Enhanced temperature difference control of distillation columns based on the averaged absolute variation magnitude

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

Keywords:Distillation column Temperature difference AAVM Temperature inferential control Process control

ABSTRACT Temperature difference control (TDC) schemes can clearly suppress the adverse influence of pressure variations on product quality control of various distillation columns(DCs)by employing temperature differences (TDs) between the sensitive stage temperature (TS ) and reference stage temperature (TR ),i.e.,TS -TR ,to infer the controlled product qualities.However,because the TDC scheme has failed to specially take the corresponding relationship between the TD employed in each control loop and the controlled product quality into account,it may suffer from relatively large steady-state errors in the controlled product qualities.To address this problem,an enhanced TDC(ETDC)scheme is proposed in the current article,in which an enhanced TD(ETD),i.e.,TS -α×TR ,is employed to replace the conventional TD for each control loop.While the locations of the sensitive and reference stages of the ETD are respectively determined according to sensitivity analysis and SVD analysis,the adjusted coefficient α is set to be the ratio between the averaged absolute variation magnitudes (AAVMs) of the TS and TR so that the relationship between the TS and TR can be appropriately coordinated.With reference to the operations of three different distillation systems,i.e.,one conventional DC distilling an ethanol(E)/butanol(B)binary mixture,one conventional DC distilling an E/propanol(P)/B ternary mixture,and one dividing-wall distillation column distilling an E/P/B ternary mixture,the performance of the ETDC scheme is assessed by compared with the conventional TDC scheme and the double TD control(DTDC)scheme.The dynamic simulation results show that the ETDC scheme is better than the conventional TDC scheme with reduced steady-state errors in the controlled product qualities and improved dynamic responses,and is comparable with the DTDC scheme despite the less temperature measurements are employed.

1.Introduction

Because temperature inferential control has the advantages of high reliability,short delay,and low price,it is more acceptable than composition control in controlling various distillation columns (DCs) [1–3].Unfortunately,temperature inferential control suffers from inevitable steady-state errors in the maintained product qualities,and this results in it can’t be used once there are very high requirements to control qualities of the controlled products.Facing this problem,sometime composition-temperature cascade control has to be employed for the purpose of entirely eliminating the steady-state errors in the controlled product purities [4–6],despite of the certain deterioration in dynamic performance brought about by the employment of composition measurements.To substantially reduce the steady-state errors but simultaneously keep the superiority of temperature inferential control on dynamic performance,temperature difference control(TDC) schemes that use a temperature difference (TD) between the sensitive stage temperature (TS) and reference stage temperature(TR),i.e.,TS-TR,to infer the controlled product purity for each control loop were proposed [7–10].In view of the fact that the adverse influences of pressure variations within DCs on the TSand TRare similar,the TD can offset these two kinds of adverse influences and thereby provide a more accurate estimation to the controlled product purity.However,because this conventional TDC scheme has failed to specially take the corresponding relationship between the TD employed in each control loop and the controlled product purity into account,the resulted steady-state errors are still a little large,particularly for ternary and multicomponent distillation systems.This raises actually a thorny but important problem waiting to be addressed in the field of temperature inferential control.

In fact,it is clear that to further enhance the conventional TDC scheme we must focus on how to increase the corresponding relationship between the TD synthesized for each control loop and the maintained product quality.The method of approaching the sensitive and reference stages may be a good choice,however,it is hard to determine how close they should be located.If they are too close,the synthesized TD will be insensitive to the corresponding manipulated variable;otherwise,the inference accuracy of the synthesized TD for the controlled product quality can’t be significantly increased.All in all,relying on the method of approaching the sensitive stage and reference stage to enhance the TD synthesized for each control loop seems to be full of uncertainty and troublesome.Furthermore,the locations of the sensitive and reference stages are better to be respectively determined by sensitivity analysis and SVD analysis since these two analysis methods have been demonstrated to be reliable for guarantying the sensitivity of the derived TD with the corresponding manipulated variable[7,11,12].Without adjusting the locations of the sensitive and reference stages of the TD,the only feasible method left for improving the TD is to introduce an additional adjusted coefficient α to coordinate the relationship between the TSand TRsubtracted in each TD,namely,to employ an enhanced TD(ETD)equaling to TS-α×TRto infer the controlled product purity in each control loop.Since the adjusted coefficient α can greatly influence the quality of the final ETD synthesized,its value must be carefully determined and this represents actually the most difficulty of configuring an enhanced TDC (ETDC) scheme.To answer the question of how to derive a high-quality ETDC scheme for a given DC will be the main mission of the current article.

2.Derivation Method of the ETDC Schemes for the Operations of Various DCs

The key of effectively deriving an ETDC scheme for a given DC lies in to synthesize a good ETD for every control loop,as well as the key of effectively synthesizing an ETD lies in to determine two appropriate locations for the sensitive stage and reference stage and an appropriate value for the adjusted coefficient α.In view of the fact mentioned in the above section that the locations of the sensitive and reference stages should be respectively determined according to sensitivity analysis and SVD analysis,the main difficulty left for synthesizing the ETD is how to determine the value of the adjusted coefficient α.Next,to clearly explain the determination method of the adjusted coefficient α,the determination of the adjusted coefficient α of an ETD synthesized for one control loop of a conventional DC (which will later be studied in Example II) is taken for example.Fig.1 shows the required variations of the TSand TRinvolved in this ETD for various variations in feed compositions while the controlled product purities are tightly maintained on their specifications.The coefficients α1,α2,and α3 represent the ratios between the required temperature variations of the TSand TRfor±10%variations in feed compositions of E,P,and B.respectively.It can be found from Fig.1 that,if there is only one type of feed composition disturbance needed to consider,the determination of the adjusted coefficient α is very simple.We just need to select the one corresponding to the considered feed composition disturbance from coefficients α1,α2,and α3 as the adjusted coefficient,and the constructed ETD can provide a relatively accurate inference to the controlled product purity(this can be reflected from the fact that the curve representing the TSis almost coincident with the curve representing the α1 × TR,α2 ×TR,or α3 ×TR).More specially,coefficient α1 should be selected as the adjusted coefficient if considering only the variations in E feed composition,coefficient α2 should be selected as the adjusted coefficient if considering only the variations in P feed composition,and coefficient α3 should be selected as the adjusted coefficient if considering only the variations in B feed composition.However,if all types of feed composition disturbances need to be considered,it is hard to decide which one of the coefficients α1,α2,and α3 should be selected because no one of them can cover all types of disturbances in feed composition.To deal with this problem,a performance metric,the averaged absolute variation magnitude (AAVM),defined in Eq.(1) is employed to measure the variation degrees of one considered variable V (such as TS,TR,TD,etc.),under the promise of that all controlled product qualities have been strictly maintained on their specifications for all kinds of disturbances in feed component composition.

Fig.1.Required variations of the TS and TR for various variations in feed compositions of E,P,and B while the controlled product purities are tightly maintained on their specifications.

where NC represents the number of feed components and λi(i=1,...,2NC) is weighting coefficient that measures the relative importance of suppressing feed composition disturbance (λiis uniformly set to be 1 in the current article because all kinds of disturbances in feed component composition are treated the same).Referencing our previous articles can find more information on the definition of the AAVM [13,14].With the aid of the AAVM,the adjusted coefficient α can be directly set to be the ratio between the AAVMs of the TSand TRfor each control loop,as shown in Eq.(2).One thing must be pointed out here,that is,two ETDs can be synthesized for each control loop because two reference stages can be selected out for one sensitive stage according to SVD analysis.For maximally improving the inference accuracies for the controlled product purities,the AAVMs of these two ETDs synthesized must be calculated and the one ETD with a smaller AAVM should be finally selected.

In the next three sections,the performance of the ETDC scheme will be assessed by compared with the conventional TDC scheme.Moreover,considering the fact that the DTDC scheme developed by the new method proposed in our recent work displays rather good steady-state and dynamic performances for the operation and control of various DCs [14],it is also chosen as a reference in the current article.

Fig.2.Steady-state design of the conventional DC distilling an E/B binary mixture and its TDC,ETDC,and DTDC schemes(Example I):(a)steady-state design,(b)TDC scheme,(c) ETDC scheme,and (d) DTDC scheme(1 atm=105Pa).

3.Example I:Operation of a Conventional DC Distilling an E/B Binary Mixture

3.1.Process design of the conventional DC

Fig.2a displays the process design of the conventional DC distilling an E/B binary mixture.A binary mixture feed consisting of 40 mol%E and 60 mol%B is fed onto stage 15 with a feed flow rate of 1 kmol·s-1.Two products with purities of 99 mol% are respectively withdrawn from stages 1 and 30.Throughout the work,the simulations are carried out in Aspen Plus and Aspen Dynamic,UNIFAC thermodynamic model is used to describe the characteristics of the feed,top pressure is set to be 1 atm(1 atm=101,325 Pa),and stage pressure drop is set to be 0.0068 atm.

3.2.Derivations of the TDC,ETDC,and DTDC schemes

To maintain the two product purities,a top control loop manipulating distillate flow rate D and a bottom control loop manipulating reboiler heat duty QREBare needed to be employed,respectively.Fig.3 shows the results of sensitivity analysis and SVD analysis for the D and the QREB.According to the results of sensitivity analysis,stages 5 and 26 should be selected as two candidate sensitive stages,because two peaks occur at these two locations.Furthermore,considering the fact that stage 5 is closer to the D and stage 26 is closer to the QREB,the former is selected as the sensitive stage of the top control loop and the latter is selected as the sensitive stage of the bottom control loop (they are marked with red pentagrams).According to the results of SVD analysis,stages 2 and 14 and stages 17 and 29 (marked with green triangles) should be respectively selected as the reference stages correspond to stages 5 and 26,because these stages display very large absolute values against the sensitive stages.Therefore,two conventional TDs of TD1=T5-T14and TD2=T5-T2can be synthesized for the top control loop,and two conventional TDs of TD1=T26-T17and TD2=T26-T29can be synthesized for the bottom control loop.Table 1 lists the AAVMs of the temperatures of the above-mentioned sensitive and reference stages and the ratios between the AAVMs of the TSand TR.According to the data given in Table 1,two ETDs of ETD1=T5-0.5535× T14and ETD2=T5-90.9811× T2and two ETDs of ETD1=T26-0.6459× T17and ETD2=-T26-73.7662 × T29are respectively synthesized for the top and bottom control loops.Table 2 gives the AAVMs of the conventional TDs and ETDs synthesized for the top and bottom control loops.The one with a smaller AAVM between the two candidate TDs/ETDs synthesized for the top and bottom control loops should be finally adopted,and the derived TDC and ETDC schemes are shown in Fig.2b and c,respectively.The DTDC scheme is given in Fig.2d.Because the three DCs studied in the current work are actually same with those studied in our previous work about effectively deriving the DTDC scheme [14],the details of the derivations of the DTDC schemes are no longer given here.

3.3.Comparison between the ETDC,TDC,and DTDC schemes

Fig.3.Results of sensitivity analysis and SVD analysis(Example I):(a)sensitivity analysis for the D,(b)sensitivity analysis for the QREB ,(c)SVD analysis for the D,and(d)SVD analysis for the QREB .

Table 1 AAVMs of the TS and TR and their ratios (Examples I to III)

A PI controller(gain=0.5 and integral time=0.3 min)is used to maintain the feed flow rate,a pure P controller is used to control the reboiler level,and a PI controller (gain=20 and integral time=12 min)is used to control the top pressure of the conventional DC[15].Two PI controllers are employed in the temperature inferential control loops,and each temperature measurement is assumed to involve a dead time equaling to 1 minute.The tuning of the temperature controller parameters is carried out by the Tyreus-Luyben tuning rule that is built in Aspen Dynamic,and at least three rounds are performed to ensure the quality of the obtained controller parameters.The controller parameters of the ETDC,TDC,and DTDC schemes are tabulated in Table 3 (the controller parameters for Examples II and III are also given in this table).It should be emphasized that the same settings of the pressure controller,feed flow rate controller,liquid level controller,dead time of temperature measurement,and tuning method for temperature inference controller are used in the following two examples.

In Fig.4,the dynamic responses of the controlled product qualities of the conventional DC controlled under the ETDC,TDC,and DTDC schemes after facing ±10% variations in feed compositions are given.Throughout the article,the curves in black correspond to the positive disturbances,and the curves in gray correspond to the negative disturbances.It is obvious that the ETDC and DTDC schemes display similar magnitudes of overshoots and settling times,and they generally show better dynamic performances than the TDC scheme.The steady-state errors in the controlled product qualities are tabulated in Table 4.The ‘‘X/Y”in the last line of the table means that in a total of Y scenarios there are X scenarios that the current control scheme shows a bigger steady-state deviation than the ETDC scheme.It can be seen that the ETDC scheme shows smaller steady-state errors in the controlled product qualities than does the TDC scheme,and shows comparable steady-state performance with the DTDC scheme.In addition,the ETDC scheme results in the smallest maximum deviation (they are underlined in Table 4) between the three considered temperature inferential control schemes.

4.Example II:Operation of a Conventional DC Distilling an E/P/B Ternary Mixture

4.1.Process design of the conventional DC

Fig.5a shows the process design of the conventional DC distilling an E/P/B/ternary mixture.A ternary mixture feed consisting of 33.3 mol%E,33.3 mol%P,and 33.4 mol%B is fed onto stage 20,and the feed flow rate is 1 kmol·s-1.A top product with 99 mol%E and a bottom product with 1 mol% E are respectively withdrawn from stages 1 and 40.

4.2.Derivations of the TDC,ETDC,and DTDC schemes

Being the same with Example I,a top control loop with D as manipulated variable and a bottom control loop with QREBasmanipulated variable are respectively employed to maintain the two product purities.According to the results of sensitivity analysis and SVD analysis given in Fig.6,two conventional TDs of TD1=T10-T21and TD2=T10-T2and two conventional TDs of TD1=T32-T20and TD2=T32-T39are respectively synthesized for the top and bottom control loops.With reference to the ratios between the AAVMs of the TSand TRgiven in Table 1,two ETDs of ETD1=T10-0.6074 × T21and ETD2=T10-55.7674 × T2and two ETDs of ETD1=T32-0.7577 × T20and ETD2=T32-1.2483 ×T39are respectively synthesized for the top and bottom control loops.By selecting the TD/ETD with a smaller AAVM for each control loop in terms of Table 2,the TDC and ETDC schemes are derived and shown in Fig.5b and c,respectively.Moreover,the DTDC scheme is given in Fig.5d.

Table 2 AAVMs of the TDs and ETDs (Examples I to III)

Table 3 Controller parameters of the TDC,DTDC,and ETDC schemes (Examples I to III)

Fig.4.Comparison between the ETDC,TDC,and DTDC schemes after facing ±10% variations in feed compositions (Example I):(a) variations in E feed composition,and (b)variations in B feed composition.

Table 4 Steady-state errors for ±10% variations in feed compositions of E and B (Example I)

4.3.Comparison between the ETDC,TDC,and DTDC schemes

Fig.7 shows the dynamic responses of the controlled product qualities of the conventional DC controlled under the ETDC,TDC,and DTDC schemes after facing ±10% variations in feed compositions.Compared with the TDC scheme,the ETDC scheme shows better dynamic performances except for the top product after facing±10%variations in feed composition of component P.Compared with the DTDC scheme,the ETDC scheme shows comparable even better dynamic performances.The steady-state errors in the controlled product qualities are listed in Table 5.The TDC and DTDC schemes have,respectively,10 and 5 scenarios that shows bigger steady-state deviations than that of the ETDC scheme for all of the 12 scenarios.As for the maximum deviation,the ETDC scheme is smaller than the TDC scheme but is bigger than the DTDC scheme.

5.Example III:Operation of a DWDC Distilling an E/P/B Ternary Mixture

5.1.Process design of the DWDC

Fig.8a displays the process design of the DWDC distilling an E/P/B ternary mixture.A ternary mixture feed consisting of 33.3 mol%E,33.3 mol%P,and 33.4 mol%B is fed onto stage P32(P denotes the prefractionator) with a feed flow rate of 1 kmol·s-1.The top,sidestream,bottom products with purities of 99 mol% are respectively withdrawn from stages 1,26,and 64.The dividing wall runs from stage 13 down to stage 47.

5.2.Derivations of the TDC,ETDC,and DTDC schemes

A top control loop manipulating D,a sidestream control loop manipulating sidestream flow rate S,a bottom control loop manipulating QREB,and a prefractionator’s control loop manipulating liquid split ratio RLare respectively employed to maintain the purities of the top,sidestream,bottom,and prefractionator’s products.According to the results of sensitivity analysis and SVD analysis given in Fig.9,two conventional TDs of TD1=T9-TP21and TD2=T9-T2are synthesized for the top control loop,two conventional TDs of TD1=T39-T26and TD2=T39-T48are synthesized for the sidestream control loop,two conventional TDs of TD1=T56-T48and TD2=T56-T63are synthesized for the bottom control loop,and two conventional TDs of TD1=TP21-TP15and TD2=TP21-TP32are synthesized for the prefractionator’s control loop.With reference to the ratios given in Table 1,two ETDs of ETD1=T9-0.5584 × TP21and ETD2=T9-68.0782 × T2are synthesized for the top control loop,two ETDs of ETD1=T39-11.8913 × T26and ETD2=T39-0.9982 × T48are synthesized for the sidestream control loop,two ETDs of ETD1=T56-0.8607×T48and ETD2=T56-23.7815 × T63are synthesized for the bottom control loop,and two ETDs of ETD1=TP21-1.1179 × TP15and ETD2=TP21-0.8495 ×TP32are synthesized for the prefractionator’s control loop.By selecting the TD/ETD with a smaller AAVM for each control loop in terms of Table 2,the TDC and ETDC schemes are derived and shown in Fig.8b and c,respectively.Moreover,the DTDC scheme is given in Fig.8d.

5.3.Comparison between the ETDC,TDC,and DTDC schemes

Fig.10 displays the dynamic responses of the controlled product qualities of the DWDC controlled under the ETDC,TDC,and DTDC schemes after facing ±10% variations in feed compositions.Compared with the TDC scheme,the ETDC scheme shows generally reduced overshoots for ±10% variations in feed composition of components E and B,but comparable dynamic performances for±10% variations in feed composition of component P.However,the dynamic performance of the DTDC scheme seems to be a little better than that of the ETDC scheme.Table 6 tabulates the steadystate errors in the controlled product qualities.The TDC and DTDC schemes have,respectively,12 and 6 scenarios that shows bigger steady-state deviations than that of the ETDC scheme for all of the 18 scenarios.As for the maximum deviation,the ETDC scheme is smaller than the TDC scheme but is bigger than the DTDC scheme.

6.Discussion

Based on the three distillation systems studied,the superiority of the ETDC scheme for the operation and control of various DCs has been fully demonstrated.The ETDC scheme not only greatly reduces the steady-state errors but also clearly improves the dynamic responses for various disturbances in feed composition compared with the conventional TDC scheme.Meanwhile,despite employing less number of temperature measurements,the ETDC scheme shows comparable steady-state and dynamic performances with the DTDC scheme.This significant improvement in control performance of the ETDC scheme comes certainly from the employment of the ETD in each control loop.More specifically,since the relationship between the TSand TRis appropriately coordinated with the aid of the adjusted coefficient α,the constructed ETD can display a stricter corresponding relationship with the maintained product purity(this can be reflected from the fact that the ETDs have smaller AAVMs than the alternative TDs with reference to Table 2),thereby can provide a more accurate estimation to the product purity to be controlled.

Fig.5.Steady-state design of conventional DC distilling an E/P/B ternary mixture and its TDC,ETDC,and DTDC schemes(Example II):(a)steady-state design,(b)TDC scheme,(c) ETDC scheme,and (d) DTDC scheme.

Although there are exist many feasible methods to enhance the temperature inferential control of DCs,such as temperature inferential control with regression estimators[16],temperature control with variable stage temperature set-point[17],and pressure compensated temperature control [18],etc.,they are usually complex in principle and hard to be applied to real industry.In contrast,the ETDC scheme proposed in the current article is rather simple in principle and its derivation does not need to take too much calculation efforts.These advantageous lead the ETDC scheme to be a powerful alternative for enhancing the temperature inferential control of DCs.

7.Conclusions

Fig.6.Results of sensitivity analysis and SVD analysis(Example II):(a)sensitivity analysis for the D,(b)sensitivity analysis for the QREB ,(c)SVD analysis for the D,and(d)SVD analysis for the QREB .

Fig.7.Comparison between the ETDC,TDC,and DTDC schemes after facing ±10% variations in feed compositions (Example II):(a) variations in E feed composition,(b)variations in P feed composition,and (c) variations in B feed composition.

Table 5 Steady-state errors for ±10% variations in feed compositions of E,P,and B (Example II)

Fig.8.Steady-state design of the DWDC separating an E/P/B ternary mixture and its TDC,ETDC,and DTDC schemes(Example III):(a)steady-state design,(b)TDC scheme,(c)ETDC scheme,and (d) DTDC scheme.

Fig.9.Results of sensitivity analysis and SVD analysis(Example III):(a)sensitivity analysis for the D,(b)sensitivity analysis for the S,(c)sensitivity analysis for the QREB ,(d)sensitivity analysis for the RL ,(e) SVD analysis for the D,(f) SVD analysis for the S,(g) SVD analysis for the QREB ,and (h) SVD analysis for the RL .

Fig.10.Comparison between the ETDC,TDC,and DTDC schemes after facing ±10% variations in feed compositions (Example III):(a) variations in E feed composition,(b)variations in P feed composition,and (c) variations in B feed composition.

In the current work,an ETDC scheme is proposed for the control of the DC,in which an ETD(TS-α × TR)is used to infer the maintained product quality in each control loop.While the locations of the sensitive and reference stages of the ETD are respectively determined according to sensitivity analysis and SVD analysis,the adjusted coefficient α is set to be the ratio between the AAVMs of the TSand TRso that the relationship between the TSand TRcan be appropriately coordinated.In terms of three different distillation systems,i.e.,one conventional DC separating an E/B binary mixture,one conventional DC separating an E/P/B ternary mixture,and one DWDC separating an E/P/B ternary mixture,the performance of the ETDC scheme is assessed by compared with the conventional TDC scheme and the DTDC scheme.According to the closed-loop evaluation results obtained,the ETDC scheme is better than the TDC scheme because of not only the reduced steady-state errors but also the improved dynamic performances.Moreover,the ETDC scheme is generally comparable with the DTDC scheme on steady-state and dynamic performances in despite of the fact that the former employs less temperature measurements.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The research funding is from China Postdoctoral Science Foundation (No.2019M650453),Fundamental Research Funds for theCentral Universities(ZY1930),National Natural Science Foundation of China (21808007,21878011,21676011,and 21576014),and Open Foundation of State Key Laboratory of Chemical Engineering(No.SKL-ChE-18B01).

Table 6 Steady-state errors for ±10% variations in feed compositions of E,P,and B (Example III)