A novel kind of multiple steady states characteristics in the dividing w all column☆

Erwei Song,Zengxi Li*,Erqiang Wang*

School of Chemical Sciences,University of Chinese Academy of Sciences,Beijing 100049,China

Keywords:Dividing wall column Multiple steady states Vapor split ratio Liquid split ratio Recycling fl ow

ABSTRACT In this article,one kind of multiple steady states(MSS)phenomenon wasinvestigated for a dividing wall column(DWC).The four-section model constructed in Aspen Plusw as employed to simulate two DWCcases:mixture of n-hexane,n-heptane and n-octane;system of methanol,ethanol and n-propanol.It can be seen that there is a range of vapor split ratio in w hich multiple solutions of re fl ux ratio exist for fi xed DWCcon fi guration w ith the same feed and product streams.The width and the curve shapesof the MSSregion,and the number of solutions change w ith the liquid split ratio.This MSSphenomenon w as further explained using the component recovery around the prefractionator and the component recycling fl ow inside the DWC.This MSSphenomenon is helpful for DWCdesign by knowing the probable existence of multiple solutions in advance.

1.Introduction

The phenomenon of multiple steady states(MSS)is common in the fi eld of chemical engineering,such asthe classical S-curve of exothermic reactor[1],and that of distillation column[2,3].Actually,MSS,caused by the nonlinear nature of these systems,is important to their design,operation,and control.The optimal solution selected from these multiple solutions should consider its steady state and dynamic property.

As for the dividing w all column(DWC)show n in Fig.1,it is getting more and more attentions in industry and academia recently[4-22].Actually,it belongs to a fully thermally coupled column developed from a Petlyuk column,but has special advantages like saving both energy and capital/space by integrating tw o separate columns into one shell,w hich can avoid the problem of pressure balance betw een the prefractionator and the main column of the Petlyuk column.

Compared w ith the traditional distillation column,DWC has more coupled design parameters and optimization variables,of w hich internal vapor split ratio Rvand liquid split ratio Rlare important parameters.They have a strong impact on the energy ef fi ciency of DWC.Xiaolong Ge et al.[22]investigated the effect of the vapor split ratio on the operability of the divided w all column,and found that the total annual cost(TAC)is sensitive to the vapor split ratio.By simulation,the vapor split ratio is show n to be an important decision variable of DWCw ith uncertain feed composition.Maralani et al.[23]studied the relationship betw een the vapor split ratio and the TACof a DWCassuming a fi xed Rlof 0.5,and found that the positioning of the dividing w all and the decision on the vapor split may affect signi fi cantly the operability of DWC.Halvorsen et al.[24-27]studied the steady state behavior of the Petlyuk distillation column and found that it requires some kind of optimizing control in order to realize its full potential for energy saving since the solution surface of the criterion function is quite narrow,and the operation is very sensitive to certain disturbances.

One kind of MSSin the Petlyuk column and DWC[28-30]has been reported,w hich belongs to “input multiplicity”since multiple solutions of Rland Rvcan be found under the same re fl ux ratio.Wayburn and Seader[28]have been the fi rst to report the existence of such MSSin an interlinked separation column,using a differential arc-length homotopy continuation method,and in their following papersexplained the reasons of MSSbased on the degree-of-freedom analysis.For given con fi guration of columns and fi xed re fl ux ratio,they obtained four different solutions of Rland Rvw hich can give the same output product streams with the same feed stream.Wang[31]reported a group feature based on detailed comparison and analysis of these multiple solutions of BTX DWCcase.

Fig.1.Schematic diagram of(a)Petlyuk column and(b)dividing wall column.

In this w ork,another kind of MSSphenomenon(there is a range of vapor split ratio in which multiple solutions of re fl ux ratio exist for fi xed DWC con fi guration w ith the same feed and product streams.The w idth and the curve shapes of the MSSregion,and the number of solutions change w ith the liquid split ratio.)in DWCw as discovered,w hich has not been mentioned in the published literatures.Using a four-section model of DWCwith the help of process simulation softw are,tw o cases w ere studied,including the ideal system of hydrocarbon mixture of n-hexane(C6),n-heptane(C7)and n-octane(C8),and somewhat non-ideal system of methanol,ethanol and n-propanol.This MSS phenomenon w as illustrated in graphs based on the simulation results,and detailed discussion and explanations w ere given according to component recovery around the prefractionator.Finally,some factors which affect the distribution of MSSw ere studied.

2.DWCModel and Simulation Cases

Currently,there is a MultiFrac module in Aspen plus,w hich can be used for Petlyuk or DWCsimulation directly.Also,standard column modules in the model library of most process simulation softw are could be used for thispurpose.DWCactually belongsto afully thermally coupled distillation system connected by internal vapor and liquid streams,w hich can be modeled as a sequence of simple distillation columns.For steady state simulation of DWC,several kinds of models can be used such as pumparound model,two section model,and foursection model[5].Here the four-section model was used,as shown in Fig.2,including four tow er sections:the common rectifying section above the dividing w all,the common stripping section below the dividing w all,the prefractionator on the left side of the dividing w all,and the side-stream section on the right side of the dividing w all.Although it is sometimes dif fi cult to initialize and get convergence,the four-section model is straight forw ard,w ell understandable and nicely suitable for dynamic simulation,since it allow s for maximum fl exibility regarding speci fi cations for different column sections,and internal vapor and liquid splits.Most process simulation softw are can be used to set up the four-section model.The Rad Frac module of Aspen plus was used here,as show n in Fig.2.T1 can be seen as the prefractionator column.Mixer and Split modules w ere used to reduce the number of tear streams and computation time,and to conveniently de fi ne the internal liquid and vapor split ratios in the Split modules respectively.Here,the vapor split ratio Rvwasde fi ned asthe percentage of the vapor from thetop of T4 going to the prefractionator T1,and liquid split ratio Rlas the percentage of the liquid from the bottom of T3 going to T1.

Table 1 Feed parameters and purity speci fi cations of product streams for Case 1

Fig.2.Four section model of DWC.

Tw o cases w ere studied,including the ideal hydrocarbon mixture of n-hexane(C6),n-heptane(C7)and n-octane(C8),and the somew hat nonideal system of methanol,ethanol and n-propanol.Feed parametersand other speci fi cations were shown in Tables 1 and 2,respectively.For the vapor-liquid equilibrium,an ideal physical property system w as used for Case 1 and that of NRTLmethod for Case 2.The speci fi cations for the four Rad Frac modules were show n in Table 3.

Table 2 Feed parameters and purity speci fi cations of product streams for case 2

Table 3 Tower speci fi cations for the tw o cases

There are fi ve degrees of freedom in the optimization of threeproduct DWCw ith given feed and pressure:re fl ux ratio,side-stream fl ow rate,bottom product fl ow rate,liquid and vapor split ratios.Ideally,the purities(or impurities)of all three product streams should be controlled.If product purities are speci fi ed,three variables are needed,and the remaining tw o,usually the vapor split ratio and liquid split ratio,can be used for energy optimization.If impurities are speci fi ed,one more freedom is needed,so only Rvor Rlis left for energy optimization.To meet the product purity speci fi cations,the“design spec/vary”function in Aspen plus w as used to satisfy the three speci fi cations by varying three variables:re fl ux ratio of T3,the molar fl ow rate of side stream S,and molar fl ow rate of bottom product stream B.

In addition,the sensitivity analysis module in Aspen plus was used to analyze the liquid split ratio and vapor split ratio simultaneously.To fi nd multiple solution branches,sensitivity analysis is an effective tool w hen the blocks and streams of the process are selected to be not reinitialized in the simulation options.Furthermore,it is usually ef ficient to begin the calculationsfor a new row evaluation w ith the results of the previous row evaluation,w hich can take less computation time.Thus,it is easy to get a series of adjacent solutions continuously w hen moving point by point on one solution branch if one solution point of this branch can be found and used as a starting point.The remaining problem is how to fi nd a starting solution point for every one solution branch,w hich can be solved by trying different initial values manually or using a modi fi ed “design spec/vary”function.In addition,it is essential to use small steps in sensitivity analysis w hen approaching the turning point of the solution branch.For numerical solution of nonlinear algebra equations for strongly coupling systems such as DWC,the simultaneous method is usually better than the sequential method,especially w ith “design spec/vary”function,so the convergence option of solving “design spec/vary”w ith tear streams simultaneously can be good to be selected in Aspen plus.

3.Results and Discussion

Based on theabovefour-section model and speci fi cations,theresults of steady state simulation of DWCwere summarized and discussed in this section.Fig.3 show s the change of calculated re fl ux ratio w ith Rvat Rl=0.05,of w hich the right graph is partial enlargement draw ing of the left graph so as to show the MSSregion better.It can be seen from the left graph that converged solutions exist w ithin some certain range of Rv,such as Rvfrom 0 to 0.46 for Rl=0.05,w hile out of this range no solutions can be found,and to the both ends of this range the calculated re fl ux ratio increases sharply.It w as also found that there is a sub-range w here three steady state solutions of re fl ux ratio exist for the same Rv,as show n in the rectangular shadow area of the right graph.This idea of MSSphenomenon w as fi rst inspired by some random simulation results that three different re fl ux ratios w ere found for thesame Rv,and then con fi rmed after three solution branches,just like that AB,BC,and CDin Fig.3,w ere obtained by sensitivity analysis.These multiple solutions have different re fl ux ratios,w hich mean different energy consumption of DWC.Actually,the lowest solution branch ABis w hat researchers and industries are interested in,since it means low energy consumption,especially the lowest fl at region such asthe OBsegment in the right graph.Thelow est solution point of re fl ux ratio is important and expected for DWCdesign,w hich is very close to point Band usually located in the MSSregion,so this MSSphenomenon may be important to remind people to design DWCusing its most appropriate solution point,although this MSSregion isrelativeshort compared w ith the w hole region.Sometimes it is possible for the DWC model to converge to these solution points on BCsince they are not far from the points on OB,and these points w ith higher energy consumption may be used for DWCdesign by mistake if people do not know this kind of MSSphenomenon in advance.Besides,it also can be seen that the optimized vapor split ratio for the minimized re fl ux ratio is about 0.15 and larger than the liquid split ratio of 0.05,which can be explained from the liquid feed of the prefractionator column,in w hich more vapor fl ow from the main column is needed to input into the bottom of the prefractionator to maintain the suitable ratio of liquid and vapor fl ow there.

Fig.3.The effect of vapor split ratio on re fl ux ratio for Case 1 at Rl=0.05.

Fig.4.The change of fl owrates of product streams w ith R v for Case 1 at Rl=0.05.

Fig.5.The fl ow rate changes of streams at both end of the prefractionator w ith R v for Case 1 at Rl=0.05.

All solution points in Fig.3 w ere calculated and converged w ith the“design spec/vary”function,so product purity speci fi cations w ould be met w ell.Calculated mole fl owrates of product streams w ere listed in Fig.4,from w hich it can be seen that w hen Rvincreases,the fl ow rate of side-stream S basically remains constant w hile that of D and B changes slightly in the opposite direction.This MSSphenomenon can be seen as an “input multiplicity”since it has the same input of feed stream and almost the sameoutput of product streams,but hasdifferent re fl ux ratios,w hich means different liquid and vapor fl ow s inside the DWCcolumn.Fig.5 show s the fl ow rate change of four streams at the top and bottom of the prefractionator w ith Rvat Rl=0.05.It can be seen that the vapor fl ow rate through the prefractionator almost remainsunchanged whilethat of theliquid phasehasadifferencebecause of the liquid feed.MSSphenomenon also exists on these curves in Fig.5 and at optimum Rvthe fl ow rates of the four streams reduce to their minimum values.

Fig.6 is similar to Fig.3 but listsmore curves at different Rl,and only the partial enlarged draw ing of the MSSregion is displayed.It can be seen that w hen Rlincreases,this MSSregion also exists and slow ly movesto the right sidealong the x axisw hich meanslarger Rvisneeded,and that the number of solutionschanges from 3 at Rl=0.4 to 2 at Rl=0.5,so there should be a critical value of Rlbetw een 0.4 and 0.5 at w hich two turning pointsare reduced to one on the MSScurve,which can also be seen from the comparison of the changing trend at Rl=0.42 w ith that at Rl=0.45 in Fig.7.In addition,the MSSregion will be reduced or degenerated to a smaller range when Rlincreases,so this phenomenon is easier to be ignored for large Rl.

Fig.7.The effect of R v on re fl ux ratio for Case1 at Rl=0.42 and Rl=0.45.The right graph is a partial enlarged drawing of MSSregion in the left graph.

Fig.6.The effect of R v on re fl ux ratio for Case 1 at different Rl.

Fig.8.The effect of R v on re fl ux ratio for Case 2 at Rl=0.1.

Fig.9.The effect of R v on re fl ux ratio for Case 2 at Rl=0.4.

Fig.10.The effect of R v on re fl ux ratio for Case 2 at Rl=0.7.

Figs.8-10 show theeffect of Rvon calculated re fl ux ratio for Case 2 at Rl=0.1,0.4 and 0.7,respectively.The right graphs of Figs.9 and 10 are partial enlargement draw ing of the MSSregion in the left graphs,respectively.It can be seen that MSSphenomenon also exists for Case 2,and is similar to that of Case 1 except that the change from three solutions to tw o solutions appears at a smaller value of Rl,such as Rlof 0.4 for Case 2.

Con fi rmed by the above results of tw o cases,this MSSphenomenon really exists.It isa common phenomenon for DWCand decided by special internal con fi guration of DWC,having no or little relationship w ith thermodynamic property.Being a ideal system or non-ideal system,only affects behavior of phase equilibrium inside the column,dif fi cult or easy of separation,and product purity that can be reached fi nally.How ever,as that stressed above,the range of liquid split ratio or vapor split ratio w ithin w hich MSSphenomenon happens,could vary w ith the non-ideality of the system.

To further evaluate the effect of product purity speci fi cations on this phenomenon,the follow ing new speci fi cations w ere used for case 1:xD,C6=0.99,xS,C7=0.99,and xB,C8=0.99,w here D,Sand Bstand for three product streams,respectively.Then a new simulation w as run and the results at Rl=0.1 w ere show n in Fig.11.It can be seen that this MSSphenomenon also exists and is similar to that of old speci fi cations in Fig.6.There is no obvious difference betw een these tw o kinds of speci fi cations,so this MSSphenomenon needs to be explained from other aspects.

Fig.11.The effect of R v on re fl ux ratio for Case 1 at new product speci fi cations.

To explain this MSS phenomenon,material balance around the prefractionator column T1 w as done to calculate the component recoveries at the T1 top,ri,pretop,w hich is de fi ned as follow s:

w here Vpretopand Lpretopstand for the mole fl ow rate of vapor and liquid stream at the T1 top,respectively,w hile yi,pretopand xi,pretopmean mole fraction of component i of stream Vpretopand Lpretop,respectively.The recovery of middle component Binsidethe prefractionator is an important parameter as pointed out in some papers[11,12].It has a relation w ith the minimized vapor fl ow in the column,w hich alw ays changes w ith Rland Rv.

Fig.12 show sthe recovery of the middle component at Rl=0.05 for case 1 and the bottom graph is partial enlargement drawing of the MSS gray region in the up graph.It can be seen that similar MSSphenomenon exists,and w hen Rvincreases,the recovery of component B increases continually and monotonously along the curve from negative value to positive value,even larger than 1.0.For a traditional distillation column,thecomponent recovery islimited to arange of 0.0 to 1.0,while this isnot necessary for DWCand there may be a recycling fl ow of component Baround the dividing w all.For the dividing w all column,negativerecovery meansno component B fl ow out of the prefractionator top,namely,the liquid stream fl ow ing from the main column into the prefractionator top w ould have more component B than the vapor stream fl ow ing out oppositely there.Thus,there w ill be a net mass fl ow of Binto the prefractionator top,w hich plus mass fl ow of Bin the feed will totally fl ow out at the prefractionator bottom.Positive recovery less than 1.0 means component B fl ow out on both top and bottom of the prefractionator.Similarly,if recovery of Bis larger than 1.0,all component Bin the feed w ill fl ow out at the prefractionator top,and meanw hile,there w ill be a net mass fl ow of component B from the main column to the prefractionator at the prefractionator bottom.

Fig.12.The effect of R v on middle component recovery at the prefractionator top for Case 1 at Rl=0.05.

Fig.13 shows the recovery of C6and C8at the prefractionator top.It can be seen that similar MSSphenomenon exists,and the C6recovery increases quickly to 1.0 w hen Rvincreases,w hile the C8recovery remains almost at 0.0 fi rstly and then increases to some larger value,w hich means that w hen Rv increases,more and more C6and C8w ill be brought up and fl ow out of the prefractionator top.The C6recovery at the top of prefractionator maintains w ithin a certain range w hile that of C7and C8will increase sharply.So based on Figs.12 and 13,it can be seen that these multiple solutions w ithin the MSSregion have different component recoveries,w hich mean different net mass fl ow of component around the prefractionator and different component recycling around the dividing w all for the w hole DWC.Normally,recycling of component around the dividing w all will waste energy,so the solution with higher re fl ux ratio will have more recycling fl owrate,while that w ith low est re fl ux ratio w ill have no recycling.Therefore,it is useful to judge the normal design solution according to the degree of recycling around the dividing w all.

Fig.13.The effect of R v on light and heavy component recovery at the prefractionator top for Case 1 at Rl=0.05.

Fig.14.The change of R v on impurity of sidestream for Case 1 at Rl=0.05.

Table 4 The calculated impuritiesof sidestream at optimized R at different Rl for Case 1

When Rvchangesat given Rl,the impurity of Dand Bw ill not change greatly since D and Bmainly consist of tw o components,C7and C6,C7and C8,respectively.While impurity of Sw ill change to a certain extent because only purity of C7isspeci fi ed for Shere,so themole fraction of C6and C8can change freely w ith thelimitation that their sum should equal to 0.01,just as show n in Fig.14.It can be seen that w ith Rvincreasing,the C8composition in S is fi rst increasing from 0 to 0.01 and then dropping dow n,w hilethetrend for C6purity isin theoppositedirection.The optimization re fl ux ratio is obtained at Rv≈0.153 w ith xS,C6≈0.0017,xS,C8≈0.0083.These calculated impuritiesof side-stream at optimized re fl ux ratio Roptfor other Rlw ere listed in Table 4.It can be seen that these impurities change with Rl,so for DWCdesign,it is important to give appropriate purity speci fi cations,especially for side-stream product when impurities are speci fi ed,which would greatly in fl uence the energy consumption.In addition,there may be tw o solutions for the same impurities of side-stream,such as for xS,C6=0.00172,xS,C8=0.00828,which can be seen from Fig.14.

Fig.15.The effect of R v and Rl on re fl ux ratio for Case 1.

Fig.15 shows the 3Dsolution surface of R(z)on the plane of Rl(x)-Rv(y)for Case 1.Re fl ux ratio can be used as an indication of energy consumption of DWC.From Fig.15,it can be seen that the bottom of the surface standing for low energy consumption is approximately in the diagonal direction of the x-y plane,so Rland Rvshould change simultaneously along the diagonal direction to maintain low re fl ux ratio and high energy ef fi ciency.

Fig.16.The change of minimum re fl ux ratio with Rl and R v for Case 1.

Fig.16 shows the calculated minimum re fl ux ratio and corresponding Rvat each Rl.It can be seen that the curve is almost in the diagonal direction,which agrees w ell with that trend in Fig.15,and there is a fl at segment corresponding to lowest re fl ux ratio in the middle region,w hich means that too large or small Rlis not suited for thisexample case.

To investigate the effect of C7purity speci fi cation of side-stream on the MSSregion,new speci fi cations of xS,C7=0.95 and xS,C7=0.90 w ere used,respectively.The simulation results are show n in Fig.17,from w hich it can be seen that with xS,C7decreasing from 0.99 to 0.90,the MSSregion moves to the right side along the x axis while remains almost the same w idth,and re fl ux ratio at both ends of the x axis drop dow n sharply.Fig.18 show s the change of mole fl ow rates of D,Sand Bw ith Rvat xS,C7=0.95 and xS,C7=0.90,respectively.It can be seen that with Rvincreasing,the fl owrate of Sremains almost unchangeable for fi xed xS,C7and changesfor different xS,C7,w hile the fl ow rate of Dand B changes to an extent for fi xed xS,C7,and w hen xS,C7decreases,the amount of fl ow rate change increases.Since decreasing of xS,C7speci fi cations means more C6or C8w ould appear in the sidestream,so w hen Rvchanges the fl ow rates of D or Bw ill change w ith a larger amount to meet the total material balance of DWC.

Fig.17.The change of re fl ux ratio with different C7 purity of sidestream for Case 1.

Fig.18.The fl owrate changesof D,S,Bwith R v at different C7 purity of sidestream for Case 1.

4.Conclusions

In this article,one kind of new multiple steady states phenomenon in a dividing w all column w as investigated using tw o example cases.The MSS region is limited w ithin some certain range of vapor split ratio,out of w hich only a single solution exists.In addition,it w as found that the shape and structure of MSSregion w ill change w ith the liquid split ratio.Although this MSSregion is relatively short compared w ith the total range of Rv,it really exists and is useful for DWCdesign since it probably includes the global minimum solution point of energy consumption and can be used to explain some strange simulation results.This MSSphenomenon w as explained according to component recovery around the prefractionator column and component recycling around the dividing w all,the solution w ith higher re fl ux ratio w ill have more serious recycling and need more energy consumption.In addition,the effect of product purity speci fi cation of side-stream on this MSSregion was studied.