Lijing Zang,Kejin Huang*,Ting Guo,Yang Yuan,Haisheng Chen,Liang Zhang,Xing Qian,Shaofeng Wang

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

Keywords:Reactive distillation column Temperature inferential control Sensitivity analysis Transient deviation Steady-state deviation

ABSTRACT Temperature inferential control(TIC)is studied for a reactive distillation column with double reactive sections(RDC-DRSs)processing a hypothetical tw o-stage consecutive reversible reaction(A+B?C+D,C+B? E+D withαD＞αB＞αC＞αA＞αE).Because of the complicated dynamic behaviors,the controlled stages by sensitivity analysis lead to great steady-state deviations(SSDs)in top and bottom product purities.Since TICinvolves considerably reduced settling times in comparison w ith direct composition control,small SSDs in product qualities correspond generally to small transient deviations(TDs)in product qualities.An objective function that measures SSDs in product qualities is formulated to represent the performance of a TICsystem and an iterative procedure is devised to search for the best control con fi guration.The application of the procedure to the RDC-DRS gives considerably suppressed TDs and SSDs in top and bottom product qualities as compared with the one by sensitivity analysis.The method is simpler in principle and less computationally intensive than the current practice.These striking outcomes show the effectiveness of the proposed principle for the development of TIC systems for complicated reactive distillation columns.

1.Introduction

Temperature inferential control(TIC)is generally advantageous over direct composition control in distillation processoperation because temperature sensing needsmuch lessexpensive analyzers,lower maintenance costs,and smaller measurement times(i.e.,the dead-time introduced to each control loop involved)than composition sensing[1-3].The former is,how ever,usually much more dif fi cult to develop than the latter.In addition to the commonly shared problem of pairing controlled variables w ith manipulated variables,the temperatures of the controlled stages must also be good indicationsof the product qualitiesto be controlled and this necessitates an effective determination of the locations of the controlled stages(e.g.,the search for the so-called sensitive stages for the rectifying section,reactive section(RS),and stripping section of a reactive distillation column(RDC)).A plenty of criteria and methods have been proposed so far to guide the discrimination process and only have a few of them frequently been employed in thedevelopment of TICsystemsfor variousdistillation columns,including:(i)slope criterion,(ii)sensitivity criterion,(iii)singular value decomposition criterion,(iv)invariant temperature criterion,and(v)minimum product variability criterion[4-8].Although these criteria usually work effectively in the determination of the locations of the controlled stages,they sometimeslead to TICsystemsthat fail to yield the potentially best control performance especially in the operation of complicated distillation systemsthrough processintegration and intensi fication,e.g.,heat-integrated distillation columns,RDCs,and dividing-wall distillation columns[9,10].This not only limits the applications of TICsystemsbut also posesnegativeeffect on thedevelopment of novel energy effi cient distillation systems.Although the determination of the controlled stages appears to be a rather obsolete topic in the area of distillation process control,it has still remained to be an important issue that is worth studying w ith great attention.

Recently,Kaymak et al.studied the operation of a RDCw ith double reactive sections(RDC-DRS)derived by Yu et al.,w hich processed a hypothetical tw o-stage consecutive reversible(HTSCR)reaction(A+B?C+D,C+B?E+D w ithαD＞αB＞αC＞αA＞αE)w ith a much higher energy ef fi ciency than its conventional counterpart(i.e.,a RDCw ith a single reactive section)[11,12].Kaymak and his cow orkers indicated that the TIC system by sensitivity analysis displayed rather poor performance and w as unable to give tight quality control of the top and bottom products.To deal with the diffi culty,they devised a composition and temperature combined control system w ith direct composition control of the top and bottom products and indirect temperature control of the stoichiometric balance betw een reactants A and B.Although the RDC-DRScould be operated smoothly w ith the composition and temperature combined control system,the involvement of composition measurements degraded de fi nitely,to a certain extent,the control system performance by substantially increasing transient deviations(TDs)and considerably extending settling times.In a more recent paper about the RDC-DRSperforming the transesteri fi cation of DMCw ith EtOH to DECand MeOH,they demonstrated,how ever,that the TIC system by sensitivity analysis could w ork effectively to regulate the qualities of the top and bottom products[13].The completely different outcomes betw een these tw o case studies reveal evidently the unique dynamic features of the RDC-DRS.Since the arrangement of multiple reactive sections stands for a much more effective methodology for the development of RDCs,it is imperative to investigate in more detail if the RDC-DRSis likely to be controlled tightly w ith a TICsystem[14-16].

The objective of the current research is to dw ell on the TICof the RDC-DRSseparating a reacting mixture involving a HTSCRreaction.Because the TIC system resulted from sensitivity analysis fails to give tight product quality control w hen facing step changes in product throughput and feed composition,an effective principle and iterative procedure are derived to fi nd the potentially best control con fi guration that yields the smallest TDs and SSDs in product qualities.Detailed comparisons are made betw een the resultant TICsystems and the one by sensitivity analysis.After a brief discussion of the obtained outcomes,the article ends w ith the conclusions that can be draw n.

2.RDC-DRSand its TICSystem by Sensitivity Analysis

2.1.Processdescription

The RDC-DRSstudied in this w ork is sketched in Fig.1a,w hich involves totally 30 stages(including condenser and reboiler)[11].Unlike its conventional analogues,it accommodates the rectifying section,upper RS,middle non-RS,low er RS,and stripping section.The speci fi c structure facilitates the coordination between the reaction operations and separation operation involved and leadsconsequently to reductions in capital investment and operating cost.While the heavy reactant A is fed onto the top of the upper RS,the light reactant Bis divided into two portionswith a fi xed ratio and introduced,respectively,into thebottom of the upper and low er RS.The top and bottom products are speci fi ed to be 95 mol%Dand 95 mol%E,respectively.The HTSCRreaction involves fi ve reacting components w ith the follow ing kinetic and thermodynamic properties.

The reaction rates for the fi rst-and second-stage reactions are

where k+mand k-m(m=1,2)are the rateconstantsof the forward and backw ard speci fi c reaction,which can be expressed as

Table 1 lists the kinetic and physical data of the HTSCRreaction,w hich are derived in terms of the common characteristics of such kind of reaction systems.Ideal vapor and liquid equilibrium relationships and constant volatilities(αD＞αB＞αC＞αA＞αE)are assumed and the vapor saturation pressure is given by

Fig.1.RDC-DRSand its steady-state behaviors:(a)RDC-DRS;(b)composition pro fi le;(c)temperature pro fi le.

Table 1 Physicochemical characteristicsand design speci fi cationsof the RDC-DRS(1 cal=4.1868 J)

The steady-state and dynamic simulations of the RDC-DRSare carried out in the environment of the commercial softw are Mathematica.Sketched in Fig.1b is the steady-state pro fi le of liquid composition.Except that the puri fi cations of products Dand Ecan be clearly seen in the rectifying and stripping sections,it is rather dif ficult to identify such tendencies in the upper RS,middle non-RS,and low er RS.This phenomenon re fl ects potentially the strong interactions betw een the three sections,w hich may even affect the operation of the rectifying and stripping sections.In terms of the steadystate pro fi le of stage temperatures(c.f.Fig.1c),a rather fl at change is observed in the upper RS,middle non-RS,and low er RS.In spite of the relatively great changes in compositions(for example,components A,D,and E)happen along the three sections,the complicated kinetics of the HTSCRreaction suppresses substantially the changes in temperature and consequently results in the intensive coupling betw een the three sections.

2.2.TICsystem by sensitivity analysis

Instead of the three-point temperature control con fi guration examined by Kaymak et al.,a more complicated one that involves four temperature control loops is adopted here in the current w ork.System pressure is maintained at 1 MPa w ith the manipulation of heat removal from condenser.The re fl ux drum level is controlled w ith the fl ow rate of top product and the reboiler level w ith the fl ow rate of bottom product.The fresh feed fl ow rate of reactant A(FA)is fl ow controlled and serves as the production rate handle.The tw o fresh feed fl ow rates of reactant B,FB1and FB2,are paired w ith the temperatures of tw o stages to be selected,serving to maintain the stoichiometric balance w ith reactant A.The re fl ux fl ow rate,RR,and reboiler heat duty,Qreb,are used to control the temperatures of tw o stages to be selected,w orking,respectively,to maintain the purity of component D in the top product and the purity of component Ein the bottom product.

Fig.2.Sensitivity analysis of the RDC-DRS.

In Fig.2 are given the results of sensitivity analysis for the RDCDRS.Here,sensitivity serves to measure the variability of stage temperatures to a change in one of the four manipulated variables(i.e.,FB1,FB2,RR,and Qreb).Note that the RDC-DRSdisplays sharply different behaviors from conventional distillation columns.In the case of an increase in reboiler duty,the conversion rate goes dow n and results in the increase in the composition of component A and the decrease in the composition of component E in the upper and low er parts of the RDC-DRS,respectively.Thus,the temperature drops along the w hole column.In the case of an increase in the feed fl ow rate of reactants A and Band the re fl ux fl ow rate,the conversion rate goes up and results in the increase in the composition of component Ealong the w hole column.Thus,the stage temperature show s unexceptionally an increase in these cases.Qrebexhibits only a negative peak on stage 26 w here RR,FB1,and FB2show,respectively,their highest peaks.While the latter three variables present,respectively,their second peaks on stage 20,RR and FB1display their third peaks on stage 10.The common positions of the peaks reveal the potentially intensive coupling betw een the controlled variables to be chosen.In accordance w ith the relative gain array(RGA)listed in Table S1 of the supplementary information and the proximity principle,w hile RR should be paired w ith T10,Qrebshould be w ith T26and FB1and FB2should,respectively,be w ith T20and T24(it is chosen from the stages near FB2).The resultant control con fi guration is sketched in Fig.3a and labeled the CS1,hereinafter.P controllers(w ith a gain of 2)are applied to the tw o liquid level control loops and PIcontrollers are for the four temperature control loops.Tw o 60s fi rst-order measurement lags are included in each temperature measurement.All temperature controllers are tuned w ith the Tyreus-Luyben rule and the obtained parameters are listed in Table 2.Because the open-loop gain of the bottom control loop is considerably smaller than those of the other control loops,a much larger Kcvalue is required in this control loop than in the other control loops.This outcome also re fl ects the unique characteristicsof the RDC-DRS.

The closed-loop responses(CLRs)of the CS1 to a±20%step change in production rate handleareillustrated in Fig.4.Throughout thearticle,unlessotherwisestated,thepositive responsesare represented by black curves and negative responses by gray curves.The FB1and FB2control loopsfail to w ork effectively to keep thestoichiometric balancebetw een reactants A and B and force the top and bottom products to exhibit rather great peak deviations.Although stable operation is secured,the top and bottom products converge fi nally to new steady-state values(SSVs)and cannot be recovered back to their set-points.Table 3 lists their absolute and relative SSDs.Whilein the caseof the+20%throughput change,the D composition in the top product and Ecomposition in the bottom product stabilize at 0.9417 and 0.952,respectively,in the case of the-20%throughput change,they level out at 0.958 and 0.9474,respectively.Given in Fig.5 are the CLRs of the CS1 to a+5%step change of B composition in FA.In addition to the rather great peak deviations in the top and bottom products,they converge to new SSVs and cannot be recovered back to their set-points.Table 3 also includes their absolute and relative SSDs.The D composition in the top product and Ecomposition in the bottom product stabilize at 0.9398 and 0.9496,respectively.The FB1and FB2control loops fail again to w ork effectively in maintaining the reactant A and B's stoichiometric balance and result in an excess of reactant B.The excess of reactant B initiates further a transition to a new steady state and this can clearly be identi fi ed from the response of the top control loop.Around the instant of 0.667 h,the Dcomposition in the top product already reaches thevicinity of an intermediatesteady state.Despite that thefour manipulated variables(i.e.,FB1,FB2,RR,and Qreb)have involved no sharp changes since that time instant,the D composition in the top product fails to settle dow n to that intermediate steady state.Instead,it still displays a long excursion with a very different speed from the one before that time instant.Thisoutcome implies that the CS1 cannot effectively deal w ith the output multiplicity of the RDC-DRS.

Fig.3.TICschemes of the RDC-DRS:(a)CS1;(b)CS2;(c)CS3.

Table 2 Controller parameters of the CS1,CS2,and CS3

It seemsthat the TICsystem by sensitivity analysiscannot hold tight operation of the RDC-DRS.The reason should de fi nitely be attributed to the strong coupling betw een the controlled stages chosen.To deal w ith the failure,one has to ascertain if alternative controlled stages can be found that is likely to result in alleviated interaction betw een the four control loops.Thus,an effective principle and procedure should be developed for that purpose.

Fig.4.Comparison between the CS1,CS2,and CS3 schemes in face of a±20%step change in F A.

3.A Novel Principle and Procedure for the Development of a TIC System

3.1.Anovel principle for the development of TICsystem

Generally speaking,the performance of a TIC system should be evaluated in terms of both TDs and SSDs in product qualities.Because temperatures are dominant variables in a distillation column,their tight control leads generally to considerably reduced settling times as compared w ith direct composition control[17].Moreover,the settling times are generally not so sensitive to the changes in the locations of the controlled stages.Such characteristics might allow the small SSDs in product qualities correspond generally to the small TDs in product qualities.It seems therefore reasonable to evaluate theperformance of a TICsystem in terms of merely SSDs in product qualities.The simpli fi cation of the evaluation metric can greatly facilitate the determination of the controlled stages since the search process is now turned from a dynamic optimization to a static one with the steady-state model as an equality constraint.Horiand Skogestad once employed a kind of metric describing the SSDs in product qualities to locate the appropriate controlled stages for tw o-product distillation columns[18].With reference to the RDC-DRS,the follow ing sum of the absolute SSDs caused by the step changes in feed fl ow rates and compositionscan be constructed astheobjective function for the search of the best controlled stages.

Table 3 SSDs of the CS1,CS2,and CS3

Fig.5.Comparison betw een the CS1,CS2,and CS3 schemes in face of a 5%step change of Bcomposition in F A.

w here L1,L2,L3,and L4 arethecontrolled stages'locationsin the RR,FB1,FB2,and Qrebcontrol loops.

The objectivefunction of Eq.(6)equally treatsthedisturbancesfrom feed fl ow rates and compositions and so do the SSDs in the top and bottom products.If necessary,w eighting coef fi cients can be included to represent their relative impacts on control system performance.

3.2.An iterative procedure for the development of TICsystem

Because the formulation of Eq.(6)isessentially amixed integer nonlinear optimization problem and its solution is rather complicated in principle and fairly time-consuming in search computation.To avoid the draw backs,w e develop an iterative procedure as given in Fig.6 for the determination of controlled stages with the application of a univariable search method reinforced w ith someheuristic evolutional principles.The initial controlled stages are determined w ith a conventional criterion(e.g.,the sensitivity analysis here)follow ed by pairing them w ith theavailable manipulated variables.If theresultant control system performance appears satisfactory in terms of the SSDs in product qualities,then the tuning of thetemperature controllerscan proceed;otherwise,the loop-by-loop adjustment of the controlled stages should be performed.During the process,the location of the controlled stage for each control loop is adjusted until the objective function of Eq.(6)has reached its minimum value.The search process stops in case that there is only negligible changes in the objective function of Eq.(6)between the two adjacent roundsof iterations.Because of the possibly drastic changes in the locations of the controlled stages,their pairing w ith the available manipulated variables should again be examined here.It should be indicated that the proposed iterative procedure is much more effective than theordinary sequential enumeration method.

Fig.6.An iterative procedure for thesynthesisand design of a TICsystem for the RDC-DRS.

Although the optimality cannot theoretically be guaranteed with the adopted uni-variable search method,it can,in most cases,lead to reasonable approximate solution to the formulation of Eq.(6).

4.Development of a TICSystem for the RDC-DRS

The RDC-DRS'steady-state model described in the supplement information is employed in the current study.With the application of the iterative procedure show n in Fig.6 to the RDC-DRS,the obtained relationshipsbetw een theobjective function of Eq.(6)and thelocations of the controlled stagesare displayed in Fig.7 for the fi rst round of iterations(k=1).Due to the effective use of the know ledge gained from the intermediate search processes,only 8 times of steady-state model calculations are actually required.In the top control loop(c.f.Fig.7a),the objective function decreases steadily by moving the controlled stages from stage 10 to stage 4.Further ascending the controlled stage causes no convergence in steady-state simulation and this signi fi es that the temperatures on the chosen stages cannot be kept constant in face of disturbances from feed fl ow rates and compositions.Thus the adjustment should end up at stage 4.In the FB1control loop(c.f.Fig.7b),the objective function decreases steadily by moving the controlled stages from stage 20 to stage 11.Further ascending the controlled stage causesno convergence in steady-state simulation and thus the adjustment endsup at stage 11.In the FB2control loop(c.f.Fig.7c),the objective function decreases steadily by moving the controlled stages from stage 24 to stage 25.For the avoidance of the same controlled stage w ith the bottom control loop,the adjustment ceases to stage 25.In the bottom control loop(c.f.Fig.7d),the objective function decreases steadily by moving the controlled stages from stage 26 to stage 29.The fi rst round of iterations(k=1)leads to a substantial reduction of the objective function and gives rise to the control con fi guration as sketched in Fig.3b and labeled the CS2,hereinafter.

Fig.7.Objective function versus the controlled stages(k=1):(a)top control loop;(b)F B1 control loop;(c)F B2 control loop;(d)bottom control loop.

Fig.8 illustrated the obtained relationships betw een the objective function of Eq.(6)and the locations of the controlled stages for the second round of iterations(k=2).Here only 5 times of steady-state model calculations are needed.In the top control loop(c.f.Fig.8a),the objective function decreases again by moving the controlled stages from stage 4 to stage 2 and no divergence problems occur any more in these cases.In the FB2control loop(c.f.Fig.8c),the objective function exhibits its minimum value at stage 26.In the other tw o control loops(c.f.Fig.8b and d),the current controlled stages keep,respectively,the objective function at its minimum value and thus no adjustments are needed any more.The second round of iterations(k=2)leads again to a substantial reduction of the objective function and gives rise to the control con fi guration as sketched in Fig.3c and labeled the CS3,hereinafter.The third round of iterations(k=3)is conducted and no adjustment of the controlled stages are necessary.Note the fact that the CS3 conforms w ell to the proximity principle and the prediction by the RGA listed in Table S2 of the supplementary information.

Fig.8.Objective function versus the controlled stages(k=2):(a)top control loop;(b)F B1 control loop;(c)F B2 control loop;(d)bottom control loop.

Also given in Fig.4 are the CLRs of the CS2 and CS3 to a±20%step change in the production rate handle.It is noted that the peak deviations are suppressed substantially in the top and bottom products w ith almost the same settling times w ith the CS1.Although new SSVs are reached,the SSDsare substantially suppressed and the detailed outcomes are also summarized in Table 3.While for the positive change,the CS2 reduces the SSDs in the top and bottom products by 0.796%and 0.196%,respectively,in comparison w ith the CS1,for the negative change,the two numbers become 0.718%and 0.276%,respectively.In comparison w ith the CS2,w hile the CS3 reduces the SSD of the top product by 0.023%and increases the SSD of the bottom product by 0.008%,for the positivechange,for the negative change,the CS3 reduces the SSD in the top product by 0.053%and increases the SSD of the bottom product by 0.01%.In other words,a more coordinated system performance is reached by the CS3 than by the CS2 betw een the RR and Qrebcontrol loops.

Also given in Fig.5 are the CLRs of the CS2 and CS3 to a 5%step change of B composition in FA.Again,the peak deviations are suppressed substantially in the top and bottom products.They still settle dow n into new SSVs,but their SSDs are again substantially suppressed in these situations.In comparison with the CS1,the CS2 reduces the SSD of the top product by 0.957%and increases the SSD of the bottom product by 0.003%.In comparison w ith the CS2,the CS3 reduces the SSDof the top and bottom products by 0.076%and 0.013%,respectively.Again,a more coordinated control performance is reached by the CS3 than by the CS2 between the RR and Qrebcontrol loops.The detailed outcomes can also be identi fi ed from Table 3.

With reference to the objective function,the CS2 and CS3 abate,respectively,its value by 88.768%and 93.176%as compared w ith the CS1,implying evidently a monotonic improvement in control system performance by the iterative procedure show n in Fig.6.

5.Discussions

The failure of the TICsystem(resulted from sensitivity analysis)in securing tight product quality control is certainly aroused by the complicated dynamic behaviorsof the RDC-DRS,i.e.,theinherent strong interaction and the high degree of process nonlinearity.With the application of the novel principle and procedure proposed in the current w ork,the draw backs of the TICsystem are gradually alleviated and the potentially best TICsystem can be approached,w hich gives rise to considerably suppressed TDs and SSDs in the top and bottom product qualities.It is noticed that even in the presence of output multiplicities(note that the very different SSVs of FB1and FB2between the CS1,CS2 and CS3 also give support to the transition to a new steady state of the RDC-DRS incorporated w ith the CS1),the proposed novel principle and procedure can still w ork effectively.These outcomes demonstrate de fi nitely their rational and effectiveness.Since merely SSDsin product qualities are involved in the objective function of Eq.(6),the derivation of control con fi guration and thetuning of controllerscan be executed in asequential manner and thisfacilitatesthesearch processto benot only simpler in principle but also less computationally intensive than the current practice.

To further evaluate the feasibility and effectiveness of the proposed principle and procedure for the development of the TICfor the RDCDRS,w e also examine here the in fl uences of different control loop search sequences and different pairings between the manipulated and controlled variables and the detailed outcomes are included in the Supporting Information.In terms of a sharply different control loop search sequence,L2,L3,L4,and L1,the obtained relationshipsbetween the objective function of Eq.(6)and the locations of the controlled stages are included in Figs.S1 and S2 of the supplementary information for the fi rst and second rounds of iterations.With totally 13 times of steady-statemodel calculations,the CS3 isreached.In termsof asharply different pairings between the manipulated and controlled variables,RR-T26,FB1-T24,FB2-T20,and Qreb-T10,the obtained relationships between theobjective function of Eq.(6)and the locationsof the controlled stages are depicted in Figs.S3 and S4 of the supplementary information for the fi rst and second rounds of iterations.With totally 14 times of steadystate model calculations,the CS3 is also reached.In terms of a sharply different control loop search sequences L2,L3,L4,and L1 and sharply different pairings betw een the manipulated and controlled variables,RR-T26,FB1-T24,FB2-T20,and Qreb-T10,the obtained relationships between theobjective function of Eq.(6)and the locationsof the controlled stages are delineated in Figs.S5 and S6 of the supplementary information for the fi rst and second rounds of iterations.With totally 13 times of steady-state model calculations,the CS3 can again be reached.In spite of the sharply different control loop search sequences and different pairingsbetw een themanipulated and controlled variables,thethreesituations converge unexceptionally to the outcome of Figs.7 and 8 with somew hat similar computational requirement.These outcomes indicate that the proposed principle and procedure can w ork effectively for the development of TICsystems for the RDC-DRSseparating the HTSCR reaction.

6.Conclusions

The complicated dynamics of the RDC-DRSpreclude the TICsystem resulted from sensitivity analysis to achieve tight product quality control.This observation,how ever,does not necessarily signify that TICis inapplicable and ineffectivein thissituation.In fact,through thedeliberate discrimination of controlled stages in the development of the TIC system,its performance can,in most cases,be substantially enhanced and tight product quality control is likely to be achieved.Because the changesin controlled stagesusually do not in fl uencevery much the settling times of a TICsystem,an objective function that measures merely the SSDs in product qualities can frequently be formulated to describe the dynamic performance of the TICsystem.An iterative procedure hasbeen devised to search for the potentially best control con fi guration that givesriseto thesmallest TDsand SSDsin product qualities.Through the comparison w ith the one by sensitivity analysis,the resultant TICsystem suppresses considerably the SSDs in product qualities and can be used to secure tight control of the RDC-DRS.Since no prerequisites at all are needed on the steady-state and dynamic characteristics of the processto be controlled,the proposed novel principle and procedure are deemed to be applicable to any other types of complicated RDCs.

Supplem entary Material

Supplementary material to thisarticle can befound onlineat https://doi.org/10.1016/j.cjche.2018.11.023.