Circumventing the Black-Hole Problem in Design and Control of

Article pubs.acs.org/IECR

Circumventing the Black-Hole Problem in Design and Control of Dividing-Wall Distillation Columns Yufeng Wang,† Kejin Huang,*,† Shujun Luan,† Wei Chen,† San-Jang Wang,‡ and David S. H. Wong§ †

College of Information Science and Technology, Beijing University of Chemical Technology, Beijing 100029, People’s Republic of China ‡ Department of Chemical and Material Engineering, Ta Hwa Institute of Technology, Hsinchu 307, Taiwan, Republic of China § Department of Chemical Engineering, National Tsing Hua University, Hsinchu 300, Taiwan, Republic of China ABSTRACT: Owing to the great degree of mass integration and energy integration between the prefractionator and the main distillation column, it is usually infeasible to achieve a four-point composition control policy (i.e., the control of the main compositions of the three products and the ratio of the two impurities in the intermediate product) in a dividing-wall distillation column (DWDC), and this restricts, to a certain extent, process applicability and flexibility. The issue is referred to as the blackhole problem in the current work and has received very little attention so far. In this paper, an attempt is made to tackle this intricate problem, and the number of stages in each section of the DWDC is used as adjustment variables to coordinate the relationship between the prefractionator and the main distillation column involved. A simple and yet effective procedure is devised to guide the structural modifications of a given DWDC and the separation of two ternary mixtures of hypothetical components, A, B, and C, and benzene, toluene, and o-xylene are chosen as illustrative examples to evaluate its feasibility and effectiveness. Through steady-state analysis and closed-loop operation studies, it is demonstrated that the black-hole problem can be completely circumvented with the careful adjustment of the number of stages in each section of the DWDC. This outcome gives evidence to the feasibility and effectiveness of the proposed philosophy and indicates a reasonable way to enhance the applicability and flexibilities of the DWDC.

1. INTRODUCTION Because of the considerably low requirement on capital investment and operating cost (e.g., frequently reported to be around 30% less in these two aspects, respectively, than its conventional counterparts), the dividing-wall distillation column (DWDC) has received increasingly more attention ever since the world’s first DWDC was successfully established by BASF in 1985.1−9 In the industrial sector, nowadays much more DWDC systems have been applied in the chemical and petrochemical process industries worldwide and the number has already been far beyond one hundred.7−11 In the academic sector, a great number of papers have been published on process design and dynamic control, and these have shown deep insights into the steady-state and dynamic behaviors of the DWDC.12−20 With regard to process synthesis and design, a key issue that must be addressed along with the pursuit of economic benefit is the coordination between the prefractionator and the main distillation column involved, which reflects essentially the unique characteristic of the DWDC in contrast to its conventional counterparts.21−24 In the aspect of process dynamics and operation, great attention needs also to be focused on this issue by means of one control loop in the prefractionator and three control loops for the top, intermediate, and bottom products in the main distillation column (i.e., the so-called three-point composition control scheme).25−27 In fact, the operation of the prefractionator with that control loop serves to adjust the relationship between the prefractionator and the main distillation column involved and can have a significant effect on the energy efficiency of the DWDC.25 Although it has been clarified that the DWDC can be operated © 2012 American Chemical Society

smoothly with the direct or indirect control of the main compositions of the three products, very few studies have been conducted so far on the four-point composition control of the process, for example, the control of the main compositions in the three products and the ratio between the two impurities in the intermediate product.28 The four-point composition control strategy is especially advantageous if the lightest and/or heaviest components are more valuable than the intermediate component. If a DWDC permits direct or indirect control of the lightest or heaviest composition in the intermediate product, more valuable products can be produced at the top and/or bottom and economical benefits are likely to be enhanced as compared with the three-point composition control strategy. Although keeping a relatively high composition in the intermediate product might not be favorable to the energy efficiency of the DWDC as pointed out by Tedder and Rudd,29 the overall economical consideration may justify the specification of two compositions in the intermediate product. Since the feasibility of applying the four-point composition control policy is essentially an issue that is closely related to the applicability and flexibility of the DWDC, it is apparently necessary and imperative to study it in more detail. Owing to the great degree of mass integration and energy integration between the prefractionator and the main distillation column involved, it becomes much more difficult Received: Revised: Accepted: Published: 14771

July 18, 2012 October 21, 2012 October 22, 2012 October 23, 2012 dx.doi.org/10.1021/ie301902a | Ind. Eng. Chem. Res. 2012, 51, 14771−14792

Industrial & Engineering Chemistry Research

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to design a four-point composition control system than a threepoint composition control system for the operation of the DWDC. Among the early researchers and practitioners, Wolff and Skogestad were the pioneers in studying this challenging issue and found out that the great difficulties stemmed essentially from the strong interaction between the prefractionator and the main distillation column involved in the DWDC.28 In particular, they noticed the existence of a special operating region in which it was impossible to achieve some desired product specifications in the intermediate product even under the extreme operating condition of an infinite boilup rate or an infinite reflux ratio. The unique operating region was therefore identified as one of the main obstacles for the feasible and smooth operation of the DWDC and described as a “hole” problem in the literature because it looked as if a real hole locating somewhere in the operating region of interest. Although the “hole” problem was encountered in the control and flexibility studies of the DWDC, it is essentially a drawback resulting from a deficient process design, more specifically, the improper interlinking between the prefractionator and the main distillation column involved. It seems to us therefore reasonable to tackle the problem in terms of new guidelines for process synthesis and design (instead of the optimization of an economical index) and this represents the main objective of the current work. The “hole” problem is renamed the blackhole problem here due to the fact that its formation mechanism has not been fully clarified. How to effectively deal with the black-hole problem is certainly a quite challenging issue in the aspects of design and operation of the DWDC. The current paper attempts to address the design and control of the DWDC with special attention focused on the circumvention of the black-hole problem in terms of a new guideline for process synthesis and design. First of all, the occurrence of the black-hole problem is demonstrated in the four-point composition control of the DWDC and the reason behind this phenomenon is analyzed. An attempt is then made to tackle this intricate problem with the sequential adjustment of the number of stages in each section of the DWDC, and a simple and yet effective procedure is devised for structural modifications. Two DWDC systems involving, respectively, the separation of two ternary mixtures of hypothetical components, A, B, and C and benzene, toluene, and o-xylene are employed as illustrative examples to evaluate the philosophy proposed. Both the steady-state analysis and closed-loop control studies are conducted in addition to the strict comparison between the two process designs with and without the black-hole problems, respectively. The features of the philosophy proposed are further analyzed and some conclusions are given in the last section of this article.

Figure 1. A typical black hole in the DWDC and its sensitivity to operating condition changes: (a) a typical black hole, (b) sensitivity to operating condition changes.

Figure 2. Structure representation of the Petlyuk distillation column and its thermodynamic equivalent, the DWDC: (a) Petlyuk distillation column, (b) DWDC.

2. BLACK-HOLE PROBLEM OF THE DWDC In terms of the separation of an equi-molar ideal ternary mixture of hypothetical components, A, B, and C, into three relatively pure components each with a composition of 99 mol % (i.e., Example I to be studied in section 4 of the current work), the relationship between the boilup rate, V/F, and the vapor split ratio, RV, is examined for a design of the DWDC involving totally 80 stages in the prefractionator and the main distillation column involved, and the obtained results are depicted in Figure 1a. Four constraints have been imposed onto the compositions of the three products, and these include the composition specification of 99 mol % for the components, A, B, and C, in the top, intermediate, and bottom products,

respectively, and the equal distribution of the components, A and C, in the intermediate product (in other words, xI, A/xI, C = 1). Here, the reflux flow rate, the intermediate product flow rate, the reboiler heat duty, and the liquid split ratio are simultaneously adjusted to satisfy stringently these four constraints. Instead of a simple relationship between the boilup rate, V/F, and the vapor split ratio, RV, as usually expected, two branches of the solution curves are finally obtained, including a left one corresponding to relatively smaller values of the vapor split ratio, RV, and a right one corresponding to relatively greater 14772

dx.doi.org/10.1021/ie301902a | Ind. Eng. Chem. Res. 2012, 51, 14771−14792


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Figure 3. A simple method for the circumvention of the black-hole problem in the DWDC.

values of the vapor split ratio, RV. Input and/or output multiplicities are found and these imply a great degree of nonlinearity between the input and output variables. These complicated phenomena should be attributed to the strong interaction between the prefractionator and the main distillation column involved and are likely to be affected

significantly by the variations in the topological structure of the DWDC. Between the left and right branches of the curve shown in Figure 1a, one can readily note that there occurs a special region between 0.5 and 0.66 (i.e., the shadowed region) of the vapor split ratio, RV, in which it is no longer feasible to maintain 14773



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stable operation of the DWDC. For example, in the case that the vapor split ratio, RV, has been assigned a value of 0.6, the DWDC can never reach the desired specifications simultaneously in the three products no matter what values are given to the reflux flow rate, the intermediate product flow rate, the reboiler heat duty, and the liquid split ratio. The situation cannot be altered even though the DWDC has been operated under the extreme operating condition of an infinite boilup rate or an infinite reflux flow rate. This special region represents actually the black hole of the DWDC. Although the black-hole problem is demonstrated here in terms of the separation of an ideal ternary mixture of hypothetical components A, B, and C, it is actually quite similar in shape to the one by Wolff and Skogestad, who studied the separation of a different ternary mixture of ethanol, propanol, and butanol.28 This fact implies that the black-hole problem is quite likely to occur in the fourpoint composition control of the DWDC and represents essentially a common drawback of this kind of processes (including certainly its thermodynamically equivalent, the Petlyuk arrangement). It can no doubt impose a stringent limitation on the applicability and flexibility of the DWDC and should be dealt with carefully in process development. Again, the intricate phenomenon is considered to be closely related to the strong interaction between the prefractionator and the main distillation column involved in the DWDC. It should be pointed out here the fact that the black-hole problem can be strongly affected by the variations in operating conditions (e.g., feed compositions, product specifications, etc.) and this arouses great uncertainties to process flexibility and operation. As shown in Figure 1b, the relationship between the boilup rate, V/F, and the vapor split ratio, RV, is depicted for two perturbed steady states along with the one for the nominal steady state. One of the perturbed steady states involves a −10% change in the feed compositions of component A, and the other a +10% change in the feed compositions of component A. The ratio between the feed compositions of components B and C has been kept the same as in the nominal operating conditions. It is readily noted that the black-hole Table 1. Physical Properties and Design Specifications of Example I parameter condenser pressure (bar) stage pressure drop (bar) feed composition (mol %) A B C feed flow rate (mol/s) feed thermal condition relative volatility A:B:C latent heat of vaporization (kJ/kmol) vapor pressure constants A(Avp/Bvp) B(Avp/Bvp) C(Avp/Bvp) product specification (mol %) A B C ratio between the compositions of components A and C in the intermediate product, A:C

value 3 0 33.3 33.4 33.3 27.8 1.0 4:2:1 29053.7

Figure 4. Initial and final process designs for Example I: (a) initial process design, (b) final process design.

12.35/3862 11.65/3862 10.96/3862

changes significantly in both shapes and locations. Since the vapor split ratio, RV, once determined, will not change in the practical situation, it is likely that the black-hole problem still occurs in the perturbed steady states even though it has been avoided in the nominal steady state with deliberately choosing the vapor split ratio, RV. This fact indicates again the possibly detrimental effect of the black-hole problem on process flexibility and control and reminds us of cautiously tackling it

99 99 99 1:1 14774



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Figure 5. Composition profiles in the initial and final process designs for Example I: (a) prefractionator in the initial process design, (b) main distillation column in the initial process design, (c) prefractionator in the final process design, (d) main distillation column in the final process design.

optimization software.31,32 This reality suggests that a two-step philosophy should be adopted for the development of the DWDC. The first step is to conduct process synthesis and design in terms of the optimization of an economical objective function, and the second step is to coordinate the relationship between the prefractionator and the main distillation column involved through a certain extent of structural modifications toward the resultant optimum process design. With this philosophy, the economical advantage of the optimal process design is usually no longer retained (although the penalty is generally rather small) and this is the expense that must be paid for alleviating the interaction between the prefractionator and the main distillation column involved in the DWDC. The relationship between the prefractionator and the main distillation column is affected by many variables and these include the thermodynamic properties of the ternary mixture separated, the four specifications imposed on the three products, the relevant design variables (i.e., the locations for the feed processed and the intermediate product withdrawn, the locations of the connecting flows between the prefractionator and the main distillation column, the total number of stages in these two distillation columns, and the vapor split ratio), and the relevant operating variables (i.e., the reflux flow rate, the intermediate product flow rate, the reboiler heat duty, and the liquid split ratio). With a given separation problem in hand, one can readily understand that only can those design

in the synthesis and design of the decentralized control system for the DWDC. Above all, the black-hole problem remains to be a major obstacle that hinders the application of the four-point composition control strategy to the DWDC, thereby constraining, to a certain extent, the applicability and flexibility of the process.

3. CIRCUMVENTION OF THE BLACK-HOLE PROBLEM THROUGH DELICATE PROCESS DESIGN 3.1. Principle of Circumventing the Black-Hole Problem. Since the strong interaction between the prefractionator and the main distillation column involved is the root cause that gives rise to the complicated behaviors of the DWDC, i.e., the input and/or output multiplicities and the black-hole problem, it is imperative to make a careful coordination between these two distillation columns in process development and this should essentially be attributed to a challenging issue of delicate steady-state design. Remember that the common practice of designing a DWDC is to base it on the optimization of an economical objective function (e.g., the total annual cost (TAC), which combines operating cost and discounted capital investment),30 it is generally, however, extremely difficult to achieve simultaneously the optimization of the economical objective function along with a satisfactory coordination between the prefractionator and the main distillation column involved, even though with the help of the currently available 14775



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Figure 6. Black hole and its sensitivity to operating condition changes (Example I): (a) ±10% changes in the feed compositions of component A, (b) ±10% changes in the feed compositions of component B, (c) ±10% changes in the feed compositions of component C. Figure 7. Variations of the black hole with the adjustment of the number of stages in sections I and II (Example I): (a) black hole in the initial process design, (b) black hole after the adjustment of the number of stages in section I, (c) no black hole after the adjustment of the number of stages in section II.

variables be used to influence the relationship between the prefractionator and the main distillation column involved. However, care should be taken because these design variables are not fully independent to each other, and it is therefore necessary to find a simple and yet effective way to represent the topological structure of the DWDC. Since the DWDC is divided actually into six sections (i.e., sections I to VI) with the locations of the feed, the intermediate product, and the connecting flows between the prefractionator and the main distillation column involved as shown in Figure 2 panels a and b (note that the former is based on a Petlyuk distillation column and the latter its thermodynamic equivalent, the DWDC), it is

reasonable to employ the number of stages in each section of the DWDC as the structural variables for process representation. Once the number of stages in each section has been fixed, the topological configuration of the DWDC is determined. In terms of this kind of process representation, a simple procedure was recently proposed and found effective for the synthesis and design of the DWDC.30 In the current work, this kind of 14776



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procedure proposed for the circumvention of the black-hole problem for a given process design of the DWDC. The given process design is termed the initial process design, hereinafter, in the current work, which includes generally those obtained by the optimization of an economical objective function with or without consideration of the four-point composition control issue in process development. Since the black-hole problem might also occur in the perturbed steady states aroused by the nonstationary changes in the nominal operating conditions (e.g., feed compositions, product specifications, etc.), it is therefore necessary to take into account these situations in the proposed procedure. This consideration serves to guarantee the smooth operation of the DWDC in the vicinity of the nominal steady state. First, the relationship between the boilup rate, V/F, and the vapor split ratio, RV, should be examined, and this can be accomplished with the intensive application of the steady-state model of the DWDC. With reference to the obtained relationship between these two variables, it is straightforward to ascertain whether or not there exists a black hole in the operating region of interest. If it is the case, two branches of the curve must exist and they never intersect with each other as shown in Figure 1a. The lower and upper bounds of the black hole can then be identified, respectively, in terms of the vapor split ratio, RV. Here, the lower bound (LB) refers to the greatest value of the vapor split ratio, RV, beyond which the given product specifications cannot be met any more in the left branch. In the case that the curve has a very great positive slope, a maximum limitation should be imposed on the boilup rate, V/F, and the corresponding vapor split ratio, RV, should be regarded as the LB, instead. Similarly, the higher bound (HB) represents the smallest value of the vapor split ratio, RV, beyond which the given product specifications cannot be met any more in the right branch. In the case that the curve has a very great negative slope, a maximum limitation should also be imposed on the boilup rate, V/F, and the corresponding vapor split ratio, RV, should be taken instead as the HB. Despite that the left and right branches of the curve might exhibit very complicated shapes, the LB and HB can be readily indentified by means of a single variable search method developed, which combines effectively the uses of two single-variable search algorithms with a fixed- and variable-step sizes, respectively. In terms of the obtained lower and upper bounds, LB and HB, the width of the black hole (WBL) can be determined with eq 1, which is to be used as a measurement index to reflect the variations of the size of the black hole during the structural adjustment of the DWDC.

Figure 8. Relationship between the boilup rate, V/F, and the vapor split ratio, RV, in the final process design (Example I): (a) ±20% changes in the feed compositions of component A, (b) ±20% changes in the feed compositions of component B, (c) ±20% changes in the feed compositions of component C.

WBL = HB − LB

(1)

Second, the number of stages is adjusted in a sequential manner from section I to section VI of the DWDC. For the adjustment of the number of stages in each section, only are several stages permitted to be added or removed in each iteration step as shown in eq 2. Although the search efficiency might not be very high with this stipulation, it is acceptable in the current work because great changes are rarely needed in the initial process design. With reference to the newly obtained DWDC, the lower and upper bounds, LB and HB, are identified, respectively, and the WBL is calculated, again. The magnitude of the variation of the WBL, that is, the DWBL in eq 3, is taken as the convergence criterion for the iterative calculation in each section. If it is greater than a given value (i.e., ε1 > 0), then the iterative adjustment of the number of

process representation is still adopted as the basis for tackling the black-hole problem. Since the adjustments of the number of stages in each section of the DWDC affect the flow rates and compositions of the interlinking between the prefractionator and the main distillation column, their relationships can thus be used to coordinate the interaction between these two distillation columns. This is essentially the mechanism of circumventing the black-hole problem with the careful structural modifications toward a give process design. 3.2. A Simple Procedure Proposed for the Structural Modifications of the DWDC. Figure 3 shows the simple 14777



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Figure 9. A decentralized control scheme for Example I: (a) control structure, (b) an equivalent control structure by Aspen Dynamics.

Nk + 1, j = NK , j ± ΔNK , j

stages should be continued in the current section, otherwise, the iterative structural modification should be moved to the next section. Along with the adjustment of the number of stages in each section, the circumvention of the black-hole problem is also judged for each newly obtained DWDC. If the WBL becomes minus and less than a given value (i.e., ε2 < 0), it signifies that the circumvention of the black-hole problem is successful and the modification of the initial process design should be stopped, otherwise, the iterative adjustment of the number of stages should be continued from section I to section VI of the DWDC (cf., eq 4).

1 ≤ ΔNK , j ≤ 5

(2)

0 < DWBLK = WBLK − 1 − WBLK < ε1

(3)

WBLK < ε2 < 0

(4)

Lastly, the circumvention of the black-hole problem should be considered in the perturbed steady states, and the principle is exactly the same with that in the nominal steady state. In general, the magnitudes of the frequently encountered disturbances, for example, the nonstationary variations in feed compositions and product specifications, etc., are known for a 14778



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first be tailored to the existing plant, which then plays an important role in the follow-up structural modifications. In what follows, two DWDC systems separating, respectively, an ideal ternary mixture of hypothetical components A, B, and C, and a real ternary mixture of benzene, toluene, and o-xylene are employed to evaluate the proposed philosophy for the circumvention of the black-hole problem.

Table 2. Controller Parameters for the Initial Process Design (Example I) controller

manipulated variable

controlled variable

KC (−)

TI (min)

CC1 CC2 CC3 CC4 FC LC11 LC12 LC2 LC3 LC4 PC1 PC2 PC3

D I QR RL F RR F1to2 F2to4 F3to4 B WCOMP,1 WCOMP,2 WCOMP,3

XD,A XI,B XB,C RA/C F LR LC1 LC2 LC3 LC4 PC3 PC2 PC4

96.63 73.31 2.43 0.07 0.5 2 2 2 2 2 20 20 20

99 34.32 75.24 118.8 0.3 9999 9999 9999 9999 9999 12 12 12

4. EXAMPLE I: A DWDC SYSTEM SEPARATING AN IDEAL TERNARY MIXTURE OF HYPOTHETICAL COMPONENTS A, B, AND C 4.1. Problem Description. Ideal vapor and liquid phase behaviors are assumed for the hypothetical ternary mixture separated, and the vapor−liquid equilibrium relationship can be expressed by Pj = xA, jPAs + x B, jPBs + xC, jPCs

yi , j = xi , jPis, j/Pj

Table 3. Controller Parameters for the Final Process Design (Example I) controller


controlled variable

KC (−)

TI (min)



XD,A XI,B XB,C RA/C F LR LC1 LC2 LC3 LC4 PC3 PC2 PC4

119.37 80.12 6 0.17 0.5 2 2 2 2 2 20 20 20

81.84 33 88.44 146.52 0.3 9999 9999 9999 9999 9999 12 12 12

1≤j≤N

i = A, B, C, and 1 ≤ j ≤ N

(5) (6)

The vapor saturation pressure is calculated via the following equation: ln Pis, j = A vp , i − Bvp , i /Tj

i = A, B, C, and 1 ≤ j ≤ N (7)

Equimolar overflow is assumed here, so the liquid and vapor flow rates are constant in each section of the DWDC. The commercial software Aspen Plus is used for the steady-state simulations of the DWDC. The physical properties and design specifications are listed in Table 1 for the DWDC system to be developed, and Figure 4a gives a design of the DWDC in terms of a simple methodology proposed in our earlier work.30 The operating pressure is fixed at 3 bar in this situation. The process has a 20-stage prefractionator and a 60-stage main distillation column with a dividing-wall running from stage 11 to stage 39. The feed is introduced onto stage 10 in the prefractionator and the intermediate product is withdrawn from stage 30 on the other side of the dividing-wall. The process design is taken as the initial process design here, and Figure 5a,b show the composition profiles of the prefractionator and the main distillation column involved in the resultant DWDC system, respectively. 4.2. Black-Hole Problem. Figure 6 depicts the relationship between the boilup rate, V/F, and the vapor split ratio, RV, for the initial process design in the nominal steady state and those perturbed ones by a ±10% change in the feed compositions of components, A, B, and C, respectively. The solid lines represent their relationship in the nominal steady state and the dotted and the dotted and dashed lines those in the perturbed steady states. As can be clearly seen from these figures, if the vapor split ratio, RV, has been set around 0.6, it is impossible to find a corresponding value of the boilup rate, V/F, so that the fourpoint composition control policy can be feasible. This is evidently a serious drawback that has resulted from a process design in the light of a pure economical consideration. 4.3. Circumvention of the Black-Hole Problem. Figure 7 shows the effect of adjusting the number of stages of the DWDC on the relationship between the boilup rate, V/F, and the vapor split ratio, RV. It is noted that the black hole disappears completely after the adjustments of the number of stages in the prefractionator and the final process design is sketched in Figure 4b with 22 and 6 stages in section I and

given process design and these can be used to infer the perturbed steady states mostly concerned. Under these perturbed operating conditions, if the black-hole problem can still be removed through the adjustment of the number of stages in each section of the DWDC, then it becomes evident that it is feasible to apply the four-point composition control policy to the resultant DWDC, which is termed the final process design, hereinafter, in the current work. Although conflicts may, in principle, occur between the structural modifications in the nominal and perturbed steady states, they can be solved with further adjustments of the number of stages in each section of the DWDC. In our studies so far, this circumstance is rarely found. The above procedure features great simplicity in principle because the adjustment of the number of stages is employed to coordinate the relationship between the prefractionator and the main distillation column involved in the DWDC. Its major drawback lies in the great intensity of model calculations conducted for the identification of the black hole during the iterative structural modifications. However, with careful reference to the previous calculated outcomes, the computation intensity can be lowered substantially. Although the proposed procedure has been aimed at guiding the synthesis and design of the DWDC, it can also be used as an effective guideline to retrofit an existing plant so as to remove the inherent black-hole problem in the case where a four-point composition control policy is to be implemented. Again, a steady-state model must 14779



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Figure 10. Regulatory responses of Example I for a ±20% step change in the feed compositions of component A, respectively: (a) composition of component A in the top product, (b) flow rate of the top product, (c) composition of component B in the intermediate product, (d) flow rate of the intermediate product, (e) composition of component C in the bottom product, (f) heat duty of reboiler, (g) ratio between the compositions of components A and C in the intermediate product, (h) liquid split ratio. Solid lines, initial process design; dashed lines, final process design; gray curves, negative responses; black curves, positive responses.

the main distillation column contained in the final process design. Although overdesign of the DWDC was already known

section II, respectively. Figure 5 panels c and d delineate, respectively, the composition profiles of the prefractionator and 14780



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Figure 11. Regulatory responses of Example I for a ±20% step change in the feed compositions of component B, respectively: (a) composition of component A in the top product, (b) flow rate of the top product, (c) composition of component B in the intermediate product, (d) flow rate of the intermediate product, (e) composition of component C in the bottom product, (f) heat duty of reboiler, (g) ratio between the compositions of components A and C in the intermediate product, (h) liquid split ratio. Solid lines, initial process design; dashed lines, final process design; gray curves, negative responses; black curves, positive responses.

to present little influences to the system performance,33 it makes the composition profiles of components A and C more

resembling to each other around the location for withdrawing the intermediate product. Figure 8 depicts the relationships 14781



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Figure 12. Regulatory responses of Example I for a ±20% step change in the feed compositions of component C, respectively: (a) composition of component A in the top product, (b) flow rate of the top product, (c) composition of component B in the intermediate product, (d) flow rate of the intermediate product, (e) composition of component C in the bottom product, (f) heat duty of reboiler, (g) ratio between the compositions of components A and C in the intermediate product, (h) liquid split ratio. Solid lines, initial process design; dashed lines, final process design; gray curves, negative responses; black curves, positive responses.

between the boilup rate, V/F, and the vapor split ratio, RV, of the resultant final process design. The solid lines represent their

relationship in the nominal steady state and the dotted and the dotted and dashed lines those in the perturbed steady states. 14782



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Table 4. Physical Properties and Design Specifications of Example II parameter condenser pressure (atm) stage pressure drop (atm) feed composition (mol %) benzene toluene o-xylene feed flow rate (mol/s) feed thermal condition relative volatility B:T:X normal boiling point (K) benzene toluene o-xylene product specification (mol %) benzene toluene o-xylene ratio between the compositionsof benzene and o-xylene in the intermediate product, B:X

value 0.37 0.0068 33.3 33.4 33.3 1000 1.0 7.1: 2.2:1 353 385 419 99 99 99 1:1

It can be noted that there is no longer a black hole in these curves, even under the perturbed steady states by a ±20% change in the feed compositions of components A, B, and C, respectively. In other words, in terms of the viewpoints of steady-state design and dynamic operation, there is now no limitation at all in specifying the vapor split ratio, RV, in the final process design. For an arbitrary value given to the vapor split ratio, RV, a corresponding boilup rate, V/F, can be found to guarantee the feasibility of applying the four-point composition control policy to the final process design. 4.4. Closed-Loop Evaluation. Figure 9a shows the decentralized control structure of the DWDC. The system pressure is regulated by the heat removal from the condenser and three inventory control systems are installed in this process. While the level of the reflux-drum is controlled by the reflux flow rate (because the initial and final process designs have a reflux ratio of 5.04 and 4.66, respectively, and both are greater than 3.0), the level of the reboiler is by the flow rate of the bottom product. The level on the stage above the dividingwall is controlled by the liquid flow rate to the main distillation column, and a ratio control loop is used to regulate the liquid flow rate to the prefractionator, thereby maintaining a constant liquid split ratio, RL. The purities of the top, intermediate, and bottom products are controlled, respectively, by the top product flow rate, the intermediate product flow rate, and the heat input to the reboiler. The liquid split ratio, RL, is employed to maintain the ratio between the compositions of components A and C in the intermediate product. Since there is no available dynamic model for the DWDC or the Petlyuk distillation column in the commercial software Aspen Dynamics, we must develop an equivalent model with the available modules in our study, and Figure 9b shows such an equivalent one along with the decentralized control structure shown in Figure 9a. As can be seen, the dynamic model consists of four columns, a rectifying column with only a condenser, two paralleled absorber columns without reboiler or condenser, and a stripping column with only a reboiler. In addition to those control loops described in Figure 9a, some additional control

Figure 13. Initial and final process designs for Example II: (a) initial process design, (b) final process design.

loops need to be included here. These include three pressure control loops at the tops of the stripping column and the two paralleled absorber columns, respectively, with the power of the corresponding compressors as manipulated variables, and two liquid level control loops at the bottoms of the two paralleled absorber columns, respectively, with the flow rates of their bottom withdrawals as manipulated variables. A 5-min dead-time element has been included in the composition measurements, 14783



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Figure 14. Composition profiles in the initial and final process designs for Example II: (a) prefractionator in the initial process design, (b) main distillation column in the initial process design, (c) prefractionator in the final process design, (d) main distillation column in the final process design.

compositions of components A and C has been kept the same as in the nominal operating conditions. For the final process design, it can effectively reject the two kinds of disturbances and keep the three products on their set-points, respectively. For the initial process design, although it can suppress the −20% step change in the feed compositions of component B, it fails in face of the +20% step change in the feed compositions of component B. In addition to the great deviation in the bottom product composition and serious oscillations in the intermediate product composition, the ratio between the compositions of components A and C in the intermediate product cannot be maintained at 1:1 in spite of a drastic variation in the liquid split ratio. Figure 12 presents the closed-loop responses of the initial and final process designs in face of a ±20% step change in the feed compositions of component C, respectively. The ratio between the feed compositions of components A and B has been kept the same as in the nominal operating conditions. Again, the final process design can effectively reject the two kinds of disturbances and keep the three products on their setpoints, respectively. The initial process design is unable to maintain the bottom product composition and the ratio between the compositions of components A and C in the intermediate product on their set-points in face of the +20% step change in the feed compositions of component C. Divergent oscillations occur actually at the instant of 50 h (cf., the small figures included in Figure 12a−h).

and the four composition controllers are tuned in a sequential manner with the application of the built-in Tyreus−Luyben tuning rule (cf., Tables 2 and 3).34 Figure 10 displays the closed-loop responses of the initial and final process designs in the face of a ±20% step change in the feed compositions of component A. The ratio between the feed compositions of components B and C has been kept the same as in the nominal operating conditions. In the beginning, the two processes are both kept at their steady states, respectively, and the step disturbances are then added at a time instant of 2.5 h. For the +20% step change in the feed compositions of component A, both the initial and final process designs are able to set back to their expected steady states, but the former is inferior to the latter in the aspects of peak deviations and settling times. For the −20% step change in the feed compositions of component A, while the final process design can still be stabilized in the expected steady state, the initial process design cannot. In addition to the great deviation in the bottom product composition and serious oscillations in the intermediate product composition, the ratio between the compositions of components A and C cannot be kept at 1:1 in the intermediate product even though with a great magnitude of variations in the liquid split ratio. Figure 11 depicts the closed-loop responses of the initial and final process designs in the face of a ±20% step change in the feed compositions of component B. The ratio between the feed 14784



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Figure 15. Black hole and its sensitivity to operating condition changes (Example II): (a) ±10% changes in the feed compositions of benzene, (b) ±10% changes in the feed compositions of toluene (c) ±10% changes in the feed compositions of o-xylene.

These simulation results demonstrate that it is reasonable to pursue the four-point composition control policy in the final process design.

Figure 16. Variations of the black hole with the adjustment of the number of stages in sections I and II (Example II): (a) black hole in the initial process design, (b) black hole after the adjustment of the number of stages in section I, (c) no black hole after the adjustment of the number of stages in section II.

5. EXAMPLE II: A DWDC SYSTEM SEPARATING A TERNARY MIXTURE OF BENZENE, TOLUENE, AND O-XYLENE 5.1. Problem Description. Table 4 summarizes the physical properties and design specifications of the DWDC system to be developed, and Figure 13a shows a process design by Ling and Luyben (Note that the total number of stages has

now been decreased arbitrarily to 66 in the current example), which is taken here as the initial process design.25 The pressure of condenser is specified to be 0.37 atm and a pressure drop of 6.8 × 10−3 atm per stage is assumed. The steady-state and dynamic models are constructed, respectively, with the commercial software Aspen Plus and Aspen Dynamics, and 14785



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the Chao-Seader method is adopted to estimate the thermodynamic properties of the ternary mixture of benzene, toluene, and o-xylene. Figures 14 panels a and 14b depict, respectively, the composition profiles of the prefractionator and the main distillation column involved in the initial process design. 5.2. Circumvention of the Black-Hole Problem. For the initial process design, a black hole is found in the relationship between the boilup rate, V/F, and the vapor split ratio, RV, as shown in Figure 15. With the application of the proposed procedure, a final process design is generated and shown in Figure 13b. In comparison with the initial process design, the number of stages has been varied from 10 to 18 in section I, and from 10 to 5 in section II. Figures 14 panels c and d present, respectively, the composition profiles of the prefractionator and the main distillation column involved in the final process design. Again, the modification of the prefractionator facilitates the composition profiles of benzene and o-xylene more analogous to each other around the location for withdrawing the intermediate product. Figure 16 demonstrates how the black hole is gradually removed with the adjustment of the number of stages in section I and section II of the initial process design, and Figure 17 displays the relationship between the boilup rate, V/F, and the vapor split ratio, RV, of the final process design in the perturbed steady states by a ± 20% change in the feed compositions of benzene, toluene, and o-xylene, respectively. It is obvious that the black hole has disappeared, completely. 5.3. Closed-Loop Evaluation. The control configuration shown in Figure 9 for Example I is employed here. However, for the initial process design, because it has a reflux ratio of 2.73 (which is less than 3.0), the top product quality is controlled by the reflux flow rate and the level of the reflux-drum by the top product flow rate. Tables 5 and 6 give the controller parameters of the initial and final process designs, respectively. In Figure 18, the closed-loop responses are compared between the initial and final process designs in face of a ±20% step change in the feed compositions of benzene, respectively. The ratio between the feed compositions of toluene and o-xylene has been kept the same as in the nominal operating conditions. It is noted that the final process design can attenuate effectively the influences of these two disturbances and keep the three products on their set-points, respectively. The initial process design appears to be uncompetitive with the final process design. Apart from the great deviations in the top product and strong oscillations in the intermediate and bottom products, a certain extent of discrepancy is observed in the ratio between the compositions of benzene and o-xylene in the intermediate product in the case of the −20% step change in the feed compositions of benzene. In Figure 19, the closed-loop responses are compared between the initial and final process designs in face of a ±20% step change in the feed compositions of toluene, respectively. The ratio between the feed compositions of benzene and o-xylene has been kept the same as in the nominal operating conditions. Again, the final process design outperforms the initial process design, because the latter exhibits much greater peak deviation in the top product and more severe oscillations in the intermediate and bottom products than the former. Moreover, the ratio between the compositions of benzene and o-xylene in the intermediate product cannot be maintained to be 1:1 by the

Figure 17. Relationship between the boilup rate V/F, and the vapor split ratio, RV, in the final process design (Example II): (a) ±20% changes in the feed compositions of benzene, (b) ±20% changes in the feed compositions of toluene, (c) ±20% changes in the feed compositions of o-xylene.

former in the case of the +20% step change in the feed compositions of toluene. In Figure 20, the closed-loop responses are compared between the initial and final process designs in face of a ±20% step change in the feed compositions of o-xylene, respectively. The ratio between the feed compositions of benzene and toluene has been kept the same as in the nominal operating conditions. Quite similar to the case of the step changes in the feed compositions of toluene, the initial process design presents much greater peak deviation in the top product and stronger oscillations in the intermediate and bottom products than the 14786



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Figure 18. Regulatory responses of Example II for a ±20% step change in the feed compositions of benzene, respectively: (a) composition of benzene in the top product, (b) flow rate of the top product, (c) composition of toluene in the intermediate product, (d) flow rate of the intermediate product, (e) composition of o-xylene in the bottom product, (f) heat duty of reboiler, (g) ratio between the compositions of benzene and o-xylene in the intermediate product, (h) liquid split ratio. Solid lines, initial process design; dashed lines, final process design; gray curves, negative responses; black curves, positive responses.

final process design, besides a certain extent of discrepancy in the ratio between the compositions of benzene and o-xylene in

the intermediate product in the case of the +20% step change in the feed compositions of o-xylene. 14787



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current work, one may readily find that the adjustment of the number of stages has been made exclusively to the prefractionator and these facts reveal the great importance of its process design. The adjustment of the number of stages in the prefractionator affects the compositions and flow rates of its connecting flows to the main distillation column and these serve as a means of coordination to compromise the interaction between the two distillation columns involved in the DWDC. The mechanism is, in principle, quite analogues to that of the 4 × 4 decentralized control of the DWDC, where the purity of the top or bottom product in the prefractionator must be controlled with the liquid split ratio in order to aid the composition control of the three products in the main distillation column.16,18,25 Although the number of stages in the main distillation column (remember the fact that these stages belong actually to sections III to VI) somewhat influences the black-hole problem, its effect is relatively small and this is why the adjustment of the number of stages is not necessary in the two example systems studied. This is, in fact, a little different from the interpretation made by Fidkowski and Krolikowski on the development of the DWDC, namely, the combination between the upper and lower parts of the main distillation column is crucial to the thermodynamic performance of the DWDC.21 It should be borne in mind that there should be a great number of design alternatives for the DWDC that can circumvent the black-hole problem since the vapor split ratio can be determined arbitrarily in the final process design. The philosophy proposed in the current work features essentially a process design with the smallest number of stages because the reduction of the number of stages is considered prior to augment of the number of stages in structural modifications. Owing to the fact that the avoidance of the black-hole problem has been included as an additional requirement for the final process design, capital investment is usually increased for more stages than which are usually needed in the initial process design. If the initial process design resulted from the optimization of an economical objective function, then the final process design should be quite close to the optimum one under the elevated design specifications (i.e., the design specifications for the initial process design plus the requirement for circumventing the black-hole problem in the nominal as well as perturbed steady states). If the tradeoff between operating cost and capital investment has been included in the proposed methodology (note that this can readily be realized in practice, which is, however, beyond the scope of the current work), the methodology can find the optimum process design under the elevated design specifications. For example, in the case of the Example I studied in the current work, its optimum process design is directly searched and the resultant scheme is found to have a TAC of 6.98 × 105 $. On the other hand, the TAC of the final process design shown in Figure 4b is also calculated and this gives rise to a value of 7.04 × 105 $. Note that these two values of the TAC differ only by 0.86% and the marginal difference is certainly due to the fact that the minimization of the TAC has not been considered in the proposed methodology. In the case of the Example II studied in the current work, quite similar outcomes are actually yielded and not shown here. Although the circumvention of the black-hole problem permits the four-point composition control of the DWDC,

Table 5. Controller Parameters for the Initial Process Design (Example II) controller


controlled variable

KC (−)

TI (min)


RR I QR RL F D F1to2 F2to4 F3to4 B WCOMP,1 WCOMP,2 WCOMP,3

XD,B XI,T XB,X RB/X F LR LC1 LC2 LC3 LC4 PC3 PC2 PC4

1.08 21.08 3.36 0.07 0.5 2 2 2 2 2 20 20 20

143.88 47.52 52.8 139.92 0.3 9999 9999 9999 9999 9999 12 12 12

Table 6. Controller Parameters for the Final Process Design (Example II) controller


controlled variable

KC (−)

TI (min)



XD,B XI,T XB,X RB/X F LR LC1 LC2 LC3 LC4 PC3 PC2 PC4

22.41 16.01 5 0.06 0.5 2 2 2 2 2 20 20 20

134.64 43.56 67.32 56.76 0.3 9999 9999 9999 9999 9999 12 12 12

These simulation results indicate again that it is reasonable to conduct the four-point composition control policy in the final process design.

6. DISCUSSION The two example systems studied in the current work have demonstrated that it is possible to circumvent the black-hole problem with the adjustment of the number of stages in each section of the DWDC. In other words, structural modifications can be employed to compromise the relationship between the prefractionator and the main distillation column involved in the DWDC. Although it is extremely difficult to present a theoretical proof on its rationale and feasibility, its applicability and effectiveness are not considered to be affected by feed compositions, product specifications, the thermodynamic properties of the ternary mixtures separated, etc. The physical interpretation is of paramount significance because it points out a potential way to enhance the applicability and flexibility of the DWDC in terms of careful and deliberate structural modifications. These two features reflect actually the common concerns on the application of the DWDC to the chemical and petrochemical process industries. In terms of the comparison between the initial and final process designs of the two example systems studied in the 14788



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Figure 19. Regulatory responses of Example II for a ±20% step change in the feed compositions of toluene, respectively: (a) composition of benzene in the top product, (b) flow rate of the top product, (c) composition of toluene in the intermediate product, (d) flow rate of the intermediate product, (e) composition of o-xylene in the bottom product, (f) heat duty of reboiler, (g) ratio between the compositions of benzene and o-xylene in the intermediate product, (h) liquid split ratio. Solid lines, initial process design; dashed lines, final process design; gray curves, negative responses; black curves, positive responses.

dynamics and controllability in the final process design, one may make further adjustments of the number of stages from

it never means necessarily the achievement of satisfactory dynamic performance. To guarantee satisfactory process 14789



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Figure 20. Regulatory responses of Example II for a ±20% step change in the feed compositions of o-xylene, respectively: (a) composition of benzene in the top product, (b) flow rate of the top product, (c) composition of toluene in the intermediate product, (d) flow rate of the intermediate product, (e) composition of o-xylene in the bottom product, (f) heat duty of reboiler, (g) ratio between the compositions of benzene and o-xylene in the intermediate product, (h) liquid split ratio. Solid lines, initial process design; dashed lines, final process design; gray curves, negative responses; black curves, positive responses.

section I to section VI and this necessitates actually an effective algorithm to simultaneously address the issues of process

design and process operation, that is, the integration between process design and process control. 14790



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7. CONCLUSIONS In this work, the black-hole problem associated with the fourpoint composition control of the DWDC has been studied. It has been clarified that it is essentially an inherent drawback caused by the strong interaction between the prefractionator and the main distillation column involved and can pose great limitations to process applicability and flexibility. The blackhole problem is quite likely to happen in those DWDC systems that result purely from economical considerations and this fact suggests that new guidelines should be developed for the synthesis and design of the DWDC. A simple and yet effective philosophy has been proposed to tackle the black-hole problem, with the number of stages as decision variables to compromise the relationship between the prefractionator and the main distillation column involved. In terms of the two DWDC systems separating, respectively, an ideal ternary mixture of hypothetical components, A, B, and C, and a real ternary mixture of benzene, toluene, and o-xylene, the rationale and effectiveness of the proposed philosophy has been evaluated in the aspects of steady-state design and dynamic operation. The obtained outcomes demonstrate that the resultant DWDC systems can get rid of completely the black-hole problem and present rather satisfactory performance in the four-point composition control of the DWDC in the face of various feed composition disturbances. The proposed philosophy is considered to be of great significance to the synthesis, design, and operation of the DWDC and can work effectively to enhance process applicability and flexibility. One of our future research topics will be centered on the examination of the applicability and effectiveness of the proposed method to those DWDC systems separating mixtures with more complicated thermodynamic properties and/or more components, for example, four or five components. Other effective strategies should also be explored to facilitate the removal of the black-hole problem from the DWDC, and these may include, for example, the arrangement of multiple feeds in the prefractionator and/or intermediate products in the main distillation column, the deliberate determination of thermal conditions of feed and/or intermediate products, etc. Moreover, it is imperative to examine the feasibility of using temperature inferential control strategies for the operation of the DWDC compensated by the proposed methodology of the current work.

■

C = hypothetical component CC = composition controller COMP = compressor D = distillate flow rate, mol s−1 DWBL = width variation of a black hole DWDC = dividing-wall distillation column F = feed flow rate, mol s−1 F1to2 = liquid flow rate from columns I to II, mol s−1 F2to4 = liquid flow rate from columns II to IV, mol s−1 F3to4 = liquid flow rate from columns III to IV, mol s−1 FC = flow rate controller HB = higher bound of a black hole I = intermediate product flow rate, mol s−1 J = section number of the DWDC K = iteration number KC = proportional gain LB = lower bound of a black hole LC = level controller M = multiplier N = number of stages ΔN = variation in the number of stages P = pressure, Pa PC = pressure controller QC = condenser duty, MW QR = reboiler duty, MW RA/C = ratio between the compositions of components A and C in the intermediate product RC = ratio controller RL = liquid split ratio RV = vapor split ratio RR = reflux flow rate, mol s−1 T = temperature, K ΔT = time delay, s TAC = total annual cost, #dollar TI = integral time, s V = vapor flow rate, mol s−1 WBL = width of a black hole W = power of a compressor, MW x = liquid composition y = vapor composition Greek Letters

ε1 = error tolerance in judging the variation of a black hole ε2 = error tolerance in judging the circumvention of a black hole

AUTHOR INFORMATION

Subscripts

Corresponding Author

A = component index B = component index or bottom product C = component index COMP = compressor D = distillate product I = intermediate product J = section number of the DWDC K = iteration number T = component index or top product X = component index

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

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The current work is financially supported by The National Science Foundation of China under Grant No. 21076015 and The Doctoral Programs Foundation of Ministry of Education of China under Grant No. 20100010110008.

■

Superscripts

■

NOTATION A = hypothetical component Avp = vapor pressure constant, Pa B = hypothetical component or bottom flow rate Bvp = vapor pressure constant, Pa·K

s = saturation

REFERENCES

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