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Pareto domain: an invaluable source of process information

  • Geraldine Cáceres Sepúlveda , Silvia Ochoa and Jules Thibault EMAIL logo
Published/Copyright: August 15, 2020
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Chemical Product and Process Modeling
From the journal Chemical Product and Process Modeling Volume 17 Issue 1

Abstract

Due to the highly competitive market and increasingly stringent environmental regulations, it is paramount to operate chemical processes at their optimal point. In a typical process, there are usually many process variables (decision variables) that need to be selected in order to achieve a set of optimal objectives for which the process will be considered to operate optimally. Because some of the objectives are often contradictory, Multi-objective optimization (MOO) can be used to find a suitable trade-off among all objectives that will satisfy the decision maker. The first step is to circumscribe a well-defined Pareto domain, corresponding to the portion of the solution domain comprised of a large number of non-dominated solutions. The second step is to rank all Pareto-optimal solutions based on some preferences of an expert of the process, this step being performed using visualization tools and/or a ranking algorithm. The last step is to implement the best solution to operate the process optimally. In this paper, after reviewing the main methods to solve MOO problems and to select the best Pareto-optimal solution, four simple MOO problems will be solved to clearly demonstrate the wealth of information on a given process that can be obtained from the MOO instead of a single aggregate objective. The four optimization case studies are the design of a PI controller, an SO2 to SO3 reactor, a distillation column and an acrolein reactor. Results of these optimization case studies show the benefit of generating and using the Pareto domain to gain a deeper understanding of the underlying relationships between the various process variables and performance objectives.

Keywords: decision variables; objective functions; pareto domain; process optimization
Corresponding author: Jules Thibault, Department of Chemical and Biological Engineering, University of Ottawa, Ottawa, Ontario, K1N 6N5, Canada, E-mail:

Funding source: Natural Sciences and Engineering Research Council of Canada

Award Identifier / Grant number: RGPIN/ 2018-04433

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendices

Appendix A

The design of the distillation column for the third case study of this investigation was obtained analytically using the Fenske–Underwood–Gilliland (FUG) method. This method consists in finding the minimum number of trays as well as the minimum reflux ratio for a given feed stream, which are then used to determine the other design parameters.

A1 N min = log [ ( x L K x H K ) D ( x H K x L K ) B ] log α L K / H K ave

A2 α L K / H K ave = ( α L K / H K D α L K / H K B ) 1 / 2

A3 α i / j = y i y j x i x j

A4 ∑ i = 1 n α i / H K x i , F α i / H K − θ = 1 − q

A5 R min = ∑ i = 1 n α i / H K x i , D α i / H K − θ − 1

A6 R = C ⋅ R min

A7 N − N min N + 1 = 0.75 [ 1 − ( R − R min R + 1 ) 0.566 ]

A8 E o = 50.3 ( α L K / H K , F ⋅ μ ) − 0.226

A9 N real = N E o

A10 N actual = N E o + 1

A11 u f = C s b F S T F F F H A ρ L − ρ G ρ G

A12 F S T = ( σ L 20 ) 0.20

A13 2 A d A T = { 0.1           for   F LV   ≤   0.1 0.1   +   F LV   −   0.1 9 for   0 . 1   ≤   F LV   ≤   1.0 0.2           for   F LV   ≥   1.0

A14 D T = 4 V ( f u f ) π ( 1 − 2 A d A T ) ρ G

A15 H P = { 0.4 for   D ≤ 0.5   m 0.5 D 0.3 for   D > 0.5   m

A16 H C = ( N actual − 1 ) ⋅ H P + ∑ Δ H

The following equations were used to determine the bare module cost of the column, the reboiler and the condenser. In Equation A17, variable A represents the capacity size parameter that will change depending on the unit. The height is the capacity parameter used for the column itself and the parameters of the equations are function of the column diameter. The capacity parameter of heat exchangers is the heat transfer area that is determined using Equation A21.

A17 C p = 10 ˆ ( K 1 + K 2 log ( A ) + K 3 ( log ( A ) ) 2 )

A18 C BM = C p ( B 1 + B 2 F M F P )

As the bare module cost is calculated for a CEPCI value of 382 [90], unless otherwise specified, it is necessary to correct for time with the latest cost index: a CEPCI of 616.4 [103]. This was done using the following relations:

A19 C new = C base ( C E new C E base )

A20 Q ˙ = n ˙ C p Δ T

A21 A = Q ˙ U Δ T L M

As for the column internals, valve trays were selected. The CEPCI value that was initially used in this case is 397. The bare module cost equation for the internals of the column is different:

A22 C BM = C P N F M F q

Table 3:

Parameters used for the capital cost (CAPEX) calculation [90].

K 1 K 2 K 3 F M F P F q B 1 B 2
Condenser 3.4338 0.1445 0.1079 1 1 - 1.8 1.5
Reboiler 3.5638 0.1906 0.1107 1 1 - 1.8 1.5
Column * * * 1 1 - 2.5 1.72
Internals 3.3322 0.4838 0.3434 1 1 1 - -

  1. *These values will depend on the diameter of the column, see Turton et al. [90].

Appendix B

Appendix C

Below are all the required parameters that were used in order to apply the Net Flow method to rank the Pareto domain obtained for each case study as seen in Section 4.

Table 4:

Decision variables and objectives for the four case studies.

Name Bounds or type of optimization Definition
PI controller
Decision variables Controller gain [0.01–10] K C
Integration time [0.01–10] τ I
Objectives ITAE Min
OZ Min
ISDU Min
SO 2 to SO 3 reactor
Decision variables Inlet gas feed temperature [K] [500–900] T 1
Length of the bed [m] [0.01–2.00] L 1
Objectives Productivity [kmol/m3 h] Max P
Conversion [%] Max X
Catalyst weight [kg] Min W
Distillation column
Decision variables Ratio of the actual reflux ratio (R) to the minimum reflux ratio (R min) [1], [2], [3], [4], [5], [6], [7], [8], [9], [10] C
Objectives Capital cost [M$] Min CAPEX
Operating cost [M$/year] Min OPEX
Acrolein reactor
Decision variables Molar flowrate of propylene [kmol/h] [91–203] F P
Molar flowrate of air [kmol/h] [433–2900] F A
Molar flowrate of steam [kmol/h] [91–3047] F S1
Temperature in R-100 [°C] [330–430] T 1
Pressure in R-100 [bar] [1.05–6] P 1
Objectives Productivity [kmol/m3 h] Max P
Conversion [%] Max X
Soft constraint on oxygen concentration Min Sum E
Table 5:

Relative weight and thresholds values used in the net flow method to classify the Pareto domain obtained for the four case studies.

Case study Objective Relative weight Thresholds
Indifference Preference Veto
PI controller ITAE 0.4 1.5 4.0 8.0
OZ 0.4 0.5 1.0 3.0
ISDU 0.2 0.005 0.01 0.03
SO3 reactor P 0.4 0.001 0.003 0.007
X 0.4 1.2 4.0 8.0
W 0.2 1500 3000 7000
Distillation column CAPEX 0.5 2000 5000 8000
OPEX 0.5 6000 9000 12000
Acrolein reactor P 0.3 0.08 0.15 0.3
X 0.3 0.2 0.5 1.0
Sum E 0.4 0.02 0.05 0.10

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Received: 2020-02-16
Accepted: 2020-07-15
Published Online: 2020-08-15

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Optimal tuning of PID controller in time delay system: a review on various optimization techniques
  4. Pareto domain: an invaluable source of process information
  5. The optimization of biodiesel production from transesterification of sesame oil via applying ultrasound-assisted techniques: comparison of RSM and ANN–PSO hybrid model
  6. Process simulation for the production of methanol via CO2 reforming of methane route
  7. Nitrate removal studies on polyurea membrane using nanofiltration system – membrane characterization and model development
https://doi.org/10.1515/cppm-2020-0012
Keywords for this article
decision variables; objective functions; pareto domain; process optimization

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Optimal tuning of PID controller in time delay system: a review on various optimization techniques
  4. Pareto domain: an invaluable source of process information
  5. The optimization of biodiesel production from transesterification of sesame oil via applying ultrasound-assisted techniques: comparison of RSM and ANN–PSO hybrid model
  6. Process simulation for the production of methanol via CO2 reforming of methane route
  7. Nitrate removal studies on polyurea membrane using nanofiltration system – membrane characterization and model development
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