Kejin Huang*,Yang YuanLiang ZhangHaisheng ChenShaofeng WangNian Liu

1College of Information Science and Technology,Beijing University of Chemical Technology,Beijing 100029,China

2Electrical&Instrument Department,Wuhuan Engineering Co.,Ltd,Wuhan 430223,China

1.Introduction

Reactive distillation columns(RDCs),being known as one of the most effective alternatives of process intensification,have received comprehensive attention over the past two decades[1–8].Although the combination between the reaction operation(RO)and the separation operation(SO)involved serves as the major impetus for reducing capital investment and operating cost as compared with their conventional counterparts,it also bears the blames for the occurrence of complicatedprocess dynamics and degraded controllability issues[9–11].In spite of the fact that deep insights were already acquired into process design,process dynamics,and process operation,little progress has been made so far on the interpretation of the interactions between process design and process dynamics and operation[12].Since this represents a key issue encountered in the development of RDCs,it is apparently imperative to gain a thorough understanding into the mechanism.In the current article,the totally reboiled reactive distillation columns(TRRDCs)are chosen to be investigated because they stand for one of the simplest combinations between the RO and the SO involved.

The cardinal purpose of this article is to investigate the dynamics and operation of the TRRDCs with special attention paid to their dependence on the special topological configuration. The interaction between the RO and the SO involved is analyzed in terms of transfer function based process models and found to be the primary reason for the occurrence of under-damped step responses. With the tight inventory control of the bottom reboiler, the under-dampness can be substantially eliminated and this yields a beneficial effect to process dynamics and controllability. In terms of two RDCs, separating, respectively, a hypothetical synthesis reaction from reactants A and B to product C and a real decomposition reaction from 1, 4-butanediol (BDO) to tetrahydrofuran (THF) and water, the unique behaviors of the TRRDCs are highlighted. Both open-loop tests and closed-loop control studies are carried out and the outcomes obtained are in good accordance with the insights acquired from the transfer function model based analysis.

2.Dynamics versus Reboiler Inventory Control of the TRRDCs

When a RDC has been developed to operate in a totally reboiled operation mode,for example,in the separation of a reversible synthesis reaction A+B? 2C(αC＞αB＞αA)as shown in Fig.1a,the lightest product C is extracted fromthe top and no discharge is fromthe bottom.Stripping section is thus not needed and the reactive section is directlyconnected to the bottom reboiler, thus exhibiting a simpler structurethan those RDCswith double outlets at the top and bottom, respectively.While the reactive section serves as an equivalent reactor for the RO involved,the rectifying section is in charge of the SOinvolved, both receivingenergy supply from the bottom rebolier.

Fig.1.A to tally reboiled reactive distillation column separatinga hypothetical synthesis reaction A+B?2 C(Example I).(a)Scheme,(b)blockdiagram of the re boilerinventory control loop,(c)control structure.

Since the quality of top product is the main controlled variable and re flux flow rate is usually the corresponding manipulated variable,it is desirable to derive the open-loop dynamic model that describes their relationship.Regarding the influenceofre flux flowrateon the quality of top product,three contributions can be identified,namely,the influences from the SO(i.e.,the rectifying section),the RO(i.e.,the reactive section),and the inventory control of bottom reboiler.Under the assumption that thein ventories of thetop condenserand bottom reboiler are controlled,respectively,with the top product flow rate and the heat duty of botto mreboiler,the corres ponding trans fer functionc ant hus be written as

whereGX,RR(S)represents the transfer function between the quality of toppro ductandre flux flowrate,GX,SO(S),thetrans fer function bet ween the quality of top product and the SO involved,GX,RO(S),the transfer function between the quality of top product and the RO involved,andGX,IC(S),the transfer function between the quality of top product and the inventory control loop of the bottom reboiler.

For theGX,SO(S)andGX,RO(S),they can simply be assumed to be first-order systems,namely,

whereKSOis the steady-state gain between the quality of top product and the SO involved,KRO,the steady-state gain between the quality of top product and the RO involved,TSO,the time constant between the quality of top product and the SO involved,andTRO,the time constant between the quality of top product and the RO involved.

Since the RO and the SO involved work together to yield the desired purity of top product,their common effects can be further simplified as

whereKSR(=KSO+KRO)is the steady-state gain between the quality of top product and the RO and the SO involved,andTSR(should be greater thanTSOandTRO),the time constant between the quality of top product and the RO and the SO involved.

For theGX,IC(S),it can be derived in terms of Fig.1b,in which the block diagram of the inventory control loop of the bottomre boiler is sketched.Here,the inventory of the bottom reboiler is described as an integrating process to the liquid flow from the bottom stage and vapor lf ow from the bottom reboiler.

whereKLHis the steady-state gain between reboiler inventory and liquid flow rate,KVH,the steady-state gain between reboiler inventory and vapor flow rate,Kreb,the controller gain of the inventory control loop of the bottom reboiler,KV,the steady-state gain between the quality of top product and vapor flow rate,TL,the time constant for liquid flowing from the top to the bottom,andTV,the time constant between top product quality and vapor flow rate.

The above transfer function can be further simplified as

It should be indicated here thatTICis mainly dominated by the term,1/KVHKreb,because of the large liquid holdup in the bottom reboiler.Note that it also holds a reciprocal relationship with the controller gain of the bottom reboilerKreb.For the steady-state gain,KIC(=KLHKV/KVH),it is generally rather small because the bottom reboiler usually exhibits relatively much slow dynamics in process operation.

Substituting Eqs.(2.3)and(3.2)into Eq.(1),one can get,

In the case of a high-purity TRRDC,the last term in the numerator of Eq.(4),KSR-KIC,is generally very small in magnitude since no product is withdrawn from the bottom. It must also be greater than zero becausethe top product quality changes proportionally with the reflux flowrate,i.e.,KSR-KIC＞0.Owing to the large amount of liquid holdup in the bottom re boiler,TICmust be considerably greater thanTSR.Thus,the two coefficients in the numerator of Eq.(4)cangener ally satisfy the inequality relationship,KSRTIC-KICTSR＞KSR-KIC,which dictates the top product quality exhibiting under-damped responses to the step changes in reflux flow rate.The larger the difference between the two coefficients is,the more serious the degree of the under-dampness becomes.With the tight inventory control of the bottom re boiler(i.e.,Krebhas been assigned to a largevalue),the difference between the set wo coefficients can be dwindled and this serves to alleviate the degree of the under dampness,hence yielding a beneficial effect to the dynamics and operation of the TRRDC.

In the next two sections,two TRRDCs,separating,respectively,a hypothetical synthesis reaction A+B?2C,and a real decomposition reaction BDO?THF+H2O,are employed to examine their unique dynamics and the effect of reboiler inventory control on process dynamicsand operation.

3.Example I:A TRRDC Separating a Hypothetical Ideal Reaction A+B?2C

3.1.Process description

The process is depicted in Fig. 1a,which performs the separation of areversible liquid-phase reaction,

Table 1Physicochemical properties and nominal operation conditions of Example I

Table 1 summaries the physicochemical properties and operating conditions.The relative volatility ranking of the three reacting components is αC＞αB＞αA.Two pure reactant feeds,FAandFB,are fed onto the top and bottom of the reactive section(i.e.,stage 11 and stage 18),respectively.The net reaction rate for componention stagejin the reactive section is given by

whereKf,jandKb,jare the forward and backward specific reaction rate constants and given by

The mass balance equations can be expressed as follows.

For the rectifying section

For the reactive section

Fig.2.Steady-state profile of Example I.(a)Temperature,(b)vapor and liquid flow rates,(c)liquid composition,(d)net reaction rates.

The TRRDC is simulated using the dynamic mode developed by Liuet al.[13].The steady-state pro files of temperature,vapor and liquid flow rates,liquid compositions,and net reaction rates are shown in Fig.2.

3.2.Open-loop process dynamics

Fig.3 depicts the open-loop transient responses of the ideal TRRDC with three kinds of controller settings for the re boiler inventory control loop(i.e.,Kreb=0.2 s-1,1.0s-1,and 2.5 s-1),when a±5%step change is imposed on reflux flow rate,respectively.For both the positive and negative perturbations in reflux flow rate,the ideal TRRDC shows under-damped responses in the quality of top product.It can readily be noted that the degree of the under-dampness changes inversely to the controller gain of the re boiler inventory control loop.

The open-loop transient responses of the ideal TRRDC with three kinds of controller settings for the reboiler inventory control loop(i.e.,Kreb=0.2 s-1,1.0 s-1,and 2.5 s-1)are shown in Fig.4 when a±5%step change is imposed on the feed flow rate of reactant B,respectively.Only slight differences can be observed with respect to the different controller gains of the re boiler inventory control loop.

3.3.Evaluation of closed-loop control performance

Fig.3.Open-loop transient responses of Example I with different Krebinthe face of a±5%step change in the re flux flow rate,respectively.(a)Positive responses,(b)negative responses.

For the sake of examining the influences of inventory control of the bottom reboiler on process dynamics and operation,the closed-loop control of the ideal TRRDC is studied here.Fig.1c presents the control scheme in which the re flux flow rate is manipulated with a proportional-plus-integral(PI)controller to control the purity of the top product C.The heat duty of reboiler and the top product flow rate are manipulated with two proportional-only(P)controllers,respectively,to control the levels of reboiler and condenser.The feed flow rate of reactantA is manipulated to control the composition on stage 5,serving to maintain the stoichiometric balance between the two reactants.Again,a P controller is employed here.The Zigeler–Nichols rule is used to calculate the controller parameters and the resultant outcomes are tabulated in Table2.Afirst-orderlag with at imeconst ant of5minis added to all composition measurements and each control valve is set to be half open at the nominal steady state.

A±0.003 step change is introduced to the set-point of the top control loop and the regulatory responses of the ideal TRRDC are shown in Fig.5.Strong impact is observed from the inventory control of the bottom reboiler.When the controller gain of the bottom reboilerKrebis 0.2 s-1,severe oscillations appear and even instability problem occurs in the case of negative change in the set-point of top control loop.With the increasingly tight tuning of the reboiler inventory control loop(e.g.,Kreb=1.0 s-1),the oscillations are gradually suppressed.When the controller gain of the bottom reboilerKrebis 2.5 s-1,the totally reboiled ideal RDC turns to exhibit rather smooth responses.

Fig.4.Open-loop transient responses of Example I with different Krebin the face of a±5%step change in the feed flow rate of reactant B,respectively.(a)Positive responses,(b)negative responses.

Fig.6 shows the regulatory responses of the ideal TRRDC when a±5%step change is imposed on the feed flow rate of reactant B,respectively.Again,sustained oscillations are observed when the controller gain of the reboiler inventory control loopKrebhas been set to be 0.2 s-1.With the increasingly tight tuning of the reboiler inventory control loop(e.g.,Kreb=1.0 s-1and 2.5 s-1),the oscillations are suppressed,implying apparently a beneficial effect to the dynamics and controllability of the ideal TRRDC.

4.Example II:A TRRDC Dehydrogenizing BDO into THF and H2O

4.1.Process description

The mechanism of the decomposition of BDO into THF and H2O is

Table 2Controller parameters for Example I

Fig.5.Servo responses of Example I with different Krebinthe face of a±0.003 step change in the set-point of top control loop,respectively.(a)Positive responses,(b)negativeresponses.

whererBDOis the rate of consumption of BDO,k,the reaction rate constant,CBDO,the composition of BDO,andKBDO,the adsorption equilibrium constant[14].

The reaction rate constant and adsorption equilibrium constant are calculated from the Arrhenius equation as follows:

Fig.6.Regulatory responses of Example I with differentKrebintheface of a±5%step change in the feed flow rate of reactant B,respectively.(a)Positive responses,(b)negativeresponses.

The BDO decomposition reaction can be conducted in a RDC,and Fig.7a gives such a process configuration.Table 3 summaries the physico chemical properties and nominal operation conditions.The relative volatility ranking of the three reacting components is αBDO＜αH2O＜αTHF,so the last two components are withdrawn from the top.The reactive section runs from stage 11 to stage 18 and BDO is added to the top of the reactive section.The reactant BDO generally accumulates at the bottom and must be recycled back to the reactive section by means of a totally reboiled operation mode.The BDO TRRDC is simulated with the commercial software Aspen Dynamics and its steady-state pro files of temperature,vapor and liquid flow rates,liquid compositions,and net reaction rates are delineated in Fig.8.

Fig.7.A totally reboiled reactive distillation column dehydrogenizing BDO into THF and H2O(Example II).(a)Scheme,(b)control structure.

4.2.Open-loop process dynamics

The open-loop transient responses of the BDO TRRDC are depicted inFig. 9when a±5% step change is imposed on the reflux flow rate. In thecase of negative response, a high degree of under-dampness is presentedwhenthe controller gain of the reboiler inventory control loopKrebis set to be 0.2 s-1and 1.0 s-1,respectively.The situation is greatly improved after the controller gain of the reboiler inventory control loopKrebis enhanced to be 3.0 s-1,in which the degree of the under dampness is attenuated substantially.In the case of positive response,although the degree of the under-dampness is not so prominent, the favorableeffects of tightening the inventory control of the bottom reboiler can still be clearly observed.

Table 3Physicochemical properties and nominal operation conditions of Example II

In Fig.10,the open-loop transient responses of the BDO TRRDC are delineated when a±5%step change is imposed on the feed flow rate of BDO,respectively.The process settles to the new steady states much more slowly in case that the controller gain of the reboiler inventory control loopKrebis set to be 0.2 s-1than in case that the controller gain of the reboiler inventory control loopKrebis set to be 1.0 s-1or 3.0 s-1.Tightening the inventory control of bottom reboiler poses apparently a beneficial effect to process dynamics and controllability,which will bene fit closed-loop operation of the BDO TRRDC.

4.3.Evaluation of closed-loop control performance

Thedetailed control schemeis given in Fig.7b andthe controller parameters are listed in Table 4.In Fig.11,the servo responses of the BDO TRRDC are illustrated when a±0.003 step change is introduced to the set-point of the top control loop,respectively.When the controller gain of the reboiler inventory control loopKrebhas been set to be 0.2 s-1,stable operation cannot be maintained in the case of the positive and negative changes in the set-point of the top control loop.When the controller gain of the reboiler inventory control loopKrebis increased to be 1.0 s-1,the situation is slightly improved.While stable operation can be maintained in the case of the positive change in the set-point of the top control loop,divergent oscillations still happen in the case of the negative change in the set-point of the top control loop.When the controller gain of the reboiler inventory control loopKrebis increased to be 3.0 s-1,quite smooth operation has been achieved in both the positive and negative changes in the set-point of the top control loop.

Fig.12 gives the regulatory responses of the BDO TRRDC when a±5%step change is imposed on the feed flow rate of BDO.When the controller gain of the reboiler inventory control loopKrebis set to be 0.2 s-1,stable operation cannot be maintained in the case of the positive and negative changes in the feed flow rate of BDO.After the controller gain of the reboiler inventory control loopKrebhas been increased to 1.0 s-1or 3.0 s,quite smooth operation has been attained.This comparison implies again a favorable effect from the tightly tuning of the reboiler inventory control loop to process dynamics and controllability.

Fig.8.Steady-state pro file of Example II.(a)Temperature,(b)vapor and liquid flow rates,(c)liquid composition,(d)net reaction rates.

5.Discussion

In the light of the two example syste ms investigated in the current article,it has been demonstrated that the TRRDCs exhibit uniformly under-damped step response.Such outcomes are actually in excellent accordance with the theoretical prediction made in Section 2 in terms of trans fer function based process models and reflect the unique behavior of this kind of RDCs.The uniqueness is actually closely related to the interaction between the RO and the SO involved as well as the topologicalstructure of the TRRDCs. It alsomakes such kind of processes sharply different from those conventional distillation columns and RDCs with double product flows at the top and bottom,respectively.Although detailed reaction kinetics and thermodynamic properties of the reaction operation involved can,to a great extent,affect process dynamics,they are unlikely to change the occurrence of the under-damped step response.Therefore,special care should be taken to deal with this unique dynamic behavior during the development of control system for the TRRDCs.

For the TRRDCs,the special operation style makes it possible and necessary to improve their process dynamics and controllability through the inventory control of the bottom rebolier.In terms of the tight inventory control of the bottom reboiler,the degree of underdampness is alleviated and this provides us an additional means to modify process dynamics and controllability(apart from process modifications).The finding also extends the tuning rule by Luyben and Yu(2008),who suggested that,irrespective of conventional distillation columns and RDCs,a value of 2 should be assigned to the controller gains of the inventory control loops of re flux-drum and reboiler.According to the current work,it is not difficult to understand that this recommendation is not always appropriate.A reasonable controller gain for the inventory control loop of the bottom reboiler can only be found by detailed studies of the TRRDCs.

6.Conclusions

In this work, the dynamics and operation of the TRRDCs has been analyzed in terms of transfer function based process models. It has been revealed that the interaction between the RO and the SO involved and the special topological structure are the main reasons that give rise to the under-damped step responses and the degree of the underdampness can be alleviated through tight inventory control of the bottom reboiler. This finding represents essentially the common behavior of the kind of RDCs and is of great significance to the development of their decentralized control systems.

Nomenclature

Ahypothetical component

Avpvapor pressure constant,Pa

Bhypothetical component

Botbottom product flow rate,mol·s-1

Bvpvapor pressure constant,Pa·K

Chypothetical component

CCcomposition controller

CBDOBDO composition,mol·L-1

Ddistillate flow rate,mol·s-1

Eactivation energy of a reaction,kJ·mol-1

Ffeed flow rate,mol·s-1

FCflow rate controller

Gtransfer function

Hstage holdup,mol

ΔHadadsorption energy,kJ·mol-1

ΔHRheat of a reaction,kJ·mol-1

ΔHVlatent heat of vaporization,kJ·mol-1

Ksteady state gain

KBDOequilibrium adsorption constant of BDO,L·mol-1

KCproportional gain

Krebcontroller gain of the reboiler inventory control loop,s-1

K0,BDOequilibrium adsorption constant of BDO asT→ ∞,L·mol-1

krate constant,L·gcat1·s-1

k0rate constant asT→ ∞,L·gcat1·s-1

Lliquid flow rate,mol·s-1

LClevel controller

nctotal number of components

NRfeed location of reactant A

NTtotal number of stages

Ppressure,Pa

PCpressure controller

PIproportional plus integral controller

qfeed thermal condition

Rideal gas law constant,J·mol-1·K-1

RRre flux flow rate,mol·s-1

rreaction rate,mol·mol-1·s-1or mol·m-3·s-1

rBDOreaction rate of BDO,mol··s-1

Ssymbol of Laplace transform

SPset-point

Ttime constant,s,or temperature,K

TIreset time,s

Vvapor flow rate,mol·s-1

xliquid composition

α pre-exponential factor

ν stoichiometric coefficient of a reaction

Fig.9.Open-loop transient responses of Example II with differentKrebin the face of a±5%step change in the re flux flow rate,respectively.(a)Positive responses,(b)negative responses.

Fig.10.Open-loop transient responses of Example II with different Krebin the face of a±5%step change in the feed flow rate of BDO,respectively.(a)Positive responses,(b)negative responses.

Table 4Controller parameters for Example II

Subscripts

b backward reaction

f forward reaction

IC inventory control loop of bottom reboiler

icomponent index

jstage index

L liquid phase

RO reaction operation

SO separation operation

SR separation operation and reaction operation

V vapor phase

Fig.11.ServoresponsesofExampleIIwithdifferentKrebinthefaceofa±0.003stepchangeintheset-pointoftopcontrolloop,respectively.(a)Positiveresponses,(b)negativeresponses.

Fig.12.Regulatory responses of Example II with different Krebin the face of a±5%step change in the feed flow rate of BDO,respectively.(a)Positive responses,(b)negative responses.

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