Simultaneous Optimization of Complex Distillation Systems with a

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Simultaneous Optimization of Complex Distillation Systems with a New PTC Model Yingjie Ma, Yiqing Luo, and Xigang Yuan Ind. Eng. Chem. Res., Just Accepted Manuscript • Publication Date (Web): 04 May 2017 Downloaded from http://pubs.acs.org on May 5, 2017

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Industrial & Engineering Chemistry Research

Simultaneous Optimization of Complex Distillation Systems with a New PTC Model Yingjie Ma, † Yiqing Luo,*, † and Xigang Yuan†,‡ †

Chemical Engineering Research Center, School of Chemical Engineering and Technology,

Tianjin University, Tianjin 300350, China ‡

State Key Laboratory of Chemical Engineering, Tianjin University, Tianjin 300350, China

ABSTRACT: Complex distillation systems, such as extractive distillation systems and thermally coupled distillation systems are necessary or energy-saving in chemical industry. However, simultaneous optimization of all the variables in these systems with the rigorous MESH model is still difficult because of both the convergence problem and integer optimization problem. In this work, based on the pseudo-transient continuation (PTC) approach, a simple and robust PTC distillation model was developed in Equation-Oriented (EO) environment. It was based on the dynamic simulation but with a pseudo hold-up correlation, which resulted in a well-posed and easily-solved differential algebraic equation (DAE) system. Then, three optimization cases were studied to validate the performance of the bypass efficiency method and the new PTC model.

Corresponding Author *Tel：+86 22 27407645. Fax: +86 22 27404496. E-mail: [email protected] (Yiqing Luo).

ACS Paragon Plus Environment

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1. INTRODUCTION Complex distillation is common in chemical industry. For the separation of azeotropic mixture, extractive distillation or pressure-swing distillation has to be used. In order to save energy, thermal couple and heat integration are usually used. In these systems, there are not only classical compromise problems like those in simple columns, such as the contradiction between the reflux ratios and stage numbers, but also strong interaction among columns because of heat and material recycles. Thus, for the design of a complex distillation system, experience is less reliable, which will be shown later in this work. In sum, in order to make full use of all the design freedoms in a complex distillation system, optimizing it from the systematic perspective is necessary. Systems are relative. How large the system should be depends on the research objective. Like many science and engineering fields, distillation is also a multiscale problem. From the fluid or even molecular scale to the sequence scale, there have been many researches. Usually, we have to make a compromise between larger scales and more accurate models due to convergence or calculation amount problem. From the perspective of application, optimizing a distillation system with fixed configuration descripted by equilibrium stage (MESH) model is real but still challenging. And this is what this work would deal with. Viswanathan and Grossmann are the pioneers on this topic. In 1990, they optimized the distillation column described by MESH equations through solving the famous MINLP (Mixed Integer Nonlinear Programming) problem with the outer approximation algorithm1. Then in 1993, the authors developed an alternate MINLP algorithm to improve the solving efficiency2 and extended this method to the distillation column that had multiple feeds3. By reformulating this


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model, Yeomans and Grossmann4, 5 developed a Generalized Disjunctive Programming (GDP) model and optimized the single column, column sequence and complex columns. Additionally, Jackson and Grossmann6 optimized a reactive distillation column with the disjunctive model. However, in a large flowsheet, mixing integer optimization with strongly nonlinear models is rather difficult to solve. As a result, some studies7, 8 have been performed to avoid integer variables. The most successful one should be the introduction of the bypass efficiency (εj) proposed by Dowling and Biegler9, 10. Using the summation of continuous variables εj at all stages to substitute the stage number in a column, the authors optimized a cryogenic air separation unit with heat integration. However, while all the above studies benefited from the Equation-Oriented (EO) modeling environment for the access of automatic or symbolic differentiation, they had to develop some complex initialization procedures to converge their models, which was rather difficult. Even for a single column, there are many material recycles between stages and stages, the top stage and condenser, and the bottom stage and reboiler, which results in a large-scale nonlinear equation system. And the inclusion of phase equilibrium equations causes stronger nonlinearity. Currently, the direct solving of MESH equations of a distillation column with generalized nonlinear solvers in EO software is nearly impossible. And for complex distillation systems, the solving difficulty will be much larger because of more recycles. It seems that only those skilled in the art can successfully apply these methods to achieve their goals11. In fact, for the simulation of distillation systems, the most popular method is the Sequential Modular (SM) method, which solves each unit in a flowsheet sequentially according to the material flow direction. On the one hand, this method can take advantage of the mature and robust algorithms for single distillation column; on the other hand, sequentially solving each unit


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is easy to encounter convergence difficulty for flowsheets with recycles, especially for the optimization problem, in which many variables vary in a large scope. The worse thing is that in the SM environment, numerical differentiation has to be solved for gradient-based optimization algorithms. However, as we know, the numerical differentiation is computationally expensive and potentially inaccurate12. Both two shortages result in frequent optimization failure in SM environment. In order to avoid the solving of gradient information, a series of derivative-free optimization techniques, such as genetic algorithm13-15, simulated annealing algorithm16 and particle swarm optimization11, have been developed to optimize flowsheets based on SM environment. But these kinds of optimization algorithms still have convergence problem. What’s more, they are less efficient and cannot guarantee the optimum. In conclusion, more or less, both the EO and SM methods have convergence difficulty, while the former is advantageous for optimization. In order to resolve the convergence problem and get the optimization advantage of EO modeling environment, a systematic pseudo-transient continuation (PTC) method was proposed by Pattison and Baldea12. Through constructing pseudo-transient processes and converting algebraic equations (AE) into differential algebraic equations (DAE) with equivalent steady-state solutions, the convergence basin of original models was largely extended. Since the final objective is to obtain the equivalent steady-state solution and the dynamic path is not important, the PTC model can be formulated into the simplest form for initialization and integration. Based on this, the main thought of the authors was to convert the original algebraic equation (AE) system into the differential algebraic equation (DAE) system whose initialization problem was to solve a series of small-scale nonlinear or even linear equation systems. With the developed PTC model libraries, they have achieved significant success in solving several large-scale problems with rigorous models.


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In sum, currently, the PTC model is the most successful and general method for the simultaneous optimization of complex systems. However, two problems motivated this work. Firstly, since excessive pursuit of easy initialization in Pattison and Baldea’s work12, their method of reformulating distillation model was complex and not intuitive. In this paper, a natural PTC distillation model based on dynamic simulation was developed. Moreover, in order to strengthen the robustness of this model, a simple pseudo hold-up correlation was proposed. Secondly, it is important for the bypass efficiency method to explore whether the optimal bypass efficiencies still prefer for binary values when the objective function is a nonlinear total annual cost (TAC) function, which is more accurate and real-life. However, among the existing reports on this method, there were only two cases10, 17 studied and both of them used simple objective functions (nonlinear total energy consumption and linear TAC, respectively). In fact, in this work, our optimization results indicated that when the complex objective function was used, there would be some problems for the bypass efficiency method. Moreover, as supports for the generality of the new PTC model, two typical complex distillation optimization cases were studied in this work: a divided wall column and a thermally coupled extractive distillation process. To the best of our knowledge, the latter one was the first time reported to be optimized. Based on the optimization results, a more complete review about our PTC model and the bypass efficiency method was given. In the remaining parts, first, we express the model equations in section 2. Section 3 shows three optimization cases to validate the model. In section 4, according to our optimization results, the performance of the developed model is reviewed. Finally, section 5 states our conclusions.


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2. MODEL EQUATIONS 2.1. Dynamic simulation equations. The main difficulty of solving MESH equations is that the equations on each stage are coupled with those of adjacent stages, which result in a large-scale nonlinear equation system. If the equations on each stage can be solved separately, we only need to solve a series of simple flash problems. Combining this thought with the idea of the PTC model, it is natural to think of the dynamic simulation of the distillation column. Related equations are as follows: Material balance (M): j

dMi dt

= Vj+1 yi + Lj-1 xi - Vj yi - Lj xi i=1, 2, .., Nc; j=1, 2, … , Ns j+1

j

j-1

j

(1)

Phase equilibrium (E): yi = ki xi

i=1,2,…, Nc; j=1, 2,…, Ns

(2)

j Summation (S): ∑Nc i=1 xi = 1

j ∑Nc i=1 yi = 1

(3)

j j

j

Enthalpy balance (H):

dHj dt

j=1, 2,… , Ns

= Vj+1 hV + Lj-1 hL - Vj hV - Lj hL j+1

j-1

j

j

j=1, 2, …, Ns

(4)

Herein, Nc and Ns are the number of components and stages, respectively. Mi and Hj refer to j

the total molar hold-up of component i in both phases on stage j and total hold-up enthalpy on j

j

j

stage j, respectively. xi , yi , ki represent liquid stream mole fraction, vapor stream mole fraction and phase equilibrium constant of component i in the steams leaving stage j respectively, while j

j

Vj, Lj, hV , hL are vapor mole flowrate, liquid mole flowrate, vapor molar specific enthalpy and liquid molar specific enthalpy of the streams leaving stage j respectively. We assume that vapor and liquid leaving and held up on stage j have the same composition, temperature and pressure. Thus, we have


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Mi = ML xi + MV yi j

j

j

j

j

j

i=1, 2, …, Nc; j=1, 2, …, Ns

j

j

j

Hj =ML hL +MV hV j

(5)

j=1, 2, …, Ns

(6)

j

where ML and MV are total liquid and vapor molar hold-up on stage j, respectively. 2.2. Pseudo hold-up correlation. Until now, we have the same equations as normal dynamic simulation. However, due to complex hydraulics correlations, the initialization and integration process of normal dynamic simulation is not trivial work. To make the formulated model more easily to be initialized and integrated, it is wise to adopt some simple pseudo correlations that will not change the steady-state results. First, for simplicity, we assume pressure drops on all stages are constant or can be specified，which is the same as the steady-state simulation. For the hold-up correlation, the simplest one should be to set both the total vapor and liquid hold-up (MV, ML) on each stage as constants. Unfortunately, it will cause a high-index problem, which is much more difficult to solve. One pseudo hold-up correlation that is simple and can make the equation system well-posed at the same time is as follows: Lj =CL ML Vj =CV MV (7) j

j

where CL and CV are constant coefficients. The values of CL and CV have some influence on the model performance. Overly large values can accelerate the speed of reaching steady state, but they will cause a stiff system. In contrast, overly small values make integration easier, but the model will require more pseudo time to reach the steady state. According to our experience, specifying the values of CL and CV as 1800 can ensure the model has good performance when the units of the flowrate and hold-up are kmol and kmol/h, respectively.


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2.3. Phase switches problem. The switch method of single-phase and two-phase equations is critical to the convergence performance of the dynamic distillation model. In this work, the robust flash submodel in Aspen Customer Modeler (ACM) was adopted. In other EO modeling software, we propose the nonsmooth model18. 2.4. Stage number. For a large flowsheet, integer decision variables are difficult to be optimized. Some studies have been done to avoid integer optimization7, 8, 19 and the most successful one should be the introduction of the bypass efficiency proposed by Dowling and Biegler9, 10. The bypass efficiency εj refers to the vapor and liquid streams fraction that enters stage j. The remaining part of the streams bypasses the stage j. Here, εj is a continuous variable between zero and one. As the authors have reported and our results will show, εj has a preference for binary values at the optimal point. As a result, with the summation of bypass efficiencies as the substitution of the stage numbers, the optimal solution will not be missed. We can incorporate the bypass efficiency in our model in the same way as previous authors9, 10, 17 have used. The concept of the bypass efficiency is depicted in Figure 1.

Figure 1. Illustration of bypass efficiency where Vj,eq and Lj,eq are pseudo streams in phase equilibrium when the vapor stream Vj+1 and liquid steam Lj-1 completely enter stage j. After applying the bypass efficiency to Vj,eq and Lj,eq


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and mixing them with the vapor and liquid bypass streams, respectively, we obtain the final vapor and liquid streams, Vj and Lj, leaving stage j. Moreover, the bypass efficiency is also applied to calculate the stage pressures. Corresponding equations are as follows: Vj = 1-εj Vj+1 + εj Vj,eq

j=1,2, …, Ns

Vj yi = 1-εj Vj+1 yi + εj Vj,eq yi j

j+1

j,eq

Vj hV = 1-εj Vj+1 hV + εj Vj,eq hV j

j+1

Lj = 1-εj Lj-1 + εj Lj,eq

j,eq

j

j-1

j,eq

Lj hL = 1-εj Lj-1 hL + εj Lj,eq hL j

j-1

i=1,2, …, Nc; j=1,2, …, Ns

j=1,2, …, Ns

j=1,2, …, Ns

Lj xi = 1-εj Lj-1 xi + εj Lj,eq xi

j,eq

(8)

(9)

(10)

(11)

i=1,2, …, Nc; j=1,2, …, Ns

(12)

j=1,2, …, Ns

(13)

j Nstrue = ∑Ns j=1 ε

(14)

Pj = Ptop + ∆P( ∑k=1 εk -1) j

j=1, 2, …, Ns

(15)

Herein, Nstrue is the true total stage number. Pj, Ptop and ∆P are the pressure on stage j, pressure above the column section and pressure drop at each stage, respectively. 3. Optimization case studies Three typical optimization cases are shown in this work. The first one is to optimize a single column, which is to prove the preference for binary values of bypass efficiency at the optimal point. The latter two complex distillation processes, a divided wall column and a ternary


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extractive distillation process are all important industrial cases. In all the cases, total annual cost (TAC) is used as evaluation index. The TAC includes capital cost and energy cost, and the capital cost is calculated according to equipment geometry sizes. Detailed TAC model is shown in the Appendix. For the optimization of the PTC model, dynamic optimization solvers have to be used since in ACM, there are no time-relaxation based optimizers12. In all the cases, the Hypsqp optimizer was adopted and the calculation was performed on the desktop with Intel Core i7 CPU @ 3.40GHZ. Additionally, considering that the nonlinear programming (NLP) problem exists multiple minimums, all the optimization cases were optimized from six different initial points with all initial stage bypass efficiencies at 0.1, 0.3, 0.5, 0.7, 0.9 and 1 respectively. 3.1. Separation of the propane-isobutane mixture. In this optimization, the purity requirements of propane and isobutane are over 98% and 99%, respectively. Other constraints are that the inlet and outlet temperature differences of the rebolier and condenser must be over 10 °C. The design variables include the column pressure P, reboil ratio BR, distillate flowrate D, and stage numbers N1 and N2 in rectifying section and stripping section respectively. This optimization has 3611 equations, 59 decision variables and 12 inequality constraints. For the calculation of the simple hydrocarbon mixture, Chao-Seader equation is suitable to describe relevant thermodynamic properties. The optimization results are in Table 1. Table 1. Optimization results of the propane-isobutane column ε = 0.1

ε = 0.3

ε = 0.5

ε = 0.7

ε = 0.9

ε = 1.0

P (bar)

11.48

11.50

11.48

11.48

11.48

11.48

BR (kmol/kmol)

2.54

2.54

2.54

2.50

2.54

2.52


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D (kmol/h)

1447.42

1442.82

1447.42

1447.42

1447.42

1447.42

N1

12

14.99

13

13

13

14

N2

19

17

18

20

18

17

Iterations

24

23

23

25

25

26

Calculation time (s)

1290

1160

964

1362

1220

1586

Optimization status

Success

Success

Success

Success

Success

Success

The number of ε with non-binary value

0

1

0

0

0

0

1e-3TAC ($/Year)

127.56

128.58

127.38

127.79

127.38

127.62

As shown in the Table 1, starting from five in six different initial points, the bypass efficiencies turned to zero or one at the optimal points, and the only one result with a non-integer bypass efficiency had an evidently larger TAC than the others. Moreover, the optimizations starting from ε = 0.5 and ε = 0.9 both obtained the smallest local minimum in the six cases, which indicated that even when the objective function was rather complex, the bypass efficiencies still tended to have binary values at the optimal point. In sum, at least, for the optimization of a simple column, the bypass efficiency method is very efficient because it can get physically realizable solution, at the same time, avoid integer optimization problem. 3.2. Divided Wall Column (DWC). In this case, a DWC was used to separate the mixture of benzene, toluene and p-xylene. Javaloyes-Antón et al.11 optimized this process through linking Aspen Hysys with particle swarm optimization (PSO) algorithm. But in order to guarantee the convergence, they approximately represented a DWC with two traditional columns without material recycles between them, which would take some deviation from the real DWC. Luyben et al.20 researched the control configuration of this DWC system but they did not explain their


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design method. Here, the PTC model with bypass efficiency was used to optimize it. In our framework, the DWC can be represented as Figure 2. In this figure, B, T and X refer to benzene, toluene and p-xylene, respectively.

Figure 2. Optimization results of DWC The feed flowrate is 500 kmol/h, and feed mole fractions are benzene 0.3, toluene 0.4 and pxylene 0.3. Optimization variables include six stage numbers (N1, N2, N3, N4, N5 and N6), vapor and liquid spilt fractions (SV, SL), reboil ratio (BR) and distillate flowrate (D), Side draw flowrate (Fs). The product purity requirements are 99.95% for all the three components. For this case, total numbers of equations, decision variables and inequality constraints are 10788, 173 and 8 respectively. As proposed by Luyben et al.20, here, Chao-Seader equation was used to calculate the thermodynamic property. Table 2 shows the optimization results:


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Table 2. Optimization results of DWC ε = 0.1

ε = 0.3

ε = 0.5

ε = 0.7

N1

13.21

13

15

13.90

14

13

N2

27

25.77

30

28

25.23

25

N3

18

21.25

17

17.00

18.13

21

N4

24

20.43

23

24.62

21

22

N5

21

26.59

24

20.38

22.36

24

N6

19

19

19

19

18

19

SV (kmol/kmol)

0.4415

0.4666

0.4145

0.4189

0.4439

0.4587

SL (kmol/kmol)

0.3760

0.3635

0.3905

0.3884

0.3760

0.3677

BR (kmol/kmol)

3.146

3.115

3.105

3.150

3.164

3.122

D (kmol/h)

150.03

150.02

150.02

150.03

150.03

150.04

Fs (kmol/h)

199.95

199.95

199.95

199.95

199.95

199.95

Iterations

68

61

65

85

55

70

Calculation time1

ε = 0.9

ε=1

11h54m21s 7h40m30s 15h3m36s 9h16m24s 9h13m20s 10h25m54s

Optimization Status

Success

Success

Success

Success

Success

Success


1

5

0

11

3

0

1e-4TAC ($/Year)

152.04

151.64

152.17

152.51

156.90

151.46

1

For the calculation time, h, m and s refer to hours, minutes and seconds, respectively.

For the DWC, four optimization cases had fractional bypass efficiencies. However, the best optimization result, TAC = 151.46e4 $/year, was gotten when all the bypass efficiencies were binary values. Relevant stream results are shown in Figure 2. Because of different economic data, we did not compare our results with those of Javaloyes-Antón et al.11


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3.3. Partially thermally coupled extractive distillation process. The third case is to separate the benzene-cyclohexane-toluene mixture. The feed flowrate is 100 kmol/h and the feed composition is 30%, 30%, 40% for benzene, cyclohexane and toluene respectively. In the mixture, benzene and cyclohexane form a homogeneous azeotrope, which makes them impossible to be separated through simple distillation. Timoshenko et al.21 proposed a partially thermally coupled extractive distillation flowsheet to separate this ternary mixture with N-Methyl pyrrolidone (NMP) as the entrainer and used a sequential, iterative procedure to optimize this process. This flowsheet is shown in Figure 3 and Figure 4. In both figures, for the convenience, B represents benzene, CH represents cyclohexane and T represents toluene. With N,N-Dimethylformamide (DMF) as the entrainer, Luyben22 compared the controllability of conventional and partially thermally coupled flowsheets. For the design of the conventional extractive distillation process, five Design Spec/Vary functions in Aspen Plus were used to achieve five specific objectives (the impurities in streams). Nevertheless, for the thermally coupled process, due to the convergence difficulty, even as a very skilled Aspen Plus user, in order to reach five specific objectives, the author had to compromise to use two Design Spec/Vary functions and manually vary the other three variables. This indicates that, Aspen Plus is much less possible to be used to optimize this flowsheet since there will be more design freedoms and larger variable ranges for an simultaneous optimization problem compared with the author’s design problem. In our framework, however, simultaneous optimization was performed easily. Some work has been done to optimize binary extractive distillation processes7, 23, but to our knowledge, it should be the first time to optimize a ternary extractive distillation process with thermal couple.


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Figure 3. Optimal results of the partially thermally coupled extractive distillation process.

Figure 4. Luyben’s results of the partially thermally coupled extractive distillation process.


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Because the entrainer DMF is added in this process, the TAC should include this part. The price of DMF is 0.856 $/kg. This case has 14474 equations, 215 decision variables and 29 inequality constraints. The decision variables include the distillate flowrates in column 1 and column 2 (D1 and D2 respectively), bottom flowrate in column 3 (B3), reboil ratio in column 1 (BR1), reflux ratio in column 3 (RR3), side vapor flowrate from column1 to column 2 (Fs), entrainer feed flowrate (Fx), entrainer feed temperature (T) and all the bypass efficiencies on stages (εj). The constraints are the minimum purity of benzene, cyclohexane and toluene, which are 99%, 99% and 99.4%, respectively. The operation pressures of three columns are 0.8 bar, 0.4 bar and 0.12 bar, respectively, which are the same as those proposed by Luyben22. In the simulation, liquid phase was described by NRTL equations and vapor phase adopted ideal gas equation of state since there were built-in NRTL parameters for all the six binary mixtures pairs in the VLE-IG databank in Aspen Properties, which guaranteed the accuracy of phase equilibrium calculation. Based on above specifications, the optimized results are shown in Table 3. Table 3. Optimization results of the extractive distillation process ε = 0.1

ε = 0.3

ε = 0.5

ε = 0.7

ε = 0.9

ε=1

N1

33

20.33

25

26

25

25

N2

24

26.62

28.99

29

29

30

N3

12

15.00

14

14

14

14

N4

16

16.11

18.76

18

17

17

N5

12

6.78

12

13

12

12

N6

15

14.27

13

14

14

15

N7

2

2.39

2

2

2

2


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D1 (kmol/h)

30.14

30.18

30.18

30.15

30.16

30.17

D2 (kmol/h)

30.04

30.10

30.06

30.06

30.06

30.06

B3 (kmol/h)

105.25

99.25

99.76

94.57

93.12

95.66

BR1 (kmol/kmol)

1.53

1.77

1.59

1.63

1.66

1.63

RR3 (kmol/kmol)

1.16

1.14

1.10

1.08

1.10

1.05

Fs (kmol/h)

59.63

75.10

57.23

56.49

57.00

57.84

Fx (kmol/h)

105.53

99.59

100.07

94.87

93.42

95.96

T (°C)

82.01

53.07

79.55

80.85

81.69

80.86

Recycle Purity1 (kmol/kmol)

0.86

0.88

0.89

0.85

0.86

0.85

Iterations

72

40

53

54

65

52

Calculation time1

10h28m58s 5h58m15s 6h28m24s

7h5m54s

8h22m35s 6h41m57s

Optimization Status

Success

Fail

Success

Success

Success

Success


0

7

27

0

0

0

1e-4TAC ($/Year)

127.31

135.29

127.29

127.00

127.07

127.19

1

Recycle Purity refers to the entrainer recycle purity.

From Table 3, we can see that one optimization failed to find the optimum. In the successful optimization results, only the optimization starting from εj = 0.5 had partial bypass. And the smallest minimum of TAC was obtained when all the initial values of εj were 0.7. Relevant stream results of the best optimization are shown in Figure 3. For the results, we want to stress that all the optimal temperatures of the entrainer feed in the successful optimizations were nearly the same as those of the recycled entrainer, resulting in very small heat duties and heat exchanger areas, as shown in Figure 3, which indicates that the entrainer precooler could be removed. Moreover, as shown in Table 3, at all the optimal points,


17


Page 18 of 26

the purities of the recycled entrainer were between 0.8 and 0.9. This indicates that it is unnecessary to recover very pure entrainer. However, engineers always specify high entrainer recovery standards when they are designing an extractive distillation process according to experience. In fact, higher purity will require higher separation ability for the entrainer recovery column, while it will decrease the recycling amount of one or several product components in the whole flowsheet. As a result, the optimal entrainer recovery purity must be a value that makes two sides compromise. Since Luyben used the built-in procedures in Aspen Plus to calculate column diameters, their diameters were different from ours. In order to make a comparison between our results and Luyben’s, we used his specifications to repeat his results in our program. And there were good matches between our recalculation results and Luyben’s original results. Figure 4 shows relevant stream results. Table 4 shows the comparison between our best optimization results and his. Table 4. Comparison between the best optimization results and Luyben’s results 1e-4Cost ($/Year)

Our results

Luyben’s

Saving (%)

Capital Cost

56.03

69.55

19.44

Energy Cost

48.86

65.70

25.63

Entrainer Cost

22.11

37.17

40.52

TAC

127.00

172.42

26.34

In Table 4, it can be seen that there were evident decreases in the capital cost (19.44%), energy cost (25.63%), entrainer cost (40.52%) and TAC (26.34%) through this optimization compared with Luyben’s results. This vividly manifested the benefits of simultaneous optimization.


18

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Moreover, since the entrainer is usually expensive, it should not be neglected from TAC during optimization. As shown in Table 4, the entrainer cost had the largest decline percentage (40.52%), leading to cost decrease of 15.06e4 $/year. 4. Discussion of the PTC distillation model performance 4.1 Robustness of the PTC distillation model As the optimization cases have shown, The PTC distillation model was very robust both for single column and complex distillation systems. This was because: on the one hand, it had easy initialization because of the decoupled equation system; on the other hand, the proposed simple pseudo hold-up correlation resulted in a simple integration process. 4.2. Bypass efficiency method It is obvious that the optimization of a single column was rather easy because it could always get physically realizable integer solutions. For the complex processes, the model in this work could also complete the optimization objective. However, these results were different from those reported by Dowling and Biegler10 and Pattison et al.17 With simple linear TAC as objective function, Pattison et al. reported that all the optimal cases had no bypass efficiency between 0 and 1 from different initial bypass efficiencies. However, with nonlinear total energy consumption as objective function, Dowling and Biegler reported that a small part of optimization cases had optimal solutions with fractional bypass efficiencies. In this work, it was more difficult to get integer solutions, such as the DWC case, or find the optimal solutions, such as the extractive distillation case (one optimization failed), which might be because the objective


19


Page 20 of 26

function in this work had stronger nonlinearity. Even so, in our work, the best optimal solution was still the one that all the bypass efficiencies were binary values. Thus, we can conclude that increasing flowsheet complexity and stronger nonlinear objective function both make the finding of optimal solution with integer stage numbers more difficult. However, the best optimum should still be attained when all the stage numbers are integers. 4.3. Optimization time For the optimization of two complex processes, around 10 and 7 hours were required respectively. There are two reasons responsible for the long calculation time. One reason is that the dynamic optimizers in ACM are mainly used to solve the dynamic optimization problem. This means that the optimizers calculate sensitivity matrix at each integration step, which makes the calculation time of each dynamic run within the optimization much longer than that of a normal dynamic simulation. To estimate this effect, we performed a small test for the extractive distillation case. With the same initial values for all the free and state variables and the same specifications for all the design variables, we performed two simulations, a normal dynamic simulation and the first step of a dynamic optimization on the same computer. The former took 144 s, and the latter took 831 s. This indicates that the calculation time of a dynamic run in a dynamic optimization was nearly six times as long as that of a normal dynamic simulation. Moreover, the dynamic run within the dynamic optimization could not make full use of already known good initial values. As we know, with optimization variables approaching the optimum, the variable values between adjacent optimization steps should be closer and closer, so the results from the last optimization step could be good initial values for the next optimization step, which could make the optimization process faster and faster. However, the dynamic optimizers


20

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in ACM begin each dynamic run from the starting point of the optimization. In fact, these two shortages can be resolved through using the time-relaxed optimization algorithms12, which is our future work to develop this kind of algorithms in ACM environment. 5. CONCLUSION In this paper, we developed a new PTC distillation model. Compared to the PTC model developed by Pattison and Baldea12, our method was more straightforward and natural to implement. It was based on the normal dynamic simulation but with a pseudo simple holdup correlation to make it robust and fast. To avoid the integer optimization problem, stage numbers were represented as the summation of the bypass efficiencies on all stages. Three optimization cases proved the robustness of our PTC distillation model and the effectiveness of the bypass efficiency method. Furthermore, the optimization results of the ternary extractive distillation process indicated that for the design of a complex flowsheet, experience was usually not reliable since there were complex compromise relationships among design variables. For example, for extractive distillation process, very pure recycle entrainer has been a common specification for the design without rigorous optimization, which, however, is far from the true optimal point. However, different from previous reports about bypass efficiency method10, 12, two optimization cases of complex processes in this work indicated that when the objective function was more complex, the bypass efficiency method was more difficult to find the optimal solution with integer stage numbers. So, for this method, when the process to be optimized and the objective function are both very complex, it is sensible to start optimization from many different initial points and perform parallel computation to all these cases.


21


Page 22 of 26

Moreover, the optimization solvers in Aspen Customer Modeler were not suitable for the optimization based on the PTC model, which has been discussed in the last section. In response, an aim of our future work is to link our model with the optimizer based on time-relaxation algorithms12. Appendix : Total Annual Cost (TAC) model

TAC =

Ccap Yback

+ Cenergy + Cex

(A1)

Ccap = Chx,cap + Ccol,cap

(A2)

Here, Ccap, Cenergy and Cex refer to the capital cost, energy cost and entrainer cost respectively. For the cases in this work, only the extractive distillation process needs to incorporate the entrainer cost. Yback represents payback period and 3 years was given here. Ccap includes exchanger cost (Chx,cap) and column cost (Ccol,cap). For the column capital cost, we only consider the vessel cost. Here are the relevant formulas, which are from Luyben’s book24. Chx,cap = 7296 A0.65

(A3)

Ccol,cap = 17,640 D1.066 L0.802

(A4)

A=

Q

(A5)

U ∆Tmean

D = max( π

Vg,1 u 4 1

, π

Vg,2 u 4 2

, …, π g,j ,…, π V

Vg,Ns

u 4 j

u 4 Ns

)

L = 1.2 × 0.61 Ns

(A6)

(A7)


22

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F = 0.8197 uj ρV

j

j = 1, 2, …, Ns

(A8)

Where A, D, L, Ns are the heat exchanger area, column diameter, column height and total stage number respectively. Vg,j, uj, ρV refer to vapor volume flowrate, maximum vapor velocity and j

vapor mass density on each stage, respectively. In order to calculate the maximum vapor velocity, in Eq. (A8), the F factor, F is set to be 1 kg0.5 m0.5/s, according to the approximate heuristic. In Eq. (A5), U is the heat coefficient, whose value can be seen in Table A1. An approximation25 of logarithmic mean temperature difference ∆Tmean is used to avoid numerical difficulty when there are the same temperature differences at both ends of heat exchangers. The relevant equation is shown in Eq. (A9). For all the variables in Eqs. (A3) – (A9), SI units are used. 1

out out in in out out in 3 Tin hot - Tcold Thot - Tcold Thot - Tcold + Thot - Tcold

∆Tmean =

2

(A9)

In our work, there are two kinds of heat utilities used and the lower pressure steam is used at first unless the heat exchange temperature differences are less than 10 °C. Cold utility is cold water. The temperatures and prices of utilities are shown in Table A1. Table A1. Parameters of economic evaluation Parameter

Value

Heat transfer coefficient Reboiler

0.568 kW/m2/K

Condenser

0.852 kW/m2/K

Heat exchanger

0.430 kW/m2/K

Energy price


23


Heat steam 1 (125 °C)

4.70 $/106 kJ

Heat steam 2 (174 °C)

5.40 $/106 kJ

Cold water (25 °C)

0.54 $/106 kJ

Page 24 of 26

AUTHOR INFORMATION Corresponding Author *Tel：+86 22 27407645. Fax: +86 22 27404496. E-mail: [email protected]. Notes The authors declare no competing financial interest. Acknowledgement The financial support is provided by the National Natural Science Foundation of China (No. 21676183).

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23. García-Herreros, P.; Gómez, J. M.; Gil, I. D.; Rodríguez, G., Optimization of the Design and Operation of an Extractive Distillation System for the Production of Fuel Grade Ethanol Using Glycerol as Entrainer. Industrial & Engineering Chemistry Research 2011, 50, (7), 39773985. 24. Luyben, W. L., Distillation Design and Control Using Aspen™ Simulation. John Wiley & Sons, Inc.: Hoboken, New Jersey, USA, 2006. 25. Chen, J., Comments on improvements on a replacement for the logarithmic mean. Chemical Engineering Science 1987, 42, (10), 2488-2489.

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Simultaneous Optimization of Complex Distillation Systems with a

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