Avoiding the Black-Hole Problem by the Arrangements of Multiple

Article pubs.acs.org/IECR

Avoiding the Black-Hole Problem by the Arrangements of Multiple Intermediate Products to Dividing-Wall Distillation Columns Jinglan Gao, Kejin Huang,* Shujun Luan, Yingjie Jiao, and Shen Li College of Information Science and Technology, Beijing University of Chemical Technology, Beijing 100029, People’s Republic of China ABSTRACT: Because of the strong coupling between the prefractionator and the main distillation column involved, the dividing-wall distillation column (DWDC) may exhibit the so-called black-hole problem in the operating range of interest when four product compositions (i.e., the main compositions in the three products and the ratio between the two impurities in the intermediate product) have been specified. In this paper, a novel strategy is proposed to solve the black-hole problem by the arrangements of multiple intermediate products to the main distillation column of the DWDC. The number, locations, and flow rates of the multiple intermediate products are designated as decision variables to coordinate the relationship between the prefractionator and the main distillation column involved, and a simple procedure is developed for the determination of their values effectively. The separations of two ternary mixtures of hypothetical components, A, B, and C and ethanol, propanol, and butanol, are chosen as illustrative examples to evaluate the feasibility and effectiveness of the proposed strategy. In terms of steady-state analysis and dynamic operation studies, it is demonstrated that the black-hole problem can be completely circumvented by the arrangements of multiple intermediate products to the DWDC. The proposed strategy is considered to be of general significance and can be applicable and effective to the synthesis and design of the DWDC separating ternary mixtures with widely different thermodynamic properties.

1. INTRODUCTION The successful establishment of the world’s first dividing-wall distillation column (DWDC) by BASF in 1985 has spurred considerable interest in the studies of this highly thermally coupled separation process.1−6 Both theoretical research and practical applications have demonstrated that the scheme can be greatly advantageous in capital investment and utility consumption over its conventional analogues, e.g., the direct and indirect separation sequences, and this has already established it to be a competitive alternative for the separation of ternary mixtures.7−9 It is noted, however, that almost all studies conducted so far have been focusing the design and operation of the DWDC involving three specifications on the three products, respectively (i.e., the specification of the main component in the top, intermediate, and bottom products, respectively).10−17 With regard to the design and operation of the DWDC involving four specifications on its three products (e.g., the above three specifications plus the ratio between the two impurities in the intermediate product), only has scarce attention been given actually.18 This style of process design and operation is sometimes encountered in practical situations because additional constraints might be imposed on the top, intermediate, and bottom products of the DWDC. For instance, due to the pursuit of economical benefit and/or the ever-strengthening environmental legislations and regulations, the light or heavy component may be required not to exceed a certain limit in the intermediate product and this necessitates certainly special efforts to deal with the additional constraint during the design and operation of the DWDC. Therefore, how to conduct process design and operation with four specifications on its three products represents essentially an important issue that is closely related to the applicability and flexibility of the DWDC. It is certainly deserved to be studied in great detail. © 2013 American Chemical Society

In sharp contrast to the situation of imposing three specifications on the three products of the DWDC, the introduction of four specifications complicates considerably process design and operation because much stronger conflicts are quite likely to occur in the latter than in the former between the prefractionator and the main distillation column involved. With reference to the separation of an equi-molar ternary mixture of ethanol, propanol, and butanol with a Petlyuk distillation column accommodating a 20-stage prefractionator and a 40-stage main distillation column (note that a similar scheme is sketched in Figure 1a, in terms of, however, the separation of an ideal ternary mixture of hypothetical components A, B, and C), Wolff and Skogestad pioneered the studies of the four-point composition control issue in 1995.18 They found that the Petlyuk distillation column might display unusual steady-state and dynamic behaviors in addition to the occurrence of input and/or output multiplicities. More specifically, in a special operating region of interest, it was found 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 were locating somewhere in the operating region of interest. Although the “hole” problem was frequently encountered in the operation of the DWDC, it is essentially an inherent drawback resulting from deficient process Received: Revised: Accepted: Published: 4178

December 4, 2012 January 27, 2013 February 28, 2013 February 28, 2013 dx.doi.org/10.1021/ie303336y | Ind. Eng. Chem. Res. 2013, 52, 4178−4201

Industrial & Engineering Chemistry Research

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resultant structural variations could help to ease the conflicts between the prefractionator and the main distillation column involved. Although the black-hole problem could be completely removed through the careful process modifications, the strategy usually resulted in a certain extent of increase in the number of stages and incurred consequently additional capital investment. Furthermore, it was not very suitable for the situation in which the DWDC was already built and put into operation. To avoid these drawbacks, we initiated finding other effective yet simple strategies for the circumvention of the black-hole problem, since the DWDC rendered redundant degrees of freedom for process synthesis, design, and operation. This stands for the main purpose of this study. In the current work, an attempt is made to disentangle the black-hole problem encountered in the design and control of the DWDC with the arrangements of multiple intermediate products because this design option can serve to coordinate the relationship between the prefractionator and the main distillation column involved. A simple and effective procedure is devised for the removal of the black-hole problem in terms of the careful determination of the number, locations, and flow rates of the multiple intermediate products. Two DWDC systems involving, respectively, the separation of two ternary mixtures of hypothetical components, A, B, and C and ethanol, propanol, and butanol, are employed as illustrative examples to evaluate the feasibility and effectiveness of the proposed strategy. Both the steady-state analysis and closed-loop operation studies are conducted in addition to the strict comparison between the process designs with one and multiple intermediate products, respectively. The salient features of the strategy proposed are further analyzed, and some concluding remarks are finally reached in the last section of this article.

2. AVOIDANCE OF THE BLACK-HOLE PROBLEM BY THE ARRANGEMENTS OF MULTIPLE INTERMEDIATE PRODUCTS TO THE DWDC 2.1. Principle of Avoiding the Black-Hole Problem of the DWDC. Consider the fact that the black-hole problem has been aroused by the intricate interplay between the prefractionator and the main distillation column involved. It is then necessary to exercise a careful coordination between these two distillation columns during process synthesis and design, and this constitutes actually a much more complicated problem than the development of a simple DWDC involving only three specifications on its three products. Since the prefractionator and the main distillation column involved are interlinked through the four connecting flows between them (cf., the bold lines in Figure 1a and b), it appears to be reasonable to influence their relationship through the compositions and flow rates of these connecting flows. Several design strategies can be used to meet this purpose, and these include, for example, the variation of the number of stages in the prefractionator and/or the main distillation column involved, the employment of multiple feeds to the prefractionator and/or multiple intermediate products withdrawn from the main distillation column, and the determination of appropriate thermal conditions for the multiple feeds processed and/or the multiple intermediate products withdrawn, etc. For the variation of the number of stages in the prefractionator and/or the main distillation column involved, it was already demonstrated to be an effective strategy for the circumvention of the black-hole problem in our early work.19 In the current work, the feasibility and effectiveness of avoiding the black-hole problem in the DWDC through the arrangements of

Figure 1. Petlyuk distillation column and its thermodynamic equivalents: (a) Petlyuk distillation column; (b) DWDC; (c) DWDC with multiple intermediate products.

development, namely, the improper interlinking between the prefractionator and the main distillation column involved. In our recent work, the “hole” problem was renamed the black-hole problem in consideration of the fact that its formation mechanism has not yet been fully clarified.19 The adjustment of the number of stages in each section of the DWDC was proposed to circumvent the black-hole problem because the 4179

dx.doi.org/10.1021/ie303336y | Ind. Eng. Chem. Res. 2013, 52, 4178−4201


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Figure 2. A simple procedure for avoiding the black-hole problem in terms of multiple intermediate products in the DWDC.

specifications (e.g., the composition of the main component and the ratio between the two impurities). A number of degrees of freedom are thus yielded with this structural modification and can serve to coordinate the relationship between the prefractionator and the main distillation column involved. These consist of the number of the intermediate products, their locations, and flow rates, with the first two being discontinuous variables and the last one continuous variables. It is thus necessary to develop an effective procedure for the determination of these variables in process synthesis and design, whose purpose is not only to seek

multiple intermediate products to the main distillation column is to be examined. A control study of a binary distillation column with three products was once reported by Tyreus and Luyben, and the location of the intermediate product was found to be a better manipulated variable than its flow rate.20 A DWDC with the arrangements of multiple intermediate products to the main distillation column is delineated in Figure 1c. Note that the multiple intermediate products are withdrawn from different stages of the main distillation column and finally mixed together to form a product that must satisfy the given 4180



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the most economical process design but also to avoid the blackhole problem possibly occurring in the case of four specifications given to the three products of the DWDC. 2.2. Avoiding the Black-Hole Problem by the Arrangements of Multiple Intermediate Products to the DWDC. Figure 2 shows a simple procedure devised for the avoidance of the black-hole problem for a given process design of the DWDC having four composition specifications on its three products. The given process design includes generally those obtained by the optimization of an economical objective function with or without the consideration of the additional composition specification on the intermediate product (i.e., the ratio between the two impurities) in process development. It is termed the initial process design, hereinafter, in the current work. 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 flow rate, feed compositions, product specifications, etc.), it is therefore necessary to take into account these situations in the proposed procedure. The avoidance of the black-hole problem in the perturbed steady states serves to ensure the smooth operation of the DWDC in the vicinity of the nominal steady state. First of all, the relationship between the boilup rate, V/F, and the vapor split ratio, RV, should be examined in the nominal steady state, 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 problem in the operating region of interest. If a black-hole problem does occur in the initial process design, then process modifications must be attempted in the light of the strategy proposed in the current work. According to our earlier work, the lower bound (LB) and higher bound (HB) of the black-hole can be determined, which reflect the positions of the two branches of the relationship between the boilup rate, V/F, and the vapor split ratio, RV.19 The width of the black-hole (WBL) is then calculated through eq 1 and can serve as a measurement index to reflect the variations of the size of the black-hole by the arrangements of multiple intermediate products to the main distillation column. WBL = HB − LB

Table 1. Physical Properties and Design Specifications of Example I parameter condenser pressure (atm) stage pressure drop (atm) 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 specifications (mol %) A B C ratio between the compositions of components A and C in the intermediate product (A:C)

value 1 0 33.3 33.4 33.3 27.8 1.0 4:2:1 29053.7 12.35/3862 11.65/3862 10.96/3862 99 99 99 1:1

products with reference to their current places and values; otherwise, a convergent arrangement of the multiple intermediate products is reached. In terms of the resultant arrangement of the multiple intermediate products, the WBLN,K is examined. If it is smaller than a given negative value (i.e., ε2 in eq 4), it means that the black-hole problem is already eliminated; otherwise, the black-hole problem is unlikely to be resolved with the guessed number of intermediate products and the process modifications should thus be attempted with an increased number of intermediate products. Note here that the choice of the ε2 can greatly affect the resultant final process designs. However, with the allowable range given to the intersection point between the two branches of the curve, its influences can be confined. With reference to the relationship between the number of intermediate products and the steady-state performance of the resultant DWDC with multiple intermediate products (the heat duty of reboiler is employed in the current work), the optimum process modification can be identified in a straightforward manner. Since the arrangement of multiple intermediate products does not involve very much capital investment, the avoidance of the blackhole problem generally adds negligible complications to the trade-off between capital investment and operating cost. This represents actually a potential advantage over the method of varying the number of stages of the DWDC proposed in our early work.19

(1)

Next, the number, locations, and flow rates of the multiple intermediate products should be determined. Because both continuous and discontinuous design variables have been involved, it is reasonable to solve the problem here through an iterative search method in a sequential manner. For a guessed number of the intermediated products, N, their locations are first searched one by one assuming an equal flow rate between the intermediated products except the one employed for keeping the mixed intermediate product on the given specification. In terms of the resultant locations of the intermediated products, their flow rates are then adjusted and this can be achieved, for instance, in terms of a modified Newton−Raphson method. For each round of the iteration, K, the lower and upper bounds, LBN,K and HBN,K, and the width of the black-hole, WBLN,K, are calculated, and they are used to estimate the magnitude of the variation of the WBLN,K, i.e., the DWBLN,K in eq 2, which is taken to be the convergence criterion for the iterative searches of the locations and flow rates of the multiple intermediate products. If it is greater than a given positive value (i.e., ε1 in eq 3), then further iterative calculations should be performed toward the determination of the locations and flow rates of the multiple intermediate

DWBLN , K = WBLN , K − 1 − WBLN , K

(2)

0 < DWBLN , K < ε1

(3)

WBLN , K < ε2 < 0

(4)

Finally, the circumvention of the black-hole problem should be examined in the perturbed steady states, and the principle is exactly the same as the one for the nominal steady state. Under the perturbed operating conditions concerned, if the black-hole problem does not occur in the resultant process design with multiple intermediate products, then it becomes evident that the design and operation of the DWDC with four product specifications has been made feasible by the above process 4181



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Figure 3. Initial and final process designs for example I: (a) initial process design; (b) final process design with two intermediate products; (c) final process design with three intermediate products; (d) final process design with four intermediate products.

examples to evaluate the proposed strategy, i.e., the rationale and effectiveness of avoiding the black-hole problem by the arrangements of multiple intermediate products to the DWDC.

modifications and the resultant process design is termed the final process design, hereinafter, in the current work. In case of the occurrence of a black-hole problem, further coordination should be sought between the prefractionator and the main distillation column involved through the adjustments of multiple intermediate products. Although conflicts may, in principle, occur between the structural modifications made in the nominal and perturbed steady states, they can be solved through the further adjustments of the number, locations, and flow rates of the multiple intermediate products of the main distillation column. In the following two sections, two example DWDC systems separating, respectively, an ideal ternary mixture of hypothetical components A, B, and C and a real ternary mixture of ethanol, propanol, and butanol are chosen as illustrative

3. EXAMPLE I: A DWDC SEPARATING AN IDEAL TERNARY MIXTURE OF HYPOTHETICAL COMPONENTS A, B, AND C 3.1. 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 4182

1≤j≤n

(5)



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Figure 4. Composition and temperature profiles in the initial and final process designs of example I: (a) composition profiles of the prefractionator in the initial process design; (b) composition profiles of the main distillation column in the initial process design; (c) temperature profiles of the prefractionator in the initial process design; (d) temperature profiles of the main distillation column in the initial process design; (e) composition profiles of the prefractionator in the final process design; (f) composition profiles of the main distillation column in the final process design; (g) temperature profiles of the prefractionator in the final process design; (h) temperature profiles of the main distillation column in the final process design. 4183



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Figure 5. Black-hole and its sensitivity to operating condition changes (example I): (a) black-hole of the initial process design; (b) ±10% changes in the feed compositions of component A.

yi , j = xi , jPis, j/Pj

i = A, B, C and 1 ≤ j ≤ n

(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)

The physical properties and design specifications are listed in Table 1. An equi-molar ternary mixture of hypothetical components A, B, and C is separated with a DWDC to be developed, and its three products are specified to be 99 mol %, respectively, for the three main components in addition to an equal distribution of the two impurities in the intermediate product (i.e., xI,A/xI,C = 1). As shown in Figure 3a, a DWDC is developed and adopted here as the initial process design, which contains a 20-stage prefractionator and a 60-stage main distillation column. The stages are numbered according to the Aspen stipulation that the top condenser is designated to be the first stage and the bottom reboiler the last one. The DWDC is simulated using the commercial software Aspen Plus, and the operating pressure is assumed to be 1 atm. No pressure drop is considered in this case. The feed is fed onto stage 10 in the prefractionator, and the intermediate product is withdrawn from stage 30 of the main distillation column. The rectifying section runs from stage 2 to stage 20 and the stripping section from stage 41 to stage 59, leaving the dividing wall from stage 21 to stage 40.

Figure 6. Variations of the black-hole with the adjustments of the locations and flow rates of the two intermediate products (example I): (a) black-hole in the initial process design; (b) black-hole versus the locations of the two intermediate products; (c) black-hole versus the flow rate of the top intermediate product.

Figure 4a−d depicts the composition and temperature profiles of the prefractionator and the main distillation column involved. Figure 5a delineates the relationship between the boilup rate, V/F, and the vapor split ratio, RV, for the initial process design. It is generated by imposing four constraints on 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). The reflux 4184



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Figure 7. Variations of the black-hole with the adjustments of the locations and flow rates of the three intermediate products (example I): (a) black-hole in the initial process design; (b) black-hole versus the locations of the three intermediate products; (c) black-hole versus the flow rates of the top two intermediate products.

Figure 8. Variations of the black-hole with the adjustments of the locations and flow rates of the four intermediate products (example I): (a) black-hole in the initial process design; (b) black-hole versus the locations of the four intermediate products; (c) black-hole versus the flow rates of the top three intermediate products.

flow rate, R, the intermediate product flow rate, I, the reboiler heat duty, QR, and the liquid split ratio, RL, are simultaneously adjusted to satisfy stringently these four constraints. It is readily seen that a black-hole problem happens in the range of 0.5−0.68 of the vapor split ratio, RV, implying an unsatisfactory coordination between the prefractionator and the main distillation column involved. Within this operation range, it is no longer possible to achieve the desired product specifications even under the extreme operating condition of an infinite boilup rate or an infinite reflux flow rate. Figure 5b shows the relationship between

the boilup rate, V/F, and the vapor split ratio, RV, in the case of a ±10% change in the feed compositions of component A. While the solid lines stand for the relationship in the nominal steady state, the dotted and dashed lines represent those in the perturbed steady states. It is noted that the size and location of the black-hole changes with the operating conditions, and this reality makes it necessary to consider its circumvention in not only the nominal steady state but also the perturbed ones resulting in the nonstationary changes in operating conditions. Similar observations can also be made in the case of a ±10% 4185



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Figure 9. Minimum boilup rate, V/Fmin, versus the number of intermediate products (example I).

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 R FRL2 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

36.27 75.50 2.06 0.11 0.5 2 2 2 2 2 20 20 20

113.52 27.72 39.6 176.88 0.3 9999 9999 9999 9999 9999 12 12 12

Table 3. Controller Parameters for the Final Process Design with Three Intermediate Products (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

25.39 70.16 3.0 0.03 0.5 2 2 2 2 2 20 20 20

151.80 42.24 20.0 25.0 0.3 9999 9999 9999 9999 9999 12 12 12

Figure 10. Relationship between the boilup rate, V/F, and the vapor split ratio, RV, in the final process design with three intermediate products in the face of a ±10% change in the feed compositions of component A.

products display a great effect on the size and location of the black-hole problem (cf., Figure 6). In the case that the top intermediate product flow rate is set to be 4.6 mol/s and the two intermediate products are located at stages 26 and 30, respectively, the black-hole problem can be completely avoided. In terms of the adjustments of the flow rate of the top intermediate product (i.e., I1 = 4.0 mol/s), the relationship between the boilup rate, V/F, and the vapor split ratio, RV, is further refined, as shown in Figure 6c. The resultant final process design is shown in Figure 3b. 3.3. Avoidance of the Black-Hole Problem by the Arrangements of Three Intermediate Products to the DWDC. Modification of the initial process design is conducted

change in the feed compositions of component B or C, and the detailed results are omitted here. 3.2. Avoidance of the Black-Hole Problem by the Arrangements of Two Intermediate Products to the DWDC. Modification of the initial process design is conducted here with the arrangements of two intermediate products to the main distillation column. Figure 6 shows the relationship between the boilup rate, V/F, and the vapor split ratio, RV, for the DWDC with two intermediate products. One can readily note that the locations and flow rates of the two intermediate 4186



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Figure 11. A decentralized control scheme for the initial process design (example I): (a) control structure; (b) an equivalent control structure by Aspen Dynamics.

here with the arrangements of three intermediate products to the main distillation column. Figure 7 shows the relationship between the boilup rate, V/F, and the vapor split ratio, RV, for

the DWDC with three intermediate products. The locations and flow rates of the three intermediate products are found again to affect considerably the size and location of the black-hole 4187



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Figure 12. A decentralized control scheme for the final process design with three intermediate products (example I): (a) control structure; (b) an equivalent control structure by Aspen Dynamics.

3.4. Avoidance of the Black-Hole Problem by the Arrangements of Four Intermediate Products to the DWDC. Modification of the initial process design is conducted here with the arrangements of four intermediate products to the

problem (cf., Figure 7). When the top two intermediate products locate at stages 26 and 28, with their corresponding flow rates as 3.0 mol/s, respectively, the black-hole problem is eventually circumvented. The resultant final process design is shown in Figure 3c. 4188



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Figure 13. Regulatory responses of example I for a ±10% step change in the feed compositions of component A: solid lines, initial process design; dashed lines, final process design; gray curves, negative responses; black curves, positive responses.

main distillation column. Figure 8 shows the relationship between the boilup rate, V/F, and the vapor split ratio, RV, for the DWDC with four intermediate products. Similar to the

situations of arranging two or three intermediate products to the initial process design, the arrangement of four intermediate products affects again the size and location of the black-hole 4189



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along with their decentralized control systems. Four pressure control loops are included at the tops of the rectifying column, the two paralleled absorber columns, and the stripping column, respectively. While the first uses the heat removal from the condenser as the manipulated variable, the rest employs the power of the corresponding compressors as their manipulated variables. Five inventory control systems are installed into these two processes. 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 5.39 and 4.67, 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 dividing wall 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 levels at the bottoms of the two paralleled absorber columns are controlled, respectively, with the flow rates of their bottom withdrawals. The purities of the top, intermediate, and bottom products are controlled, respectively, by the top product flow rate, the intermediate product flow rates, and the heat input to the reboiler. Note that two ratio control schemes (i.e., K1 and K2) are employed here between the three intermediate products, since they were demonstrated to be effective for the operation of paralleled distillation columns.24 Here, K1 and K2 are assigned to be the steady-state ratios of the top two intermediate products to the bottom one, respectively. The liquid split ratio, RL, is employed to maintain the ratio between the compositions of components A and C in the intermediate product. It is assumed that all composition sensors have a 5 min dead-time element and all composition control loops make use of proportionalplus-integral (PI) controllers. The composition control systems are tuned by the built-in Tyreus−Luyben rule in a sequential manner.21−23 Namely, the QR loop is first tuned, since it greatly affects all of the other variables. Second, with the QR loop on automatic, the D loop is tuned using the same procedure. Third, with the QR and D loops on automatic, the I loop is tuned. Finally, with all three loops on automatic, the RL loop is tuned. The resultant parameters are summarized in Tables 2 and 3, respectively. Figure 13 depicts the closed-loop responses of the initial process design and the final process design with three intermediate products in the face of a ±10% step change in the feed compositions of component A, respectively, at a time instant of 2.5 h. The ratio between the feed compositions of components B and C has been kept the same as in the nominal operating conditions. For the +10% step disturbance in the feed compositions of component A, both the initial process design and the final process design with three intermediate products are capable of settling down to the expected steady state, but the latter exhibits smaller peak deviations and relatively shorter settling times. For the −10% step disturbance in the feed compositions of component A, the final process design with three intermediate products can handle the disturbance quite well, and the three product purities and the ratio between the compositions of components A and C in the intermediate product are kept closely to the given set-points. The initial process design, however, cannot be stabilized in the expected steady states; e.g., the ratio between the compositions of components A and C in the intermediate product is unable to be maintained at 1:1 even after a very long settling time. In particular, a sudden change happens in the three product compositions at a time instant of 21.86 h, which is anticipated to be caused by the intricate conflicts between the prefractionator

Table 4. Physical Properties and Design Specifications of Example II parameter condenser pressure (atm) stage pressure drop (atm) feed composition (mol %) ethanol (E) propanol (P) butanol (B) feed flow rate (kmol/s) feed thermal condition relative volatility E:P:B normal boiling point (K) ethanol (E) propanol (P) butanol (B) product specifications (mol %) ethanol (E) propanol (P) butanol (B) ratio between the compositions of ethanol and butanol in the intermediate product (E:B)

value 1 0 33.3 33.3 33.4 1 1.0 4:2:1 352 370 392 99 98 99 1:1

problem considerably (cf., Figure 8). When the top three intermediate products locate at stages 26, 27, and 28, with their corresponding flow rates as 2.2, 2.3, and 2.3 mol/s, respectively, the black-hole problem turns to disappear completely and the resultant final process design is shown in Figure 3d. Since the final process designs with two, three, and four intermediate products appear all to be effective to remove the black-hole problem, cautious discrimination between them should then be made on the basis of their steady-state performance. Figure 9 describes the relationship between the minimum boilup rate, V/Fmin, and the number of intermediate products. It is obvious that the final process design with three intermediate products should be chosen here because it requires the lowest energy requirement among the four schemes considered. Its composition and temperature profiles are shown in Figure 4e−h. Figure 10 shows the relationships between the boilup rate, V/F, and the vapor split ratio, RV, of the final process design with three intermediate products in the perturbed steady states by a ±10% change in the feed compositions of components A, B, and C, respectively. It can readily be noted that there is no longer a black-hole problem in these curves. 3.5. Closed-Loop Evaluation. The dynamics and controllability of the obtained final process design with three intermediate products should be examined here in order to further ascertain the removal of the black-hole problem. For comparison purposes, the closed-loop operation of the initial process design is also included here. The decentralized control structures for the initial process design and the final process design with three intermediate products are sketched in Figures 11a and 12a, respectively.21−23 They are devised in terms of RGA analysis. Because there is no available dynamic model for the DWDC or the Petlyuk distillation column in the commercial software Aspen Dynamics, an equivalent dynamic model must be constructed, and this can be accomplished using a stripping column with only a reboiler, two paralleled absorber columns without a reboiler or condenser, and a rectifying column with only a condenser. Figures 11b and 12b show the Aspen Dynamics implementation of the initial and final process designs 4190



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Figure 14. Initial and final process designs for example II: (a) initial process design; (b) final process design with two intermediate products; (c) final process design with three intermediate products; (d) final process design with four intermediate products.

here with a DWDC to be developed, and Table 4 lists the physical properties and design specifications. The operating pressure is set at 1 atm, and the flow rate of the feed processed is 1 kmol/s. Wolff and Skogestad once studied the problem and presented a process design, as shown in Figure 14a.18 It is taken here as the initial process design. While the main distillation column consists of 40 stages, the prefractionator contains 20 stages, allowing the dividing wall to be from stage 11 to stage 30. The steady-state simulation is performed with the commercial software Aspen Plus, and the Redlich−Kwong-UNIFAC model is used to estimate the thermodynamic properties of the ternary mixture separated. The composition and temperature profiles of the prefractionator and the main distillation column are shown in Figure 15a−d. The relationships between the boilup rate, V/F, and the vapor split ratio, RV, in the nominal steady state and in the face of a ±10% change in the feed compositions of ethanol are described in Figure 16a and b. One can readily note that the

and the main distillation column involved. For the case of a ±10% step change in the feed compositions of components B and C, the final process design with three intermediate products is still found to be superior in dynamic performance to the initial process design and the detailed results are not shown here. The steady-state and closed-loop simulation results obtained so far demonstrate that the adjustment of the initial process design with the introduction of three intermediate products to the main distillation column can be effective in removing the black-hole problem of the DWDC separating an ideal ternary mixture of hypothetical components A, B, and C.

4. EXAMPLE II: A DWDC SEPARATING A TERNARY MIXTURE OF ETHANOL, PROPANOL, AND BUTANOL 4.1. Problem Description. The separation of an equi-molar ternary mixture of ethanol, propanol, and butanol is conducted 4191



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Figure 15. Composition and temperature profiles in the initial and final process designs of example II: (a) composition profiles of the prefractionator in the initial process design; (b) composition profiles of the main distillation column in the initial process design; (c) temperature profiles of the prefractionator in the initial process design; (d) temperature profiles of the main distillation column in the initial process design; (e) composition profiles of the prefractionator in the final process design; (f) composition profiles of the main distillation column in the final process design; (g) temperature profiles of the prefractionator in the final process design; (h) temperature profiles of the main distillation column in the final process design. 4192



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Figure 16. Black-hole and its sensitivity to operating condition changes (example II): (a) black-hole of the initial process design; (b) ±10% changes in the feed compositions of component A.

black-hole problem exists in the operating region of interest at the nominal as well as perturbed steady states. 4.2. Avoidance of the Black-Hole Problem by the Arrangements of Two Intermediate Products to the DWDC. Figure 17 shows the relationship between the boilup rate, V/F, and the vapor split ratio, RV, for the DWDC with two intermediate products. It is readily found that the black-hole problem gradually disappears with the careful adjustments of the locations and flow rates of the two intermediate products. A final process design is finally generated and shown in Figure 14b, with the top intermediate product flow rate as 0.12 kmol/s and the two intermediate products located at stages 16 and 20, respectively. 4.3. Avoidance of the Black-Hole Problem by the Arrangements of Three Intermediate Products to the DWDC. Figure 18 shows the relationship between the boilup rate, V/F, and the vapor split ratio, RV, for the DWDC with three intermediate products. With the locations of the three intermediate products at stages 18, 19, and 20 and the flow rates as 0.11 kmol/s for the top two intermediate products, the black-hole problem vanishes completely, and this results in a final process design as shown in Figure 14c. 4.4. Avoidance of the Black-Hole Problem by the Arrangements of Four Intermediate Products to the DWDC. Figure 19 shows the relationship between the boilup rate, V/F, and the vapor split ratio, RV, for the process design with four intermediate products. With the locations of the four intermediate products at stages 17, 18, 19, and 20 and the flow

Figure 17. Variations of the black-hole with the adjustments of the locations and flow rates of the two intermediate products (example II): (a) black-hole in the initial process design; (b) black-hole versus the locations of the two intermediate products; (c) black-hole versus the flow rate of the top intermediate product.

rates as 0.08 kmol/s for the top three intermediate products, the black-hole problem disappears completely and this gives rise to a final process design as shown in Figure 14d. In Figure 20, the relationship between the minimum boilup rate, V/Fmin, and the number of intermediate products of the final process designs obtained is sketched. It is obvious that the minimum operation cost is achieved with the arrangements of two intermediate products to the DWDC and the corresponding process design should be adopted here. The composition and temperature profiles of the final process design with two intermediate products are shown in Figure 15e−h. Figure 21 4193



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Figure 18. Variations of the black-hole with the adjustments of the locations and flow rates of the three intermediate products (example II): (a) black-hole in the initial process design; (b) black-hole versus the locations of the three intermediate products; (c) black-hole versus the flow rates of the top two intermediate products.

Figure 19. Variations of the black-hole with the adjustments of the locations and flow rates of the four intermediate products (example II): (a) black-hole in the initial process design; (b) black-hole versus the locations of the four intermediate products; (c) black-hole versus the flow rates of the top three intermediate products.

shows the relationship between the boilup rate, V/F, and the vapor split ratio, RV, of the final process design with two intermediate products in the face of a ±10% change in the feed compositions of ethanol, propanol, and butanol, respectively. It is evident that there is no longer a black-hole problem in the final process design with two intermediate products. 4.5. Closed-Loop Evaluation. The examination of dynamics and controllability is conducted using the same control structures, as shown in Figures 8 and 9, and the controller parameters are listed in Tables 5 and 6, respectively, for the initial process design and the final process design with two intermediate products. In Figure 22, the closed-loop responses are compared

between these two process designs in the face of a ±10% step change in the feed compositions of ethanol at a time instant of 2.5 h. The ratio between the feed compositions of propanol and butanol has been kept the same as in the nominal operating conditions. For both circumstances, the final process design with two intermediate products appears to reject the disturbances effectively and can set back smoothly to the expected steady state. The initial process design, however, cannot keep the bottom product and the ratio between the compositions of propanol and butanol in the intermediate 4194



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Figure 20. Minimum boilup rate, V/Fmin, versus the number of intermediate products (example II).

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


controlled variable

KC (-)

TI (min)



XD,E XI,P XB,B RE/B F LR LC1 LC2 LC3 LC4 PC3 PC2 PC4

95.57 23.37 4.52 0.05 0.5 2 2 2 2 2 20 20 20

112.20 48.84 44.88 228.36 0.3 9999 9999 9999 9999 9999 12 12 12

Table 6. Controller Parameters for the Final Process Design with Two Intermediate Products (Example II) controller


controlled variable

KC (-)

TI (min)



XD,E XI,P XB,B RE/B F LR LC1 LC2 LC3 LC4 PC3 PC2 PC4

98.69 41.91 4.08 0.13 0.5 2 2 2 2 2 20 20 20

120.12 55.44 47.52 62.04 0.3 9999 9999 9999 9999 9999 12 12 12

Figure 21. Relationship between the boilup rate, V/F, and the vapor split ratio, RV, in the final process design with two intermediate products in the face of a ±10% change in the feed compositions of ethanol.

intermediate products to the main distillation column can be effective at removing the black-hole problem of the DWDC separating a nonideal ternary mixture of ethanol, propanol, and butanol.

product on their set-points, leaving quite large steady-state offsets. For the case of a ±10% step change in the feed compositions of propanol and butanol, the final process design with two intermediate products is still found to be superior in dynamic performance to the initial process design and the detailed results are not given here. The steady-state and closed-loop simulation results obtained so far demonstrate again that the adjustment of the initial process design with the introduction of multiple

5. DISCUSSION In terms of examples I and II studied in the current work, it has been demonstrated that the arrangement of multiple intermediate products to the main distillation column can be an effective strategy for the avoidance of the black-hole problem occurring in the DWDC involving four specifications on its three 4195



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Figure 22. Regulatory responses of example II for a ±10% step change in the feed compositions of ethanol: solid lines, initial process design; dashed lines, final process design; gray curves, negative responses; black curves, positive responses.

arrangements of multiple intermediate products should be considered as the primary impetus. Through the four connecting flows between the prefractionator and the main distillation

products. Although it is extremely difficult to gain a deep interpretation into its intricate mechanism, the variations in the liquid and vapor compositions and flow rates caused by the 4196



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Figure 23. Initial and final process designs for the modified example I: (a) initial process design; (b) final process design with two intermediate products; (c) final process design with three intermediate products; (d) final process design with four intermediate products.

column involved, a strong coordination function is generated between these two distillation columns and works to eliminate eventually the black-hole problem in the initial process design. It is interesting to note the fact that in addition to the removal of the black-hole problem the introduction of multiple intermediate products to the main distillation column can also pose a strong effect on the steady-state performance of the DWDC. As shown in Figures 12 and 20 for examples I and II, respectively, an upward parabolic relationship exists between the boilup rate, V/F, and the number of intermediate products for the resultant final process designs. Both of the optimum final process designs require a relatively lower utility requirement than the initial process designs, and these should definitely be attributed to the performed structural modifications, which not only lead to a refined relationship between the prefractionator and the main distillation column involved but also offer

additional degrees of freedom (i.e., the number, locations, and flow rates of the multiple intermediate products) for process optimization. It is these two changes that permit a certain degree of improvement in the steady-state performance of the optimum final process designs, which outweighs actually the unfavorable effect from the mixing of all of the intermediate products. The mechanism can also be used to explain the formation of an upward parabolic relationship between the boilup rate, V/F, and the number of intermediate products for the resultant final process designs. Comparing the composition profiles between the initial process design and the final process design with multiple intermediate products (cf., Figures 4 and 15), one can find that almost no changes occur between these two process designs, and this differs considerably from our earlier proposed method of adjusting the number of stages of the DWDC to eliminate the black-hole problem.19 The difference implies also 4197



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Figure 24. Variations of the black-hole with the adjustments of the locations and flow rates of the two intermediate products (modified example I): (a) black-hole in the initial process design; (b) black-hole versus the locations of the two intermediate products; (c) black-hole versus the flow rate of the top intermediate product.

Figure 25. Variations of the black-hole with the adjustments of the locations and flow rates of the three intermediate products (modified example I): (a) black-hole in the initial process design; (b) black-hole versus the locations of the three intermediate products; (c) black-hole versus the flow rates of the top two intermediate products.

the possibility of gaining improvement in the steady-state performance of the DWDC. It is worth examining here the applicability and flexibility of the proposed strategy by changing the thermodynamic properties and operating conditions of example I. In the case that the relative volatilities of the hypothetical components A, B, and C are arbitrarily modified to be 2:1.5:1 (through the variations of the vapor saturation pressure constants), the circumvention of the black-hole problem is conducted. The optimum design of the DWDC is derived in terms of the minimization of total annual

cost and shown in Figure 23a. With the application of the strategy proposed in the current work, the resultant final process designs with multiple intermediate products are yielded and shown in Figures 23b, c, and d, respectively. Figures 24−26 describe the variations of the black-hole problem with the adjustments of the locations and flow rates of the multiple intermediate products. One can readily find that the inherent black-hole problem can be completely removed from the initial process design with the effective arrangement of two, three, and four intermediate products to the initial process design. Figure 27 4198



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Figure 27. Minimum boilup rate, V/Fmin, versus the number of intermediate products (modified example I).

6. CONCLUSIONS The black-hole problem occurring in the design and operation of the DWDC with four specifications on its three products (i.e., the main compositions in the three products and the ratio between the two impurities in the intermediate product) has been addressed in this work. Since the number of intermediate products, their locations, and their flow rates can influence the coupling between the prefractionator and the main distillation column involved, it is reasonable to choose them as decision variables for process modifications, and a simple and yet effective procedure has been devised for the determination of their number, locations, and flow rates. In terms of two example DWDC systems separating, respectively, two ternary mixtures of hypothetical components, A, B, and C and ethanol, propanol, and butanol, the proposed strategy has been evaluated in the aspects of steady-state design and closed-loop operation. The obtained results have revealed its feasibility and effectiveness in tackling the black-hole problem of the DWDC. Though further research work should be conducted on the validation of the proposed strategy, it can be considered to be of general significance for the synthesis, design, and operation of the DWDC separating widely different ternary mixtures. It can also be viewed as an effective guideline for enhancing the applicability and flexibility of the DWDC. Current work is now focused on the studies about the removal of the black-hole problem from the DWDC with the arrangements of multiple feeds to the prefractionator as well as the deliberate determination of the thermal conditions for the multiple feeds and/or multiple intermediate products. It seems also imperative to explore the feasibility and effectiveness of using temperature inferential control strategies for the operation of the DWDC compensated by the methodologies proposed so far in our research works.

Figure 26. Variations of the black-hole with the adjustments of the locations and flow rates of the four intermediate products (modified example I): (a) black-hole in the initial process design; (b) black-hole versus the locations of the four intermediate products; (c) black-hole versus the flow rates of the top three intermediate products.

depicts the relationship between the minimum boilup rate, V/Fmin, and the number of intermediate products, and the optimum process design appears now to be a DWDC with two intermediate products. In the case that example I is modified to involve the separation of a ternary mixture of the hypothetical components A, B, and C with feed compositions other than the equi-molar ones, the proposed strategy is still found to work well.25 These additional studies serve to be good demonstrations about the applicability and flexibility of the proposed strategy.

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AUTHOR INFORMATION

Corresponding Author

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

The authors declare no competing financial interest. 4199



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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.

j = stage index m = main distillation column N = number of intermediate products in the DWDC p = prefractionator P = propanol R = reboiler

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NOTATION A = hypothetical component Avp = vapor pressure constant, bar B = hypothetical component or bottom flow rate, mol/s Bvp = vapor pressure constant, bar·K C = hypothetical component CC = composition controller COMP = compressor D = distillate flow rate, mol/s DWBL = width variation of a black-hole DWDC = dividing-wall distillation column F = feed flow rate, mol/s FC = flow rate controller HB = higher bound of a black-hole I = total flow rate of the multiple intermediate products, mol/s K = iteration number or the ratio between the flow rates of the intermediate products KC = proportional gain L = liquid flow rate, mol/s LB = lower bound of a black-hole LC = level controller M = multiplier n = number of stages in the DWDC N = number of intermediate products in the DWDC P = pressure, bar PC = pressure controller Q = heat duty, MW q = feed thermal condition R = reflux flow rate, mol/s RA/C = ratio between the compositions of components A and C in the intermediate product RC = ratio controller RL = liquid split ratio RR = reflux ratio RV = vapor split ratio S = stage for withdrawing the intermediate products T = temperature, K ΔT = time delay, s TI = integral time, s V = vapor flow rate, mol/s W = power of a compressor, MW WBL = width of a black-hole x = liquid composition y = vapor composition

Superscript

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s = saturation

REFERENCES

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Greek Letters

ε1 = error tolerance in judging the variation of a black-hole ε2 = error tolerance for judging the existence of a black-hole Subscripts

A = component index B = component index or bottom product C = component index or condenser COMP = compressor D = distillate product E = ethanol i = component index I = intermediate product 4200



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(24) Tsai, W.; Huang, H.; Yu, C. Control Structure Design for Parallel Processes: Application to Heat-Integrated Distillation. Chem. Eng. Res. Des. 2005, 83, 153. (25) Gao, J. Avoiding the Black-Hole Problem by the Arrangements of Multiple Intermediate Products to Dividing-Wall Distillation Columns. MS Thesis, Beijing University of Chemical Technology, 2013.

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Avoiding the Black-Hole Problem by the Arrangements of Multiple

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