Quantifying Potential Energy Savings of Divided Wall Columns Based

Ind. Eng. Chem. Res. 2011, 50, 1473–1487

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Quantifying Potential Energy Savings of Divided Wall Columns Based on Degree of Remixing† Yuan-Chieh Ho, Jeffrey D. Ward,* and Cheng-Ching Yu Department of Chemical Engineering, National Taiwan UniVersity, Taipei 106-17, Taiwan

A new framework is introduced to analyze divided wall columns. The framework provides a simple and intuitive understanding of why different types of divided wall columns save energy when they are compared with corresponding conventional distillation sequences. Furthermore, analytical expressions are derived to quantify the potential energy saving of divided wall columns based on remixing. The method is illustrated with some real mixture examples often seen in the literature. 1. Introduction Distillation is a widely used method for separating mixtures. The main disadvantage of distillation is its high energy consumption, and therefore saving energy is an important goal in distillation system design. A divided wall column (DWC)1 is a column with a single shell and a dividing wall, which is thermodynamically equivalent to a thermally coupled distillation column. DWC is an alternative to conventional distillation sequences that may provide significant energy savings. Some studies show the energy saving of DWC can be up to 30% with respect to the conventional indirect and direct sequences.2–6 In a pioneering series of contributions, Fidkowski and Krolikowski7,8 applied Underwood’s9 classical method to analyze and design divided wall columns. They developed expressions for the minimum vapor flow rate in divided wall systems and showed that divided wall systems always have a lower minimum vapor flow rate than the corresponding twocolumn systems. Since then, other researchers have also applied Underwood’s method to divided wall systems. Carlberg and Westerberg10,11 presented a simpler analysis of divided wall systems also based on Underwood’s method. Fidkowski and co-workers12,13 compared the efficiency and minimum energy requirement of thermally coupled columns and conventional distillation sequences and also proposed some more operable arrangements of thermally coupled columns. Halvorsen and Skogestad14 extended the analysis to consider nonsharp separations and arbitrary feed quality. The energy efficiency of divided wall columns compared with conventional distillation sequences can be explained in terms of the remixing of species in the first column of the conventional sequence. For example, in the conventional indirect sequence, the heaviest species is separated first, and a second column separates the lightest species from the intermediate species. In the first column of such a sequence, there is a point in the column profile where there is very little species C, and species B is enriched relative to its composition at the top of the column. However this separation between species A and B is not exploited and instead species B is “remixed” with species A and the two species are drawn out together from the top of the column. Divided wall columns reduce the remixing effect and thus save energy.2,6 This result has been discussed by several authors. Hernandez15 compared the behavior of conventional and coupled indirect * To whom correspondence should be addressed. E-mail: [email protected]. † Y.-C. Ho and J. D. Ward dedicate this paper to C. C. Yu, who was the principal investigator on this project.

distillation columns and showed the significant reduction of remixing in a thermally coupled column. Kim16 discussed the remixing in fully thermally coupled columns for different designs. However, in spite of these contributions, it remains difficult to understand the relationship between remixing and energy saving intuitively or analytically. In this contribution we introduce a new abstract concept, the fictitious wall, which facilitates an intuitive understanding of why divided wall columns save energy based on the remixing effect. Also, starting with Underwood’s method, we develop new simple analytical expressions that relate the remixing in the first column of the two-column process to the vapor saving in the divided wall system. Results are illustrated using both systems with fictitious species with constant relative volatilities and systems with real species. 2. Minimum Energy Consumption of Different Types of Divided Wall Columns 2.1. Three Different Types of Divided Wall Columns. In this section, three different types of divided wall columns are introduced. The three divided wall columns are DWCL, DWCU, and DWCM, which correspond to the indirect sequence (IS), direct sequence (DS), and prefractionator (PF), respectively. The divided wall columns and their corresponding conventional distillation sequences are shown in Figure 1. DWCL means the wall is at the lower section of the divided wall column. Similarly, DWCU and DWCM mean the walls are at the upper and middle section in the divided wall columns, respectively. 2.2. Underwood’s Method for Minimum Energy Consumption. In this paper, the minimum vapor flow is regarded as a proxy for the minimum energy consumption for different types of distillation columns. The minimum vapor flow is obtained by Underwood’s method9 as first shown by Fidkowski and Krolikowski.7,8 Take DWCL and IS for example. For a ternary system, the calculation steps by Underwood’s method are shown in Table 1. During the calculation, the model of DWCL is replaced by a side-stripper since the two schemes are thermodynamicly equivalent. See Figure 2. The steps can be more easily to be understood in this way. For DWCL, qD1 means the overall feed quality to the second column and is negative because the liquid flow flows back from the second column. 2.3. Observations. From Table 1, steps 1-3 are the same for DWCL and IS. However, steps 4-6 are different because of the different feed quality to the second column. The minimum vapor flow in the first column is the same for DWCL and IS if

10.1021/ie901907e  2011 American Chemical Society Published on Web 07/12/2010

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Ind. Eng. Chem. Res., Vol. 50, No. 3, 2011

Figure 1. The evolution of divided wall columns from conventional distillation sequences: (a) DWCL from IS; (b) DWCU from DS; (c) DWCM from PF.

the feed condition and product specifications are the same. The work of the first column is to separate B and C sharply. Since the work of the first column of IS and DWCL are the same, the minimum vapor demand is the same. Therefore, the difference between DWCL and IS is the minimum vapor flows in the second column. The same results can also be observed in DWCU with DS and DWCM with PF. 3. Derivation of Analytical Expressions 3.1. Interpretation of Divided Wall Columns. Before the derivation of analytical expressions, a new framework is introduced to analyze divided wall columns. The derivation is based on the framework, and the reason that divided wall columns save energy can also be explained simply in this way. Take DWCL for example and assume the split is sharp (see Figure 3a). V1 and V ′2 are the vapor flows in the left and right section, respectively. Now assume that there is a fictitious wall in the upper section from the top of the wall to the top of the column. The fictitious wall maintains separation between the vapor flows V1 and V ′2. Only liquid can flow through the fictitious wall. The fictitious wall also splits the condenser into two smaller fictitious condensers which have the same reflux ratio. What the two small condensers do is what the original condenser does. Now that the concept of the fictitious wall has been established, the next step is to split the DWCL into two fictitious columns. When the feed flow goes into the DWCL, the first column separates product C and some of the product A, and the remaining liquid is sent to the second column. In

other words, V1 is not wasted in the first column and it goes to the smaller condenser to produce some product A. The first column is like a sidestream column. For the section below the sidestream in the first column, its work is to separate component B and C and the reflux ratio is for the B/C split. For the section above the sidestream, the reflux ratio is changed to purify the component A. The sidestream F ′2 containing component B and some component A is sent to the second column from the stage at the top of the wall. The concept can also be expressed by eq 1: V'2 ) V - V1 ) (1 + R)D - V1 ) V1 (1 + R) D ) (1 + R)(D - D1′) (1) 1+R

(

)

To explain why DWCL saves energy, IS and DWCL (after the fictitious split) are compared (see Figure 3b). From the observations on the calculation using Underwood’s method, the vapor flows in the first column of the two schemes are equal, that is, V1 ) V ′1. The work of the first column of IS is to purify component C and send the rest of the components A and B into the second column. However, the work of first column of DWCL is not only to purify component C but also to distill some of component A and send the rest to the second column. Consequently, the flow rate F ′2 is smaller than F2, and the mole fraction of B in F ′2 is larger than that of F2. For the same amount of flow B2 (bottom product flow containing only component B), the minimum vapor demand of the second column of DWCL is smaller when compared with IS since xB in F ′2 is larger than xB in F2.


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Table 1. The Steps of Underwood’s Method to Calculate Minimum Vapor Flows of IS and DWCL IS 1st column 1.

2.

DWCL

solve

RBZB RCZC RAZA + + 1-q) RA - θ RB - θ RC - θ

get

RA > θ1 > RB > θ2 > RC

choose θ2

IS V1Smin ) -B1

(

RAxB1,A RA - θ2

1st column 1.

get

+

RBxB1,B RB - θ 2

2.

+ RCxB1,C RC - θ2

choose θ2

4.

get 5.

choose φ1

then total

RA > θ1 > RB > θ2 > RC

DWCL V1Smin ) -B1

(

RAxB1,A RA - θ2

+

RBxB1,B RB - θ2

+ RCxB1,C

)

then

1 - q D1 )

RAxB1,A RA - φ

+

RBxB1,B RB - φ

+

RCxB1,C

4.

RA > φ1 > RB > φ2 > RC

IS V2Smin ) -B2

(

RAxB2,A RA - φ1

qD1 ) -L1Rmin /D1 solve

RC - φ get

+

RBxB2,B RB - φ1

5.

+ RCxB2,C RC - φ1

6.

RBZB RCZC RAZA + + RA - θ RB - θ RC - θ

RC - θ2

2nd column 3.

qD1 ) 1 solve

1-q)

)

then 2nd column 3.

solve

IS IS IS VTmin ) V1Smin + V2Smin

Likewise, the reason why DWCU saves energy when compared with DS can be explained (see Figure 4a). DWCU is split into two columns after introducing the concept of the fictitious wall. Compare DS and DWCU (after the fictitious split) in Figure 4b. From the observations on the calculation using Underwood’s method, the vapor flows in the first column of the two schemes are equal, i.e. V1 ) V ′1. The work of the first column of DS is to purify component A and send the rest of the components B and C into the second column. However, the work of first column of DWCU is not only to purify component A but also to distill some of component C and send the rest to the second

choose ξ1

1 - q D1 )

RAxB1,A RA - ξ

+

RBxB1,B RB - ξ

+

RCxB1,C RC - ξ

RA > ξ1 > RB > ξ2 > RC

DWCL V2Smin ) -B2

)

(

RAxB2,A RA - ξ1

+

RBxB2,B RB - ξ1

+ RCxB2,C RC - ξ1

6.

then total

)

DWCL DWCL DWCL VTmin ) V1Smin + V2Smin

column. Consequently, the flow rate F ′2 is smaller than F2 and the mole fraction of B in F ′2 is larger than that in F2. For the same amount of flow D2 (top product flow containing only component B), the minimum reflux ratio of the second column of DWCU is smaller when compared with DS since xB in F ′2 is larger than xB in F2. In other words, the minimum vapor demand of the second column of DWCU is smaller when compared with DS. In the same way, DWCM is split into two columns in Figure 5a and compared with PF in Figure 5b. The minimum vapor flows in the first column of the two schemes are equal, that

Figure 2. The model of IS and DWCL for calculating the minimum vapor flow by Underwood’s method.

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Figure 3. (a) The concept of fictitious wall introduced to DWCL. (b) Comparison of IS and DWCL (after split).

is, V1 ) V ′1. The work of the first column of PF is to split component A and component C, then send all the component A with some component B into the upper section of the second column and all the component C with some component B into the lower section of second column. However, the work of the first column of DWCM is not only to split A/C but also to distill some of component A and component C from the top and bottom, respectively. Assume that the same amount of component B goes into the second column ′ and F ′2L are from the upper feed, then the flow rates F 2U smaller than F2U and F2L, respectively. By the same reasoning used in DWCL and DWCU, V 2U ′ is smaller than V2U for the minimum vapor demand to split the upper feed, and V ′2L is smaller than V2L for the minimum vapor demand to split the upper feed. Therefore, V ′2 is smaller than V2 for the determined minimum vapor demand in the second column of DWCM and PF. 3.2. Derivations. 3.2.1. DWCL. So far, DWCL is split into two fictitious columns. The minimum vapor demand in the first column is the same for both IS and DWCL. Therefore, the minimum vapor demand in the second column is the difference between the two configurations. Assume all three species are to be sharply split. For IS, there are only two components going into the second column, and the minimum reflux ratio of the second column can be obtained easily from eq 3, since the feed composition to the second

column is known. Equation 3 is valid only for a sharp split and saturated liquid feed. The relevant derivation can be found in Glinos and Malone.17 ZF2,A )

ZA ) xD1,A, ZF2,B ) 1 - ZF2,A ZA + ZB

(2)

1 (RAB - 1)ZF2,A

(3)

RIS 2min )

For DWCL, if the feed composition to the second column is known, the minimum reflux ratio can be obtained in the same way. L RDWC 2min )

1 (RAB - 1)Z'F2,A

(4)

Therefore, determining the feed composition is the next step. For the section above the feed stage in the first column of DWCL, assume that there are only components A and B since component C has been sharply split. The small condenser distils some pure A. Because the two condensers are split from the original condenser of the DWCL, the reflux ratios are the same. Therefore, assume the reflux ratio of the first column is equal to the minimum reflux ratio of the second column. Then, by material balance:


ZA ZF′ 2,A )

1 + R1 - R1 ZA + Z B L RDWC 2min

ZB

(5)

Combining eqs 4 and 5,

[

ZF′ 2,A2 - 1 +

]

ZB 1 + Z′ + R1(ZA + ZB) R1(RAB - 1) F2,A ZA /(1 - ZC) ) 0 (6) R1(RAB - 1)

Details of the derivations of eqs 5 and 6 are given in the Appendix. The root of eq 6 is the feed composition to the second column if there are infinite number of stages in the second column (at minimum reflux ratio), and the feed composition can be found directly from Underwood’s formula: ZF′ 2,A )

-b - √b2 - 4ac 2a

ZF′ 2,B ) 1 - ZF′ 2,A

(7) (8)

Now, the minimum reflux ratio of the second column can be calculated by eq 4. Equation 6 is mentioned by Glinos and Malone18 for designing a sidestream column, but the derivation of eq 6 here is different from that of Glinos and Malone.

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For the first column, if the reflux ratio and boil up ratio below the sidestream stage of DWCL is the same as that of IS, as shown in Figure 6, the composition profile from the bottom to the top of the first column of the two schemes should be similar for the same feed condition and product specifications. We want to find out which stage in the first column of IS is the start point for DWCL to set the sidestream and change the reflux ratio to purify component A. The feed composition calculated by eq 6 is compared with the composition profile of first column of IS (see Figure 7). The calculated composition Z′F2 ) [0.308 0.692 0], is compared with the composition profile of the first column of IS. It can be seen that Z′F2 is close to the composition of the stage where the mole fraction of B is the maximum in the rectifying section (xM ) [0.311 0.684 0.005], at the eighth stage). The result can be understood by the following explanation. Set a sidestream at the eighth stage of the first column of IS and use an infinite number of stages above the stage to purify the top product which contains only component A to simulate the first column of DWCL. In this way, the composition of the sidestream will be very close to the composition of original eighth stage, especially the mole fraction of the lightest component (A). To test this explanation, the sidestream column is simulated and compared with the first column of the IS in Figure 8. The vertical gray line indicates the eighth stage of the first column of IS. Composition profiles are similar below the eighth stage and are different above the stage. We found that the mole fraction of A of Z′F2 is very close to that of xM at different feed compositions and reflux ratios. This

Figure 4. (a) The concept of a fictitious wall introduced to DWCU. (b) Comparison of DS and DWCU (after split).

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Figure 5. (a) The concept of fictitious wall introduced to DWCM. (b) Comparison of PF and DWCM (after split).

of DWCL can be calculated. Now, xM,A can be approximated by ZF′ 2A for estimation. L RDWC 2min )

Figure 6. Reflux ratios and boil-up ratios of first column of IS and DWCL, respectively.

observation means that the feed composition to the second column of DWCL is near the composition at the stage where the mole fraction of B is the maximum in the rectifying section. This can also explains why DWCL reduces remixing when compared with IS. Without the benefit of this observation, the root of eq 6 must be solved so that the minimum reflux ratio of the second column

1 1 = (RAB - 1)ZF′ 2,A (RAB - 1)xM,A

(9)

Therefore, from the composition profile of the first column of IS, the minimum reflux ratio of second column of IS and DWCL can be estimated by reading xD1,A and xM,A, respectivly. To quantify the potential energy saving of DWCL, it is assumed that the vapor flow in the first column of IS and DWCL are the same (see Figure 11a). The darker section means the same operating conditions, which means the same vapor flow, same reflux ratio, same boil-up ratio, etc. The assumption is based on the fact that the same work is done by the first column, that is, B/C sharp split. Therefore, we assume the same conditions in first column and compare the minimum vapor demands of the second column. Divide the minimum vapor flow in the second column of DWCL by that of IS. L VDWC 2min VIS 2min

)

DWCL L (1 + RDWC 2min )D2 IS (1 + RIS 2min)D2

(

RAB )1-

(RAB

xD1,A - xM,A

)

1 - xM,A - 1)xD1,A + 1 (10)

The term below is the vapor saving ratio of DWCL in the second column at minimum vapor demand (VSRDWCL).


RAB VSRDWCL )

(

(RAB

xD1,A - xM,A

)

1 - xM,A - 1)xD1,A + 1

(11)

xM,A depends on the operation of the first column, the feed composition, and the relative volatilities of the species. Normally, the mole fraction of B in xD1 and xM represents the degree of remixing. However, eq 11 is based on the mole fraction of A instead of B. The reason is that the error is smaller when the equation is based on the mole fraction of A. Comparing Z′F2 with xM, it is generally found that the values of the mole fraction of A are closer than those of the mole fraction of B. For this example, the feed composition to the second column of DWCL (Z′F2 ) [0.308 0.692 0]) is calculated using Underwood’s method and is compared with the point in the composition profile of the first column of the IS where the composition of B is maximum (xM ) [0.311 0.684 0.005], at the eighth stage). Since these compositions are similar we anticipate that the composition xM can be used in place of Z′F2 in the calculation of the vapor-saving ratio and therefore that the vapor saving can be related to the composition profile in the first column of the IS. Since the compositions are nearly identical, we anticipate that using the composition xM to calculate the VSR will introduce only a negligible error compared to using the composition based on Underwood’s method (Z′F2). To confirm this, we calculated the VSR using the composition determined using Underwood’s method for this example. We find that the calculated VSRDWCL using ZF′ 2A is 62.2% as compared with 62.1% when the VSR is calculated using xM,A, for a deviation of only 0.1 percentage points. If the value of the two compositions (xD1,A, xM,A) and the relative volatility between A and B are known, then the potential of energy saving of DWCL in the second column can be estimated from eq 11. Generally, xD1,A is greater than xM,A so that VSRDWCL is greater than zero; that is, DWCL consumes less energy than IS. The fact that DWCL consumes less energy than IS can be shown after calculating the minimum energy requirement of both systems as was first shown by Fidkowski and Krolikowski.8 However here VSRDWCL is related analytically to the remixing in the IS. Note that the value of VSRDWCL is dependent on the vapor flow in the first column because xM,A is dependent on the vapor flow in the first column. Furthermore, the more vapor flow (V1) that is used in the first column, the more top product flow (D1) is in the first column of DWCL and the more energy is saved in the second column of DWCL (see eq 1). Note the value of VSRDWCL only indicates the energy saving of DWCL based on the minimum energy consumption in the second column of

Figure 7. The composition profile of first column of IS.

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Figure 8. Composition profile of first column of IS and DWCL.

DWCL and IS, not the total minimum energy consumption of DWCL and IS. 3.2.2. DWCU. Similarly, DWCU can also be split into two fictitious columns. (see Figure 4a). For the section below the feed stage in the first column, assume that there are only components B and C since component A has been sharply split. The small reboiler of the first column produces some pure C. Assume that the boil-up ratio of the first column is equal to the minimum boilup ratio of the second column. Then, by material balance:

ZC -

ZF′ 2,C

1 + R1

ZA U SDWC 2min ) 1 + R1 ZB + ZC - DWC ZA S2min U

The minimum boil-up ratio of the second column is

(12)

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U SDWC 2min )

(1 - ZF′ 2,C) +

1 (RBC - 1)

(13)

ZF′ 2,C

Combining eqs 12 and 13 gives

[

ZB + (RBC - 1) RBCZC RBCZC ZF′ 2,C + ) 0 (14) (RBC - 1) (RBC - 1)

(1 + ZAR1)ZF′ 2,C2 - (1 + ZAR1) +

]

[

]

The root of the eq 14 is the feed composition to the second column if there are infinite number of stages in the second column (at minimum reflux ratio). Comparing the composition calculated with the composition profile of the first column of DS, the composition Z′F2 is near the composition at the stage where the mole fraction of B is the maximum in the stripping section, especially the mole fraction of the heaviest component (C) of Z′F2. To explain the observation, the first column of DWCU is simulated in the same manner as DWCL (see Figures 9 and 10). Therefore, the minimum reflux ratio in the second column of DWCU can also be approximated. U RDWC 2min )

Figure 9. Reflux ratios and boil-up ratios of the first column of DS and DWCU, respectively.

1 1 = (RBC - 1)(1 - ZF′ 2,C) (RBC - 1)(1 - xM,C) (15)

Before quantifying the potential energy saving of DWCU, the same assumption has to be made as for DWCL; that is, the operating conditions of the first column of DS and DWCU are the same (see Figure 11b. The darker section means the same vapor flow, same reflux ratio, same boil-up ratio, etc.). We keep the same conditions in the first column and compare the minimum vapor demands of the second column in each configuration. Divide the minimum vapor flow in the second column of DWCU by that of DS:

U VDWC 2min VDS 2min

)

U (1 + RDWC 2min )D2 (1 + RDS 2min)D2

)1-

(RBC

(

xB1,C - xM,C

)

1 - xM,C - 1)(1 - xB1,C) + 1 (16)

The term below is the vapor saving ratio of DWCU in the second column at minimum vapor demand (VSRDWCU). It is also a simple equation

VSRDWCU )

(RBC

(

xB1,C - xM,C

)

1 - xM,C - 1)(1 - xB1,C) + 1

(17)

The equation is based on mole fraction of C instead of B, for the same reason discussed in section 3.2.1. 3.2.3. DWCM. The assumptions are similar to these for DWCL and DWCU, including the assumption that the operating conditions in the first columns are the same. In addition, we assume only a small amount of component A exists in the bottom product and only a small amount of component C exists in the top product of the first column so that the derivation can be based on two components in the rectifying and stripping section.

Figure 10. Composition profile of the first column of DS and DWCU.

The derivation for DWCM is more complex because of the two feed flows into the second column. The method to calculate the minimum vapor flow in the second column has been discussed in many papers, for example Halvorsen and Skogestad.14 For the second column of PF, the normal steps are to calculate the minimum vapor flows for separating the upper and lower feed, respectively, and then choose the larger one as the minimum vapor flow the column. For DWCM (after being fictitiously split), U L M VDWC 2min ) max(V2Rmin, V2Smin)

(18)

Similar to the normal steps to calculate the minimum vapor flow, the VSR for the remixing at the upper and lower section in the first column of PF must be quantified first. VSR1 represents the potential energy savings if the condenser is thermally coupled.


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VSR2 represents the potential energy savings if the reboiler is thermally coupled.

(

RAB VSR1 )

VSR2 )

(RAB

(

U xD1,A - xM,A

)

U 1 - xM,A - 1)xD1,A + 1

L xB1,C - xM,C

(19)

)

L 1 - xM,C - 1)(1 - xB1,C) + 1

(RBC

(20)

The DWCM is split into two fictitious columns and V1 is assumed to be the same in the first column. Then the vapor demand for the liquid feeds into the upper and lower sections can be described by the vapor flows of the PF. Therefore, the VSR for DWCM could be written as below. DWCM

VSR

)1)1-

M,U M,L , VDWC ) max(VDWC 2min 2min

PF,L max(VPF,U 2min, V2min) PF,L max(VPF,U 2min(1 - VSR1), V2min(1 - VSR2))

max )1-

(

PF,L max(VPF,U 2min, V2min)

VPF,U 2min (1 VPF,L 2min

)

- VSR1), (1 - VSR2)

( )

max

VPF,U 2min VPF,L 2min

,1

(21)

For simplification, let

γ)

VPF,U 2min VPF,L 2min

)

(

xD1,A +

(

1 - xB1,C

)

1 D RAB - 1 1 1 + B RBC - 1 1

)

(22)

Then, the vapor saving ratio of DWCM in the second column at minimum vapor demand (VSRDWCM) becomes VSRDWCM ) 1 -

max(γ(1 - VSR1), (1 - VSR2)) (23) max(γ, 1) DWCM

So, there are four steps to calculate the VSR : U 1. Use RAB, xD1,A and xM,A to calculate VSR1, eq 19. L 2. Use RBC, xB1,C and xM,A to calculate VSR2, eq 20. 3. Use top product flow D1 and bottom product flow B1 to calculate γ, eq 22. 4. Use VSR1, VSR2, and γ to calculate VSRDWCM, eq 23. 3.3. Observations. From eq 11, if xD1,A f 1, then VSRDWCL will approach 1. This means DWCL will save a lot of energy. If xD1,A f 0, then VSRDWCL will become very small. Therefore, the larger xD1,A is, the more energy DWCL saves. The reason is that if the mole fraction of component A in the distillate of the first column of the IS is larger, more component A would be distilled in the first column of the DWCL (with the fictitious wall) and less component A would be fed to the second column of DWCL. Therefore, DWCL will save more energy in the second column compared with the conventional IS. From eq 17, if xB1,C f 1, then VSRDWCU will approach 1. If xB1,C f 0, then VSRDWCU will become very small. Therefore, the larger xB1,C is, the more energy DWCU saves.

Figure 11. The assumptions of VSR of divided wall columns: (a) DWCL; (b) DWCU; (c) DWCM.

4. Limitations and Extensions The equations are simple because of the use of Underwood’s method for sharp splits. Owing to the assumption of sharp split, the product purity is the most important limitation (see Figure 12). The effect of product purity on the error of the VSR equation is clear. In general, the higher the product purity is, the smaller is the error. Furthermore, Z′F2 is close to xM only for a sharp split and infinite number of stages in the second column of DWC. The equations are not recommended for estimating the precise VSR for systems without high purity specifications. The recommended method is to calculate the minimum vapor flows in the second column by Underwood’s method for the product specifications based on the operating condition of the first column. Furthermore, constant molar flow rates and constant relative volatilities are assumed in the derivations. Therefore, the equations are not recommended for the system with significant variations in molar flow rates and relative volatilities. The vapor-saving ratio is defined based on the energy consumption in the second column only. Although it would be

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overstate the savings relative to the case where the ratio is defined on the basis of the combined vapor consumption of both parts of the process. The engineer should keep this in mind when evaluating whether thermal integration is attractive for a particular application. 5. Real Systems

Figure 12. The error of VSRDWCU at different purity specifications and feed compositions: R ) [4 2 1].

Figure 13. Composition profile of the first column of IS for case 1.

Figure 14. Composition profile of the first column of DS for case 1.

possible to modify the expressions for the vapor-saving ratio so that they expressed the ratio of the combined vapor consumption for the nonintegrated processes to the combined vapor consumption for the integrated process, the expression for the VSR would be much more complicated and much of the insight would be lost. The VSR as we have defined it will

Figure 15. Composition profile of the first column of PF for case 1.

To test the performance of the equations, some real systems often seen in the literature are used as examples. Procedure. The feed flow rate, feed composition, and product mole fraction specifications of the example are F ) 45.4 kmol/ h, Z ) [0.4 0.2 0.4] and 0.99, respectively. The software is Aspen Plus and the procedure is described below. 1. Use the DSTWU in Aspen Plus to calculate the minimum number of stages (Nmin) of each column of conventional distillation sequences (IS, DS, and PF). 2. Set the actual number of stages of column as 2Nmin + 1 and estimate the feed stage, reflux ratio, and reboiler duty of conventional distillation sequences by DSTWU. 3. Adjust the reflux ratio and reboiler duty to approach the product specifications. 4. Scale 10 times up the number of stages of the second column to represent the infinite number of stages and obtain the minimum vapor flow in the second column. 5. The number of stages and the feed stage of the divided wall columns (DWCL, DWCU, and DWCM) are set the same as conventional distillation sequences (IS, DS, and PF). 6. Adjust the vapor flow in first column of divided wall columns to be the same as that of corresponding conventional distillation sequences. 7. Choose the geometric mean of the relative volatility in the top and bottom of the first column of conventional distillation sequences as the constant relative volatility. 8. Read the data from the composition profile in the first column of conventional distillation sequences, estimate the VSR by corresponding equations, and compare them with the results of simulation by Aspen Plus. Literature Examples. Case 1: Benzene, Toluene, and Ethyl Benzene. Benzene, toluene, and ethyl benzene is a commonly studied system. Figures 13, 14, and 15 show the composition profile in the first column of IS, DS, and PF and the VSR of corresponding divided wall columns. The results of simulation by Aspen Plus are shown in Tables 2, 3, and 4. For DWCL, the VSR predicted by eq 11 is 0.67 while the actual vapor saving ratio is 70%. For DWCU, the VSR predicted by eq 17 is 0.30 while the actual value is


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Table 2. Comparison of IS and DWCL for Case 1. (Predicted VSR ) 0.67) IS

no. of stages feed stage reflux ratio boilup rate (kmol/h) liquid flow from col. 2 (kmol/h) reboiler duty (kW) vapor saving in col. 2

DWCL

col. 1

col. 2 col. 2∞ col. 1

col. 2 col. 2∞

24 16 1.49 63.27

22 8 1.48 42.28

22 8 3.89 17.88

220 80 0.96 33.54

24 16 63.27 39.47

220 80 3.45 10.06

628.20 392.13 311.02 628.24 165.78 93.26 58% 70%

Table 3. Comparison of DS and DWCU for Case 1. (Predicted VSR ) 0.30) DS

no. of stages feed stage reflux ratio boilup rate (kmol/h) vapor flow from col. 2 (kmol/h) reboiler duty (kW) vapor saving in col. 2

Table 5. Comparison of IS and DWCL for Case 2. (Predicted VSR ) 0.81)

DWCU

IS

col. 1

col. 2 col. 2∞ col. 1 col. 2 col. 2∞

26 10 1.49 41.43

24 15 4.63 46.99

240 150 3.31 36.18

26 10 1.46

24 15 4.28 84.93

240 150 1.99 65.84

41.43 410.21 466.53 359.19

843.22 653.68 7% 33%

Table 4. Comparison of PF and DWCM for Case 1. (Predicted VSR ) 0.70) PF

no. of stages feed stage

Figure 17. Composition profile of the first column of DS for case 2.

no. of stages feed stage reflux ratio boilup rate (kmol/h) liquid flow from col. 2 (kmol/h) reboiler duty (kW) vapor saving in col. 2


69 42 4.02 128.53

18 7 1.27 38.24


25 17

46 8 42 1.24 35.93 25

46 8 42 3.36 71.63 25

69 42

18 7 7.78 128.53 14.33 29.14

460 80 420 2.47 57.54 250

reflux ratio 1.01 boilup rate (kmol/h) 47.02 sidestream stage liquid flow from col. 2 7.06 (kmol/h) vapor flow from col. 2 47.02 (kmol/h) reboiler duty (kW) 470.80 356.76 297.41 711.13 571.29 vapor saving in col. 2 32% 65%

Figure 16. Composition profile of the first column of IS for case 2.

45.39 78%

no. of stages feed stage reflux ratio boilup rate (kmol/h) vapor flow from col. 2 (kmol/h) reboiler duty (kW) vapor saving in col. 2

DWCU

col. 1


20 9 2.29 55.30

69 42 12.18 111.87

690 420 9.55 89.55

20 9 2.28

69 42 8.89 138.56

690 420 6.53 118.31

55.30 391.80 801.99 641.98

993.32 848.12 26% 30%


33%. For DWCM, the VSR predicted by eq 19, 20, 22 and 23 is 0.70 while the actual value is 65%. The results of the VSR prediction are good. Case 2: n-Butane, Isopentane, and n-Pentane. Figures 16, 17, and 18 show the composition profile in first column of IS, DS, and PF and the VSR of corresponding divided wall columns. The results of simulation by Aspen Plus are shown in Tables 5, 6, and 7. The VSR of each divided wall column is close to the actual value. For DWCL, the VSR is 0.81 while the actual vapor saving ratio is 78%. For DWCU, the VSR is 0.26 while the actual value is 30%. For DWCM, the VSR is 0.19 while the actual value is 16%. The results of the VSR prediction are good.

180 70 7.31 6.58


col. 1

25 17

180 70 0.79 30.26

921.40 263.75 208.71 921.40 98.83 63%

DWCM

460 80 420 0.87 29.96 250

DWCL

col. 1

no. of stages Feed stage

DWCM

col. 1


36 16

85 7 52 5.39 102.16 14

850 70 520 4.47 87.50 140

36 16

85 7 52 6.83 126.87 14

850 70 520 5.79 110.25 140


Case 3: n-Pentane, n-Hexane, and n-Heptane. Figures 19, 20, and 21 show the composition profile in the first column of IS, DS, and PF and the VSR of the corresponding divided wall columns. The results of simulation by Aspen Plus are shown in Table 8, 9, and 10. For DWCL, the VSR is 0.69 while the actual vapor saving ratio is 69%. For DWCU, the VSR is 0.30 while the actual value is 49%. For DWCM, the VSR is 0.72 while the actual value is 46%. In this example, only the prediction of VSR of DWCL is good. However, the predictions of the other two schemes are not good. Discussion. The results of prediction are good for case 1 and 2. However, the predictions of DWCL and DWCM are not good for case 3. The error of the predictions might be due to inconstant molar flows, inconstant relative volatilities, product

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Figure 18. Composition profile of the first column of PF for case 2. Table 8. Comparison of IS and DWCL for Case 3. (Predicted VSR ) 0.69) IS

no. of stages feed stage reflux ratio boilup rate (kmol/h) liquid flow from col. 2 (kmol/h) reboiler duty (kW) vapor saving in col. 2 Figure 19. Composition profile of the first column of IS for case 3.

DWCL

col. 1

col. 2 col. 2∞ col. 1

col. 2 col. 2∞

21 12 1.03 52.88

18 6 1.16 36.35

18 6 3.32 18.02

180 60 0.72 29.14

21 12 52.88 27.22

468.56 291.29 233.47 468.54 144.47 73.31 50% 69%


no. of stages feed stage reflux ratio boilup rate (kmol/h) vapor flow from col. 2 (kmol/h) reboiler duty (kW) vapor saving in col. 2 Figure 20. Composition profile of the first column of DS for case 2.

purity specifications, and some nonideal behavior for the real system. Also, the inconstant relative volatilities of a real system in the conventional distillation sequences and divided wall columns may also be different. The deviation of prediction of DWCU for cases 1, 2, and 3 are -0.03, -0.04, and -0.19. The intermediate component and

Figure 21. Composition profile of the first column of PF for case 2.

180 60 2.80 9.14

DWCU

col. 1


24 12 2.34 54.40

21 12 3.41 35.83

210 120 2.39 27.73

24 12 2.22

21 12 1.52 72.91

210 120 1.00 68.48

54.40 477.87 317.45 245.73

645.98 606.72 48% 49%

the heaviest component for each system are the main components in the second column. Figure 22 shows the relative volatility of the intermediate component with respect to the heaviest component for each system in the first column of DS. The figure shows significant variations in a big range for case 3. Therefore, we suspect that the inconstant relative volatilities of case 3 cause the deviation.


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no. of stages feed stage

DWCM

col. 1


23 10

39 9 32 0.90 29.14 17

390 90 320 0.52 23.45 170

23 10

39 9 32 2.34 54.80 17

390 90 320 1.95 48.89 170


Figure 22. Relative volatility of the intermediate component (B) with respect to the heaviest component (C) for each case in the first column of DS.

The deviation of prediction of DWCM for cases 1, 2, and 3 are 0.05, 0.03, and 0.26. Figure 23 shows the relative volatilities of components with respect to the heaviest component for each system in the first column of PF. The figure shows significant variations in a big range for case 3. Therefore, we suspect that the inconstant relative volatilities of case 3 cause the obvious deviation. The deviation of prediction of DWCL for cases 1, 2, and 3 are -0.03, 0.03, and 0. The lightest component and the intermediate component for each system are the main components in the second column. Figure 24 shows the relative volatility of the lightest component with respect to the intermediate component for each system in the first column of IS. For case 3, the relative volatility fluctuates throughout the column. However, the range of variation is not big. Therefore, the deviation of prediction is not obvious for case 3. However, there are many factors that may contribute to this error. We cannot be certain that the vapor-liquid nonideality is the only factor.

Figure 23. Relative volatilities of the lightest component and the intermediate component with respect to the heaviest component for each case in the first column of PF: (a) case 1, (b) case 2, (c) case 3.

6. Conclusion A new framework, based on the concept of a fictitious wall that splits a divided wall column into two fictitious columns is introduced to analyze divided wall columns. The framework provides another way to see how divided wall columns work and why they save energy when compared with corresponding conventional distillation sequences. Beginning with the seminal work of Fidkowski and Krolikowski,7,8 other researchers have used Underwood’s method to analyze thermally coupled columns, and they have also shown that energy saving was somehow related to reducing remixing. In this work we have developed new quantitative relationships between the degree of remixing and the vapor saving. This is

Figure 24. Relative volatility of the lightest component (A) with respect to the intermediate component (B) for each case in the first column of IS.

useful not only as a method for a shortcut estimation of the energy savings at high product purity specifications but also because of the intuition that it provides about the reason that divided wall columns save energy.

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Finally, some real mixture examples often seen in the literature are used to examine the performance of the analytical expression. The predictions for cases 1 and 2 are close to the actual values. For case 3, only the prediction of DWCL is precise while the predictions of DWCU and DWCM are not precise. There might be some error when the equations for VSR are applied to real mixtures; however, the VSR provides a value as a reference when conventional distillation sequences and a divided wall column are compared.

)

L (1 + RDWC 2min )ZA - (1 + R1)(ZA + ZB) L (1 + RDWC 2min )(ZA + ZB) - (1 + R1)(ZA + ZB) L (1 + RDWC 2min )ZA - (1 + R1)(ZA + ZB)

)

Nomenclature Rij ) relative volatility of component i with respect to component j Ri ) relative volatility of component i with respect to component C θ,φ,ξ ) root of Underwood’s equation Bi ) bottom product flow rate of column i Di ) top product flow rate of column i LiRmin ) minimum liquid flow in the rectifying section of column i NT ) total number of stages NF ) feed stage qi ) liquid fraction of flow i R ) reflux ratio j Rimin ) minimum reflux ratio of column i of scheme j S ) reboil up ratio j Simin ) minimum reboil up ratio of column i of scheme j V ) vapor flow j ViSmin ) minimum vapor flow in the stripping section of column i of scheme j j VTmin ) total minimum vapor demand of scheme j xj,i ) mole fraction of component i in liquid flow j xM,i ) the mole fraction of component i of the liquid at the stage where the mole fraction of component B is the maximum in rectifying section of first column for IS (in stripping section of first column for DS) j,U xM,i ) the mole fraction of component i of the liquid at the stage where the mole fraction of component B is the maximum in the upper section of secheme j j,L xM,i ) the mole fraction of component i of the liquid at the stage where the mole fraction of component B is the maximum in the lower section of secheme j Zi ) mole fraction of component i in original feed flow Zj,i ) mole fraction of component i in feed flow j

Appendix

The Derivation of Equation 5: Assume the reflux ratio of the 1st column of DWCL is equal to the minimum reflux ratio of the 2nd column of DWCL. Then, the distillate of the 2nd column D′1 can be written as

D'1 )

FZA - D'1 F(ZA + ZB) - D'1 1 + R1 FZA F(ZA + ZB) L 1 + RDWC 2min ) 1 + R1 F(ZA + ZB) F(ZA + ZB) L 1 + RDWC 2min

ZA,F ′ 2)

V1 1+

L RDWC 2min

)

1 + R1 L 1 + RDWC 2min

F(ZA + ZB)

(24)

And the feed composition to the 2nd column ZA,F ′ 2 would be

L (1 + RDWC 2min )(ZA + ZB) - (1 + R1)(ZA + ZB) 1 + R1 ZB ZA - DWC R2min L - R1 ) ZA + Z B

(25) The Derivation of Equation 6: Combining eqs 4 and 5 gives

(

ZA,F ′ 2(ZA + ZB) - ZA -

( (

ZA,F ′ 2(ZA + ZB)

(RAB

)

1 + R1 ZB ) 0 1 - R1 - 1)ZA,F ′ 2 (26)

) )

1 - R1 - ZA (RAB - 1)ZA,F ′ 2 1 - R1 + (1 + R1)ZB ) 0 (27) (RAB - 1)ZA,F ′ 2

ZA,F ′ 2(ZA + ZB) - ZA,F ′ 2(ZA + ZB)((RAB - 1)ZA,F ′ 2)R1 - ZA + ((RAB - 1)ZA,F ′ 2)ZAR1 + (1 + R1)ZB((RAB - 1)ZA,F ′ 2) ) 0 (28) [R1(ZA + ZB)(RAB - 1)]ZF′ 2,A2 + [(ZA + ZB) + ZAR1(RAB - 1) + (1 + R1)ZB(RAB - 1)]ZF′ 2,A - ZA ) 0 (29)

[

ZF′ 2,A2 - 1 +

]

ZB 1 + Z′ + R1(ZA + ZB) R1(RAB - 1) F2,A ZA /(1 - ZC) ) 0 (30) R1(RAB - 1)

Literature Cited (1) Wright, R. O.; Elizabeth, N. J., Fractionation Apparatus, US Patent 2,471,134, 1949. (2) Triantafyllou, C.; Smith, R. The Design and Optimization of Fully Thermally Coupled Distillation-Columns. Chem. Eng. Res. Des. 1992, 70 (2), 118–132. (3) Hernandez, S.; Jimenez, A. Design of Energy-Efficient Petlyuk Systems. Comput. Chem. Eng. 1999, 23 (8), 1005–1010. (4) Wolff, E. A.; Skogestad, S. Operation of Integrated 3-Product (Petlyuk) Distillation-Columns. Ind. Eng. Chem. Res. 1995, 34 (6), 2094– 2103. (5) Glinos, K.; Malone, M. F. Optimality Regions for Complex Column Alternatives in Distillation Systems. Chem. Eng. Res. Des. 1988, 66 (3), 229–240. (6) Schultz, M. A.; Stewart, D. G.; Harris, J. M.; Rosenblum, S. P.; Shakur, M. S.; O’Brien, D. E. Reduce costs with dividing-wall columns. Chem. Eng. Prog. 2002, 98 (5), 64–71. (7) Fidkowski, Z.; Krolikowski, L. Thermally Coupled System of Distillation-ColumnssOptimization Procedure. AIChE J. 1986, 32 (4), 537– 546. (8) Fidkowski, Z.; Krolikowski, L. Minimum Energy Requirements of Thermally Coupled Distillation Systems. AIChE J. 1987, 33 (4), 643–653.

Ind. Eng. Chem. Res., Vol. 50, No. 3, 2011 (9) Underwood, A. J. V. Fractional Distillation of Multicomponent Mixtures. Chem. Eng. Prog. 1948, 44 (8), 603–614. (10) Carlberg, N. A.; Westerberg, A. W. Temperature Heat Diagrams for Complex Columns. 2. Underwoods Method for Side Strippers and Enrichers. Ind. Eng. Chem. Res. 1989, 28 (9), 1379–1386. (11) Carlberg, N. A.; Westerberg, A. W. Temperature Heat Diagrams for Complex Columns. 3. Underwoods Method for the Petlyuk Configuration. Ind. Eng. Chem. Res. 1989, 28 (9), 1386–1397. (12) Agrawal, R.; Fidkowski, Z. T. Are Thermally Coupled Distillation Columns Always Thermodynamically More Efficient for Ternary Distillations. Ind. Eng. Chem. Res. 1998, 37 (8), 3444–3454. (13) Agrawal, R.; Fidkowski, Z. T. More Operable Arrangements of Fully Thermally Coupled Distillation Columns. AIChE J. 1998, 44 (11), 2565–2568. (14) Halvorsen, I. J.; Skogestad, S. Minimum Energy Consumption in Multicomponent Distillation. 2. Three-Product Petlyuk Arrangements. Ind. Eng. Chem. Res. 2003, 42 (3), 605–615.

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(15) Hernandez, S.; Pereira-Pech, S.; Jimenez, A.; Rico-Ramirez, V. Energy Efficiency of an Indirect Thermally Coupled Distillation Sequence. Can. J. Chem. Eng. 2003, 81 (5), 1087–1091. (16) Kim, Y. H. Structural Design and Operation of a Fully Thermally Coupled Distillation Column. Chem. Eng. J. 2002, 85 (2-3), 289–301. (17) Glinos, K.; Malone, M. F. Minimum Reflux, Product Distribution, and Lumping Rules for Multicomponent Distillation. Ind. Eng. Chem. Process Des. DeV. 1984, 23 (4), 764–768. (18) Glinos, K. N.; Malone, M. F. Design of Sidestream DistillationColumns. Ind. Eng. Chem. Process Des. DeV. 1985, 24 (3), 822–828.

ReceiVed for reView December 3, 2009 ReVised manuscript receiVed June 1, 2010 Accepted June 7, 2010 IE901907E

Quantifying Potential Energy Savings of Divided Wall Columns Based

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