A New Shortcut Design Method and Economic Analysis of Divided

ARTICLE pubs.acs.org/IECR

A New Shortcut Design Method and Economic Analysis of Divided Wall Columns Kai-Ti Chu, Loic Cadoret, Cheng-Ching Yu, and Jeffrey D. Ward* Dept. of Chem. Eng., National Taiwan University, Taipei 106-17, Taiwan

bS Supporting Information ABSTRACT: Divided wall columns (DWCs) can save energy and capital costs compared with traditional distillation columns; however, the design of DWCs is more difficult because there are more degrees of freedom. This paper describes a novel short-cut method that can be used to rapidly determine near-optimal values of important design parameters, including the reflux ratio, number of stages in all sections, and split liquid and vapor ratios for the three most common types of DWCs. The method is based on the development of a rational and efficient net flow model and the application of the methods of Fenske, Underwood, and Gilliland and the Kirkbride equation. The method is applied to two real systems, and the results are compared with results from rigorous simulations and optimization. The results show that the shortcut method leads to a process similar to a feasible actual process, and the total annual cost (TAC) based on the design method is close to the minimum (optimum) total annual cost. The results also show that the method also provides good initial values for rigorous optimization. The method is also applied to a fictitious process consisting of three components with constant relative volatilities, for different values of the ease of separation index (ESI), overall split difficulty, and feed composition. The results indicate that the method works well for a variety of process conditions and that the minimum vapor flow rate is a good surrogate for the total cost of process operation.

1. INTRODUCTION Divided wall columns (DWCs), in which three or more highpurity product streams are collected from a single column by means of inserting one or more dividing walls, have been known of since at least 1949.1 Divided wall columns can save capital costs because of a reduced number of columns and heat exchangers and can also save energy by reducing the remixing effect. Some authors have reported energy savings as much as 30%26 using DWCs. However, because they are more integrated and have more degrees of freedom than a conventional column, they are more difficult to design, operate, and control. Most researchers that have investigated divided wall columns have focused on the calculation of the minimum vapor requirement79 and have assumed that the minimum vapor flow rate can be used as a surrogate for the total cost for the purposes of comparing process alternatives. The problem of determining the number of stages in the various column sections has received less attention. Triantafyllou and Smith10 introduced a design procedure for three columns model which means the DWC is divided into three sections. They applied the traditional shortcut method based on the Fenske, Underwood, and Gilliand equations to get the initial number of trays for each column. Although it is easy to get number of trays, matching the compositions of net flow requires time-consuming iterations. Amminudin et al.11 proposed a new method based on the equilibrium stage composition concept. That is, they apply the operation leaves method to find the compositions in each section. Then, they use the compositions to get all the design variables. However, it takes long time to tune all the design variables to get the optimal design. Kim12 proposed a new method. The first step is to calculate the limiting requirements for minimum reflux ratio and minimum number of trays. Then, they set the number of trays equal to twice the minimum r 2011 American Chemical Society

number of trays. However, the rules for conventional distillation design methods may not be suitable for every system. For example, Lek et al.13 investigated several conventional distillation systems. They found that the optimal reflux ratio can be 1.04 to 1.64 times the minimum reflux ratio. This means that the optimal design depends on the system and objective function. Muralikrishna et al.14 proposed a method based on boundary maps for minimum TAC. But they fixed the liquid split and vapor split. For a better design, the liquid split and vapor split should also be design variables. Recently, Sotudeh and Shahraki15 proposed another shortcut method. In their method, the compositions in the upper and lower zones of the prefractionator are design variables. The number of trays of each section in the main column can be calculated, and the minimum number of trays in the main column is obtained by manipulating the variables. Only the number of trays in the side section is calculated, and the number of trays in the prefractionator is set to be the same as in the side section. However, this method may not always work because we cannot be sure that the prefractionator will have enough trays. Also, they did not explain how to decide the vapor split and liquid split exactly. Furthermore, the authors did not compare their results with results from rigorous simulations. Ramirez-Corona et al.16 proposed a method based on NLP. They apply this method for several real systems. Their results show that the nominal reflux for DWC should be set between 1.05 and 1.1 times the minimum reflux ratio. DWCs can be classified into one of three types based on the position of the dividing wall. The three configurations are DWCL, Received: February 1, 2011 Accepted: June 10, 2011 Revised: June 8, 2011 Published: June 10, 2011 9221

dx.doi.org/10.1021/ie200234p | Ind. Eng. Chem. Res. 2011, 50, 9221–9235

Industrial & Engineering Chemistry Research

Figure 1. Total mole net flow model by Holland et al.

DWCU, and DWCM, which are evolved from indirect sequence, direct sequence, and prefractionator sequence, respectively, and where the wall is located in the lower, upper, and middle sections of the column, respectively. Most researchers to date have focused on the design of DWCM systems. Furthermore, the vapor split disturbance is a problem for DWCU and DWCM. In practical operation, the location of the divided wall is set in order to achieve the desired vapor split. However it is difficult to control the vapor split after the column is built. In this regard, DWCL has a benefit. In this work, we develop a novel design method which can be applied to all three types of DWCs. The method allows for the determination of near-optimal values of all important design variables, including the reflux ratio, the number of trays in all column sections, and values of the liquid and vapor split ratios. We apply the method to two real case studies with different ease of separation indexes (ESI) and find that it works well in both cases. The method is also applied to a fictitious system with three components of constant relative volatility for different values of the ease of separation index and overall split difficulty. The method was found to work well in most cases. Finally, results from the optimized processes are used to check the assumption that minimum vapor flow is a good surrogate for total cost for divided wall systems. The results suggest that minimum vapor flow is in fact a good surrogate for total cost in the sense that the column configuration (DWCU, DWCL, DWCM) with the lowest minimum vapor flow rate almost always has the lowest combined capital and operating cost when the process is optimized. The remainder of this manuscript is organized as follows: The shortcut design procedure for DWCM is described in section 2.

ARTICLE

Figure 2. Component net flow model of DWCM.

In section 3, the method is validated using both an ideal system and two real case studies. In section 4, the method is applied to investigate the economics of ideal three component systems with constant relative volatilities. Finally, conclusions are presented in section 5.

2. SHORTCUT DESIGN 2.1. Model Design and Assumptions. In this work, we consider that the relative volatility of each species is constant and that the column is symmetric; that is, it has the same number of trays on both sides of the dividing wall. In the calculation of the number of trays, the compositions in the top section and bottom section are necessary. Most researchers use these compositions as design variables; however, this causes additional complexity in the design procedure. Instead, we build a component net flow model so that the compositions in each section can be calculated straightforwardly, as described in this section. Holland et al.17 presented all of the possible net flow models for DWC systems. The most efficient one is shown in Figure 1. Because the component net flow model is needed for this design method, we evolve the total mole net flow model into a component net flow model. Figure 2 shows the component net flow model. The column is divided into five sections. Section 1 corresponds to the prefractionator. Section 2 is a rectifing section. Section 4 is a stripping section. Section 3_1 is the upper section on the right side of the dividing wall, and section 3_2 is the lower section on the right side of the dividing wall. Suppose that A is the lightest component, B is the middle component, and C is the heaviest 9222

dx.doi.org/10.1021/ie200234p |Ind. Eng. Chem. Res. 2011, 50, 9221–9235


ARTICLE

component. The top product in section 1 is mostly A and B with a little C, and the bottom product is mostly B and C with a little A. The assumption of a sharp split in section 1 is not suitable for a realistic design because it requires an infinite number of trays. But what are the recoveries of A, B, and C in section 1? Consider the top part of the column first. Since the top product stream D2 contains essentially no C, the net flow rate of species C from the top of section 1 to the bottom of section 2 must essentially be equal to the flow rate of species C from the bottom to section 2 to the top of section 3_1. On the right side of the dividing wall, component C that enters from the bottom of section 2 can either exit through the sidedraw flow S or pass out of the bottom of section 3_2 and into section 4. However, if there is a significant net flow of species C from section 3_2 to section 4, then the column is operating inefficiently because an unnecessarily large amount of species C is being boiled up in section 1 only to be condensed and returned to section 4 via section 2 (it is traveling the “long way around” the column). Therefore, we assume that the net flow of species C at the top of section 1 is equal to the flow of species C in the sidedraw. Likewise, at the bottom of the column, essentially no species A is present in the bottom stream W4; therefore, the net flow rate of species A from section 1 to section 4 is approximately equal to the net flow rate of species A from section 4 to section 3_2. And again, the column would be operating inefficiently if there was a significant net flow of species A upward in section 3_1. Therefore, we assume that the net flow of species A at the bottom of section 1 is equal to the flow of species A in the sidedraw. These assumptions allow the engineer to determine approximate values of the composition at key points in the column if the feed flow rate (F), feed composition (ZA, ZB, ZC), and product purity specifications (XD2, XS, XW4) are known. Consider the schematic diagram shown in Figure 3. β is defined as the fraction of B sent to the top of the prefractionator: β¼

XB, D1 D1 FZB

Figure 3. Schematic diagram for DWCM.

XA, S3 _1 ¼

ð1Þ

The compositions at points D1, W1, S3_1, and S3_2 can be obtained by mass balance: XA, D1 ¼

ZA F SXA, S D1

ð2Þ

βZB F D1

ð3Þ

XB, D1 ¼

XC, D1 ¼ 1 XA, D1 XB, D1

XB, W1 ¼

ð5Þ

ð1 βÞZB F W1

ð6Þ

XC, W1 ¼ 1 XA, W1 XB, W1 XB, S3 _1 ¼

βZB F D2 XB, D2 S3 _ 1

XC, S3 _1 ¼ 1 XA, S3 _1 XB, S3 _1 XB, S3 _2 ¼

ð7Þ ð8Þ

ð9Þ ð10Þ

ð1 βÞZB F W4 XB, W4 S3 _ 2

ð11Þ

XC, S S S3 _ 2

ð12Þ

XC, S3 _2 ¼

ð4Þ

SXA, S W1

XA, W1 ¼

XA, S S S3 _ 1

XA, S3 _2 ¼ 1 XB, S3 _2 XC, S3 _2

ð13Þ

These approximate values are used in the shortcut design as described below. 2.2. Minimum Vapor Flow Calculation. With the component net flow model established, the next step in the calculation is to estimate the minimum vapor flow rate in each column section using Underwood’s method,18 which relies on the assumptions of constant molar flow rate and constant relative volatility. The calculation of minimum vapor flow rate for divided wall columns has been discussed by several authors79 and is briefly summarized here. Again, Figure 3 shows a schematic diagram depicting the notation. For calculating minimum vapor flow, the relative volatilities (RA, RB, RC) and feed quality (q) are also needed. 9223



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In section 1, the equation for the Underwood roots is: RZ

∑Ri i iθ i ¼ fA, B, Cg

ð1 qÞ ¼

ð14Þ

which has solutions RA > θ1 > RB > θ2 > RC

ð15Þ

The minimum vapor flows in section 1 are calculated on the basis of the roots, and the larger one is chosen: Vmin, 1 ¼

RA XA, D1 D1 RB XB, D1 D1 RC XC, D1 D1 + + RA θ RB θ RC θ

Vmin, 1 ¼ MaxfVmin, 1 ðθ1 Þ, Vmin, 1 ðθ2 Þg

ð16Þ ð17Þ

In section 2, V1 and L1 are combined to form a net flow D1. The feed quality of D1 can be calculated: q0 ¼

Lmin, 1 D1

Figure 4. Schematic diagram of vapor flow versus β.

Because section 3_1 and section 3_2 are combined, the maximum vapor flow must be chosen as Vmin,3: Vmin, 3 ¼ MaxfVmin, 3 _1 , Vmin, 3 _2 g

ð18Þ

The minimum vapor flow for DWCM is chosen as

The minimum vapor flow in section 2 can be calculated in the same way: D1 ð1 q0 Þ ¼

RX

∑ Ri i i,D1θ0 1 i ¼ fA, B, Cg D

RA > θ01 > RB > θ02 > RC Vmin, 2 ¼

RX

∑ Ri i i,D2θ0 2 i ¼ fA, Bg D

Vmin, 2 ¼ MaxfVmin, 1 ðθ01 Þ, Vmin, 1 ðθ02 Þg

ð19Þ ð20Þ ð21Þ ð22Þ

The minimum vapor flow for section 3_1 is as follows: Vmin, 3 _1 ¼ Vmin, 2 ð1 q0 ÞD1

ð23Þ

In section 4, V 1 and L1 are also combined to form net flow W1. The feed quality of W1 can be calculated: q00 ¼

L̅ min, 1 W1

W1 ð1 q00 Þ ¼

∑

RA > θ001

Vmin, 4 ¼

Ri Xi, W 1 W1 i ¼ fA, B, Cg Ri θ00

ð25Þ

> RB > θ002

ð26Þ

> RC

R2, min ¼

MaxfVmin, 4 ðθ001 Þ, Vmin, 4 ðθ002 Þg

ð28Þ

Turning our attention to section 3, the minimum vapor flow for section 3_2 can be found by Vmin, 3 _2 ¼ Vmin, 4 + ð1 q00 ÞW4 Also, the sum of Vmin,1 and Vmin,3_2 is equal to Vmin,4: Vmin, 3 _2 ¼ Vmin, 4 Vmin, 1

Vmin, 2 1 D2

ð33Þ

Step 2. The minimum number of trays is calculated using the Fenske equation. The minimum number of trays for section 1 is calculated: ln½ðZi =Zj ÞD1 ðZj =Zi ÞW1 i, j ¼ fA, B, Cg ln Rij

ð34Þ

There are three answers, and the largest one is chosen: ð27Þ

∑

ð32Þ

The minimum vapor flow calculation depends on β. Figure 4 shows the relationship between minimum vapor flow and β. The minimum vapor flow is constant between two special points called “beta preferred” (βP)19 and “beta balanced” (βb),20 and Vmin,Petlyuk is minimized in this section. βP is the value of β for which Vmin,1 is a minimum. βb is the value of β for which Vmin,3_1 is equal to Vmin,3_2. For calculating the minimum vapor flow in each section, these two points will be used as reference points. 2.3. Design Method. After the compositions are determined using the component net flow model and the minimum vapor rate is determined using Underwood’s method, the design is completed by dividing the DWC into several parts and applying the methods of Fenske, Underwood, and Gilliland21 and the Kirkbride22 equation. This part of the design method is described step by step in this subsection. Step 1. The minimum reflux ratio is calculated:

N1, min ¼

Ri Xi, W4 W4 i ¼ fB, Cg Ri θ00

V̅ min, 4 ¼

Vmin, Petlyuk ¼ MaxfVmin, 2 , V̅ min, 4 + ð1 qÞFg

ð24Þ

The minimum vapor flow for section 4 can also be calculated:

ð31Þ

ð29Þ ð30Þ

N1, min ¼ MaxfNAB, min , NBC, min , NAC, min g

ð35Þ

Section 2 and section 3_1 should be combined for the purpose of calculating the minimum number of stages: ðN2 + N3 _1 Þmin ¼

ln½ðZi =Zj ÞD2 ðZj =Zi ÞS3 _1 ln Rij

i, j ¼ fA, Bg ð36Þ

Section 4 and section 3_2 should also be combined: ðN4 + N3 _2 Þmin ¼ 9224

ln½ðZi =Zj ÞS3 _2 ðZj =Zi ÞW4 ln Rij

i, j ¼ fB, Cg ð37Þ



ARTICLE

Table 3. Simulation Results in Ideal System for DWCM for R2 = 1.2 and R2,min when β = βP

Figure 5. Graph of SV or SL vs N1 and N3 for DWCM.

ZA/ZB/ZC =

ZA/ZB/ZC =

ZA/ZB/ZC =

ZA/ZB/ZC =

Δvariable

0.2/0.6/0.2

0.6/0.2/0.2

0.2/0.2/0.6

0.33/0.33/0.33

ΔXA,D2 ΔXB,S

0.6% 0.1%

0.38% 1.2%

ΔXC,W4

1.9%

2.3%

1.5%

ΔR2

0.66%

3.5%

11.29%

1.5%

5.4%

16.8%

0.34%

2.1%

ΔBR4 ΔSL

1%

ΔSV

0.5%

0.21%

0.99% 3.7%

1%

0.69% 1.6% 2.3% 10% 12.3% 0% 0%

Table 4. Simulation Results in Ideal System for DWCM for R2 = 1.2 and R2,min when β = βb

Figure 6. Shortcut design procedure for DWCM.

Table 1. Feed Conditions and Specifications in Ideal Case Study 2

0:99 0:005 6 F = 1 (kmol/s) q = 1 X ¼ 6 4 0:01 0:99 0 0:005

3 0 7 0:01 7 5 0:99

RA/RB/RC = 7.1/2.2/1

14.02

AVP3 (mm Hg) BVP (K)

13.24 2768.55

ZA/ZB/ZC =

0.2/0.2/0.6

0.33/0.33/0.33

ΔXA,D2

0.93%

0.84%

0.97%

0.93%

ΔXB,S ΔXC,W4

0.25% 1.8%

0.22% 2.3%

5.3% 2.1%

1.4% 2.3%

ΔR2

7.9%

4.7%

ΔBR4

1.8%

1.9%

ΔSL

0%

0.23%

ΔSV

0%

0.15%

3.9% 10% 0.3% 0%

1.45% 6.4% 0.4% 0%

ZA/ZB/ZC =

ZA/ZB/ZC =

ZA/ZB/ZC =

ZA/ZB/ZC =

Δvariable 0.2/0.6/0.2

0.6/0.2/0.2

0.2/0.2/0.6

0.33/0.33/0.33

ΔXB,S ΔXC,W4

RA/RB/RC: 7.1/2.2/1 15.20

ZA/ZB/ZC =

0.6/0.2/0.2

ΔXA,D2

Table 2. Thermodynamic Parameters for Ideal System (rA/rB/rC: 7.1/2.2/1)

AVP2 (mm Hg)

ZA/ZB/ZC =


feed compositions (ZA/ZB/ZC): 0.2/0.6/0.2, 0.6/0.2/0.2, 0.2/0.2/0.6, 0.33/0.33/0.33

AVP1 (mm Hg)

ZA/ZB/ZC = Δvariable 0.2/0.6/0.2

0.99% 0.7% 3%

0.78%

0.99%

0.99%

2.2% 4.3%

8.6% 3.1%

3.5% 4.3%

ΔR2

0.5%

ΔBR4

4.8%

ΔSL

0.2%

0.76%

0.17%

0.3%

ΔSV

0.02%

0.3%

0.02%

0.29%

20% 14.8%

2.1%

11%

9.5%

14%


Tb(A) (K)

323.15

ZA/ZB/ZC =

ZA/ZB/ZC =

ZA/ZB/ZC =

ZA/ZB/ZC =

Tb(B) (K)

374.35

Δvariable 0.2/0.6/0.2

0.6/0.2/0.2

0.2/0.2/0.6

0.33/0.33/0.33

Tb(C) (K)

419.03 ln PSi = AVP,i BVP,i/Tj

Antoine equation:

L2 D2

0.92%

0.87%

0.99%

0.98%

ΔXB,S

0.4%

0.9%

8.3%

2.5%

2.1% 3%

3.4% 11%

ΔXC,W4 ΔR2

The ratio of N2,min to N3_1,min and the ratio of N4,min to N3_2,min are degrees of freedom. These values will be determined in the design procedure. Step 3. The reflux ratio and minimum reflux ratio for each section should be defined before using the FenskeUnderwood Gilliland method to calculate the number of trays. The reflux ratio and minimum reflux ratio of section 2 are defined as R2 ¼

ΔXA,D2

ð38Þ

3.3% 9.1%

ΔBR4

2.8%

ΔSL

0.38%

0.7%

0.3%

0.62%

ΔSV

0.05%

0.28%

0.02%

0.35%

10%

R2, min ¼ 9225

3% 3% 10%

Lmin, 2 D2

12%

ð39Þ



ARTICLE

Table 7. Feed Conditions in the Ethanol, n-Propanol, n-Butanol System

Table 10. Simulation Results in EPB System When ZA/ZB/ZC = 0.3/0.3/0.4

F = 300 (kmol/h) q = 1

ZA/ZB/ZC = 0.3/0.3/0.4

RA/RB/RC= 4.199/2.1459/1 ESI=0.911 type

Ptop = 1 atm, Pbottom = 1 atm, thermodynamic model: NRTL feed compositions (ZA/ZB/ZC): 0.1/0.8/0.1, 0.6/0.2/0.2, 0.3/0.3/0.4

Table 8. Simulation Results in EPB System When ZA/ZB/ZC = 0.1/0.8/0.1 ZA/ZB/ZC = 0.1/0.8/0.1 type

DWCL

type

DWCU

type

DWCM

R2 ΔXA,D2

1.012R2,min 5%

BR3 ΔXA,D1

1.015BR3,min 6.2%

R2 ΔXA,D2

1.028R2,min 3.1%

ΔXB,W3

0.4%

ΔXB,D2

0.2%

ΔXB,S

0.8%

ΔXC,W1

0.8%

ΔXC,W3

0.79%

ΔXC,W4

3.1%

ΔR2

5.9%

ΔR1

10%

ΔR2

12.8%

ΔBR1

4%

ΔBR3

1.57%

ΔBR4

3%

ΔBR3

4%

ΔR2

7.3%

ΔSL

35%

ΔSL

2.5%

ΔSV

1.58%

ΔSV

12.4%

ΔNtotal

5

ΔNtotal

5

ΔNtotal

1

optimal

optimal

TAC

TAC 3.04 106

($/year) ΔTAC

1.3%

($/year)

DWCL

type

DWCU

type

DWCM

R2

1.018R2,min

BR3

1.029BR3,min

R2

1.053R2,min

ΔXA,D2

5.5%

ΔXA,D1

6.2%

ΔXA,D2

5.2%

ΔXB,W3 ΔXC,W1

3.8% 0.9%

ΔXB,D2 ΔXC,W3

3.8% 0.99%

ΔXB,S ΔXC,W4

4.3% 0.99%

ΔR2

0.6%

ΔR1

1.5%

ΔR2

20%

ΔBR1

4.2%

ΔBR3

6.5%

ΔBR4

7.7%

ΔBR3

19.5%

ΔR2

7.6%

ΔSL

20%

ΔSL

8.27%

ΔSV

5.3%

ΔSV

23%

ΔNtotal

19

ΔNtotal

7

ΔNtotal

14

optimal

optimal

optimal

TAC

2.03

TAC

TAC

($/year)

106

($/year)

1.90 106 ($/year) 1.85 106

ΔTAC XW3

2.8%

ΔTAC

3.9%

[0.0039

XD2

[0.00946

ΔTAC

2.9%

XS [0.00217

0.99

0.99

0.99

0.0061]

0.00054]

0.00783]

optimal TAC 2.83 106

ΔTAC

2.3%

($/year) ΔTAC

2.28 106 1.6%

Table 11. Feed Conditions in the Ethanol, n-Propanol, nButanol System F = 1 (kmol/s), q = 1

XW3

XD2

[0.0024

XS

[0.0081

0.99

0.99

0.0076]

0.0019]

[0.0024

RA/RB/RC = 3.7545/1.7271/1, ESI = 1.258

0.99 0.0076]

Ptop= 1.75 bar, Pbottom = 1.75 bar, thermodynamic model: PengRobinson feed compositions (ZA/ZB/ZC): 0.1/0.8/0.1, 0.6/0.2/0.2, 0.3/0.3/0.4

Table 9. Simulation Results in EPB System When ZA/ZB/ZC = 0.6/0.2/0.2

Table 12. Simulation Results in BTE System When ZA/ZB/ ZC = 0.1/0.8/0.1

ZA/ZB/ZC = 0.6/0.2/0.2 type

DWCL

type

DWCU

ZA/ZB/ZC = 0.1/0.8/0.1 type

type

DWCM

DWCL

type

DWCU

type

DWCM

R2

1.026R2,min

BR3

1.016BR3,min

R2

1.042R2,min

R2

1.007R2,min

BR3

1.0045BR3,min

R2

1.016R2,min

ΔXA,D2

3.1%

ΔXA,D1

3.3%

ΔXA,D2

2.4%

ΔXA,D2

6.4%

ΔXA,D1

5.5%

ΔXA,D2

3%

ΔXB,W3

5.6%

ΔXB,D2

5.6%

ΔXB,S

6.4%

ΔXB,W3

0.6%

ΔXB,D2

0.6%

ΔXB,S

0.3%

ΔXC,W1 ΔR2

0.99% 10.5%

ΔXC,W3 ΔR1

0.99% 6.1%

ΔXC,W4 ΔR2

0.99% 5.8%

ΔXC,W1 ΔR2

0.8% 3.4%

ΔXC,W3 ΔR1

0.96% 8.4%

ΔXC,W4 ΔR2

0.99% 2.4%

ΔBR1

11.6%

ΔBR3

10.2%

ΔBR4

6.5%

ΔBR1

5.9%

ΔBR3

10.25%

ΔBR4

11.1%

ΔBR3

12.5%

ΔR2

7.3%

ΔSL

4.2%

ΔBR3

0.98%

ΔR2

11.4%

ΔSL

4.6%

ΔSL

2.7%

ΔSV

3.2%

ΔSV

3.3%

ΔSL

6.6%

ΔSV

5%

ΔSV

4%

ΔNtotal

13

ΔNtotal

13

ΔNtotal

20

ΔNtotal

8

ΔNtotal

4

ΔNtotal

4

optimal

optimal

TAC ($/year)

2.12 106

optimal TAC ($/year)

2.02 106

TAC ($/year)

TAC ($/year)

2.37 107

TAC ($/year)

2.14 107

TAC ($/year)

ΔTAC

1.59%

ΔTAC

1.7%

ΔTAC

ΔTAC

2.5%

ΔTAC

0.54%

ΔTAC

optimal

XW3

[0.0005 0.99 0.0095]

XD2

[0.0077 0.99 0.0023]

XS

1.92 106 2.76%

XW3

[0.00336 0.99 0.00664]

optimal

[0.00072 0.99 0.0093]

9226

XD2

optimal

[0.0072 0.99 0.0028]

XS

1.87 107 3.68% [0.00065 0.99 0.00935]



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Table 13. Simulation Results in BTE System when ZA/ZB/ZC = 0.6/0.2/0.2

Table 15. Feed Conditions and Specifications in Economic Analysis

ZA/ZB/ZC= 0.6/0.2/0.2 type

DWCL

type

DWCU

F = 1 (kmol/s), q = 1, XA = 0.99, XB = 0.99, XC = 0.99 type

DWCM

ESI

RA/RB/RC

R2

1.016R2,min

BR3

1.0086BR3,min

R2

1.024R2,min

ESI < 1

4/2.4/1, 9/3.6/1, 2.25/1.8/1

ΔXA,D2

3.5%

ΔXA,D1

3%

ΔXA,D2

2.5%

ESI = 1

4/2/1, 9/3/1, 2.25/1.5/1

ΔXB,W3 ΔXC,W1

5.9% 0.99%

ΔXB,D2 ΔXC,W3

6.3% 0.99%

ΔXB,S ΔXC,W4

6.8% 0.99%

ESI > 1

4/1.4/1, 9/2.1/1, 3/1.2/1

ΔR2

1%

ΔR1

8.2%

ΔR2

0.67%

ΔBR1

11.8%

ΔBR3

12.9%

ΔBR4

10.8% 0.08%

ΔBR3

7.25%

ΔR2

6.9%

ΔSL

ΔSL

4.8%

ΔSV

1.4%

ΔSV

2.2%

ΔNtotal

24

ΔNtotal

19

ΔNtotal

13

Table 16. Thermodynamic Parameters for Ideal System (ESI < 1) RA/RB/RC:

optimal

optimal

TAC

TAC

1.37

TAC

1.29

($/year)

107

($/year)

107

1.59 107

($/year) ΔTAC XW3

2.2%

ΔTAC

[0.000004

XD2

optimal

1.58%

ΔTAC XS

[0.0055

[0.002

0.99

0.99

0.99

0.009996]

0.0045]

0.008]

Table 14. Simulation Results in BTE System When ZA/ZB/ ZC = 0.3/0.3/0.4

DWCL

type

DWCU

type

2.25/1.8/1

15.20

15.20

15.20

AVP2 (mm Hg)

14.68

14.28

14.97

AVP3 (mm Hg)

13.81

13.00

14.38

2768.55

2768.55

2768.55

Tb(A) (K) Tb(B) (K)

323.15 343.63

323.15 361.85

323.15 331.79

Tb(C) (K)

385.53

434.61

356.93

Table 17. Thermodynamic Parameters for Ideal System (ESI = 1) RA/RB/RC:

ZA/ZB/ZC = 0.3/0.3/0.4 type

9/3.6/1

AVP1 (mm Hg)

BVP (K)

1.38%

4/2.4/1

DWCM

4/2/1

9/3/1

2.25/1.5/1

AVP1 (mm Hg)

15.20

15.20

15.20

AVP2 (mm Hg)

14.50

14.10

14.79

13.81 2768.55

13.00 2768.55

14.38 2768.55

R2

1.0095R2,min

BR3

1.0087BR3,min

R2

1.015R2,min

AVP3 (mm Hg) BVP (K)

ΔXA,D2

5.6%

ΔXA,D1

3.8%

ΔXA,D2

5.1%

Tb(A) (K)

323.15

323.15

323.15

ΔXB,W3

3.2%

ΔXB,D2

3.7%

ΔXB,S

4.1%

Tb(B) (K)

351.60

370.68

339.20

ΔXC,W1 ΔR2

0.99% 4.7%

ΔXC,W3 ΔR1

0.99% 3.3%

ΔXC,W4 ΔR2

0.99% 10.3%

Tb(C) (K)

385.54

434.60

356.93

ΔBR1

7%

ΔBR3

8.2%

ΔBR4

2.3%

ΔBR3

2.48%

ΔR2

4.4%

ΔSL

3.75%

ΔSL

1.7%

ΔSV

1%

ΔSV

2.2%

ΔNtotal

23

ΔNtotal

20

ΔNtotal

7

optimal

optimal

TAC ($/year)

TAC ($/year)

ΔTAC XW3

1.76 107 1.7% [0.00066 0.99

1.3% [0.0072 0.99

0.00934]

RA/RB/RC:

optimal 1.57 107

ΔTAC XD2

Table 18. Thermodynamic Parameters for Ideal System (ESI > 1)

0.0028]

TAC ($/year) ΔTAC XS

1.59 107

[0.0004 0.99

L1 D1 Lmin, 1 ¼ D1

R1, min

3/1.2/1

15.20

15.20

15.20

AVP2 (mm Hg)

14.15

13.74

14.28

AVP3 (mm Hg)

13.81

13.00

14.10

2768.55 323.15

2768.55 323.15

2768.55 323.15

Tb(B) (K)

368.27

389.27

361.85

Tb(C) (K)

385.53

434.61

370.61

0.0096]

The definitions for section 1 are R1 ¼

9/2.1/1

AVP1 (mm Hg)

BVP (K) Tb(A) (K)

0.45%

4/1.4/1

ð40Þ

should be calculated individually, because only part of the liquid flow from section 2 will go to section 3_1. The reflux ratio and minimum reflux ratio of section 3_1 are defined as R31 ¼

ð41Þ

R31 , min ¼

Although section 2 and section 3_1 are combined when calculating the minimum tray number, the actual tray number 9227

L3 _1 D3

ð42Þ

Lmin , 3 _1 D3

ð43Þ



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Figure 7. Minimum vapor flow and minimum TAC (RA/RB/RC = 4/2.4/1) for (a, b) DWCM, (c, d) DWCL, (e, f) DWCU.

The actual tray number of section 3_2 and section 4 should also be calculated individually for the same reason. The definitions for section 4 are shown as R4 ¼

L4 W1 W3

R4, min ¼

Lmin, 4 W1 W3

ð44Þ ð45Þ

The definitions for section 3_2 are R3 _ 2 ¼ R3 _2, min ¼

L3 _2 S3 _ 2

ð46Þ

Lmin, 3 _2 S3 _ 2

ð47Þ

This means that the inlet reflux ratio changes when the liquid split (SL) or vapor split (SV) are changed. SL and SV are defined as L1 L̅ 2

ð48Þ

V̅ 1 SV ¼ V4

ð49Þ

SL ¼

SL and SV are dependent variables due to the constant mole flow assumption. If more vapor flow and liquid go into section 1, then there are fewer trays in section 1 (see Figure 5). Finally the values of SL and SV are chosen so that there are the same number of

trays in section 1 and section 3 and the smallest number of trays for section 1 and section 3. Step 4. Then, we calculate the tray number by using the FenskeUnderwoodGilliland method for N1, N2, N3_1, N3_2, and N4. " # N Nmin R Rmin 0:5688 ¼ 0:75 1 ð50Þ N +1 R+1 Step 5. The Kirkbride equation is used to determine the feed location for section 1: " #0:206 N1 _ 1 D1 XA, D1 D1 ¼ ð51Þ N1 _ 2 W1 XC, W1 W1 The ratio of N2 to N3_1 and the ratio of N3_2 to N4 should also be determined by the Kirkbride equation: " #0:206 N2 D2 XA, D2 D2 ¼ ð52Þ N3 _1 S3 _1 XB, S3 _1 S3 _1 " #0:206 N3 _2 S3 _2 XB, S3 _2 S3 _2 ¼ N4 W4 XC, W4 W4

ð53Þ

The procedure is shown in Figure 6. As depicted, the procedure is to determine (N2 + N3_1)min and (N4 + N3_2)min using eqs 36 and 37, then determine N2,min, N3_1,min, N4,min, and N3_2,min by assuming certain ratios N2,min/N3_1,min and 9228



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Figure 8. TAC comparisons for (a, b) RA/RB/RC = 4/2.4/1, (c, d) RA/RB/RC = 9/3.6/1, (e, f) RA/RB/RC = 2.25/1.8/1.

N4,min/N3_2,min. Then, the actual number of trays in each section is determined using the Gilliland equation. If the resulting number of trays in each section is not consistent with the Kirkbride equation, then the ratios N2,min/N3_1,min and N4,min/ N3_2,min are adjusted iteratively until consistency is achieved. Finally, the reflux ratio is used as a design variable to find the design with the minimum TAC. 2.4. Modifications. The same method, including the development of the component flow model; determination of the minimum vapor flow rate; and the application of the methods of Fenske, Underwood, and Gilliland and the Kirkbride equation can be applied, with modifications, for the design of DWCL and DWCU. Additional details are given in the Supporting Information. Figure 9. The 15% savings line for three sets of relative volatility (ESI < 1).

3. VALIDATION OF THE METHOD 3.1. Ideal System. As a first step in testing the usefulness of our method, we made a comparison between DWC processes designed using our shortcut method and “actual” process designs completed using rigorous simulation in Aspen. The designs are

based on an ideal three-component system with particular (constant) values of the relative volatilities, different feed compositions, and a particular choice for the product purity specifications. The general system parameters are given in Table 1, and 9229



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Figure 10. Minimum vapor flow and minimum TAC (RA/RB/RC = 4/2/1) for (a, b) DWCM, (c, d) DWCL, (e, f) DWCU.

the Antoine coefficients of the fictitious species are given in Table 2. The verification procedure was to first determine the number of trays in each section, the value of the liquid split or vapor split or both, the reflux ratio, the sidedraw flow rate, and the boilup ratio using the shortcut design. Then, the number of trays in each section was fixed in Aspen, and the internal flow rates were adjusted until the product purity specifications were met. The process designed in Aspen is termed the “actual” design. The procedure was repeated for different feed compositions as shown in Table 1, and for different values of R/Rmin and BR/BRmin between 1.002 and 1.2. The results indicate that the deviation in the sidedraw flow rate is always small; therefore it is not reported. We report and discuss the deviations in the other variable in order to see how well the shortcut method works. Table 3 shows the detailed results for DWCM when βP is selected as the reference point for R2 = 1.2Rmin. Tables 46 show the results when βb is selected as the reference point for R2 = 1.2Rmin, R2 = 1.002Rmin, and R2 = 1.05Rmin. For DWCM, the choice of the value of β is an important part of the design. The results show that the maximum deviation of variables is about 17% when βP is chosen, and the maximum deviation of variables is about 20% when βb is selected. The results for R = 1.2Rmin show that there is an over-design problem when choosing βP as the reference point. That is, for the same feed conditions, the shortcut method predicts more trays and more energy consumption when βP is chosen. The conclusion is that βb is the better reference point for describing the net flow model of DWCM. From the calculation of the minimum vapor flow, we know that the bigger minimum vapor flow between section 2 and section 4 is chosen as the minimum vapor flow of DWCM. The physical meaning of βb is that the minimum vapor flow of section 2 is the same as that in section 4 and the minimum vapor flow of section 3_1 is the same as that in section 3_2. In the design methods, we apply the FenskeUnderwoodGilliland method

to calculate the number of trays. If the minimum vapor flow for each section is overestimated, the tray number for the section will be calculated inaccurately. That is why the method performs better when the reference point is βb. From the results of DWCL and DWCU (see Supporting Information), all the deviations of variables are around 10%. The method performs well for these two configurations for both high and low reflux ratios. Detailed simulation results for the ideal systems are shown in the Supporting Information. 3.2. Real System. Two real systems taken from the literature are used to test the method in real situations, that is, when the relative volatility and molar overflow are not constant. A comparison between the shortcut design, actual design, and optimal design are presented. To identify the optimal design, all design variables including the number of trays in each section are adjusted to minimize TAC. In the shortcut design, only the reflux ratio or boilup ratio is adjusted to minimize the TAC. We investigate the deviations of all of the design variables, as well as the deviation of TAC, which is defined as ΔTAC ¼

TACact TACopt TACopt

ð54Þ

All of the rigorous simulations were done in Aspen Plus. The use of “Spec-Vary” in Aspen Plus was used to get the specifications. The calculation of TAC is according to the method of Kaymak and Luyben.23 In this work, we calculate the equipment cost for the column shell, trays, and heat exchangers but not the dividing wall. The payback time is 3 years. The local optimization procedure for DWCM is listed below. R2, BR4, and S are the variables that are manipulated to get the product specifications. (1) Start from the actual design (2) Change N2 and tune SL and SV until TAC is minimized (3) Change N4 and tune SL and SV until TAC is minimized (4) Change N3 (N1 = N3) and tune SL and SV until TAC is minimized 9230



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Figure 11. TAC comparisons for (a, b) RA/RB/RC = 4/2/1, (c, d) RA/RB/RC = 9/3/1, (e, f) RA/RB/RC = 2.25/1.5/1.

(5) Change N3_1 and tune SL and SV until TAC is minimized (6) Change N1_1 and tune SL and SV until TAC is minimized (7) Go back to step 5 until TAC is minimized (8) Go back to step 4 until TAC is minimized (9) Go back to step 3 until TAC is minimized (10) Go back to step 2 until TAC is minimized The procedures of optimization for DWCL and DWCU are listed in the Supporting Information. 3.2.1. Case 1: Ethanol, n-Propanol, n-Butanol System. In this system, ethanol is the lightest component, n-propanol is the middle component, and n-butanol is the heaviest component. Details about the feed conditions are shown in Table 7. The ease of separation index (ESI)24 is smaller than 1 for this system. The product purity specification for each stream is 99%. Three feed compositions are considered. Tables 810 show the results for the three different column types at different feed compositions. R2 and BR3 are the initial values from the shortcut design for the three configurations. Rows 28 show the deviations of key process variables between the shortcut design and the actual design. The numbers of trays for shortcut calculation and actual design are the same, and ΔNtotal (in row 9) is the difference

Figure 12. The 15% savings line in three sets of relative volatility (ESI = 1).

between the numbers of trays in the shortcut design and optimal design. Rows 10 and 11 show the total annual cost of the optimized process and the deviation in total annual cost between the shortcut design and the optimal design. In the shortcut calculation, the nonkey component compositions in the middle component product stream are set as 0.005. 9231



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Figure 13. Minimum vapor flow and minimum TAC (RA/RB/RC = 4/1.4/1) for (a, b) DWCM, (c, d) DWCL, (e, f) DWCU.

These compositions will usually be different in the optimal design. XW3 and XD2 are compositions of the middle component flow for DWCL and DWCU. Row 13 shows the values of the middle product stream compositions for the optimal design. 3.2.2. Case 2: Benzene, Toluene, Ethylbenzene System. In the second system, benzene is the lightest component, toluene is the middle component, and ethylbenzene is the heaviest component. The detailed feed conditions are shown in Table 11. The ESI is larger than 1 for this system. The product purity specification for each stream is 99%. Tables 1214 show the results for all three column types for different feed compositions. Detailed simulation results for both case 1 and case 2 are shown in the Supporting Information. 3.3. Discussion. The deviations of TAC are all below 5% for both systems and for all three configurations. This means that the shortcut method works well for finding a configuration which is close to minimum TAC. The maximum deviation of the reflux ratio is about 20% in the EPB system and 13% in the BTE system. The deviations for the number of trays are small when the feed contains more of the middle component in both systems but large when there is more of the lightest or heaviest component in the feed. When comparing the results of the actual design and optimal design, we can find that the deviations of reflux ratio and boilup ratio are small. The over-design problem occurs in the shortcut method when more of the lightest and heaviest components are in the feed. The reflux ratio is close to the minimum reflux ratio, and the boilup ratio is close to the minimum boilup ratio for all feed conditions and all three configurations. This suggests that the energy cost dominates the economics. RamirezCorona et al.16 also found the same result. That is why a large deviation in the number of trays does not cause a large deviation in TAC. In the shortcut calculation, the light and heavy component compositions in the sidedraw stream are set to be 0.005; however, in the optimal design, they may be different, as long as their sum is 0.01, so that the intermediate boiling product purity is 0.99. Thus, in theory, this could be an additional design

variable. However, because the intermediate boiling product purity is high in this study, the deviation in TAC is not significant if this design variable is neglected. However, it may be worthwhile to consider this design variable if the intermediate boiling product purity is lower.

4. ECONOMIC ANALYSIS Most research on the design of divided wall columns concentrates on DWCM design. However DWCU and DWCL should also receive attention. In this work, we give some guidelines to choose a DWC configuration by comparing the minimum vapor flow and minimum TAC of different configurations. The shortcut method presented in this work will also be applied to estimate the minimum TAC. An ideal system will be assumed in order to simplify the calculation. Three ESI conditions (ESI < 1, ESI = 1, and ESI > 1) are considered. The detailed feed conditions and specifications are given in Table 15. Tables 1618 show the thermodynamic properties of the fictitious species. When comparing the three different column types, it is important to remember that there is a vapor split disturbance problem for DWCU and DWCM. At the design stage, the position of the dividing wall can be set so that the desired vapor split is achieved. For example, if the wall is set in the center of the column, approximately half of the vapor flow from below will go to the left side and half will go to the right side (if the number of stages and other process properties are approximately the same on both sides of the wall). If the wall is set closer to the left side, more vapor will go to the right side. However, after the column has been built, it is difficult to manipulate or control the vapor split. Thus, DWCL has an advantage in operation. It may be preferable to use DWCL when the TAC of the three types is similar. 4.1. ESI < 1. Figure 7 shows the relationship between feed composition and minimum vapor flow and the relationship between feed composition and minimum TAC for three types of DWCs. The trend in these two figures is almost the same, which suggests that energy consumption is a good surrogate for total cost. 9232



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Figure 14. TAC comparisons for (a, b) RA/RB/RC = 4/1.4/1, (c, d) RA/RB/RC = 9/2.1/1, (e, f) RA/RB/RC = 3/1.2/1.

That is, heat duty is more important than capital cost. The lowest vapor flow and TAC occur under the conditions with more of the C component in the feed. This is not surprising because the separation between B and C is easier than the separation between A and B when ESI is smaller than 1. Next, we define an index TAC_DWCL/TAC_DWCU. If the value is less than 1, it means that DWCL has a lower minimum vapor flow or TAC. Panels a, c, and e in Figure 8 show this ratio for different feed compositions. Panel a is the base case, panel c is for an easy overall split, and panel e is for a difficult overall split. The results show that the difference between DWCL and DWCU is not very significant (11%∼7%). To compare DWCM with DWCL and DWCU, we define an additional index TAC_DCWM/min(TAC_DWCL, TAC_DWCU). That is, the cost of DWCM is compared with the best choice between DWCL and DWCU. DWCM has a lower TAC in most regions, and the difference is more significant (27%∼8%). The 15% savings line for the three overall split difficulties is plotted in Figure 9. If the overall split is easier, the savings zone becomes larger. In summary, from the point of view of economics, DWCM is usually preferred when ESI < 1.

4.2. ESI = 1. Figure 10 shows the effect of feed composition on the minimum vapor rate and total annual cost for the three different column designs when ESI = 1. The lowest vapor flow and TAC occurs when there is more of the C component and A component in the feed. The difficulty of separation between B and C is the same as the difficulty of separation between A and B when the ESI is equal to 1, so a lower TAC can be obtained when there is more of the C component and A component in the feed. As before, the trend of Vmin and TAC is very similar, suggesting that Vmin is a good surrogate for the total process cost. Figure 11 shows the comparison between the cost of the three column designs using the same metrics as in Figure 8. Panels a and b are for the base case, panels c and d are for the case of an easy overall split (Rmax = 9), and panels e and f are for the case of a difficult overall split (Rmax = 2.25). The results indicate that the difference between DWCL and DWCU is not significant (4.4% to 8%). When comparing DWCM with the best choice between DWCL and DWCU, DWCM has a lower TAC in most regions. And the difference is more significant (30% to 2%). Figure 12 shows the 15% savings line for the three different overall split difficulties. If the overall split is more difficult, the savings zone 9233



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Table 19. The TAC Comparisons for the EPB System and BTE System

using DWCL because of the operational advantage, especially when the ESI is smaller than 1.

feed composition ZA/ZB/ZC:

TAC comparison index

0.1/0.8/0.1 DWCL/DWCU DWCM/min(DWCL or DWCU) 0.6/0.2/0.2 DWCL/DWCU DWCM/min(DWCL or DWCU) 0.3/0.3/0.4 DWCL/DWCU DWCM/min(DWCL or DWCU)

EPB system

BTE system

(ESI < 1)

(ESI > 1)

1.07

1.11

0.80

0.87

1.05

1.15

0.95

0.94

1.07

1.12

0.97

1.01

becomes larger. Therefore, we again conclude that DWCM has the lowest cost when ESI = 1. 4.3. ESI > 1. Finally, we consider the case where ESI > 1. Figure 13 shows the effect of feed composition on minimum vapor rate and TAC for three types of DWCs. The difficulty of separation between B and C is more difficult than the difficulty of separation between A and B when ESI is bigger than 1, so a lower TAC is obtained with more of the A component in the feed. Again, the trend in Vmin is similar to the trend in TAC. Figure 14 shows a comparison between different DWC configurations using the same metrics as in Figures 8 and 11. Again, panels a and b are for the base case, panels c and d are for the case of an easy overall split (Rmax = 9), and panels e and f are for the case of a difficult overall split (Rmax = 2.25). The results show that the difference between DWCL and DWCU is not very significant (4% to 13%). Comparing the results between DWCM and the lowest value between DWCL and DWCU, DWCM has a lower TAC in most regions except panel f for the difficult overall split. However, the difference is only moderately significant (11% to 17%). When ESI > 1, either DWCM or DWCU will usually have the lowest cost. 4.4. Discussion. The results from the previous section provide guidance for the choice of DWC configuration based on TAC for different feed conditions and relative volatilities. In all cases, the minimum vapor flow is a good surrogate for the total process cost for the purposes of comparing design alternatives. Furthermore, DWCM has a lower TAC for most feed conditions, especially when there is more of the middle component in the feed. However, the ease or difficulty of column operation should be considered as well. Recall that DWCL has an advantage in this respect because there is no vapor split. Table 19 shows the TAC comparisons for the two real systems discussed. All of the values of TAC are for the optimal design. The EPB system has ESI < 1, and the BTE system has ESI > 1. When ESI < 1, the TAC for DWCL and DWCU is almost the same. The TAC savings with DWCM is more significant (20%) when more of the middle component is in the feed. Thus, the results from the real case studies are consistent with the economic analysis of the ideal systems. The analysis of ideal systems indicates that the total process cost is higher when ESI < 1 than when ESI > 1. This is also borne out by the real case studies. Furthermore, the TACs of DWCL and DWCU are lower than the TAC for DWCM when there is more of the heaviest component in the feed. Therefore, we suggest choosing DWCM when the middle component is the most prevalent in the feed for all values of ESI. For the case where there is more of the heaviest and/or lightest component in feed, the engineer may wish to consider

5. CONCLUSION This work provides a novel shortcut method that can be used for the design of the three types of DWCs. An efficient net flow model is constructed in order to simplify the shortcut calculation. Several ideal case studies are used to validate the method. The results show that βb is the best reference point for the design of DWCM. The method also works well for both high and low reflux ratios. In the real case study, the shortcut method is used to provide the initial conditions for determining the design with the minimum TAC. The results also show that the shortcut design can provide a configuration close to the optimal design. The method is also applied to perform an economic analysis of divided wall systems for different relative volatilities and feed compositions. The results suggest that DWCM is the least expensive when there is more of the middle component in the feed. For the case with more of the heaviest and/or lightest component in the feed, DWCL may be preferable in view of the column operation consideration, especially when the ESI is smaller than 1. Finally, results from two real case studies are consistent with the results developed by analyzing ideal systems. ’ ASSOCIATED CONTENT

bS

Supporting Information. Additional information about the application of this method to DWCU and DWCL systems, as well as further information about the results of the case study simulations. This information is available free of charge via the Internet at http://pubs.acs.org/.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: jeff[email protected].

’ NOMENCLATURE Rij = relative volatility of component i with respect to component j Ri = relative volatility of component i β = the fraction of B sent to the top of prefractionator BRi = boilup ratio of section i BRi,min = minimum boilup ratio of section i Di = top product flow rate of column i F = feed flow rate Wi = bottom product flow rate of column i Li = liquid flow in the upper zone of column i Lmin,i = minimum liquid flow in the upper zone of column i Li = liquid flow in the upper zone of column i Lmin,i = minimum liquid flow in the lower zone of column i Ni = number of trays of section i Ni,min = minimum number of trays of section i NTotal = total number of trays q = feed quality Ri = reflux ratio of section i Rimin = minimum reflux ratio of section i S = sidedraw flow rate SL = liquid split SV = vapor split Vi = vapor flow in the upper zone of section i Vmin,i = minimum vapor flow in the upper zone of section i 9234


Industrial & Engineering Chemistry Research V i = vapor flow in the lower zone of section i V min,i = minimum vapor flow in the lower zone of section i xi,j = mole fraction of component i in liquid flow j Zi = mole fraction of component i in feed flow θ, ϕ, ξ = root of Underwood’s equation ΔXi,j = deviation of composition for component i in product flow j

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Recycle Systems for Different Chemical Equilibrium Constants. Ind. Eng. Chem. Res. 2004, 43, 2493. (24) Tedder, D. W.; Rudd, D. F. Parametric Studies in Industrial Distillation 0.1. Design Comparisons. AlChE J. 1978, 24, 303.

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A New Shortcut Design Method and Economic Analysis of Divided

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