Design and operation of dividing-wall distillation columns. 1. Diminishing the black-hole problem through over-design

Chemical Engineering and Processing 75 (2014) 90–109

Contents lists available at ScienceDirect

Chemical Engineering and Processing: Process Intensification journal homepage: www.elsevier.com/locate/cep

Design and operation of dividing-wall distillation columns. 1. Diminishing the black-hole problem through over-design Wei Chen a , Kejin Huang a,∗ , Haisheng Chen a , Chunying Xia a , Guosong Wu b , Kun Wang a a b

College of Information Science and Technology, Beijing University of Chemical Technology, Beijing 100029, People’s Republic of China Beijing Branch, Lubricant Company, Sinopec Corporation, Beijing 100085, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 16 August 2013 Received in revised form 19 October 2013 Accepted 17 November 2013 Keywords: Dividing-wall distillation column Three product specifications Black-hole problem Process design Process flexibility

a b s t r a c t For a dividing-wall distillation column (DWDC) with three specifications on its top, intermediate, and bottom products, respectively, owing to the complex interactions between the prefractionator and the main distillation column involved, it is usually infeasible to enhance substantially the composition of the intermediate product from its nominal operating condition and this confines terribly the flexibility and operability of the DWDC. The issue reflects an inherent drawback of the DWDC and is termed the black-hole problem in the current work. In this paper, an attempt is made to diminish the black-hole problem through over-design and the number of stages in each section of the DWDC is employed as decision variable to balance the interactions between the prefractionator and the main distillation column involved during process synthesis and design. Three illustrative examples are used to evaluate the feasibility and effectiveness of the proposed procedure and it is demonstrated that the black-hole problem can be effectively diminished in terms of careful adjustments of the number of stages in each section of the DWDC. The proposed philosophy represents a novel way to balance process design and process flexibility and is considered to be of general significance to the design and operation of the DWDC. © 2013 Elsevier B.V. All rights reserved.

1. Introduction The material and thermal coupling between the prefractionator and the main distillation column enables the dividing-wall distillation column (DWDC) to be much more cost effective and thermodynamically efficient than its conventional alternatives (e.g., the direct and indirect separation sequences) in the separation of ternary mixtures [1–6]. This has spurred considerable interests in the studies of its design and operation strategies over the last thirty years [7–15]. Although deep insights were already acquired into the steady-state and dynamic behaviors of the DWDC, some inherent and yet complicated issues have remained to be addressed so far and this has posed great concerns to the application of the DWDC to the chemical and petrochemical process industries [16,17]. One of such issues is the so-called “hole” problem indicated by Wolff and Skogestad in their pioneering work of 1995 [18]. In the case of the DWDC with four product specifications (i.e., the control of the main compositions in the top, intermediate, and bottom products and the ratio of the two impurities in the intermediate product), discontinuities (or no feasible solutions) might occur between the structural and operating design variables in the operating region of interest, making it impossible to achieve the desired degree of

∗ Corresponding author. Tel.: +86 10 64434801; fax: +86 10 64437805. E-mail address: [email protected] (K. Huang). 0255-2701/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cep.2013.11.007

separation even under the extreme operating condition of an infinite boilup rate or an infinite reflux ratio. Recently, we studied the problem and renamed it the black-hole problem because its formation mechanism has not been fully clarified [19–21]. It was found that the black-hole problem could be completely removed through the adjustments of the number of stages in each section of the DWDC, the arrangements of multiple intermediate products to the main distillation column involved, or the employment of feed splitting strategy in the prefractionator involved. Not only could the steady-state behaviors but also dynamics and controllability of the DWDC be improved substantially, signifying the great importance to tackle the black-hole problem during process synthesis and design [19–21]. In the case of the DWDC with three specifications on its top, intermediate, and bottom products, respectively (i.e., the control of the main compositions in the three products), although discontinuities are rarely found in the nominal steady state, it is frequently infeasible to enhance, to a certain extent, the purity of the intermediate product even under the extreme operating condition of an infinite boilup rate or an infinite reflux ratio. This phenomenon differs sharply from the circumstance of conventional distillation columns because their top and bottom products can, in principle, reach any desired purities provided that a high enough reflux ratio and boilup rate have been adopted. This deficiency restricts terribly the flexibility and operability of the DWDC and represents definitely a serious issue that should be studied with great caution in process design and operation. Wolff and Skogestad

W. Chen et al. / Chemical Engineering and Processing 75 (2014) 90–109

Fig. 1. Black-hole problem (Example I): (a) in terms of the MAPTP and (b) in terms of the MAPIP.

noticed the problem but pursued no further studies [18]. Dünnebier and Pantelides also encountered the same problem in their studies of process design and attributed simply its occurrences to the inability of limiting the compositions of the light and heavy components in the intermediate product [22]. Despite the fact that the problem appears to be quite different in nature from the blackhole problem of the DWDC with four product specifications on its three products, it reflects actually one of the potential drawbacks of the DWDC that have been resulted from the material and thermal coupling between the prefractionator and the main distillation column involved, so we still term it the black-hole problem for the DWDC with three specifications on its top, intermediate, and bottom products, respectively, in the current work. The primary purpose of the current work is to address the blackhole problem of the DWDC with three specifications on its top, intermediate, and bottom products, respectively, in terms of overdesign. The remainder of the article is organized as follows. Firstly, the black-hole problem of the DWDC with three product specifications is illustrated along with the elucidation of the reasons behind this complex phenomenon. Secondly, a generalized procedure is developed to diminish the black-hole problem in terms of overdesign and the number of stages in each section of the DWDC is adopted as decision variables to coordinate the conflicts between the prefractionator and the main distillation column involved. Thirdly, the separations of three ternary mixtures of hypothetical components, A, B, and C, benzene, toluene, and o-xylene, and ethanol, propanol, and butanol, are chosen as illustrative examples

91

Fig. 2. A generalized philosophy proposed for diminishing the black-hole problem of the DWDC: (a) in terms of the MAPTP and (b) in terms of the MAPIP. Bold line: optimum process design in the corresponding steady state, solid line: process design in the NSS, dotted line: process design in the MASS, and dashed line: process design desired.

to evaluate the proposed philosophy. Thorough comparisons are also conducted between the resultant process design and those based on the given nominal and maximally achievable steady states. Last but not least, the proposed philosophy is further analyzed in the discussion section followed by a brief summary of the current work in the last section of this article. 2. Diminishing the black-hole problem of the DWDC through over-design 2.1. Black-hole problem of the DWDC with three specifications on its top, intermediate, and bottom products, respectively Generally speaking, chemical processes are developed in their nominal steady states in terms of the minimization of an economical objective function (plus the consideration of their environmental impacts). Even though redundancy considerations have not yet been taken into account during process synthesis and design, the resultant process designs can usually operate in a relatively wide operating region. For instance, a binary distillation column can, in principle, produce a top and bottom products with widely different compositions through the adjustments of reflux and reboil flow rates simultaneously. For the DWDC with three specifications on its top, intermediate, and bottom products, respectively, the situation can be quite different. Owing to the high degree of material and thermal coupling between the prefractionator and the main distillation column involved, some

92

W. Chen et al. / Chemical Engineering and Processing 75 (2014) 90–109

Fig. 3. A generalized procedure proposed for diminishing the black-hole problem of the DWDC.

unusual steady-state behaviors including, for example, nonlinearity and input and/or output multiplicities are quite likely to occur within the DWDC and narrow consequently the variability of the intermediate product quality. This can inevitably pose serious influences to the flexibility and operability of the DWDC. For example, in the case of the separation of an equi-molar ideal ternary mixture of hypothetical components, A, B, and C, into three relatively pure components with compositions of 99 mol%, respectively (i.e., the Example I to be studied in the next section of the current work), if the DWDC is derived in the nominal steady state (NSS) through the consideration of only economical factors, the maximally achievable purities of the three products (MAPTP, an index defined under

the assumption of equal compositions of the main components in the top, intermediate, and bottom products, which can simply be determined in terms of a single variable search method) are only 99.25 mol% (c.f., Fig. 1a), representing certainly a too much narrow operating region in the right of the NSS. The process is evidently infeasible to operate in the operating region between 99.25 mol% and 100 mol% even under the extreme operating condition of an infinite boilup rate or an infinite reflux ratio. The region (i.e., the shadowed area) is then the black hole of the DWDC with three specifications on its top, intermediate, and bottom products, respectively, which can restrain terribly process flexibility and operability.

W. Chen et al. / Chemical Engineering and Processing 75 (2014) 90–109

The black-hole problem can also be represented in terms of the maximally achievable purity of the intermediate product (MAPIP). It is an index closely related to the material and thermal coupling between the prefractionator and the main distillation column involved and calculated in the following manner. Assume further that the DWDC is required to being able to produce the top, intermediate, and bottom products with compositions up to 99.5 mol%, respectively (It is termed the maximally achievable steady state (MASS), hereinafter, in the current work, which is evidently an appropriate index to reflect the flexibility and operability of the DWDC and usually given as a requirement for process development), it is then reasonable to fix strictly the top and bottom products at 99.5 mol% through the careful adjustments of the reflux flow rate and boilup rate. Under this specific condition, the maximal value of the intermediate product is searched and this yields the MAPIP. In the case of the hypothetical example, the MAPIP is found to be only 98.95 mol%, considerably smaller than the required one of the MASS, and the operating region between the MAPIP and 100 mol% is another kind of representation of the black hole. The relationship between the composition of the intermediate product and total annual cost (TAC) is depicted in Fig. 1b. The TAC is defined to include operating cost plus discount capital investment by a payback time of three years (c.f., Appendix A). The formation of the black-hole problem is no doubt related to the complex interactions between the prefractionator and the main distillation column involved in the DWDC. Since the black-hole problem cannot be completely removed from the DWDC with three specifications on its top, intermediate, and bottom products, respectively (unless with infinite capital investment), it is then only feasible to diminish it through careful process design. Thus, how to diminish effectively the black-hole problem represents an important issue to be addressed for the synthesis and design of the DWDC. 2.2. Diminishing the black-hole problem through over-design One simple method is available to diminish the black-hole problem of the DWDC with three specifications on its top, intermediate, and bottom products, respectively. Namely, instead of designing the DWDC in the NSS, design it in the MASS. Since the DWDC usually operates in the NSS, the MASS method leads inevitably to a larger TAC than the process design in the NSS, representing therefore the most conservative strategy. To suppress the deficiency, we may rely on the over-design of the DWDC with the aim to balance the relationship between the prefractionator and the main distillation column involved and this represents essentially a common requirement for all kinds of process intensification [23,24]. A detailed illustration of the principle is sketched in Fig. 2a. Here, the bold line represents the performance of the optimum process design in the corresponding steady state (i.e., the one derived from the minimization of the TAC), which indicates the bottom line of all kinds of process development. The solid line stands for the performance of the process design in the NSS. Because of the complex interactions between the prefractionator and the main distillation column involved, the NSS process design is quite likely to present a great black-hole problem with the MAPTP substantially smaller than the MASS. The dotted line represents here the performance of the optimum process design in the MASS. Although it leads to a small black-hole problem with the MAPTP even greater than the MASS, a large TAC can generally be expected around the NSS. The wider the operating region between the NSS and MASS is required for the DWDC to be developed, the sharper the NSS and MASS process designs differ in the TAC around the NSS. Therefore, it is necessary and quite likely to derive a process design that represents a careful compromise between these two process designs and one of the possible solutions is demonstrated here by the dashed

93

A

I II V ABC

B VI

III

IV C

(a)

A

I

ABC

V

II

VI

III

B

IV C

(b) Fig. 4. Petlyuk distillation column and its thermodynamic equivalent, the DWDC: (a) Petlyuk distillation column and (b) DWDC.

line. As can be seen, the desired process design can not only have a diminished black-hole problem in comparison with the process design in the NSS, but also become more cost effective than the MASS process design around the NSS. The derivation of such a process design needs certainly an effective procedure for guidance and over-design in terms of a careful coordination between the prefractionator and the main distillation column involved seems to be a good candidate in this aspect. In terms of the MAPIP, the same principle can also be sketched and shown in Fig. 2b. By coordination between the prefractionator and the main distillation column involved, the desired process design is likely to give an intermediate product with its composition equal or even higher than the intermediate composition of the maximally achievable steady state (ICMASS, i.e., the maximally achievable composition of the intermediate product specified for process development) while still representing a careful trade-off between the two process designs in the NSS and MASS, respectively.

2.3. A generalized procedure proposed for diminishing the black-hole problem Fig. 3 shows a generalized procedure proposed for diminishing the black-hole problem of the DWDC with three specifications on its top, intermediate, and bottom products, respectively. It involves mainly three steps and is explained in detail as follows.

94

W. Chen et al. / Chemical Engineering and Processing 75 (2014) 90–109

QC = 1.157 MW

P = 3 bar

D = 9.322 mol/s XD, A = 0.99 XD, B = 0.01 XD, C = 0

RR = 3.269

2 11

RL = L1/L11 = 0.32 q=1 F = 27.8 mol/s A/B/C = 0.333/0.333/0.334

1

12

19

37 RV = V37/V42 = 0.556

I = 9.191 mol/s XI, A = 0.00316 XI, B = 0.99 XI, C = 0.00684

23

41 42 55 QR = 1.157 MW V = 39.792 mol/s

B = 9.287 mol/s XB, A = 0 XB, B = 0.01 XB, C = 0.99

Fig. 5. Initial process design (Example I).

Firstly, process synthesis and design should be performed in the NSS in terms of the minimization of an economical objective function. The resultant process design is termed the initial process design, hereinafter, in the current work. Based on the operating region required for the DWDC to be developed, the MASS can easily

Table 1 Physical properties and design specifications of Example I. Parameter Condenser pressure (bar) Stage pressure drop (bar) Feed compositions (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

Value 3 0 33.3 33.3 33.4 27.8 1.0 4:2:1 29,053.7 13.04/3862 12.34/3862 11.65/3862 99 99 99

Table 2 Economical basis of process design for Example I. Parameter Condenser Heat transfer coefficient (kW/K m2 ) Temperature difference (K) Reboiler Heat transfer coefficient (kW/K m2 ) Temperature difference (K) Cooling water cost ($/ton) Steam cost ($/ton) Payback period (yr)

Value 0.852 13.9 0.568 34.8 0.06 25 (0.985 + 0.015PS ) 3

be identified. With the application of the steady-state model of the DWDC, the MAPIP (or the MAPTP in the case of equal compositions in the three products) can be determined and used to ascertain whether or not the initial process design meets the given flexibility and operability requirement. If it is the case, then process design is finished, otherwise, over-design should be performed toward the initial process design to diminish the black-hole problem. Secondly, structural modifications should be conducted through the careful adjustments of the number of stages in the initial process design. Since the DWDC is divided actually into six sections (i.e., sections I–VI) with the locations of the feed to the prefractionator, the intermediate product from the main distillation column, and the connecting flows between these two distillation columns as shown in Fig. 4a and b (note here that the former is based on a Petlyuk distillation column and the latter its thermodynamic equivalent, the DWDC), the number of stages in each section of the DWDC can be used as the decision variable for process modifications and our earlier studies already demonstrated that it was effective to coordinate the relationship between the prefractionator and the main distillation column involved [19]. During each round of iteration, the section whose number of stages should be adjusted is firstly determined and this can be accomplished in terms of adding one stage to each section of the DWDC and then examining in which section this leads to the greatest MAPIP. In the chosen section, only are one or two stages permitted to be added each time (Eq. (1)) and this serves to avoid a too conservative process design. Eq. (2) is employed to check whether or not the adjustment of the number of stages should still be conducted in the current section (ε1 > 0). The difference between the ICMASS and MAPIP, as shown in Eq. (3), is taken as the convergence criterion for process modifications. If the difference is smaller than a given value (i.e., ε2 > 0), then the iterative adjustment of the number of stages should be continued in the current section according to the proposed procedure, otherwise, it indicates that the black-hole problem has already been diminished to a degree that the modification of the initial process design should be stopped. Here, ε1 and ε2 are two important parameters that can influence the performance of the iterative search. While too small a value of ε1 may slow down the search process,

W. Chen et al. / Chemical Engineering and Processing 75 (2014) 90–109

95

Fig. 6. Diminishing the black-hole problem through the adjustments of the number of stages in sections I to VI (Example I): (a) black-hole problem in the initial process design with the MASS as 99.5 mol%, (b) black-hole problem in the initial process design with the MASS as 99.6 mol%, (c) black-hole problem in the intermediate process design with the MASS as 99.5 mol%, (d) black-hole problem in the intermediate process design with the MASS as 99.6 mol%, (e) black-hole problem in the final process design with the MASS as 99.5 mol%, and (f) black-hole problem in the final process design with the MASS as 99.6 mol%.

Table 3 Comparison between the initial and final process designs and those based on the MASS (Example I). Scenario

Initial process design MASS1 = 99.5 mol% Process design in the MASS1 Final process design MASS2 = 99.6 mol% Process design in the MASS2 Final process design

TAC (105 $/yr) NSS

Comparison (%)

5.8908

100

−

6.0236 5.9635

102.25 101.23

6.1065 6.2296

100 102.02

6.0716 6.0205

103.07 102.20

6.1646 6.3838

100 103.56

MASS

Comparison (%) −

96

W. Chen et al. / Chemical Engineering and Processing 75 (2014) 90–109

Fig. 7. Intermediate and final process designs (Example I): (a) intermediate process design with the MASS as 99.5 mol%, (b) intermediate process design with the MASS as 99.6 mol%, (c) final process design with the MASS as 99.5 mol%, (d) final process design with the MASS as 99.6 mol%.

too great a value of ε2 may generate an unnecessarily conservative solution. The resultant process design is termed the intermediate process design, hereinafter, in the current work. NJ,K+1 = NJ,K + NJ,K

1 ≤ NJ,K ≤ 2

(1)

MAPIPJ,K+1 − MAPIPJ,K ≤ ε1

(2)

MAPIPJ,K+1 − ICMASS≥ε2

(3)

Lastly, the optimization of the intermediate process design should be performed because its economical optimality can no longer be guaranteed due to the structural modifications performed in the second step. Note here that the decision variables are still the number of stages in each section of the intermediate process design and so is the economical objective function to be minimized. Other operating variables, including the liquid split ratio, RL , and the vapor split ratio, RV , should remain unvaried because their changes may result in great deviations from the optimality of the initial process design [25,26]. With the prerequisite of being feasible to operate in

the MASS (i.e., the reboiler heat duty in the MASS should be smaller than the given maximally allowable value), the intermediate process design is re-optimized in the NSS and this can be achieved with the intensive use of the steady-state model of the DWDC. The optimization is converged in case that the following criterion is satisfied. |TAC(M + 1) − TAC(M)| ≤ ε3 TAC(M)

(4)

where ε3 is a predetermined error tolerance. The resultant process design is termed the final process design, hereinafter, in the current work. Since the areas of the heat exchangers in the condenser and reboiler change with the operating conditions, this gives rise to difficulties in the calculation of the TAC for the other steady states. Here, the capital investment of the heat exchangers is exclusively estimated based on the MASS and this serves actually to be a rather conservative estimation of the capital investment for the final process design in the NSS.

W. Chen et al. / Chemical Engineering and Processing 75 (2014) 90–109

0.8

1 A

B

C

0.8

0.6

Mole fraction (-)

Mole fraction (-)

97

0.4

0.2

0.6 A

B

C

0.4 0.2 0

0 0

10

20 Stage of prefractionator

0

30

10

20 30 40 Stage of main distillation column

(a)

50

(b) 1

0.8 B

C

0.8 Mole fraction (-)

Mole fraction (-)

A 0.6

0.4

0.2

0.6 A

B

0.2

0

0

0

10

20 Stage of prefractionator

30

0

10

20 30 40 Stage of main distillation column

(c)

50

(d)

0.8

1 A

B

C

0.8

0.6 Mole fracion (-)

Mole fraction (-)

C

0.4

0.4

0.2

0.6 A

0.4

B

C

0.2 0

0 0

10

20 Stage of prefractionator

30

40

0

10

20 30 40 Stage of main distillation column

(e)

50

60

(f)

Fig. 8. Composition profiles of the initial and final process designs in the NSS (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 with the MASS as 99.5 mol%, (d) main distillation column in the final process design with the MASS as 99.5 mol%, (e) prefractionator in the final process design with the MASS as 99.6 mol%, and (f) main distillation column in the final process design with the MASS as 99.6 mol%.

In the following three sections, three DWDC systems separating, respectively, an ideal ternary mixture of hypothetical components, A, B, and C, a real ternary mixture of benzene, toluene, and o-xylene, and a real ternary mixture of ethanol, propanol, and butanol, are chosen as illustrative examples to evaluate the generalized philosophy proposed, i.e., the feasibility and effectiveness of diminishing the black-hole problem in terms of the adjustments of the number of stages in each section of the DWDC. Thorough comparisons are also made between the initial process design, final process design and the most conservative process design based on the given MASS. In all of the three examples, ε1 , ε2 , and ε3 are set to be 10−4 , 10−5 , and 10−4 , respectively.

3. Example I: a DWDC separating an ideal ternary mixture of hypothetical components A, B, and C 3.1. Process description Ideal vapor and liquid phase behaviors are assumed for the ternary mixture of hypothetical components A, B, and C, and the vapor–liquid equilibrium relationship can be expressed by

Pj = xA,j PAs + xB,j PBs + xC,j PCs

1 ≤ j ≤ NT

(5)

98

W. Chen et al. / Chemical Engineering and Processing 75 (2014) 90–109 Table 4 Physical properties and design specifications of Example II. Parameter

Fig. 9. Comparison between the initial and final process designs and those in the MASS (Example I): bold line: optimum process design in the corresponding steady state, solid line: initial process design, dotted line: process designs in the MASS (99.5 mol% and 99.6 mol%), dashed line: final process designs with the MASS as 99.5 mol% and 99.6 mol%, respectively.

Condenser pressure (atm) Stage pressure drop (atm) Feed compositions (mol%) Benzene (B) Toluene (T) O-xylene (X) Feed flow rate (mol/s) Feed thermal condition Relative volatility B:T:X Normal boiling points (K) Benzene (B) Toluene (T) O-xylene (X) Product specifications (mol%) Benzene (B) Toluene (T) O-xylene (X)

yi,j =

Pj

i = A, B, C, and 1 ≤ j ≤ NT

(6)

The vapor saturation pressure is calculated via s = Avp,i − ln Pi,j

Bvp,i Tj

i = A, B, C, and 1 ≤ j ≤ NT

0.37 0 30 30 40 1000 1.0 7.1:2.2:1 353 385 419 99 99 99

Table 5 Economical basis of process design for Examples II and III. Parameter

s xi,j Pi,j

Value

Condenser Heat transfer coefficient (kW/K m2 ) Temperature difference (K) Reboiler Heat transfer coefficient (kW/K m2 ) Temperature difference (K) Steam cost ($/106 kJ) Payback period (yr)

Value 0.852 13.9 0.568 34.8 4.7 3

(7) 3.2. Black-hole problem

The commercial software Aspen Plus is used for the steady-state simulations of the DWDC to be developed. Remember the fact that the Petlyuk distillation column is equivalent to the DWDC in thermodynamic efficiency in case that no heat transfer is allowed to across the dividing-wall, the module of the Petlyuk distillation column is used instead to predict the steady-state behaviors of the DWDC in the current work. The physical properties and design specifications are listed in Table 1 for the DWDC to be developed. The feed separated is an equi-molar ternary mixture of hypothetical components A, B, and C, with a flow rate of 27.8 mol/s, and the three products are specified to be 99 mol%, respectively. It is further required here that the DWDC should be able to reach the purities of 99.5 mol% and 99.6 mol% (i.e., the MASS1 and MASS2 ), respectively, in its top, intermediate, and bottom products. The sizing relationships are taken from Wang et al. [27] and reproduced in Appendix A for quick reference and the economic factors are listed in Table 2. Fig. 5 gives the optimum design of the DWDC based on the minimization of the TAC with a simple optimization method proposed in our earlier work [27]. This process design is regarded as the initial process design here. The operating pressure is set to be 3 bar and no pressure drop is considered in this example. The initial process design contains 37 stages in the prefractionator and 56 stages in the main distillation column. In accordance with the Aspen Plus notation, the condenser is designated as stage 1 and the reboiler stage 56. The rectifying and stripping sections have 10 and 14 stages, respectively, and the dividing-wall runs from stage 12 to stage 41 in the main distillation column. The feed processed is introduced onto stage 19 of the prefractionator and the intermediate product is withdrawn from stage 23 of the main distillation column. The reflux ratio is 3.269, and the reboiler heat input is 1.157 MW. The liquid split ratio, RL , is 0.32 and the vapor split ratio, RV , is 0.556, which means that the dividingwall is actually not located at the middle of the cross-sectional area of the DWDC.

Fig. 6a and b depicts the relationship between the TAC and the composition of the intermediate product for the two situations mentioned above. It can readily be noted that both of the MAPIPs are much smaller than the ICMASS (i.e., 99.5 mol% and 99.6 mol%), indicating the necessity of diminishing the black-hole problem through over-design. Note also here the fact that the magnitudes of the black holes are greatly affected by the given MASS. 3.3. Diminishing the black-hole problem through over-design With the application of the generalized procedure proposed in the current work, two intermediate process designs are generated and shown in Fig. 7a and b, respectively, which can now reach the purities of 99.5 mol% and 99.6 mol%, respectively, in their three products. It can readily be found that these two process designs have been generated through the addition of several stages to sections I and V of the initial process design. While for the intermediate process design that can achieve the purities of 99.5 mol% in its three products, 5 and 2 stages are added, respectively, for the intermediate process design that can achieve the purities of 99.6 mol%, 5 and 4 stages are appended instead. Fig. 6c and d depicts the relationship between the TAC and the composition of the intermediate product for the two intermediate process designs. It is obvious that the black-hole problems have been diminished as compared with those of the initial process design. Fig. 7c and d sketches the resultant final process designs and Fig. 6e and f depicts the relationships between the TAC and the composition of the intermediate product. Note that these two final process designs are still able to reach the purities of 99.5 mol% and 99.6 mol%, respectively, in their three products and these outcomes demonstrate the effectiveness of the generalized strategy proposed in the current work. While the final process design that can achieve the purities of 99.5 mol% is derived by the removal of 2 and 1 stages, respectively, from sections I and VI of

W. Chen et al. / Chemical Engineering and Processing 75 (2014) 90–109

99

Fig. 10. Initial process design and its black-hole problem (Example II): (a) initial process design and (b) black-hole problem in terms of the MAPTP.

the corresponding intermediate process design, the final process design that can achieve the purities of 99.6 mol% is by the reduction of 1 stage from sections III and VI. It is worth mentioning here the fact that opposite strategies are adopted actually in the derivation of the intermediate and final process designs that can achieve the purities of 99.5 mol% in its three products and this is closely related to the strong non-linearity of the DWDC. Fig. 8 compares the steady-state profiles of liquid compositions of the initial and the two final process designs in the NSS. Remixing effect occurs in the prefractionator of the initial process design, but disappears completely in the final process designs. Fig. 9 delineates the performance of the initial and two final process designs and those based on the given MASS in the operating region between the NSS and MASS. It is readily found that in the NSS the final process designs represent a careful compromise between the initial process design and those based on the given MASS. In particular, they have acquired an extended feasible operating region in comparison with the initial process design. Table 3 gives a detailed comparison between the initial and final process

designs and those based on the given MASS. The results indicate that the flexibility and operability of the final process designs have been strengthened with an additional increase of the TAC by 1.23% and 2.20%, respectively, as compared with the initial process design. Although in the MASS these final process designs lead to a 2.02% and 3.56% larger TAC than those process designs based on the given MASS, they yield a reduced TAC by 1.02% and 0.87%, respectively, in the NSS. Since the NSS represents the major operating condition of the DWDC, the resultant final process designs are generally more economically favorable than those process designs based on the given MASS. 4. Example II: a DWDC separating a ternary mixture of benzene, toluene, and o-xylene 4.1. Process description Table 4 summarizes the physical properties and design specifications of the DWDC to be developed. The ternary mixture

100

W. Chen et al. / Chemical Engineering and Processing 75 (2014) 90–109

Fig. 11. Diminishing the black-hole problem through the adjustments of the number of stages in sections I to VI (Example II): (a) black-hole problem in the initial process design with the MASS as 99.6 mol%, (b) black-hole problem in the initial process design with the MASS as 99.8 mol%, (c) black-hole problem in the intermediate process design with the MASS as 99.6 mol%, (d) black-hole problem in the intermediate process design with the MASS as 99.8 mol%, (e) black-hole problem in the final process design with the MASS as 99.6 mol%, and (f) black-hole problem in the final process design with the MASS as 99.8 mol%.

separated has a composition of 30/30/40 mol% with a flow rate of 1 kmol/s and the three products are specified to be 99 mol%, respectively. The DWDC to be developed should also reach the purities of 99.6 mol% and 99.8 mol% (i.e., the MASS1 and MASS2 ), respectively, in its top, intermediate, and bottom products. The pressure of condenser is set to be 0.37 atm and no pressure drop is considered here. The DWDC is still simulated with the commercial software Aspen Plus and the Chao–Seader method is adopted to represent the thermodynamic properties of the ternary mixture separated. The economical basis shown in Table 5 is employed for process synthesis and design here, and the initial process design is generated in terms of the minimization of the TAC using the optimization

algorithm proposed in our earlier work [27]. The resultant process design is sketched in Fig. 10a. Owing to the employment of different optimization algorithm and the inclusion of stage cost, this resultant initial process design is slightly different from the one by Ling and Luyben [11], and so do the capital investment and objective function. The initial process design accommodates, respectively, 23 and 44 stages in the prefractionator and the main distillation column involved, with the dividing-wall locating between stage 9 and stage 30 in the main distillation column. The feed is located at stage 14 in the prefractionator and the intermediate product is withdrawn from stage 17 in the main distillation column. The liquid split ratio, RL , is 0.307 and the vapor split ratio, RV , is 0.625.

W. Chen et al. / Chemical Engineering and Processing 75 (2014) 90–109

101

Fig. 12. Intermediate and final process designs (Example II): (a) intermediate process design with the MASS as 99.6 mol%, (b) intermediate process design with the MASS as 99.8 mol%, (c) final process design with the MASS as 99.6 mol%, (d) final process design with the MASS as 99.8 mol%.

4.2. Black-hole problem Fig. 10b depicts the relationship between the TAC and the purities of the three products and the MAPTP is found to be 99.29 mol% (apparently smaller than the given 99.6 mol% and 99.8 mol%) in this case. Fig. 11a and b displays the relationship between the TAC and the composition of the intermediate product for the two circumstances and the MAPIP is found to be 98.82 mol% and 93 mol%, respectively. Since they are both smaller than the ICMASS (i.e., 99.6 mol% and 99.8 mol%), it is then necessary to consider the alleviation of the black-hole problem through over-design. 4.3. Diminishing the black-hole problem through over-design Fig. 12a and b gives the intermediate process designs that can reach the purities of 99.6 mol% and 99.8 mol%, respectively. While the intermediate process design that can achieve the purities of 99.6 mol% in its three products is resulted in by the addition of 2, 3, and 3 stages to sections I, II, and V of the initial process design, the intermediate process design that can achieve the purities of 99.8 mol% in its three products is by 2, 4, and 4 stages to the same sections, respectively. Fig. 11c and d shows the relationship between the TAC and the composition of the intermediate product and the black-hole problems have apparently been diminished as compared with those of the initial process design. Fig. 12c and d shows the resultant final process designs and Fig. 11e and f depicts

their black-hole problems. One can readily understand that these two final process designs can reach the purities of 99.6 mol% and 99.8 mol%, respectively, in its three products and these outcomes highlight again the simplicity and effectiveness of the generalized strategy proposed in the current work. While the final process design that can achieve the purities of 99.6 mol% in its three products is generated by deleting 1 stage from sections I, IV, and V, and 2 stages from section II of the corresponding intermediate process design, the final process design that can achieve the purities of 99.8 mol% in its three products is by removing 1 stage from sections I, III, IV, and V. Fig. 13 compares the steady-state profiles of liquid compositions of the initial and two final process designs in the NSS. Remixing effect is found again in the prefractionator of the initial process design, but can no longer be observed in the final process designs. Fig. 14 depicts the performance of the initial and two final process designs and those based on the given MASS in the operating region between the NSS and MASS and Table 6 gives a detailed comparison between them in these two steady states, respectively. It is demonstrated that the flexibility and operability of the final process designs have been acquired with an additional increase of the TAC by 0.75% and 1.58%, respectively, as compared with the initial process design. Although in the MASS they result in a 2.93% and 3.62% larger TAC than those process designs based on the MASS, they give a reduced one by 0.72% and 1.19%, respectively, in the NSS.

102

W. Chen et al. / Chemical Engineering and Processing 75 (2014) 90–109

1

0.8 T

X 0.8 Mole fraction (-)

Mole fraction (-)

B 0.6

0.4

0.2

0.6 B

T

0.2

0

0 0

5

10 15 Stage of prefractionator

20

0

10 20 30 Stage of main distillation column

(a) 1 B

T

X

0.8

0.6

Mole fraction (-)

Mole fraction (-)

40

(b)

0.8

0.4

0.2

0.6 B

T

X

0.4 0.2 0

0 0

5

10 15 Stage of prefractionator

20

0

25

10 20 30 Stage of main distillation column

(c)

40

(d) 1

0.8 B

T

X 0.8

0.6

Mole fraction (-)

Mole fraction (-)

X

0.4

0.4

0.2

0.6 B

0.4

T

X

0.2 0

0 0

5

10 15 Stage of prefractionator

20

0

25

(e)

10

20 30 Stage of main distillation column

40

(f)

Fig. 13. Composition profiles of the initial and final process designs in the NSS (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 with the MASS as 99.6 mol%, (d) main distillation column in the final process design with the MASS as 99.6 mol%, (e) prefractionator in the final process design with the MASS as 99.8 mol%, and (f) main distillation column in the final process design with the MASS as 99.8 mol%.

Table 6 Comparison between the initial and final process designs and those based on the MASS (Example II). Scenario

Initial process design MASS1 = 99.6 mol% Process design in the MASS1 Final process design MASS2 = 99.8 mol% Process design in the MASS2 Final process design

TAC (106 $/yr) NSS

Comparison (%)

MASS

Comparison (%)

7.2962

100

−

7.4034 7.3509

101.47 100.75

7.5151 7.7353

100 102.93

7.4984 7.4114

102.77 101.58

7.6444 7.9213

100 103.62

−

W. Chen et al. / Chemical Engineering and Processing 75 (2014) 90–109

103

5. Example III: a DWDC separating a ternary mixture of ethanol, propanol, and butanol 5.1. Process description

Fig. 14. Comparison between the initial and final process designs and those in the MASS (Example II): bold line: optimum process design in the corresponding steady state, solid line: initial process design, dotted line: process designs in the MASS (99.6 mol% and 99.8 mol%), dashed line: final process designs with the MASS as 99.6 mol% and 99.8 mol%, respectively.

Table 7 summarizes the physical properties and design specifications of the DWDC to be developed. The mixture separated is an equi-molar one with a flow rate of 83.33 mol/s, and the three products are specified to be 99 mol%, respectively. The DWDC should be operable in case that its top, intermediate, and bottom products have the purities of 99.4 mol% and 99.6 mol% (i.e., the MASS1 and MASS2 ), respectively. The pressure of the condenser is set to be 1 atm with a pressure drop of 0.0068 atm per stage. The DWDC is still simulated with the commercial software Aspen Plus and the UNIFAC method is adopted to estimate the thermodynamic properties of the ternary mixture separated. The economical basis shown in Table 5 is still employed here for process synthesis and design. The initial process design is generated in terms of the minimization

Fig. 15. Initial process design and its black-hole problem (Example III): (a) initial process design, (b) black-hole problem in terms of the MAPTP.

104

W. Chen et al. / Chemical Engineering and Processing 75 (2014) 90–109

Fig. 16. Diminishing the black-hole problem through the adjustments of the number of stages in sections I to VI (Example III): (a) black-hole problem in the initial process design with the MASS as 99.4 mol%, (b) black-hole problem in the initial process design with the MASS as 99.6 mol%, (c) black-hole problem in the intermediate process design with the MASS as 99.4 mol%, (d) black-hole problem in the intermediate process design with the MASS as 99.6 mol%, (e) black-hole problem in the final process design with the MASS as 99.4 mol%, (f) black-hole problem in the final process design with the MASS as 99.6 mol%.

of the TAC using the optimization algorithm proposed in our earlier work [27] and sketched in Fig. 15a, which accommodates, respectively, 38 and 57 stages in the prefractionator and the main distillation column involved. The dividing-wall runs from stage 11 to stage 41 in the main distillation column. The feed processed is introduced onto stage 19 in the prefractionator and the intermediate product is withdrawn from stage 23 in the main distillation column. The liquid split ratio, RL , is 0.317 and the vapor split ratio, RV , is 0.57. 5.2. Black-hole problem Fig. 15b depicts the relationship between the TAC and the purities of the three products and the MAPTP is found to be 99.24 mol% (apparently smaller than the given 99.4 mol% and 99.6 mol%) in

these situations. Fig. 16a and b shows the relationship between the TAC and the composition of the intermediate product for the two circumstances and the MAPIP is 99.04 mol% and 96.66 mol%, respectively. Since they are both smaller than the ICMASS (i.e., 99.4 mol% and 99.6 mol%), it is thus necessary to consider the alleviation of the black-hole problem through over-design. 5.3. Diminishing the black-hole problem through over-design Fig. 17a and b gives the resultant intermediate process designs, which can reach the purities of 99.4 mol% and 99.6 mol%, respectively, in their three products. While the intermediate process design that can achieve the purities of 99.4 mol% is evolved from the initial process design by adding 2 and 1 stages to sections I and V, respectively, the intermediate process design that can achieve the

W. Chen et al. / Chemical Engineering and Processing 75 (2014) 90–109

105

Fig. 17. Intermediate or final process designs (Example III): (a) with the MASS as 99.4 mol%, (b) with the MASS as 99.6 mol%.

purities of 99.6 mol% is by appending 3 and 5 stages to the same sections. Fig. 16c and d shows the relationship between the TAC and the composition of the intermediate product and the blackhole problems have been diminished in comparison with those of the initial process design. Optimization studies indicate that these intermediate process designs are also the final process designs and Fig. 16e and f depicts their black-hole problems. Fig. 18 compares the steady-state profiles of liquid compositions of the initial and the two final process designs in the NSS. Remixing effect is found again in the prefractionator of the initial process design, but has been avoided in the final process designs.

Fig. 19 delineates the performance of the initial and two final process designs and those based on the given MASS in the operating region between the NSS and MASS and Table 8 gives a detailed comparison between them in these two steady states, respectively. It is demonstrated that the flexibility and operability of the final process designs have been enhanced with an additional increase of the TAC by 0.85% and 1.73%, respectively, as compared with the initial process design. Although in the MASS they result in a 1.38% and 1.66% larger TAC than those process designs based on the MASS, they give a reduced one by 0.39% and 0.81%, respectively, in the NSS.

106

W. Chen et al. / Chemical Engineering and Processing 75 (2014) 90–109

0.8

1 P

B

0.8

Mole fraction (-)

Mole fraction (-)

E 0.6

0.4

0.2

0.6 E

0 0

10

20 Stage of prefractionator

30

0

10

20 30 40 Stage of main distillation column

(a)

50

(b) 1

0.8 E

P

B

0.8

0.6

Mole fraction (-)

Mole fraction (-)

B

0.2

0

0.4

0.2

0.6 E

0.4

P

B

0.2 0

0 0

10

20 Stage of prefractionator

0

30

10

20 30 40 Stage of main distillation column

(c)

50

(d)

0.8

1 E

P

B

0.8

0.6

Mole fraction (-)

Mole fraction (-)

P

0.4

0.4

0.2

0.6 E

P

B

0.4 0.2

0

0 0

10

20 30 Stage of perfractionator

40

(e)

0

10

20 30 40 Stage of main distillation column

50

60

(f)

Fig. 18. Composition profiles of the initial and final process designs in the NSS (Example III): (a) prefractionator in the initial process design, (b) main distillation column in the initial process design, (c) prefractionator in the final process design with the MASS as 99.4 mol%, (d) main distillation column in the final process design with the MASS as 99.4 mol%, (e) prefractionator in the final process design with the MASS as 99.6 mol%, and (f) main distillation column in the final process design with the MASS as 99.6 mol%.

6. Discussion The three examples studied in the current work have demonstrated that the black-hole problem can be effectively diminished through careful adjustments of the number of stages in each section of the DWDC. Although this frequently accompanies a certain extent of capital cost as compared with the initial process design, the feasible operating region of the DWDC can be extended, thereby presenting favorable effects to process flexibility and operability. In comparison with the process design in the MASS, the resultant final process design appears to secure only a slight reduction in the TAC around the NSS and the improvement in system performance depends heavily on the required operating region between the NSS and MASS. The resultant final process design is also expected to

have better process dynamics and controllability than the process design in the MASS and the relevant outcomes will be presented in the second paper of this series. During process synthesis and design, with the determination of the maximally allowable reboiler heat duty in the MASS, a careful trade-off between the diminution of the black-hole problem and the additional capital cost can be made. The simulation studies corroborate also the fact that the black-hole problem is inherently related to the complex interactions between the prefractionator and the main distillation column involved. With the consideration of only economical factors in process synthesis and design, it is unlikely to lead to a satisfactory process design with a well coordinated relationship between the prefractionator and the main distillation column involved. This fact reminds us of the great importance in making a careful trade-off between

W. Chen et al. / Chemical Engineering and Processing 75 (2014) 90–109 Table 7 Physical properties and design specifications of Example III. Parameter

Value

Condenser pressure (atm) Stage pressure drop (atm) Feed compositions (mol%) Ethanol (E) Propanol (P) Butanol (B) Feed flow rate (mol/s) Feed thermal condition Relative volatility E:P:B Normal boiling points (K) Ethanol (E) Propanol (P) Butanol (B) Product specifications (mol%) Ethanol (E) Propanol (P) Butanol (B)

1 0.0068 33.3 33.3 33.4 83.33 1.0 4:2:1 352 370 392 99 99 99

Table 8 Comparison between the initial and final process designs and those based on the MASS (Example III). Scenario

Initial process design MASS1 = 99.4 mol% Process design in the MASS1 Final process design MASS2 = 99.6 mol% Process design in the MASS2 Final process design

TAC (106 $/yr) NSS

Comparison (%)

MASS

Comparison (%)

1.2300

100

−

1.2452

101.24

1.2592

100

1.2404

100.85

1.2766

101.38

1.2613

102.54

1.2812

100

1.2513

101.73

1.3025

101.66

−

107

along with the minimization of an economical objective function during process synthesis and design; however, the problem becomes a complicated mixed-integer nonlinear programming problem and can be extremely difficult and time-consuming to solve in practice. This is why the black-hole problem has been resolved in terms of the sequential procedure proposed in the current work. Although the split ratio of the middle component in the prefractionator and the vapor and liquid split ratios can be chosen as the decision variables for the diminution of the black-hole problem, they are likely to cause great deviations from the most economical process design as already indicated by Halvorsen and Skogestad [25,26]. Instead, if the number of stages in each section of the DWDC has been adopted as the decision variable, such kind of situations could be avoided. Thus, the sequential procedure proposed in the current work should be regarded as a generalized methodology for the synthesis and design of the DWDC. Despite the fact that intensive application of the steady-state model is unavoidable, the improvement in the steady-state performance of the DWDC should justify the efforts and time necessitated in process synthesis and design. It is worth analyzing here the remixing effect occurring in the prefractionators of the initial process designs in Examples I–III (as shown in Figs. 8, 13 and 18). The deficiency is certainly attributed to the uncompromised interactions between the prefractionator and the main distillation column involved. With the careful adjustments of the number of stages in each section of the DWDC, the remixing effect vanishes in the final process designs and this provides an additional benefit from diminishing the black-hole problem in process synthesis and design.

7. Conclusions

process economics and process flexibility and operability in all stages of process development. This issue, however, has long been ignored and very few studies have been conducted so far in this regard. Since the flow rates and compositions of the interlinking streams govern essentially the interactions between the prefractionator and the main distillation column involved, all structural and operating design variables are considered to affect the blackhole problem of the DWDC. Theoretically speaking, all these variables should be employed to tackle the black-hole problem

Fig. 19. Comparison between the initial and final process designs and those in the MASS (Example III): bold line: optimum process design in the corresponding steady state, solid line: initial process design, dotted line: process designs in the MASS (99.4 mol% and 99.6 mol%), dashed line: final process designs with the MASS as 99.4 mol% and 99.6 mol%, respectively.

The black-hole problem involved in the DWDC with three specifications on its top, intermediate, and bottom products, respectively, is an inherent drawback of process intensification and quite likely to present detrimental effects to process flexibility and operability. In spite of the fact that the complete removal of this issue is impossible (unless with an infinite number of stages to compromise the interactions between the prefractionator and the main distillation column involved), it can be alleviated through over-design. The current work attempts to diminish the blackhole problem in terms of careful structural modifications and a generalized procedure has been proposed to balance the interactions between the prefractionator and the main distillation column involved through the careful adjustments of the number of stages in each section of the DWDC. The philosophy is of special significance to the design and operation of the DWDC and should be considered as a generalized strategy for process development. The separations of three ternary mixtures of hypothetical components, A, B, and C, benzene, toluene, and o-xylene, and ethanol, propanol, and butanol, are chosen as illustrative examples to evaluate the feasibility and effectiveness of the proposed philosophy. It has been demonstrated that the black-hole problem can be effectively diminished through structural modifications in terms of careful adjustments of the number of stages in each section of the DWDC. Although this is frequently achieved at the expanse of additional capital cost, the feasible operating region of the DWDC can be extended as expected, thereby posing favorable influences to process flexibility and operability. Apart from the enhancement of process flexibility and operability by the alleviation of the black-hole problem, the resultant DWDC is also anticipated to have improved process dynamics and controllability because the interactions between the prefractionator and the main distillation column involved have been well refined in the final process design. This represents essentially another

108

W. Chen et al. / Chemical Engineering and Processing 75 (2014) 90–109

potential advantage of the generalized philosophy proposed for the over-design of the DWDC. To confirm this interpretation, we are currently involved in studying the dynamics and controllability of the resultant DWDC in terms of the detailed comparison against the initial process design as well as the one based on the MASS. The outcomes are to be presented in the second paper of this series. Acknowledgements The current work is financially supported by The National Science Foundation of China under the grant number of 21076015 and The Doctoral Programs Foundation of Ministry of Education of China under the grant number of 20100010110008. Appendix A. Sizing and economical basis for distillation columns Assuming an F factor of 1 in engineering units, the diameter of a distillation column is calculated from the following equation d = 0.01735 ×

 M × T 0.25 W P

× VT0.5

(A.1)

While the diameters of the distillation columns in Example I are calculated with the above equation, those of Examples II and III are obtained with the aid of the Aspen Plus. The height of a distillation column is calculated by the equation H =N×2×

1.2 3.281

(A.2)

The heat transfer areas of the condenser and reboiler are calculated using the following equations AC = QC × 1.055 × 2.54 × 106 /3600/0.7457/(UC × TC )

(A.3)

AR = QR × 1.055 × 2.54 × 106 /3600/0.7457/(UR × TR )

(A.4)

In terms of the above size estimations, the capital investment (CI) of a distillation column is estimated using the following equations Column shell cost = 17640 × d1.066 × H 0.802

(A.5)

Tray cost = 229 × d1.55 × N

(A.6)

+ A0.65 ) Total heat exchanger cost = 7296 × (A0.65 R C

(A.7)

CI = Column shell cost + Tray cost + Total heat exchanger cost (A.8) The operating cost (OC) of a distillation column is estimated using the following equations OC = (QC × cooling water cost + QR × steam cost) × operating time per year

(A.9)

The TAC includes operating cost and discounted capital investment expressed with a payback time TAC = OC −

CI ˇ

Appendix B. Nomenclature

Notations A hypothetical component heat transfer area of a condenser, m2 AC

(A.10)

heat transfer area of a reboiler, m2 vapor pressure constant, Pa hypothetical component, benzene, butanol, or bottom product flow rate, mol/s Bvp vapor pressure constant, Pa K hypothetical component C CI capital investment, $ d column diameter, m D distillate flow rate, mol/s DWDC dividing-wall distillation column E ethanol F feed flow rate, mol/s H column height, m component index i I intermediate product flow rate, mol/s ICMASS intermediate composition of the maximally achievable steady state j stage index J section number of the DWDC iteration number K L liquid flow rate, mol/s M iteration number MAPIP maximally achievable purity of the intermediate product MAPTP maximally achievable purities of the three products MASS maximally achievable steady state average molecular weight, g/mol MW N sectional number of stages N variation in the number of stages NSS nominal steady state total number of stages of the DWDC NT OC operating cost, $/yr P propanol or pressure, Pa q feed thermal condition QC condenser duty, MW reboiler duty, MW QR RL liquid split ratio RR reflux ratio vapor split ratio RV T toluene or temperature, K TAC total annual cost, $/yr TC temperature driving force of a condenser, K temperature driving force of a reboiler, K TR UC overall heat transfer coefficient of a condenser, kW/K m2 overall heat transfer coefficient of a reboiler, kW/K m2 UR V vapor flow rate, mol/s maximal vapor flow rate of a distillation column, mol/s VT x liquid composition o-xylene X y vapor composition AR Avp B

Greek letters ˇ payback period, yr error tolerance in judging the variation of the MAPIP ε1 ε2 error tolerance in judging the position of a black hole error tolerance in judging the variation of the TAC ε3 Subscripts component index A component index, benzene, butanol, or bottom product B C component index distillate product D E ethanol intermediate product I section number of the DWDC J K iteration number propanol P

W. Chen et al. / Chemical Engineering and Processing 75 (2014) 90–109

T X

toluene oxylene

Superscript s saturation References [1] O. Annakou, P. Mizsey, Rigorous comparative study of energy-integrated distillation schemes, Industrial and Engineering Chemistry Research 35 (1996) 1877–1885. [2] M. Emtir, E. Rev, Z. Fonyo, Rigorous simulation of energy integrated and thermally coupled distillation schemes for ternary mixture, Applied Thermal Engineering 21 (2001) 1299–1317. [3] M.A. Schultz, D.G. Stewart, J.M. Harris, S.P. Rosenblum, M.S. Shakur, D.E. O’Brien, Reduce costs with dividing-wall columns, Chemical Engineering Progress 98 (2002) 64–71. [4] B. Kolbe, S. Wenzel, Novel distillation concepts using one-shell columns, Chemical Engineering and Processing: Process Intensification 43 (2004) 339–346. [5] M. Errico, G. Tola, B.G. Rong, D. Demurtas, I. Turunen, Energy saving and capital cost evaluation in distillation column sequences with a divided wall columns, Chemical Engineering Research and Design 87 (2009) 1649–1657. [6] Y.C. Ho, J.D. Ward, C.C. Yu, Quantifying potential energy savings of divided wall columns based on degree of remixing, Industrial and Engineering Chemistry Research 50 (2011) 1473–1487. [7] C. Triantafyllou, R. Smith, The design and optimisation of fully thermally coupled distillation columns: process design, Chemical Engineering Research and Design 70 (1992) 118–132. [8] K. Muralikrishna, V.K.P. Madhavan, S.S. Shah, Development of dividing wall distillation column design space for a specified separation, Chemical Engineering Research and Design 80 (2002) 155–166. [9] J.A. Caballero, I.E. Grossmann, Design of distillation sequences: from conventional to fully thermally coupled distillation systems, Computer and Chemical Engineering 28 (2004) 2307–2329. [10] N. Sotudeh, B.H. Shahraki, A method for the design of divided wall columns, Chemical Engineering and Technology 30 (2007) 1284–1291. [11] H. Ling, W.L. Luyben, New control structure for divided-wall columns, Industrial and Engineering Chemistry Research 48 (2009) 6034–6049. [12] H. Ling, W.L. Luyben, Temperature control of the BTX divided-wall column, Industrial and Engineering Chemistry Research 49 (2010) 189–203. [13] M.R. Hernández, J.A. Chinea-Herranz, Decentralized control and identifiedmodel predictive control of divided wall columns, Journal of Process Control 22 (2012) 1582–1592.

109

[14] S. Luan, K. Huang, N. Wu, Operation of dividing-wall columns. 1. A simplified temperature difference control scheme, Industrial and Engineering Chemistry Research 52 (2013) 2642–2660. [15] N. Wu, K. Huang, S. Luan, Operation of dividing-wall columns. 2. A double temperature difference control scheme, Industrial and Engineering Chemistry Research 52 (2013) 5365–5383. [16] S. Hernández, A. Jiménez, Controllability analysis of thermally coupled distillation systems, Industrial and Engineering Chemistry Research 38 (1999) 3957–3963. [17] M. Serra, M. Perrier, A. Espuna, L. Puigjaner, Study of the divided wall column controllability: influence of design and operation, Computers and Chemical Engineering 24 (2000) 901–907. [18] E.A. Wolff, S. Skogestad, Operation of integrated three-product (Petlyuk) distillation columns, Industrial and Engineering Chemistry Research 34 (1995) 2094–2103. [19] Y. Wang, K. Huang, S. Luan, W. Chen, S.J. Wang, D.S.H. Wong, Circumventing the black-hole problem in design and control of dividing-wall distillation columns, Industrial and Engineering Chemistry Research 51 (2012) 14771–14792. [20] J. Gao, K. Huang, S. Luan, Y. Jiao, S. Li, Avoiding the black-hole problem by the arrangements of multiple intermediate products to dividing-wall distillation columns, Industrial and Engineering Chemistry Research 52 (2013) 4178–4201. [21] W. Chen, S. Li, K. Huang, J. Gao, G. Wu, Suppressing the black-hole problem through feed splitting in dividing-wall distillation columns, Industrial and Engineering Chemistry Research (2013) (under second review). [22] G. Dünnebier, C.C. Pantelides, Optimal design of thermally coupled distillation columns, Industrial and Engineering Chemistry Research 38 (1999) 162–176. [23] K. Huang, S.J. Wang, L. Shan, Q. Zhu, J. Qian, Seeking synergistic effect—a key principle in process intensification, Separation and Purification Technology 57 (2007) 111–120. [24] K. Huang, Q. Lin, H. Shao, C. Wang, S. Wang, A fundamental principle and systematic procedure for process intensification in reactive distillation columns, Chemical Engineering and Processing: Process Intensification 49 (2010) 294–311. [25] I.J. Halvorsen, S. Skogestad, Optimal operation of Petlyuk distillation: steadystate behavior, Journal of Process Control 9 (1999) 407–424. [26] I.J. Halvorsen, S. Skogestad, Shortcut analysis of optimal operation of Petlyuk distillation, Industrial and Engineering Chemistry Research 43 (2004) 3994–3999. [27] P. Wang, H. Chen, Y. Wang, L. Zhang, K. Huang, S.J. Wang, A simple algorithm for the design of fully thermally coupled distillation columns (FTCDC), Chemical Engineering Communications 199 (2012) 608–627.