% at the instantof 0.5 h, respectively. With the enhancement of operatingpressure, the process exhibits increasingly improved dynamicresponses in the top and bottom product qualities no matter

whether negative or positive changes have been encountered in

the feed flow rate of methanol. This can easily be identifiedthrough the comparison of maximum deviations and settlingtimes. In particular, the MTBE reactive distillation column,

when operated at 6.5 atm, fails to overcome the positivedisturbance and exhibits divergent oscillations in the top and

bottom product qualities. After operating pressure has beenraised to 6.8 or 7.1 atm, the oscillations completely disappeared.The IAE is calculated and listed in Table 9, which justifies theabove observations.

The closed-loop simulation results evidence that theenhancement of operating pressure benefits the dynamics andcontrollability of the MTBE reactive distillation column in theleft subregion.

5.3.2. Comparison of Process Dynamics and Controlla-bility between Operating Pressures, 7.1, 8.5, and 10.1 atm.Figure 17 details the regulatory responses of the MTBE reactivedistillation column operated under different operating pressures(i.e., 7.1, 8.5, and 10.1 atm, respectively) when the feed flow

Figure 16. Regulatory responses of example III for a 5% step change in the feed flow rate of methanol, respectively (6.5/6.8/7.1 atm): (a)composition of NBUT in the top product, (b) reflux flow rate, (c) composition of isobutylene on stage 11, (d) feed flow rate of C4, (e) compositionof MTBE in the bottom product, and (f) heat duty of reboiler. Gray curves, negative responses; black curves, positive responses.




14/17

rate of methanol has been upset by5% at the instant of 0.5 h,respectively. Of the dynamic responses at the three operatingpressures, one can easily conclude that the one at 8.5 atmshould be the best in light of process dynamics and

controllability. In case the operating pressure has been fixedat 10.1 atm, the process exhibits inverse responses in the topand bottom product qualities, and it even fails to overcome thenegative disturbance, implying sharply deteriorated processdynamics and controllability under this operating condition. InTable 9, the IAE is also calculated and listed for thesecircumstances, which shows a good accordance with the aboveobservations.

The closed-loop simulation results obtained indicate anonmonotonic effect of operating pressure on the dynamicsand controllability of the MTBE reactive distillation column inthe right subregion.

6. DISCUSSION

In terms of the hypothetical ideal reactive distillation column,the methyl acetate reactive distillation column, and the MTBEreactive distillation columns studied, we have demonstrated the

existence of a nonmonotonic relationship between operatingpressure and process dynamics and controllability. Morespecifically, whereas the enhancement of operating pressure isfavorable to process dynamics and controllability in the lowerpart of its feasible region, it is likely to present an adverse effectin the higher part of its feasible region. Although it is extremelydifficult to prove this intricate phenomenon on a puretheoretical basis, the findings of the current work areconsidered to be of general significance, revealing a relevantand yet uncommon feature of the reactive distillation columnsinvolving kinetically controlled exothermic reactions. Becausefor any chemical processes unique process dynamics andcontrollability are generally caused by the interactions between

Figure 17. Regulatory responses of example III for a 5% step change in the feed flow rate of methanol, respectively (7.1/8.5/10.1 atm): (a)composition of NBUT in the top product, (b) reflux flow rate, (c) composition of isobutylene on stage 11, (d) feed flow rate of C4, (e) compositionof MTBE in the bottom product, and (f) heat duty of reboiler. Gray curves, negative responses; black curves, positive responses.




15/17

conflicting or competing factors, it seems reasonable to thinkthat the conflicting effects of operating pressure on reactionrate and chemical equilibrium constant are responsible for such

a complicated phenomenon.It is worth mentioning here the minor difference in therelationship between operating pressure and process dynamicsand controllability for the three reactive distillation columnsstudied. For the hypothetical ideal reactive distillation column,the relationship between operating pressure and processdynamics and controllability shares a strict correspondence

with the one between operating pressure and steady-stateperformance. For instance, at the operating pressure of 9 bar,the hypothetical ideal reactive distillation column exhibits theoptimum steady-state performance with the most favorableprocess dynamics and controllability. For the other two reactivedistillation columns, the situations are, however, different. Inthe case of the methyl acetate reactive distillation column, while

the optimal steady-state performance is gained at the operatingpressure of 2.1 atm, the most favorable process dynamics andcontrollability appears actually at an operating pressure lessthan it (i.e., around 1.5 atm). In the case of the MTBE reactivedistillation column, while the optimal steady-state performanceis gained at the operating pressure of 7.1 atm, the mostfavorable process dynamics and controllability appears actuallyat an operating pressure greater than it (i.e., around 8.5 atm).The deviations in operating pressure are anticipated to have

been caused by the high degree of nonlinearities in the vaporliquid equilibrium and enthalpycomposition relationships ofthe reacting mixture separated. It should, however, be borne inmind that the high degree of nonlinearities affects, on the first

hand, the combination between the reaction operation and theseparation operation involved, from which the deviation inoperating pressure finally resulted.

Understanding the unique effect of operating pressure onprocess dynamics and controllability is especially important forthe developments of reactive distillation columns involvingkinetically controlled exothermic reactions, because it offers apotential way to trade-off steady-state performance and processdynamics and controllability with operating pressure as a potentialdecision variable at the early stage of process synthesis and design.If, for a reactive distillation column involving a kinetically con-trolled exothermic reaction, operating pressure gives a corre-spondent effect on steady-state performance and process dynamicsand controllability, as in the case of the hypothetical ideal reac-tive distillation column studied, then its favorable operating pres-sure should be located around the point of the most economicaldesign, the gray region shown in Figure 1. If, for a reactive distilla-

tion column involving a kinetically controlled exothermic reaction,operating pressure does not present a correspondent effect onsteady-state performance and process dynamics and controllability,as in the cases of the methyl acetate or MTBE reactive distillationcolumns studied, then its favorable operating pressure should

be located between the two optimum points for steady-state per-formance and process dynamics and controllability, respectively,the gray regions shown in Figure 18a and b.

7. CONCLUSIONS

In this work, the influences of operating pressure on thedynamics and controllability of the reactive distillation columnsinvolving a kinetically controlled exothermic reaction have been

Figure 18. Optimal points for steady-state performance and process dynamics and controllability versus operating pressure for the reactivedistillation columns involving kinetically controlled exothermic reactions: (a) optimal point for steady-state performance is at the right side of theone for process dynamics and controllability, and (b) optimal point for steady-state performance is at the left side of the one for process dynamicsand controllability.




16/17

examined in terms of the three reactive distillation systemsstudied, including an ideal reactive distillation column perform-

ing a hypothetical exothermic reaction, A + B k

k

b

fC + D, and

two real ones producing, respectively, methyl acetate fromacetic acid and methanol and MTBE from isobutylene andmethanol. It has been found that operating pressure can presentnonmonotonic influences on process dynamics and controll-ability, just like its impact on the steady-state performance.

While the enhancement of operating pressure benefits processdynamics and controllability in the lower part of its feasibleregion, it turns to deteriorate process dynamics andcontrollability in the higher part of its feasible region.

The intricate behavior of the reactive distillation columnsinvolving kinetically controlled exothermic reactions isessentially governed by the conflicting effects of operatingpressure on reaction rate and chemical equilibrium constantand is useful to identify a favorable region of operating pressurefrom the viewpoint of process dynamics and controllability.Because the region may or may not coincide with the one for

process synthesis and design, operating pressure can thereforeserve as a potential decision variable to trade-off process designand operation, helping to make a process design withsatisfactory steady-state performance, process dynamics, andcontrollability.

Future work will be centered on analyzing the effect ofoperating pressure on the dynamics and controllability of thereactive distillation columns involving kinetically controlledexothermic reactions in terms of multivariable control theories.

APPENDIX A: STEADY-STATE MODEL OF THEHYPOTHETICAL IDEAL REACTIVE DISTILLATIONCOLUMN

For the ideal reactive distillation column performing ahypothetical exothermic reaction, A + B

k

k

b

fC + D, a steady-

state model is developed in terms of the principle of mass andenergy conservation in conjunction with the given vaporliquidequilibrium relationship. As shown in eqs A1 and A2, twoadditional constraints have been imposed on the top and

bottom product qualities within the steady-state model, whichguarantee a predetermined conversion rate of reactants andthus lay a fair basis for the comparative studies of variousprocess designs operated under different operating pressures.The steady-state model is solved using a modified NewtonRaphson method, and the satisfaction of component mass

balance equations, (i.e., eq A3), as well as the attainment of theproduct specifications (i.e., eqs A1 and A2), is taken as theconvergence criterion. The commercial software Mathematicais employed to compile and debug the program of the steady-state model. It appears to be quite robust and can approach asolution fairly quickly for the various hypothetical ideal reactivedistillation columns operated under different operatingpressures.

| | x xtop topsp

(A1)

| | x xbot botsp

(A2)

| + +

+ |

+ +L x V y L x V y F z

r

j i j j i j j i j j i j j i j j m

i j

1 , 1 1 , 1 , , , ,

, (A3)

APPENDIX B: DYNAMIC MODEL OF THEHYPOTHETICAL IDEAL REACTIVE DISTILLATIONCOLUMN

In terms of the principle of mass and energy conservation inconjunction with the given vaporliquid equilibrium relation-ship, a first-principle dynamic model is developed for the idealreactive distillation column performing a hypothetical exother-

mic reaction, A + B k

k

b

fC + D. Besides the reaction kinetics,

vaporliquid equilibrium relationship, and energy balanceequations described by eqs 29, the following equations areincluded.

Total mass balance equation on stage j (1 j n):

= + + ++ =

M

tV V L L F r

d

d

jj j j j j j m

i

nc

i j1 1 ,1

,

(B1)

Component mass balance equation on stage j (1 i nc,1 j n):

= +

+ +

+ +

M x

tV y V y L x L x

F z r

d

d

j i jj i j j i j j i j j i j

j j m i j i j

,1 , 1 , 1 , 1 ,

, , , (B2)

Stage hydraulic equation on stage j (2 j n 1):

=

L L

M Mj j

j jss

(B3)

It is noted that a linearized stage hydraulic equation isemployed here for the simplification of the complicatedcalculations. The dynamic model is solved using thecommercial software Mathematica, and the solution can wellrepresent the general behavior of the hypothetical ideal reactivedistillation column. It is therefore reasonable to use it as asubstitute to study process dynamics and controllability.

AUTHOR INFORMATIONCorresponding Author

*Tel.: +86 10 64434801. Fax: +86 10 64437805. E-mail:[email protected].

ACKNOWLEDGMENTS

The project is financially supported by the Doctoral ProgramsFoundation of Ministry of Education of China (Grant no.20100010110008) and thereby is acknowledged. We also thankthe State Key Laboratory of Chemical Resource Engineering atBeijing University of Chemical Technology for providingcomputing facilities.

NOTATION

A = componenta = liquid activity

Avp = vapor pressure constant, PaB = component

Bvp = vapor pressure constant, Pa K


mailto:[email protected]:[email protected]


17/17

C = componentCC = composition controllerD = component

E = activation energy of a reaction, kJ kmol1

F = feed flow rate of reactants, kmol s1

FC = flow rate controllerH = stage holdup, kmolH

R= thermal heat of a reaction, kJ kmol1

HV = heat of vaporization, kJ kmol1

k = specific reaction rate, kmol s1 kmol1

KC = proportional gainKCU = ultimate gainKeq = equilibrium constantL = liquid flow rate, kmol s1

LC = level controllerM = stage holdup, kmolmcat = catalyst weight, kgn = number of stagesnc = number of components

P = pressure, PaPCU = ultimate period, sqacid = ion-exchange capacity of the catalyst, equiv(H

+) kg1

Qreb = reboiler duty, kWR = ideal gas law constant, kJ kmol1 K1

r = net reaction rate, kmol s1

RR = reflux rate, kmol s1

t = time, sT = temperature, KTI = integral time, sV = vapor flow rate, kmol s1

Vnt = bottom vapor flow rate, kmol s1

x = liquid compositiony = vapor compositionz = feed composition

Greek Letters

= Kronecker function = error tolerance = stage hydraulics time constant, s = stoichiometric coefficients of a reaction

Subscripts

A = component indexb = backward reactionbot = bottom productB = component indexC = component indexD = component indexf = forward reactionIB = isobutylenei = component index

j = stage indexm = feed stage indextop = top product

Superscripts

s = saturation or steady statesp = product specification

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dx doi org/10 1021/ie202241p | Ind Eng Chem Res 2012 51 369237083708

influences of pressure on the operation of reactive distillation

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