Simultaneous optimization of heat-integrated extractive distillation with

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Simultaneous optimization of heat-integrated extractive distillation with a recycle feed using pseudo-transient continuation models Fenggang Cui, Chengtian Cui, and Jinsheng Sun Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b02728 • Publication Date (Web): 18 Oct 2018 Downloaded from http://pubs.acs.org on October 29, 2018

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Industrial & Engineering Chemistry Research

Simultaneous optimization of heat-integrated extractive distillation with a recycle feed using pseudo-transient continuation models Fenggang Cui, Chengtian Cui, Jinsheng Sun* School of Chemical Engineering and Technology, Tianjin University, Tianjin, 300350, P. R. China

Abstract This paper proposed a heat-integrated extractive distillation with a recycle feed (HIEDRF), including extractive distillation column (EDC), solvent recovery column (SRC), and concentrator. Heat integration was easily achieved by coupling SRC condenser and EDC reboiler. However, optimizing the complex HIEDRF is challenging, because the associated variables in both distillation and heat integration systems must be considered simultaneously. To solve this multi-variable optimization problem, we developed a new pseudo-transient continuation (PTC) model. Based on a mixed-steady state-time relaxation optimization algorithm, the associated PTC model was solved using Aspen Custom Modeler (ACM). Two case studies-acetone/methanol and methanol/toluene, were investigated. For the acetone/methanol system, the HIEDRF reduced energy consumption by around 27% and total annualized cost (TAC) by around 16% with respect to both conventional extractive distillation system (CEDS) and extractive distillation with a recycle feed (EDRF). For the methanol/toluene system, the reductions were around 21% and 10%, respectively. 1. Introduction To separate azeotropes or close-boiling mixtures, special distillation technologies are required, such as azeotropic distillation, pressure-swing distillation, and extractive distillation.1-3 Of these options, pressure-swing distillation is unfeasible if the mixtures are insensitive to pressure. For azeotropic distillation, the addition of an entrainer forms a new azeotrope containing one or more of the original components, which can be removed as the distillate.4 However, in some cases, the multiple steady states and parametric sensitivity make design and control difficult.5-7 Compared with the other two options, extractive distillation is the most widely used on an industrial scale due to its higher energy efficiency and flexibility of solvent candidates.8-10 Despite the advantages of extractive distillation for separating azeotropes or close-boiling mixtures, it is an energy-intensive process. In spite of the high capital costs, double-effect heat integration and heat pump technologies are commonly used to reduce energy consumption and total annualized cost (TAC).11-13 You et al.12, 13 studied the acetone/methanol system from a thermodynamic perspective and compared the energy saving potential of double effect schemes and heat pump assisted schemes. In their results, the partial heat integration process obtained the most reduction in both TAC and energy consumption. Li et al.14 investigated the use of sensible heat of heavy solvent by combining heat integration technology and intermediate heating. In the results of Li et al.14, the scheme that used the solvent as heat source of side and bottom reboilers was the most energy efficient scheme. Gao et al.

*

15

introduced a new combination of

Corresponding author.

E-mail address: [email protected] (J. Sun).

1

ACS Paragon Plus Environment


pressure-swing and extractive distillation schemes to separate a methyl acetate/methanol/water mixture. In Gao et al.’s work, 15 mechanical vapor recompression was also tested, and results showed that the use of a double pressure-swing column with extractive distillation scheme has advantages over other schemes. Recently, a novel extractive distillation with a recycle feed (EDRF) has been applied to several separation processes.16,

17

The EDRF design is shown in Figure 1 (a). For convenience, A, B, and S

represent the light component, the heavy component, and the solvent, respectively. Highly pure A is recovered from the EDC distillate, and a small portion of A is mixed with EDC bottoms, feeding to the SRC. In the SRC, the A-B mixture with enriched B is recovered from the distillate, and then feeds into the concentrator. The high purity solvent is obtained from the SRC bottom and recycled to the EDC. In the concentrator, the A-B mixture with enriched A is recovered from the distillate, and then recycled to the feed location of the EDC, and high purity B is recovered from the bottom. Li and Bai16 proposed the EDRF for the first time and used it in the ethanol dehydration process. Ebrahimzadeh et al.17 studied the separation of CO2/ethane using EDRF, and obtained a considerable reduction of both TAC and energy consumption compared with the conventional extractive distillation system (CEDS). Li and Bai16 found that the use of EDRF produced ethanol with a higher purity, and the introduction of an additional column did not cause a TAC increase. Hence, the EDRF seems to be an effective alternative for extractive distillation. However, Li and Bai16 fixed the operating pressure of all columns at 1 atm and did not consider the possibility of heat integration.

D1 A

S

D2

D3

A+B

A+B

Makeup

Concentrator

SRC

EDC

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F A+B

B1

B2 S

A+B+S

B3 B

(a)

2


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D1 A

S

D2

D3

A+B

A+B

Makeup

Concentrator

SRC

EDC

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


F A+B

B1

B2 S

A+B+S

B3 B

(b) Figure 1. The flowsheet of (a) EDRF (b) HIEDRF

The optimization of extractive distillation schemes and other complex distillation schemes based on rigorous model is a significant challenge. Previous attempts at optimization of distillation system with heat integration can be divided into two categories: sequential methods and simultaneous methods. As reported by Duran and Grossmann18 and Ma et al.19,

20,

the simultaneous methods, which optimize all decision

variables in a system at the same time, can obtain a better result compared with sequential methods. However, there are limited reports on simultaneous optimization of complex distillation systems based on equilibrium stage due to the high nonlinearity of the model equations and difficulty in convergence. Yeomans and Grossmann21 proposed nonlinear disjunctive models for the synthesis of heat integrated distillation sequences. However, those models were formulated with shortcut methods and simplified assumptions, leading to much uncertainty in their results. Stichlmair and Frey22 optimized reactive distillation processes by solving the MINLP (mixed integer nonlinear programming) problem. However, this method is difficult to carry out in complex flowsheets due to the integer optimization problem. Wang et al.23 proposed a simultaneous optimization and integration method for a gas turbine and air separation system to increase net power generation. However, to improve the model flexibility, they constrained the relationship between the heat duty of high-pressure column condenser and low-pressure column reboiler. Shahandeh et al.24 used genetic algorithms to optimize heat pump assisted distillation columns for the separation of methanol and water. However, stochastic algorithms are inefficient. Recently, Pattison and Baldea25 and Ma et al.19,

20

developed simultaneous optimization using a

pseudo-transient continuation (PTC) model in an equation-oriented (EO) environment. The main concept of the PTC model is the conversion of the original algebraic equations (AE) into differential algebraic equations (DAE), where the steady state solution of DAE is identical to the solution of AE. On the one hand, the automatic or symbolic differentiation is easy to access due to the EO environment. On the other hand, the convergence problem can be easily solved by the initialization of the DAE system and robustness of the variables integration method. Hence, the PTC model is shown to be an effective method for the simultaneous optimization of complex systems. Although the PTC model has advantages for the simultaneous optimization of complex systems, there 3


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is still room for improvement. The model proposed by Pattison and Baldea25 is difficult to perform due to excessive requirements of linear equations and changes in material balance equations of flash tank. Ma et al.19 suggested a PTC model based on the bulit-in dynamic optimization algorithm of Aspen Custom Modeler (ACM). However, the dynamic optimization algorithm is not suitable for simultaneous optimization, which suffers long calculation time. As reported by Ma et al. 19, optimizing a three-product divided wall column (DWC) requires about 10 hours, which is inefficient. Two problems motivated our present work. First, to enhance the energy utilization of the EDRF, we proposed a heat-integrated extractive distillation with a recycle feed (HIEDRF). Heat integration was achieved by coupling SRC condenser and EDC reboiler. The flowsheet of HIEDRF is indicated in Figure 1 (b). Second, the HIEDRF presented here is very difficult to optimize due to the entangled design variables associated with the two recycle streams. In order to overcome difficulties in optimizing HIEDRF and reduce the calculation time of the PTC model, we developed a new PTC model based on a mixed-steady state-time relaxation algorithm using ACM. This work is organized as follows. In Section 2, we give a brief comparison between the time relaxation-based algorithm and the dynamic optimization algorithm, and introduce the mixed-steady state-time relaxation optimization algorithm proposed in this work. Section 3 gives the TAC calculation model. Section 4 gives the PTC model equations of the distillation column and heat exchanger. In Section 5, two case studies are used to verify the PTC model and assess the energy saving potential of HIEDRF. In Section 6, brief discussions of the bypass efficiency and calculation time are given. Section 7 presents the conclusion of this work. 2. Background: optimization techniques using PTC model Pattison and Baldea25 developed the PTC model for the first time based on the time relaxation algorithm using gPROMS in an EO environment. In the work of Pattison and Baldea25, they first integrated the DAE system to steady state, and then computed the corresponding objective function and constraints at steady state. The optimizer determines if the optimal solution was obtained. If not, new optimization variables are computed, and the integration of the DAE system starts from the last steady state solution to a new steady state solution, until the optimal solution is obtained. Based on this algorithm, they successfully optimized several typical chemical processes, which had been reported in their work. 25-28 Inspired by the work of Pattison and Baldea25, Ma et al.19 developed a new PTC model based on the built-in dynamic optimization algorithm in ACM. The overall optimization framework of both time relaxation algorithm and dynamic optimization algorithm is similar. The differences are mainly: (1) the dynamic optimization algorithm computes the sensitivity matrix and objective function at each integration step, whereas the time relaxation algorithm computes them only at steady state; (2) At each integration step, the dynamic optimization algorithm integrates the DAE system from the initial point, whereas the time relaxation algorithm integrates the DAE system from the last steady state solution. Hence, the dynamic optimization algorithm requires much longer calculation time compared with the time relaxation algorithm. To reduce the calculation time and integration steps in the environment of ACM, one approach is to combine the steady state simulation and time relaxation based algorithm. Hence, a mixed-steady state-time relaxation algorithm is proposed in this paper. This algorithm calculates the steady state process at first, and 4


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then transfers the steady state condition into pseudo transient if the problem is not converged. The objective function and corresponding constraints are calculated only at the converged condition of the steady state simulation. The optimizer also determines whether the optimal value of objective function was obtained. If not, new optimization variables are obtained and then integrate the DAE system from the results of last steady state solution. On the one hand, compared with the pseudo transient process, the calculation of steady state process requires less time. On the other hand, during the iteration steps, the results of the previous step can be used as initial value of the next step. Hence, this algorithm could reduce much calculation time compared with the bulit-in dynamic optimization algorithm of ACM. In this optimization framework, the AE system is solved in the environment of steady state run mode of ACM, and the DAE system is solved using the dynamic optimizer Feasopt, which is a NLP solver based on sequential quadratic programming (SQP) programming. The optimization framework of the mixed-steady state-time relaxation algorithm is shown in Figure 2. Start

Give initial value of optimization variables

Transfer into DAE system and integrate to steady state

Solve AE system

Convergence?

No

Yes Calculate objective function and constraints

Compute new optimization variables

No

Optimal solution? Yes End

Figure 2. Optimization framework of the mixed-steady state-time relaxation algorithm

3. Economic evaluation In this analysis, TAC was chosen as the objective function, which includes the total annualized capital cost and the total annualized operating cost. 5



TAC =

𝐶𝑐𝑎𝑝 𝑌𝑏𝑎𝑐𝑘

Page 6 of 24

+ 𝐶𝑜𝑝𝑒

(1)

Here, 𝐶𝑐𝑎𝑝 and 𝐶𝑜𝑝𝑒 are total annualized capital cost and total annualized operating cost, respectively. 𝑌𝑏𝑎𝑐𝑘 represents the payback period, the value of which is 3 years. 3.1 Total annualized capital cost The capital cost includes distillation column shells, trays, and heat exchangers. 3.1.1 Distillation column In order to calculate the capital cost of the column shells and trays, the diameter and height must be determined. For the calculation of the diameter and height, the following listed equations are used, which are from Luyben’s book.29 𝑉𝑔,1

𝑉𝑔,2

𝑉𝑔,𝑗

𝑉𝑔,𝑁𝑠 (2)

) 𝐷𝑐 = max ( 𝜋 , 𝜋 , …, 𝜋 …, 𝜋 𝑢1 𝑢2 𝑢𝑗 𝑢𝑁𝑠 4 4 4 4

(3)

𝐻𝑐 = 1.2 ∙ 0.61𝑁𝑠 F = 0.8197𝑢𝑗 𝜌𝑉𝑗

(4) 𝜌 𝐷 𝐻 𝑁 𝑉 𝑢 Here, 𝑐, 𝑐, 𝑠, 𝑔,𝑗, 𝑗, and 𝑉𝑗 represent column diameter in meter, column height in meter, number of total stages, vapor volume flowrate in m3, maximum vapor velocity in m/s, and vapor mass density of each stage in kg/m3, respectively. In order to calculate the maximum vapor velocity, the F factor must be used, as defined in eq 4, and the value of F factor is 1 kg0.5m0.5/s. The capital cost of distillation column shells and trays are estimated as follows29: 𝐶𝑐𝑜𝑙 = 17640𝐷1.066 𝐻0.802 𝑐 𝑐 𝐶𝑡𝑟𝑎𝑦 =

(5)

229𝐷1.55 𝑐 𝑁𝑠

(6)

Here, 𝐶𝑐𝑜𝑙 and 𝐶𝑡𝑟𝑎𝑦 represent the capital cost in US$ of distillation column shells and trays, respectively. 3.1.2 Heat exchanger The capital cost of the heat exchanger is the function of the heat transfer area, and the relevant equation is as follows 29: 𝐶ℎ𝑒𝑥 = 7296𝐴0.65

(7)

Here, 𝐶ℎ𝑒𝑥 and A are capital cost of heat exchanger in US$ and heat transfer area in m2, respectively. The heat transfer area is calculated from the following equation29: 𝑄 A= 𝐾∆𝑇𝑚𝑒𝑎𝑛

(8)

Here, 𝑄 is the heat duty of the heat exchanger in kW, 𝐾 is the overall heat transfer cofficient in kW/(℃ ∙ 𝑚2), and ∆𝑇𝑚𝑒𝑎𝑛 is the logarithmic mean temperature difference between the hot and cold streams. In order to avoid a numerical problem if the temperature differences at both sides are the same, the method proposed by Chen30 was adopted in this work. The corresponding equation is shown in eq 9. ∆𝑇𝑚𝑒𝑎𝑛 = [

1/3 𝑜𝑢𝑡 𝑜𝑢𝑡 𝑖𝑛 𝑖𝑛 𝑜𝑢𝑡 𝑜𝑢𝑡 𝑖𝑛 (𝑇𝑖𝑛 ℎ𝑜𝑡 ― 𝑇𝑐𝑜𝑙𝑑)(𝑇ℎ𝑜𝑡 ― 𝑇𝑐𝑜𝑙𝑑)(𝑇ℎ𝑜𝑡 ― 𝑇𝑐𝑜𝑙𝑑 + 𝑇ℎ𝑜𝑡 ― 𝑇𝑐𝑜𝑙𝑑)

2

]

(9) 6


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The overall heat transfer coefficients are listed in Table 1.29

3.2 Total annualized operating cost The total annualized operating cost includes hot utility, cold utility, electricity and solvent cost. In this study, the cold utility is cooling water, and three types of hot utilities are used. The low-pressure steam is used at first unless the minimum heat exchange temperature difference, which is specified at 10℃, cannot be satisfied. The average costs of utilities are listed in Table 2. 29 The solvent cost is calculated as follows: 𝐶𝑠𝑜𝑙 = 𝐹𝑠 ∙ 𝑀𝑠 ∙ 𝐶𝑝𝑒𝑟

(10)

Here, 𝐶𝑠𝑜𝑙 is the solvent cost in $/h. 𝐹𝑠, 𝑀𝑠 and 𝐶𝑝𝑒𝑟 are molar flowrate of the solvent in kmol/h, relative molecular mass of the solvent in kg/kmol, and unit price of the solvent in $/kg. For the first case of this work, water was selected as the solvent, the cost of which was not considered. For the second case, 1, 2-dimethylbenzene was selected as the solvent, the unit price of which was 1.06$/kg. The total annualized operating cost is given as follows 29:

[∑(𝐶

𝐶𝑜𝑝𝑒 = 𝐴𝑂𝑇

ℎ𝑢

∙ 𝑄𝑟) +

∑ (𝐶

𝑐𝑢

]

∙ 𝑄𝑐) + 𝐶𝑒𝑙𝑒𝑐 ∙ 𝑊𝑐 + 𝐶𝑠𝑜𝑙

(11)

Here, 𝑄𝑟, 𝑄𝑐, and 𝑊𝑐 represent the energy consumption of steam, cooling water, and electricity in kW, respectively. 𝐶ℎ𝑢, 𝐶𝑐𝑢, and 𝐶𝑒𝑙𝑒𝑐 represent utility costs of steam, cooling water, and electricity in US$, respectively. 𝐴𝑂𝑇 is the annual operating time in h/a, and here the value of 𝐴𝑂𝑇 is 8000 h/a. 4. Model equations 4.1. PTC distillation model Ma et al.19 developed their distillation model with the assumption of pseudo-dynamic process. However, to strengthen the robustness of the model, they used a pseudo-hold-up correlation. The coefficient of the pseudo-hold-up correlation was determined through trial and error, a process that is not mathematically rigorous and is time-consuming. In order to solve this problem, we developed a PTC distillation model with the concept of equivalent stream. Compared with the PTC distillation model proposed by Ma et al.19, this model is more explicit and easier to carry out. 4.1.1 PTC model equations of column tray The solving of MESH equations is very difficult due to the highly coupled and strongly nonlinearity nature. However, from the perspective of separation, a column tray can be modeled as a flash tank. If the equation of each tray can be solved separately, then only several flash problems need to be solved. Hence, the MESH equations are significantly simplified. Based on this method, we substituted all of the streams entering into a tray with an equivalent stream. The model of the tray is shown in Figure 3.

7



Page 8 of 24

Vj yj,i

Lj-1 xj-1,i Fj

Stage j Lj xj,i

Vj+1 yj+1,i

Figure 3. Model of distillation column tray

The steady state material balance equations are formulated as follows: 𝑉𝑖𝑛,𝑖 = 𝐹𝑗𝑦𝐹,𝑖(1 ― 𝑞) + 𝑉𝑗 + 1𝑦𝑗 + 1，𝑖

(12)

𝑉𝑜𝑢𝑡,𝑖 = 𝑉𝑗𝑦𝑗,𝑖

(13)

𝐿𝑜𝑢𝑡,𝑖 = 𝐿𝑗𝑥𝑗,𝑖

(14)

𝐿𝑖𝑛,𝑖 = 𝐹𝑗𝑥𝐹,𝑖𝑞 + 𝐿𝑗 ― 1𝑥𝑗 ― 1,𝑖

(15)

𝐹𝑖𝑛,𝑖 = 𝑉𝑖𝑛,𝑖 + 𝐿𝑖𝑛,𝑖

(16)

𝐹𝑜𝑢𝑡,𝑖 = 𝑉𝑜𝑢𝑡,𝑖 + 𝐿𝑜𝑢𝑡,𝑖

(17)

𝐹𝑜𝑢𝑡,𝑖 = 𝐹𝑖𝑛,𝑖

(18)

Based on the steady state material balance equations above, the dynamic simulation equations are as follows: Material balance (M): 𝜏𝑇

𝑑𝐹𝑖𝑛,𝑖 𝑑𝑡

= 𝐹𝑖𝑛,𝑖 ― 𝑉𝑗𝑦𝑗,𝑖 ― 𝐿𝑗𝑥𝑗,𝑖

(19)

Phase equilibrium (E): 𝑦𝑗,𝑖 = 𝐾𝑗,𝑖𝑥𝑗,𝑖

(20)

Summation (S):

∑𝑥

𝑗,𝑖

=1

𝑖=1

∑𝑦

𝑗,𝑖

=1

𝑖=1

(21) (22)

Enthalpy balance (H): 𝜏𝑇

𝑑𝐻𝐹,𝑖𝑛 𝑑𝑡

= 𝐻𝐹,𝑖𝑛 ― 𝐻𝑗,𝑉 ― 𝐻𝑗,𝐿

(23)

Here, index i and j correspond to the stage number and component, respectively. 𝑉𝑗, 𝐿𝑗, 𝑥𝑗,𝑖, and 𝑦𝑗,𝑖 correspond to the molar flow of vapor phase out of stage j, molar flow of liquid phase out of stage j in kmol/h, mole fraction of component i in stream 𝐿𝑗, and mole fraction of component i in stream 𝑉𝑗, respectively. 𝑉𝑖𝑛,𝑖 and 𝐿𝑖𝑛,𝑖 corrsepond to the component i in vapor phase and liquid phase enter into stage, respectively. 𝑞, 𝑥𝐹,𝑖, and 𝑦𝐹,𝑖 correspond to liquid fraction of the feed stream, composition of the 8


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liquid phase of the feed stream, and composition of the vapor phase of the feed stream, respectively. 𝐻𝐹,𝑖𝑛, 𝐻𝑗,𝑉, and 𝐻𝑗,𝐿 correspond to the molar enthalpy of 𝐹𝑖𝑛, 𝑉𝑗 and 𝐿𝑗 in kJ/kmol, respectively. The stream 𝐹𝑖𝑛,𝑖 resepresents the equivalent stream that is equal to the summation of the streams that enter into a column tray. Compared with the work of Ma et al.19, the mass and enthalpy feeds in the DAE equation system of our work are considered as differential variables. This avoids the tedious process to determine the coefficient of pseudo-hold-up correlations. 4.1.2 Bypass efficiency In order to avoid optimizing integer variables, the concept of bypass efficiency was adopted in our present work, as proposed by Dowling and Biegler.31 After incorporation of the bypass efficiency into the PTC model, the initial MINLP problem is then simplified as a NLP problem. The bypass efficiency 𝜀𝑗 means the fraction of vapor and liquid streams that completely enter into stage j, and is a continuous variable between zero and one. At the optimal point, 𝜀𝑗 has a preference for binary values, as verified by Dowling and Biegler31. Hence, at the optimal point, the value of the addition of bypass efficiency is equal to the number of stages. The relevant equations are as follows: 𝑉𝑗𝑦𝑗,𝑖 = (1 ― 𝜀𝑗)𝑉𝑗 + 1𝑦𝑗 + 1,𝑖 + 𝜀𝑗𝑉𝑗,𝑒𝑦𝑗,𝑖,𝑒

(24)

𝑉𝑗𝐻𝑗,𝑉 = (1 ― 𝜀𝑗)𝑉𝑗 + 1𝐻𝑗 + 1,𝑉 + 𝜀𝑗𝑉𝑗,𝑒𝐻𝑉,𝑗,𝑒 𝐿𝑗𝑥𝑗,𝑖 = (1 ― 𝜀𝑗)𝐿𝑗 ― 1𝑥𝑗 ― 1,𝑖 + 𝜀𝑗𝐿𝑗,𝑒𝑥𝑗,𝑖,𝑒 𝐿𝑗𝐻𝑗,𝐿 = (1 ― 𝜀𝑗)𝐿𝑗 + 1𝐻𝑗 + 1,𝐿 + 𝜀𝑗𝐿𝑗,𝑒𝐻𝐿,𝑗,𝑒

(25) (26) (27)

Here, the subscript e refers to the streams in vapor or liquid equilibrium at the stage j. Hence, the total number of stages is given by: 𝑁𝑠

𝑁𝑡 =

∑𝜀

(28)

𝑗

𝑗=1

Moreover, we assume the pressure drop of each tray is a constant, so the pressure of stage j can be calculated as follows: 𝑗

𝑗

𝑃 =𝑃

𝑡𝑜𝑝

+ ∆𝑃(

∑𝜀

𝑘

― 1)

(29)

𝑘=1

Here, 𝑁𝑡 is the actual total number of trays as calculated from the summation of bypass efficiency at the optimal point, 𝑃𝑡𝑜𝑝 is the top pressure of the distilltion column in kPa, and ∆P is the pressure drop of each tray. 4.2 PTC heat exchanger model The PTC model of the two-stream countercurrent heat exchanger is presented in this section, and this model is illustrated in Figure 4. The steady state equations are as follows: 𝐹ℎ𝑖𝑛 = 𝐹ℎ𝑜𝑢𝑡

(30)

𝐹𝑐𝑖𝑛

(31)

=

𝐹𝑐𝑜𝑢𝑡

𝐻ℎ𝑖𝑛 ― 𝐻ℎ𝑜𝑢𝑡 = 𝐾𝐴∆𝑇

(32)

𝐻𝑐𝑜𝑢𝑡 ― 𝐻𝑐𝑖𝑛 = 𝐾𝐴∆𝑇

(33)

9



Page 10 of 24

(𝑇ℎ𝑖𝑛 ― 𝑇𝑐𝑜𝑢𝑡)(𝑇ℎ𝑜𝑢𝑡 ― 𝑇𝑐𝑖𝑛)(𝑇ℎ𝑖𝑛 ― 𝑇𝑐𝑜𝑢𝑡 + 𝑇ℎ𝑜𝑢𝑡 ― 𝑇𝑐𝑖𝑛) ∆T = [ ]1/3 2

(34)

Here, 𝐹, 𝐻, 𝐾, 𝐴, 𝑇 and ∆𝑇 are the molar flowrate in kmol/h, the molar enthalpy in kJ/kmol, the heat transfer coefficient in kW/(℃ ∙ 𝑚2) , the heat transfer area in m2, the temperature of the stream, and the logarithmic mean temperature difference, respectively. The index ℎ, 𝑐, 𝑖𝑛 and 𝑜𝑢𝑡 represent the hot stream, cold stream, the inlet stream of the heat exchanger, and the outlet stream of the heat exchanger, respectively. Tinh

Tinc

Toutc

Touth

Figure 4. Heat exchanger model

The aim of the PTC heat exchanger model is to make the energy balance equation dynamic, and keep the material balance equation as in the steady state equation. In this paper, we adopted the PTC heat exchanger model proposed by Ma et al.20, and made a small amendment, relevant equations are as follows:

𝜏𝑇 𝜏𝑇

𝐻ℎ𝑖𝑛𝑑𝑇ℎ𝑜𝑢𝑡 𝑇ℎ0 𝑑𝑡 𝐻𝑐𝑖𝑛𝑑𝑇𝑐𝑜𝑢𝑡

𝑇𝑐0 𝑑𝑡 (𝑉ℎ𝑜𝑢𝑡, 𝐻ℎ𝑜𝑢𝑡) = (𝑉𝑐𝑜𝑢𝑡, 𝐻𝑐𝑜𝑢𝑡) =

𝐹ℎ𝑖𝑛 = 𝐹ℎ𝑜𝑢𝑡

(35)

𝐹𝑐𝑖𝑛

(36)

=

𝐹𝑐𝑜𝑢𝑡

= 𝐻ℎ𝑖𝑛 ― 𝐻ℎ𝑜𝑢𝑡 ― 𝐾𝐴∆𝑇 = 𝐻𝑐𝑖𝑛 ― 𝐻𝑐𝑜𝑢𝑡 + 𝐾𝐴∆𝑇

(37) (38)

𝑓(𝑇ℎ𝑜𝑢𝑡, 𝑃ℎ𝑜𝑢𝑡, 𝑧ℎ𝑜𝑢𝑡)

(39)

𝑓(𝑇𝑐𝑜𝑢𝑡, 𝑃𝑐𝑜𝑢𝑡, 𝑧𝑐𝑜𝑢𝑡)

(40)

Here, V, P, and z are the molar vapor fraction, the pressure in kPa, and the molar composition, respectively. 𝑇ℎ0 and 𝑇𝑐0 are the estimated constants of the inlet temperature of the hot stream and the cold stream, respectively. Ma et al.20 did not describe how to set the initial values of 𝑇ℎ0 and 𝑇𝑐0. However, according to our experience, the initial value of constants can significantly affect calculation time. If the values are directly set as 𝑇ℎ𝑖𝑛 = 𝑇ℎ0, 𝑇𝑐𝑖𝑛 = 𝑇𝑐0, it takes much more time to reach a steady state solution. In this work, in order to reduce the calculation time for the dynamic energy balance equation, the initial values of 𝑇ℎ0 and 𝑇𝑐0 were set as follows, values that are based on our experience. 𝑇ℎ0 = 0.85𝑇ℎ𝑖𝑛

(41)

𝑇𝑐0

(42)

=

1.2𝑇𝑐𝑖𝑛

4.3 Recycle streams in PTC model In terms of the extractive distillation systems studied here, the presence of the recycle stream makes 10


Page 11 of 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


the convergence of the extractive distillation system more challenging. In sequential modular (SM) simulators, the concept of the tear stream is used to solve this problem. In the PTC model, Pattison and Baldea25 proposed a similar method to address this problem. We applied the method proposed by Pattison and Baldea25, and the equations are listed as follows: 𝑑𝑋𝑡𝑒𝑎𝑟 𝜏𝑟 = ― 𝑋𝑡𝑒𝑎𝑟 + 𝑋 𝑑𝑡

(43)

Here, 𝑋𝑡𝑒𝑎𝑟 represents the upstream states of the tear stream including flowrate, temperature, pressure, and composition, and 𝑋 represents the downstream states of the tear stream of the respective variables. 5. Case studies The optimization procedure was applied to two case studies- of the methanol/acetone system and the toulene/methanol system. The CEDS and EDRF were also optimized as the benchmark. All of the case studies were carried out on the personal computer with Core i7 CPU at 2.60 GHz. The NLP model has several different minimums, hence, the optimization was implemented from six different initial points with the bypass efficiency of all stages at 0.2, 0.4, 0.5, 0.6, 0.8, and 1. Additionally, to give a rigorous comparison between the mixed-steady state-time relaxation algorithm and dynamic optimization algorithm, all of case studies presented here were also optimized with dynamic optimization algorithm, and the corresponding results and comparisons are shown in Appendixes A and B. 5.1 Separation of methanol and acetone At atmospheric pressure, the binary system forms a minimum boiling point homogeneous azeotrope at 328 K with a composition of 22.4 mol% methanol and 77.6 mol% acetone. Water was selected as the solvent, and the UNIQUAC thermodynamic model was used to describe the vapor-liquid equilibrium (VLE) of this system. The feed flowrate was 540 kmol/h and the composition was 50 mol% acetone and 50 mol% methanol. The product molar purity of methanol and acetone was 0.995. 5.1.1 Optimization of CEDS According to Luyben32, the operating pressure of a distillation column can be reduced as long as cooling water is available as the cold utility. Hence, for this case study, the operating pressures of EDC and SRC were fixed at 85 kPa and 60 kPa, respectively. The decision variables include the reflux ratio of both columns (RR1, RR2), the bottom flowrate of both columns (B1, B2), the number of total stages of both columns (N1, N2), the feed locations (NF1, NF2, NFS), the flowrate of the makeup solvent (FS), the solvent feed temperature (T), and bypass efficiency for all stages. The optimization results are shown in Table 3. As shown in Table 3, of the six local optimal results, the best results were obtained at the point ε=0.4. The corresponding flowsheet is shown in Figure 5. For convenience, A, M, and W represent acetone, methanol, and water, respectively.

11



-8270 kW P1=85 kPa 51 ℃ D1 271.2 kmol/h 99.5 mol% A 0.12 mol% M 0.38 mol% W

2 Makeup 2.2 kmol/h 100 mol% W

S 510.4 kmol/h 99.95 mol% W

20

F

41

-982 kW

540 kmol/h 50 mol% M 50 mol% A

Page 12 of 24

-5034 kW P2=60 kPa 52 ℃ D2 271 kmol/h 0.43 mol% W 0.07 mol% A 99.5 mol% M

2

23 ID2=1.256m RR2=0.96

ID1=2.341m RR1=2.61 35

51 B1 8473 kW 84 ℃

B2

779.2 kmol/h 34.39 mol% M 65.58 mol% W 0.03 mol% A

5367 kW 90 ℃

508.2 kmol/h 99.95 mol% W 0.05 mol% M

Figure 5. Optimal results of CEDS for separation of methanol and acetone

5.1.2 Optimization of EDRF In this scheme, to ensure that the cooling water can be used as cold utility, the operating pressures of EDC, SRC, and concentrator were fixed at 85 kPa, 65kPa, and 70 kPa, respectively. The decision variables include the total number of stages of the three columns (N1, N2, N3), the feed locations of solvent and azeotrope mixtures (NF1, NFs, NF2, NF3), the reflux ratio of the three columns (RR1, RR2, RR3), the bottom flowrates of EDC and SRC (B1, B2), the distillate flowrate of the concentrator (D3), solvent feed temperature, and bypass efficiency. The optimization results are shown in Table 4. As shown in Table 4, in this optimization process, three results were obtained with nonbinary bypass efficiencies, and the optimal results were obtained with all binary bypass efficiency at the point ε = 0.8. A detailed discussion of the bypass efficiency is given in Section 6. The optimal results are shown in Figure 6. -8064 kW P1=85 kPa 51 ℃ S

Makeup 0.4 kmol/h 100 mol% W

15

D2

2


33

RR1=1.342 ID1=1.869 m

RR3=1.39 ID3=0.783 m

23

10 B2

B1 7936 kW 80 ℃

803.7 kmol/h 34.74 mol% M 2.25 mol% A 63.01 mol% W


5

RR2=0.857 ID2=1.647 m

48

D3

2


14 F

-927 kW

D1 270.5 kmol/h 99.5 mol% A 0.4 mol% M 0.1 mol% W

2

506.7 kmol/h 99.95 mol% W

-319 kW P3=70 kPa 52 ℃

-5013 kW P2=65 kPa 55 ℃

5017 kW 83 ℃

506.3 kmol/h 99.95 mol% W 0.05 mol% M Trace A

B3 829 kW 58 ℃


Figure 6. Optimal results of EDRF for separation of methanol and acetone

12


Page 13 of 24

5.1.3 Optimization of HIEDRF In this work, the operating pressures of the EDC and concentrator were fixed at 85 kPa and 70 kPa, respectively, in order to ensure that the cooling water can be used as cold utility. The operating pressure of SRC is considered as a decision variable to achieve heat integration. In our optimization procedure, the heat duties of the EDC reboiler and SRC condenser were calculated separately, and then integrated. Hence, the following equation applies: Q=min (Qr1, Qc2); where, Qr1 and Qc2 are EDC reboiler duty and SRC condenser duty, respectively, and Q is the heat duty of the reboiler-condenser exchanger. Thus, during the optimization procedure, the relationship between Qr1 and Qc2 is not constrained. In addition to the operating pressure of SRC (P2), other decision variables in the HIEDRF are the same as those in EDRF. The corresponding optimization results are shown in Table 5. As shown in Table 5, the optimal results were obtained at the point ε = 0.5 with all binary bypass efficiencies. The optimal results are shown in Figure 7, the condenser duty of SRC is 4887 kW, and the reboiler duty of EDC is 8037 kW. The EDC reboiler duty is much larger than SRC condenser duty, thus, the compulsory equality of EDC reboiler and SRC condenser results in a larger heat duty of SRC reboiler. Hence, it is obvious that partial heat integration is more economical than full heat integration. As shown in Figure 7, the consumption of hot utility in EDC reboiler is 3150 kW after heat integration.

-4887 kW

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-8071 kW P1=85 kPa 51 ℃ 2

507 kmol/h 99.95 mol% W

S

17

Makeup 0.55 kmol/h 100 mol% W

2

D2 301.55 kmol/h 92.6 mol% M 7.3 mol% A 0.1 mol% W

14 F

-1344 kW

D1 270.5 kmol/h 99.5 mol% A 0.39 mol% M 0.11 mol% W

P2=430 kPa 107 ℃

34


RR1=1.353 ID1=1.872 m

808 kmol/h 34.7 mol% M 2.6 mol% A 62.7 mol% W


6

RR3=1.54 ID3=0.858 m

27 B1 3150 kW 79 ℃

2

RR2=0.861 ID2=1.344 m

47

-354 kW P3=70 kPa 49 ℃ D3

11 B2

5983 kW 144 ℃

506.45 kmol/h 99.95 mol% W 0.05 mol% M Trace A

B3 877 kW 59 ℃


Figure 7. Optimal results of HIEDRF for separation of methanol and acetone

5.1.4 Comparison of results The comparison of TAC and energy consumption is given in Table 6. As can be seen in Table 6, for this case, the reduction in energy consumption and TAC of EDRF is limited. Hence, the heat integration design could enhance the energy utilization and economical efficiency of EDRF significantly. The HIEDRF could reduce 27.7% total reboiler energy consumption and 16.9% TAC compared with CEDS. Compared with EDRF, reductions were 27.4 and 16.3%, respectively. For all the three schemes, low-pressure steam is used as the hot utility for all reboilers. Hence, the reduction in heat duty directly corresponds to operating cost savings. 13



Page 14 of 24

5.2 Separation of toluene and methanol The second case study is the separation of toluene and methanol. At atmospheric pressure, the binary system forms a minimum boiling point homogeneous azeotrope at 337.02 K with a composition of 88.7 mol% methanol and 11.3 mol% toluene. Modla33 investigated the extractive distillation process for this system using intermediate entrainer for the first time. In this work, we aimed to investigate the economical and energy saving potential of HIEDRF compared with CEDS using a heavy entrainer. The feed flowrate was 100 kmol/h, the composition of the feed was 50 mol% methanol and 50 mol% toluene, and 1, 2-dimethylbenzene was used as the solvent. The NRTL model was selected to predict the thermodynamic property of this system. The purity of the product was specified at 99.9 mol% toluene and 99.9 mol% methanol. 5.2.1 Optimization of CEDS For the CEDS of the methanol/toluene system, the operating pressures of EDC and SRC were fixed at 60 kPa and 50 kPa, respectively. The optimization method and decision variables are the same as those of CEDS for the separation of methanol/acetone system presented in Section 5.1.1. The optimization results are shown in Table 7. As shown in Table 7, the optimal results were obtained at the point ε=0.2. The corrseponding flowsheet is shown in Figure 8. In Figure 8, for convenience, M, T, and OX are methanol, toluene, and 1, 2-dimethylbenzene, respectively.

2 Makeup 0.04 kmol/h 100 mol% OX -427.32 kW

S 100.67 kmol/h 100 mol% OX F 100 kmol/h 50 mol% M 50 mol% T

5

25

-564.18 kW P1=60 kPa 52 ℃ D1 50.02 kmol/h 99.9 mol% M 0.06 mol% T 0.04 mol% OX

2

-753.5 kW P2=50 kPa 86 ℃ D2 50.02 kmol/h 99.9 mol% T 0.07 mol% OX 0.03 mol% M

8 ID2=0.834m RR2=0.79

ID1=0.970m RR1=1.59 17

31 B1 813.51kW 116 ℃

150.65 kmol/h 33.19 mol% T 66.8 mol% OX 0.01 mol% M

B2 870.26 kW 125 ℃

100.63 kmol/h 100 mol% OX

Figure 8. Optimal results of CEDS for separation of methanol and toluene

5.2.2 Optimization of EDRF For the optimization of the EDRF, the operating pressures of EDC, SRC, and concentrator were fixed 14


Page 15 of 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


at 60 kPa, 45 kPa, and 40 kPa, respectively. The whole optimization procedure was the same as the scheme presented in Section 5.1.2. The optimization results are given in Table 8. As shown in Table 8, the optimal results were obtained at the point ε=0.6. The corresponding flowsheet is shown in Figure 9. -544.8 kW P1=60 kPa 52℃

-605.2 kW P2=45 kPa 65℃ 4

D1 50.01 kmol/h 99.9 mol% M 0.08 mol% T 0.02 mol% OX

18

RR1=0.75 ID1=0.885m

2 S Makeup 0.02 kmol/h 100 mol% OX

90.06 kmol/h 99.99 mol% OX

2


2

-402.3 kW

RR2=0.396 ID2=0.845m

RR3=1.16 ID3=0.362m

9

30

9 B2

B1 693.2 kW 114℃

7.85 kmol/h 58.3 mol% M 41.5 mol% T 0.2 mol% OX

5

4 F 100 kmol/h 50 mol% M 50 mol% T

-52.9 kW P3=40 kPa 60℃ D3


804.3 kW 121℃

B3 50.01 kmol/h 99.9 mol% T 0.08 mol% M 0.02 mol% OX

184.1 kW 84℃

90.04 kmol/h 99.99 mol% OX 0.01 mol% T

Figure 9. Optimal results of EDRF for separation of methanol and toluene

5.2.3 Optimization of HIEDRF For the optimization of this scheme, the operating pressures of EDC and concentrator were fixed at 60 kPa and 40 kPa, respectively. The operating pressure of SRC was used as a decision variable to achieve heat integration between the SRC condenser and the EDC reboiler. The optimization procedure used was the same as the scheme presented in Section 5.1.3. The optimization results are shown in Table 9. As shown in Table 9, the optimal results were obtained at the point ε=0.5. The corresponding flowsheet is shown in Figure 10. -592.7kW P2=215 kPa 137℃ -62.4 kW P3=40 kPa 55℃ D3

-547.2 kW P1=60 kPa 52℃ 4


21

RR1=0.78 ID1=0.952m

2 S Makeup 0.02 kmol/h 100 mol% OX

-613.2 kW

90.06 kmol/h 99.99 mol% OX

7 F 100 kmol/h 50 mol% M 50 mol% T


B1 153.69 kmol/h 5.69 mol% M 35.72 mol% T 58.6 mol% OX

2


5

RR2=0.417 ID2=0.755m

RR3=1.27 ID3=0.352m

12

29

129.1 kW 112℃

D2

2

10 B2

994.3 kW 170℃

90.04 kmol/h 99.99 mol% OX Trace M 0.01 mol% T

B3 195.1 kW 84℃

50.01 kmol/h 99.9 mol% T 0.08 mol% M 0.02 mol% OX

Figure 10. Optimal results of HIEDRF for separation of methanol and toluene

The heat duty of the SRC condenser and the EDC reboiler are 592.7 kW and 721.8 kW, respectively. 15



Page 16 of 24

Hence, in this scheme, partial heat integration is also better than full heat integration. As shown in Figure 10, the consumption of hot utility in EDC reboiler is 129.1 kW after heat integration. 5.2.4 Comparison of results The comparison of the results between CEDS, EDRF and HIEDRF is shown in Table 10. The energy saving performance of EDRF for this case is similar to the first case. The HIEDRF could reduce 21.7% total reboiler energy consumption and 10.2% TAC compared with CEDS. Meanwhile, compared with EDRF, the reductions were 21.5% and 8.9% respectively. For this case study, the reboiler temperature of SRC in HIEDRF is 170℃, so medium-pressure steam must be used. Compared with CEDS and EDRF, the increase of steam quality causes the reduction in operating cost not as significant as that of energy consumption.

6. Discussions of bypass efficiency and calculation time 6.1 Bypass efficiency Dowling and Biegler31 used nonlinear energy consumption as the objective function, and without the consideration of exchanger matches, obtained optimal results with all of the binary bypass efficiencies. Pattison et al.27, 28 also adopted the concept of bypass efficiency in their PTC model, and without nonbinary bypass efficiencies at the optimal point. However, Pattison et al.

27, 28

used a linear TAC as the objective

function, which significantly simplified the programming. In this work, for EDRF and HIEDRF in both case studies, we obtained three results with nonbinary bypass efficiencies. This problem was also reported by Ma et al.20, and the reasons for this problem are as follows: (1) The strong nonlinearity of the objective function and complexity of the flowsheet cause the optimal results to have fractional bypass efficiencies. (2) Heat integration causes a nonlinear trade-off between the increase in capital cost and decrease in operating cost. In order to solve this problem, Ma et al.20 proposed a rounding procedure in their work, which is not mathematically rigorous. In this work, for both case studies, the best results were obtained with binary bypass efficiencies, hence, we do not include a rounding procedure. 6.2 Calculation time As can be seen in Appendix B, the mixed-steady state-time relaxation algorithm reduced calculation time significantly compared with the dynamic optimization algorithm. Three factors contribute to the shorter calculation time. First, the mixed-steady state-time relaxation algorithm solves steady state simulation at first, which requires less time compared with dynamic optimization. To investigate this issue, we made a test for the CEDS of the first case study. With the same initial values for all variables and the same specifications, we did two simulations-a steady state simulation and the first step of a dynamic optimization-with the same computer. The former required 27 s, and the latter required 150 s. Second, the dynamic optimization algorithm calculates the constraints and objective function at each step, while the mixed-steady state-time relaxation algorithm only calculates them at the final steady state 16


Page 17 of 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


condition. This indicates that the former costs more time at each integration step compared with the latter. Third, the dynamic optimization algorithm begins each integration step from the initial point, while the mixed-steady state-time relaxation algorithm iterates from the result of the last step. Hence, for the same problem, the former requires more integration steps compared with the latter. 7. Conclusions In this paper, we successfully optimized the HIEDRF using the new mixed-steady state-time relaxation based PTC model. In terms of the HIEDRF, as shown in Tables 6 and 10, significant energy savings and TAC reduction were obtained compared with both CEDS and EDRF. In the flowsheet of the HIEDRF, the inclusion of a concentrator caused a remarkable decrease of the reflux ratio for EDC and SRC, which directly corresponds to reduction in capital cost and heat duty. On the other hand, the number of trays and diameter of concentrator were much smaller than those of EDC and SRC. Hence, the existence of the concentrator did not cause increase in TAC. Additionally, the operation of HIEDRF is easier than that of CEDS, because it is unnecessary to recover all light key components from the top of EDC. The new PTC model developed in this work combines the advantages of both time relaxation algorithm and steady state simulation. Compared with the dynamic optimization algorithm, this algorithm is more efficient for simultaneous optimization due to the significant reduction in calculation time. However, during the calculation process, the frequent transformation between steady state and pseudo-transient is inconvenient. Hence, this algorithm is still less efficient compared with the time relaxation based algorithm. Future work will be aimed at exploiting time relaxation based algorithms in ACM. Appendix A: Optimization results using dynamic optimization algorithm For all the case studies, the initial guesses, objective function, and constraints of the optimization procedure using dynamic optimization algorithm are the same with that using mixed-steady state-time relaxation algorithm. The corresponding optimization results are shown in Tables A1 to A6. Table A1. Optimization results of CEDS for separation of methanol and acetone using dynamic optimization algorithm ε = 0.2

ε = 0.4

ε = 0.5

ε = 0.6

ε = 0.8

ε=1

N1

49

51

50

50

52

53

N2

40

37

34

36

40

37

NF1

37

40

35

39

39

42

NF2

26

22

21

20

26

25

NFS

19

20

17

21

18

24

RR1

2.79

2.68

2.75

2.73

2.62

2.59

RR2

0.79

0.85

0.87

0.86

0.80

0.84

B1 (kmol/h)

801.2

778.9

779.3

802.3

802.1

802.5

B2 (kmol/h)

506.9

507.7

509.2

507.5

506.5

508.7

FS (kmol/h)

1.6

2.1

3.2

2.5

2.4

2.6

T (°C)

45.7

45.6

46.9

46.7

45.3

46.8

17



Page 18 of 24

Calculation time

3h12m24s

2h57m44s

3h10m28s

2h49m57s

2h55m12s

2h52m38s

TAC ($/a)

2,571,224

2,447,763

2,494,181

2,523,384

2,556,919

2,498,898

18


Page 19 of 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table A2. Optimization results of EDRF for separation of methanol and acetone using dynamic optimization algorithm ε = 0.2

ε = 0.4

ε = 0.5

ε = 0.6

ε = 0.8

ε=1

N1

44

43.7

48.7

50

49

44.7

N2

27

25.9

26.2

25

24

24.7

N3

12

12

11

11

11

11

NF1

30

28.4

33.8

33

33

29.4

NF2

13

13

13.3

13

14

13

NFS

15

15

15

15

15

15

NF3

6

6

5

5

5

5

RR1

1.354

1.363

1.339

1.337

1.339

1.351

RR2

0.848

0.847

0.844

0.853

0.857

0.855

RR3

1.33

1.32

1.35

1.34

1.39

1.37

B1 (kmol/h)

803.9

805.5

807.7

807.1

803.7

808.2

B2 (kmol/h)

508.1

504.7

502.6

509.3

506.3

502.9

D3 (kmol/h)

28.5

26.1

25.8

24.7

27.5

28.3

T (°C)

47.4

47.5

46.7

46.2

46.9

46.6

Calculation time

6h55m37s

6h49m23s

6h51m45s

7h11m12s

6h57m28s

6h49m35s

TAC ($/a)

2,439,832

2,440,443

2,437,529

2,442,027

2,430,545

2,441,043

Table A3. Optimization results of HIEDRF for separation of methanol and acetone using dynamic optimization algorithm ε = 0.2

ε = 0.4

ε = 0.5

ε = 0.6

ε = 0.8

ε=1

P2 (kPa)

430.1

432.9

431

434

434.2

432.5

N1

45.9

48

49

50

52.6

51.5

N2

27.2

27

28

29

28.6

30.1

N3

14.7

15

12

17

16.4

15.6

NF1

31

34

34

37

40

39

NF2

13

14

14

14

15

16

NFS

17

17

17

16

15

15

NF3

6.8

7

6

10

9.2

7.6

RR1

1.481

1.392

1.384

1.372

1.353

1.366

RR2

0.892

0.892

0.883

0.875

0.876

0.859

RR3

1.73

1.68

1.52

1.46

1.54

1.62

B1 (kmol/h)

806.73

805.27

808.06

808.58

810.22

809.67

B2 (kmol/h)

505.54

507.62

506.59

507.05

507.18

509.68

D3 (kmol/h)

34.1

32.9

31.9

30.5

31.1

30.2

T (°C)

46.7

47.2

47.5

46.1

47

47.1

Calculation time

8h45m18s

9h10m24s

9h17m37s

9h12m44s

9h11m15s

8h52m22s

TAC ($/a)

2,057,456

2,052,573

2,033,483

2,061,332

2,075,381

2,069,501

Table A4. Optimization results of CEDS for separation of methanol and toluene using dynamic optimization algorithm ε = 0.2

ε = 0.4

ε = 0.5

ε = 0.6

ε = 0.8

ε=1

19



Page 20 of 24

N1

33

34

28

32

30

31

N2

17

18

21

19

17

20

NF1

25

29

23

26

24

26

NF2

8

7

11

8

10

11

NFS

5

5

4

5

5

5

RR1

1.62

1.55

1.77

1.69

1.75

1.71

RR2

0.89

0.85

0.72

0.80

0.88

0.77

B1 (kmol/h)

151.34

150.16

151.59

151.67

150.69

150.91

B2 (kmol/h)

100.72

101.55

100.58

100.44

101.03

101.65

FS (kmol/h)

0.07

0.05

0.13

0.06

0.03

0.11

T (°C)

41.2

40.9

42.3

43.1

40.8

40.7

Calculation time

3h21m13s

3h11m37s

3h17m51s

2h57m46s

2h48m14s

3h14m52s

TAC ($/a)

1,282,289

1,285,161

1,291,721

1,300,983

1,301,161

1,291,090

Table A5. Optimization results of EDRF for separation of methanol and toluene using dynamic optimization algorithm ε = 0.2

ε = 0.4

ε = 0.5

ε = 0.6

ε = 0.8

ε=1

N1

28.9

34.7

30

31

32

30.9

N2

12.5

12.4

13

10

12

13

N3

12

13

11

10

10

12

NF1

18.4

24.1

18

18

20

19

NF2

6

7

7

4

5

7

NFS

5

6

5

5

5

5

NF3

6

7

4

4

4

6

RR1

0.81

0.61

0.77

0.76

0.72

0.76

RR2

0.352

0.354

0.347

0.402

0.363

0.348

RR3

0.99

0.97

1.01

1.18

1.17

1.02

B1 (kmol/h)

151.7

153.8

148.9

150.2

149.6

150.5

B2 (kmol/h)

89.69

90.46

90.23

90.05

89.95

90.07

D3 (kmol/h)

8.67

9.02

8.75

7.92

8.96

9.83

T (°C)

41.5

42.3

43.1

41.9

42.2

42.4

Calculation time

6h42m55s

6h53m12s

7h16m27s

6h57m48s

6h52m40s

6h45m33s

TAC ($/a)

1,275,301

1,272,456

1,277,863

1,263,949

1,266,729

1,270,341

Table A6. Optimization results of HIEDRF for separation of methanol and toluene using dynamic optimization algorithm ε = 0.2

ε = 0.4

ε = 0.5

ε = 0.6

ε = 0.8

ε=1

P2 (kPa)

218

213.7

216

219.7

214.5

217

N1

30

34.5

31

32.6

35

30.8

N2

15

12.6

13

13.9

11

14

N3

12

13

11

11

10

12

NF1

18

25

21

22

25

21.7

NF2

9

6

7

7.6

6

8

20


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NFS

4

5

4

4

5

5

NF3

7

8

5

5

5

7

RR1

1.21

0.89

1.13

1.07

0.84

1.18

RR2

0.424

0.462

0.455

0.447

0.496

0.432

RR3

1.24

1.21

1.18

1.19

1.20

1.23

B1 (kmol/h)

152.37

153.25

152.84

152.73

153.55

153.77

B2 (kmol/h)

90.05

90.04

90.05

89.95

90.26

91.12

D3 (kmol/h)

12.69

10.92

13.27

14.47

13.18

12.91

T (°C)

42.1

41.2

42.2

40.5

42.7

42.9

Calculation time

8h52m16s

8h42m31s

9h11m50s

9h27m12s

8h55m49s

8h58m47s

TAC ($/a)

1,181,861

1,179,993

1,151,723

1,177,088

1,175,135

1,178,146

Appendix B: Comparison of calculation time and integration steps between mixed-steady state-time relaxation algorithm and dynamic optimization algorithm on average Table B1. Comparison of calculation time and integration steps for the separation of methanol and acetone Dynamic optimization algorithm Calculation time

Integration steps

Mixed-steady state-time relaxation algorithm Calculation time

Integration steps

CEDS

182min

73

18min

42

EDRF

416min

83

49min

47

HIEDRF

532min

107

91min

64

Table B2. Comparison of calculation time and integration steps for the separation of methanol and toluene Dynamic optimization algorithm Calculation time

Integration steps

Mixed-steady state-time relaxation algorithm Calculation time

Integration steps

CEDS

189min

76

19min

42

EDRF

415min

83

45min

46

HIEDRF

541min

110

83min

66

Supporting Information The parameters for the calculation of total annual operating cost, detailed optimization results, and comparisons of different schemes are given in the supporting information. This information is available free of charge via the Internet at http://pubs.acs.org/. Notes The authors declare no competing financial interest. This research did not receive any specific grant from funding agencies in the public, commercial or not-for-profit sectors. Acknowledgments The authors appreciate Rhein Language Embellishment Company for providing language help.

21



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Table of Contents graphic:

23



D1 A

S

D2

D3

A+B

A+B

Makeup

Concentrator

SRC

EDC

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 24

F A+B

B1 A+B+S

B2 S

B3 B

24


Simultaneous optimization of heat-integrated extractive distillation with

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