Simultaneous design and control of catalytic distillation columns using

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Simultaneous design and control of catalytic distillation columns using comprehensive rigorous dynamic models David E. Bernal, Carolina Carrillo-Diaz, Jorge M. Gómez, and Luis Ricardez Sandoval Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b04205 • Publication Date (Web): 30 Jan 2018 Downloaded from http://pubs.acs.org on February 3, 2018

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Industrial & Engineering Chemistry Research

Simultaneous design and control of catalytic distillation columns using comprehensive rigorous dynamic models David E. Bernal1, Carolina Carrillo-Diaz2, Jorge M. Gómez*2, Luis A. Ricardez-Sandoval3 1

Department of Chemical Engineering, Carnegie Mellon University 5000 Forbes Avenue, Pittsburgh,

PA 15232, United States of America. 2

Grupo de Diseño de Productos y Procesos, Departamento de Ingeniería Química, Universidad de los

Andes. Carrera 1 No. 18A-10, Bogotá 111711, Colombia. 3

Department of Chemical Engineering, University of Waterloo, Waterloo, ON, N2L 3G1 Canada

KEYWORDS: optimal design; optimal control; dynamic optimization; catalytic distillation; process intensification

*

Corresponding author. E-mail: [email protected]

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ABSTRACT

This work addressed the rigorous modeling of catalytic distillation columns by simultaneously considering relevant phenomena such as pressure drop across the column, tray capacity constraints, nonideal behavior of both liquid and vapor, and the column’s hydrodynamics, yet to be considered in earlier modeling studies for catalytic distillation. The developed model is applied for the simultaneous optimal design and control of catalytic distillation units, taking into account economic and set-point tracking objective functions balanced through a proposed weighting parameter estimation methodology. Results for a column for the production of ethyl tert-butyl ether show the advantages of a more comprehensive model regarding design and control decisions in comparison with previous studies: design specifications were met during the entire time horizon without sacrificing economic profitability.

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Introduction Most of the chemical processes include two of the most important operations: chemical reaction and thermodynamic separations. These operations are often carried out in separate equipment. Distillation is the most popular liquid mixture separation technique in the chemical and oil & gas industries. The energy requirement of this operation can represent up to 40% of the entire plant’s energy requirements 1. Recycle streams are often used between the reaction and separation sections to increase yield and conversion, minimize undesired products synthesis, and improve energy efficiency. Instead of carrying out the reaction and separation into independent units, these operations can be performed on a single equipment

2,3

. The implementation of these combined processes represents one of the most promising

methodologies in process intensification. Economic and environmental considerations have led the industry to develop this type of processes 4, which offer considerable benefits compared to the traditional multi-unit scheme. Integrating reaction and product purification on a single multifunctional unit leads to considerable improvements compared to the traditional sequential approach such as: overcoming chemical equilibrium limitations, increasing product selectivity, and using the heat of reaction for the separation 5. A key limitation of this integration is that the operational window is reduced considerably as the conditions of the reaction and the separation must be satisfied simultaneously 1. This process intensification concept is referred to as Reactive Distillation (RD) and when a heterogeneous catalyst is involved, it is referred to as Catalytic Distillation (CD). The operation of distillation with a chemical reaction is of paramount importance in various applications; therefore, its feasible, safe and economic operation must be ensured. This unit is very sensitive to perturbations, i.e. a change in the operational conditions (e.g. changes in the feed composition of mass flowrate) may drastically affect the process dynamics, thus affecting product



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purity, energy consumption and/or throughput 6. Therefore, it is crucial to understand the dynamics of CD columns to improve the performance and economics of these systems. In order to provide insight into the transient behavior of the CD column, several studies have presented mechanistic (first-principles) models for this unit under different assumptions. Table 1 summarizes the recent contributions regarding the modeling of RD columns. On the one hand, a few studies have considered the hydraulics of sieve and valve tray columns in both homogenous and heterogeneous distillation by means of using empirical correlations, which have been validated with experimental data and simulations 7–13. On the other hand, other contributions have considered the nonidealities of the liquid phase behavior idealities of the vapor phase

6,17,18

14–16

, while fewer studies have taken into account the non-

. Furthermore, the pressure drop across the column has not been

accounted for in all these previous studies

16,17,19–27

; those studies that take this condition into account

consider assumptions that may not entirely capture the actual system’s behavior, e.g. constant pressure drop from tray to tray 28,29. Table 1. State of the art summary of relevant catalytic distillation column models Author Bennett et al. 30

Year 1983

Kister et al.8 Bennet et al.9 Tomazi31

1988 1995 1997

Van Baten et al. 10 Ross et al. 6

2000 2001

Bansal et al. 14

2002

Georgiadis et al.15

2002

Mortaheb et al. 11 Van Sinderen et al.12 Kolodziej et al.13 González-Rugerio et al.17 Patrut et al.16

2002 2003

Problem Pressure drop, liquid holdup, and liquid flow over weirs correlations for sieve tray distillation columns proposed based on experiments. Correlations to prevent entrainment in sieve tray distillation columns based on experimental data. Correlations to prevent entrainment in sieve tray distillation columns based on experiments. Design of a batch distillation column considering MESH equations, hydraulic correlations and stage efficiency for both sieve and bubble cap trays. Test of the accuracy of Bennet et al. correlations using computational fluid dynamics. Distillation column model considering non-ideality in both liquid and vapor phases, tray efficiency, liquid hydraulics, pressure drop, and correlations to prevent flooding and entrainment. RD column model considering volume constraints, tray efficiency, non-ideal liquid phase, pressure drop and hydrodynamics of liquid and vapor to prevent flooding and entrainment. Ethyl acetate production in a RD column modeled considering non-ideal liquid phase, tray efficiency, liquid hydraulics and pressure drop correlations; using modified MESH equations. Development of clear liquid height correlations in heterogeneous distillation based on experimental data. Correlations to prevent flooding and entrainment in sieve and valve trays based on experimental data.

2004 2014

Effects of structured packing for catalysts on the hydrodynamics and mass transfer in CD columns. CD column for the synthesis of tert-amyl ethyl ether (TAEE) considering non-ideal liquid and vapor.

2014

CD column model for the production of Dimethyl ether (DME) using MESH equations, and non-ideality of the liquid phase.


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Domingues et al.18

2017

This research

2017

ETBE catalytic distillation column model using modified MESH equations, pressure drop correlations, and deactivation of catalyst kinetics; considering stage efficiency and non-ideality in both vapor and liquid phases. ETBE catalytic distillation column model considering modified MESH equations, non-ideality in both liquid and vapor phases, tray efficiency, pressure drop, hydrodynamics of liquid and vapor to prevent effects such as flooding and weeping-dumping

These phenomena, i.e. hydraulics, non-idealities of liquid and vapor phases, pressure drop, and the coexistence of separation and reaction, are quite nonlinear and highly complex, e.g. they may involve multiple steady states which may lead to multiple solutions 32. To the authors’ knowledge, a mechanistic model that takes into account these phenomena simultaneously has not been presented in the literature. Having access to a highly detailed rigorous model that can simultaneously consider these phenomena is key to gain insight on the transient behavior in CD columns and to understand how these phenomena may impact the design, safety, performance, and economics of these units. Traditionally, chemical process design is performed sequentially, i.e. a steady-state analysis determines the process design followed by a dynamic analysis, which aims to evaluate the process controllability. However, it is often found that process design imposes limitations on the achievable performance of the system in closed-loop. Overdesign factors are then considered to account for the process dynamics at the expense of higher investment costs. Integration of design and control is an attractive method to perform optimal process design while taking into account the process dynamic behavior. Multiple methodologies have been proposed to address design and control. Table 2 presents a summary of the recent works in this field; review papers on the subject of integration of design and control for chemical processes are available, e.g. Sakizlis et al. 22, Ricardez-Sandoval et al. 33, Yuan et al.34. With regards to the integration of design and control for CD and RD columns, relevant works are also available, e.g. Georgiadis et al. 15, Panjwani et al. 23, Lai et al. 24, Miranda et al. 35, Mansouri et al. 26,36, Aneesh et al. 37. While the optimal design and control of a CD column is desirable since it allows the maximization of profitability while ensuring product specifications in the presence of external perturbations, the solution 4 ACS Paragon Plus Environment


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of this type of problems has a considerable mathematical complexity because of the interactions and trade-offs that exist between the separation and reaction processes. To the authors’ knowledge, comprehensive rigorous models have not been yet used to address the optimal design and control of CD systems. Based on the above, the aim of this work is to present first a rigorous mechanistic process model that overcomes some of the assumptions considered in previous CD models. The phenomena considered by the present model may offer new attractive design and control perspectives for CD columns. Hence, the optimal design and control of a CD column while using the proposed mechanistic model is also considered in this work. A systematic framework that assigns weights to set-point tracking and economic objectives in the integration of design and control formulation is considered. To the authors’ knowledge, a weight selection method for objective functions involving set-point tracking and costs has not been previously implemented in the integration of design and control studies. The design and control schemes obtained by the present approach have been compared to the sequential design approach and to integrated design and control formulations that have used CD models previously reported in the literature. A case study involving the synthesis of ethyl tert-butyl ether (ETBE) is considered to gain insight into the optimal design and control of CD columns under the effect of disturbances. The rigorous mechanistic model developed in this work to describe the dynamic behavior of a CD column designed for the production of ethyl tert-butyl ether (ETBE) is presented in the next section. This is followed by the conceptual mathematical formulation proposed in this work to address the optimal design and control of CD columns. The results of the simultaneous optimal design and control and its comparison to the sequential and integrated approaches using other CD models are presented afterward. Concluding remarks and future work are discussed for closure.


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Table 2. State of the art summary for control and design in distillation columns Author Mohideen et al. 19 Sneesby et al. 38 Bansal et al. 39 Russel et al. 40

Year 1996 1999 2000 2000

Kookos et al.20 Ross et al. 6 Bansal et al. 14 Georgiadis et al. 15 Raghunathan et al. 21 Sakizlis et al.22

2001 2001 2002 2002 2004 2004

Panjwani et al. 23 Lai et al. 24 Miranda et al.35 López-Negrete et al.

2005 2007 2008 2009

Problem Comparison of the sequential and simultaneous design and control for distillation. Two-point control configuration for an ETBE reactive distillation column. Optimal control of a methanol-water distillation. Systematic computer-aided analysis approach integrating control and design of processes applied to heatintegrated distillation. Control and Design of a binary distillation column and of a coupled reactor-distillation. Sequential optimal control and design of a water, propanol and isopropanol distillation. Optimal control and design in a benzene-toluene distillation. Comparison of sequential and simultaneous optimal design and control (ODCP) in reactive distillation. Dynamic optimization of a batch distillation column using rigorous models. Simultaneous optimal design and control of a binary distillation column using two different controller types. Economic comparison of the sequential and simultaneous design and control of a reactive distillation. Design and control of reactive distillation systems using direct search total annual cost minimization Optimal design and control (ODCP) of a CD column. Simultaneous optimal control and feed position for distillation.

41

Simon et al. 42 Alvarado-Morales et al. 43 Moghadam et al. 30 Ramos. et al 44 Sanchez-Sanchez et al. 45 Trainor et al. 46 González-Rugerio et al.17 Alcántara-Avila et al.28 Chung et al. 25 Segovia-Hernández et al.47 Mansouri et al. 36 Mansouri et al. 26

2009 2010 2012 2013 2013 2013 2014 2015 2015 2015 2016 2016

Aneesh et al. 37

2016

Bildea et al.27

2017

Diangelakis et al.48

2017

Choice of the physical equipment and optimal control of batch distillation column. New simultaneous design and control methodology based on the selection of a process group/separation technique for the synthesis of bioethanol and of succinic acid. Optimal control of a CD column through a linear quadratic (LQ) regulation. Optimal control of an extractive distillation column. New methodology for integrated design and control under uncertainty applied for a ternary distillation column. Optimal design and control of a ternary distillation column under uncertainty. Optimal design of a CD column for the synthesis of tert-amyl ethyl ether (TAEE) Optimal design of a silane CD columns with and without intermediate condensers using the process simulation software Aspen Plus. Conceptual process design and associated control strategies for the manufacture of n-butyl levulinate. Summary of optimal design using deterministic and stochastic techniques for RD columns. Computer-aided framework for integrated design and control of binary elements reactive distillation. Hierarchical decomposition-based framework for integrated design and control of reactive distillation involving multiple elements. Summary of advanced distillation technologies (including RD and CD columns) and optimization based approaches to simultaneous design and control. Optimal design of a Dimethyl ether (DME) CD column using the process simulation software Aspen Plus. Simultaneous optimal design and design dependent model predictive control for the binary separation of benzene and toluene.



Domingues et al.18 Medina-Herrera et al. 29 Moraru et al. 49 This research

2017 2017 2017 2017

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Optimal design of an ETBE catalytic distillation column considering controllers as auxiliary equipment. Optimal design of a multiproduct CD column for the synthesis of silanes based on rigorous simulations carried out in the process simulation software Aspen Plus. Optimal design and control using the process simulation software Aspen Plus. Simultaneous optimal design and optimal control of an ETBE catalytic distillation column based on rigorous models.

Catalytic distillation column model This section presents the rigorous mechanistic model for the ethyl tert-butyl ether (ETBE) CD column. This chemical compound is an oxygenating ether for fuels used as an alternative to the methyl tert-butyl ether (MTBE). The ETBE is classified as semi-renewable as it can be synthesized from the etherification of bioethanol and isobutene in the presence of an acid catalyst, as shown in Equation 15. For brevity, the detailed mathematical modeling of the reaction is presented in the Supporting Information. The synthesis process of this chemical compound is modeled as a CD through the reaction between ethanol and isobutene over an acid catalyst. The proposed CD column has 2 feed flows, one preferably of pure ethanol, and the other a mixture of butenes, typically of 30% n-butene and 70% isobutene molar composition. Note that the operating conditions for this process require consideration of non-ideality in the vapor phase, which increases the model’s complexity. This aspect has been explicitly considered in this work whereas previous studies have neglected this condition for this specific multicomponent system 18,35,50–52.

= + ⇌

(1)

The present process model considers that the CD column has separation and reactive stages in crossflow sieve trays. Figure 1 presents a flowsheet of the CD column under consideration. The separation

and reactive stages are considered as equilibrium stages, indexed by starting from the top of the

column (i.e. the condenser is the stage 1) to the bottom (i.e. the reboiler is the stage ). The components are indexed by up to the total number of components . Component molar balances (with or without the chemical reaction), phase equilibrium, summation equations, heat balances and


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hydraulic relationships for each stage are specified to determine composition, temperature, and flow profiles (i.e. MESH equations). The following assumptions have been made in the present CD column model: •

Thermodynamic equilibrium at each stage.

•

Adiabatic operation.

•

Total condenser and partial reboiler.

•

No pressure drop in the reboiler.

•

Constant mass accumulation in the condenser and the reboiler (liquid phase only). Total Condenser

Qc 1

2 Feed 1

Feed 2

Distillate D

RR

Dc

Ln-1

Vn

Ln

Vn-1

Fn

NT-1 NT QR

Partial reboiler

Bottoms B

Figure 1. Sketch of a catalytic distillation column with 2 feeds

The complete notation for the CD column model is presented in the nomenclature section of this study. The following sets are introduced here to make the formulation of the model more concrete. Set definitions

= {1, … , } = {1, … , } = 1 ⊆

(2.a) (2.b) (2.c)



= ⊆ ⊆ = ∖ ∖ ∖

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(2.d) (2.e) (2.f)

where is the set of components, indexed in ; is the set of equilibrium stages, indexed in ;

represents the condenser, the reboiler, the stages with catalyst and the separation tray stages.

The conservation balance equations applied on the CD column are described in Appendix A. These equations model the dynamic behavior of a CD column with a single reaction and take into account both the vapor and the liquid holdup at each stage. Hydraulic relationships for each stage are also included in this rigorous mechanistic CD column model and are presented in Appendix B; these equations are needed to specify the composition, temperature, and flow profiles across the column. Special attention should be drawn to the thermodynamic equilibrium, stage capacity and pressure drop equations, described next. Thermodynamic equilibrium

The thermodynamic equilibrium equations enhance the phenomenological understanding of the column since all components in both liquid and vapor phases are considered non-ideal, i.e. !",# = $",# %",# , ∈ , ∀ ∈

(3)

The equilibrium constant is given by the ratio of the vapor and liquid composition of a component in an equilibrium stage. According to the vapor and liquid equilibrium assumption, three different systems can be considered: ideal system, ideal vapor system, and real system. To improve the prediction capabilities of the present CD model, the equilibrium constant is calculated using the actual system, which employs a correction of non-ideality for the vapor and liquid phases following the gamma-phi formulation 53, i.e. $",# =

(#)*+ ,",# , ∈ , ∀ ∈ (" -",#

(4)


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The thermodynamic model used to describe the non-ideality of the liquid phase is the Non-RandomTwo-Liquid (NRTL) method. This model has been successfully applied to previous studies that considered strongly non-ideal liquid phase distillation systems for optimal control

44,54

. The constants

used for this model were adjusted from the vapor-liquid equilibrium data generated by the UNIFAC Dortmund, which is the model used to describe the non-ideality of the system considered in this work 55. Stage capacity

./," .2," 3ℎ5 67 + = , ∀ ∈ 0̅/," 0̅2," 4

Pressure behavior

(" = ("9: + ∆(" , ∀ ∈

∆(" = 30

(19.a)

(19.b)

Note that the rest of the butenes in the feed was completed with isobutene. As shown in Equation 19, the disturbance amplitude is the difference between the specifications defined for the two operational scenarios, which are described in the next section. For comparison purposes, the sequential optimal design and control of a CD column for the synthesis ETBE was performed and is further explained next. Sequential design and control approach The sequential design and control of the CD column was conducted as follows: an optimal steadystate design problem is formulated first. This problem minimizes the annualized cost of investment and operation of the column (Equation 10) while taking into account the steady-state process model equations (catalytic distillation column model with no accumulation terms), the product specification constraints (Equation 13), and the tray hydrodynamic constraints (see Appendix B). This problem determines the design variables of the CD column at steady-state, i.e. column diameters, stage height, downcomer height and sieve tray areas. These variables are then fixed and used to search for optimal trajectories in the manipulated variables that minimize the tracking objective function and the operating cost function, i.e. an optimal control problem. The latter is subject to product specification constraints (Equation 13) and the dynamic catalytic distillation column model. Two different operational scenarios depending on the molar composition of the butenes in the feed stream are considered: the nominal case establishes that the molar fraction of isobutene in the feed is 0.3, with the rest being n-butene. Meanwhile, the conservative-case scenario consists of a molar fraction 18 ACS Paragon Plus Environment


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of 0.25 of isobutene in the feed. It should be noted that the highest amount of isobutene allows a higher production of ETBE. The choice of these two scenarios was made based on the fact that the nominalcase scenario was designed with an isobutene feed molar composition equal to the initial condition of all the disturbances while the conservative-case scenario was designed with an isobutene feed molar composition equal to the lowest, and therefore most challenging in terms of control, as it will be discussed below. Previous work by Miranda et al.35 is used to compare the results from the present approach. In that work, the fundamental process model equations used to address the sequential design and control of a CD column for ETBE production do not take into account key phenomena that may impact the operation of the column, e.g. pressure drop across the column and tray capacity constraints. Also, control and economic objective functions were employed, with no further explanation regarding each objective’s weight. In contrast, the present work introduces a new comprehensive mechanistic process model that explicitly accounts for those simplifications in modeling key process phenomena. Also, the process dynamic variability is measured in this work using both tracking objectives, i.e. deviations with respect to a reference signal, and an economic objective. Hence, a weighted sum of these objectives was employed as an objective function in the optimal design and control problem formulation. Using the formulation of the optimal steady-state design stated above, the problem resulted in an NLP formulation with 1,158 equations and 1,139 continuous variables. For comparison purposes, the results obtained for each case are presented in Table 4 along with those reported by Miranda et al.

35

for a similar nominal case study. The differences in the CD modeling

approaches employed in this work and that used by Miranda et al.

35

the optimal design problem. The results presented by Miranda et al.

resulted in different solutions for 35

returned an infeasible solution

according to the rigorous mechanistic CD process model described in this work, i.e., the ETBE bottoms mole fraction is below the specified minimum of 0.83. Also, the reported data resulted in downcomer 19 ACS Paragon Plus Environment

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flooding, since the stage height was smaller than the height required to overcome the pressure difference across the downcomer. The design specifications obtained with the present formulations returned acceptable solutions, i.e. increasing the column diameter satisfies the tray capacity constraints, though it represents an increase in the overall cost. Also, the resulting feed flow rates were higher than those obtained by Miranda’s, but this fact was balanced with a lower molar reflux ratio, as shown in Table 4. Table 4. Optimal design results for each scenario Isobutene feed composition Solution source Column Diameter £¤ Stage height ¥¦ Downcomer height ¥§ Feed rate ¨© Reboiler Duty ª«¬¨ Molar reflux ratio «« ETBE bottoms composition , Isobutene conversion Entrainment flooding? Downcomer flooding? Weeping? Profit ETBE Annualized Investment cost Operating cost Total Cost

Units [mol/mol] [m] [m] [m] [mol/min] [kJ/min] [-] [mol/mol] [mol/mol]

[$/year] [$/year] [$/year] [$/year]

Conservative-Case 0.25 Current work 0.1214 0.0500 0.0166 7.515 284.24 1.525 0.83000 0.78038 No No No 21,501 10,037 42,887 31,424

Nominal Case 0.3 Miranda et al. Current work 0.09533 0.1110 0.0292 0.0500 0.0082 0.0166 4.907 5.774 265.68 249.96 3.659 2.238 0.60215 0.83000 0.87508 0.86027 No No Yes No No No 26,003 21,836 10,030 10,034 32,514 35,898 16,541 24,097

* The results in italics do not correspond to data given by Miranda et al. 35, instead, they were obtained by the formulation presented in this work. The values in bold are highlighted since they reflect a violation of the process constraints.

The results also represent a reasonable behavior between the nominal and the worst-case scenarios. The higher the fraction of isobutene in the feed, the lower the feed flow required

51,67

(as the reaction

generating ETBE is one to one with isobutene, as shown in Equation 1). This decrease in feed flow rate resulted in a smaller column diameter and a lower reboiler duty. With less flowrate of the butenes mixture, a higher conversion can be achieved, which requires a higher reflux ratio. The profit from the ETBE was proportional to the feed flow, the investment cost to the column diameter; the operating cost strongly depends on the feed flow and heat duties. Note that a feed stream with a low fraction of isobutene resulted in a more conservative design with a larger column diameter and higher operating



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costs. Compared to the nominal case, the conservative-case specified a CD column that is 30.4% higher in total costs. Once the optimal design of the CD column has been specified, the next step is to ensure the dynamic operability of the system. This was achieved by fixing the design parameters and the initial condition for the manipulated variables with the solutions obtained from the steady-state optimal design formulation. This problem was posed as an NLP formulation with 42,099 equations and 42,272 continuous variables. Weights specification

To completely determine the objective function of the optimal control formulation, the weighting parameters need to be specified first. The procedure to determine such weighting parameters is described in the methodology to estimate the weights section. The weighting parameters are determined by Equation 18, where the coefficient

®¯°±² ³±´µ

was set to 5. This decision was made using the

methodology for determining the weighting parameters from Ramos et al. 44 to avoid the assumption of equally important tracking and economic objectives for the objective function as some control variables are easier to adjust thus making the assumption of equal importance invalid. Also, this selection allows the objective to approach a Lyapunov function as the convexity of the objective is strongly dominated by the quadratic terms in the tracking objective term. The results of the weighting parameters specified for the objective functions used in the nominal case and the conservative case are presented in Table 5. As mentioned above, the weighting parameters are all relative to the costs; therefore, its weighting factor value is one for all the scenarios. Note that the weighting parameter for the ETBE composition tracking is the highest among all the different objectives considered, followed by the reflux quadratic deviation and the reboiler duty for both the sinusoidal and the step disturbance in the two design scenarios. Since the conservative-case design scenario required higher energy transfer to the reboiler to keep the steady-state operation feasible with a poorer isobutene feed, the weighting parameters reflect the need of


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a less aggressive control in the reboiler duty for all disturbances. That trend is followed by the ETBE bottoms composition tracking objective, where for all sinusoidal disturbances the weight of this objective is lower in the conservative-case than the nominal-case scenario. The reflux tracking objective is heavily penalized in the objective for the conservative-case design scenario for all but the step disturbance.

Table 5. Weighting parameters ¶ for each objective in the OCP objective function obtained from the utopia tracking procedure with each disturbance for each design scenario Objective

Design scenario

ETBE bottoms composition tracking [(mol/mol)2] »

Z^ , − ,, b l¼

º

Reboiler duty quadratic deviation [(kJ/min)2] »

Z [ª«¬¨ − ª«¬¨, j l¼

«¬½

»

º

Reflux quadratic deviation [-]

Z^«««¬½ − «« b l¼

º

Sinusoidal disturbance · = 1/2 ·=¹ · = 3/2

Step disturbance

Nominal

3.69E+05

2.30E+05

4.70E+05

5.30E+03

Conservative

2.55E+03

3.57E+04

1.77E+04

6.35E+04

Nominal

2.11E-01

2.06E-01

1.99E-01

1.93E-01

Conservative

1.87E-01

1.67E-01

1.65E-01

1.42E-01

Nominal

5.79E+02

5.92E+02

5.86E+02

9.47E+02

Conservative

8.49E+02

8.03E+02

7.95E+02

7.56E+02

Sinusoidal disturbance

The profiles for the ETBE molar composition in the bottoms and the manipulated variables (reboiler duty and molar reflux ratio) for each of the three sinusoidal disturbances are presented in Figure 2, Figure 3, and Figure 4. The profiles include their reference values and regardless of the case scenario, they follow the same trend with a deviation with respect to their steady-state condition. This deviation has the same explanation as in the optimal design problem: the lower the molar fraction of isobutene in the feed stream, the more effort required to achieve the purity of ETBE in bottoms, therefore the higher reboiler duty.



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Figure 2. ETBE in bottoms and manipulated variables profiles for the OCP solution to the sinusoidal disturbance and frequency ω=0.5



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The feed flow rate of butanes is also different between the two scenarios, being higher for the conservative-case scenario; therefore, a lower conversion of the product is achieved requiring a lower molar reflux ratio. The main results on the sequential optimal design and control problem for the nominal and the conservative-case scenarios are reported in Table 6. Since this is an optimal open loop control formulation, and there is an economic term in the objective function, the control shuts down abruptly the reboiler duty and the molar reflux ratio as both manipulated variables are taken into account in the minimization of the objective function. While the reboiler heat duty is directly considered in the objective function, the reflux ratio is indirectly included by taking the condenser cost into account. In the case of the conservative-case scenario, this action is more noticeable because the manipulated variables are farther from their reference values and this abrupt change in their responses is favored by both parts of the objective function: the tracking performance and the economic objectives. 24 ACS Paragon Plus Environment


Page 26 of 63

Table 6. Optimal solution results of the sequential design and control with sinusoidal and step disturbances with the nominal- and conservative-case design scenarios Unit

Entrainment flooding?

Downflow flooding?

Weeping? »

ETBE bottoms composition tracking

Z^ , l¼ »

− ,, b

º

[(mol/mol)2]

Reboiler duty quadratic deviation

Z [ª«¬¨ − ª«¬¨, j l¼

«¬½

»

º

Z^«««¬½ − «« b l¼

No

No

No

No

Conservative

No

No

No

No

Nominal

No

No

No

No

Conservative

No

No

No

No

Nominal

Yes

Yes

Yes

Yes

Conservative

Yes

Yes

Yes

Yes

Nominal

1.07E-9

2.63E-9

6.56E-10

0

Conservative

6.89E-5

3.20E-7

1.27E-6

5.69E-8

Nominal

1083.53

1011.55

986.059

3244.35

Conservative

1154.81

1366.81

1379.56

2002.02

Nominal

0.5208

0.4046

0.3993

0.0557

Conservative

0.8444

0.9285

0.9480

0.7592

Nominal Conservative Nominal Conservative Nominal Conservative

14.364 15.928 22.456 26.828 101,149 113,177

14.399 16.068 22.455 26.829 101,430 114,284

14.431 16.112 22.455 26.830 101,686 114,630

12.153 15.383 22.473 26.837 83,326 108,746

Step disturbance

[(kJ/min)2]

Reflux quadratic deviation

º

Nominal

Sinusoidal disturbance · = 1/2 · = ¹ · = 3/2

Design scenario

[-]

Profit ETBE

[$/hr]

Operating cost

[$/hr]

Annualized Profit

[$/year]

Step disturbance

The solution to the optimal control formulation using the step disturbance described in Equation 19.b is presented in Table 6 and Figure 5. The response to this disturbance shows that the reference points obtained by the conservative-case scenario in the optimal design problem are farther away from the manipulated variables’ profiles than in the case of the nominal-case scenario. This can be clearly observed in the last time steps where both manipulated variables decrease abruptly towards the reference values. The results obtained for the step disturbance showed that solving the optimal control problem based on different design parameters directly affects the profitability of the process. As shown in Table 6, despite the fact that the conservative-case design scenario incurs higher operating costs, it generates higher profits. The specifications of the reboiler duty, ETBE bottoms, and reflux deviations show that 25 ACS Paragon Plus Environment

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the process design was more challenging to control when the conservative-case scenario was considered, that is, more aggressive actions in the manipulated variables are needed to meet the product constraints. Comparing between the different solutions obtained for the three disturbance specifications, the results show that the faster the disturbance, the higher the profit.

Figure 5. ETBE in bottoms and manipulated variables profiles for the OCP solution to the step disturbance.

A key issue affected by the sequential design and control problem is the operability of the equipment. The optimal design problem required the design parameters to satisfy the tray hydrodynamic constraints specified in the Appendix B. Figure 6 shows the pressure over the weir on the second stage and the velocity through the holes in the tray for the last stage before the reboiler. As the stage next to the condenser has the largest vapor flow rate in the column, it is expected to be the stage where the downcomer flooding may occur. On the other side of the column, since the bottom stages have the largest liquid flows, the weeping is more likely to occur. The liquid and vapor velocity profiles are within their hydrodynamic limits and are not shown for brevity. These results indicate that the two cases 26 ACS Paragon Plus Environment


Page 28 of 63

of sequential design and control violate the tray capacity limits during operation. Specifically, none of the sequential design and control solutions are able to avoid the flow of liquid through the tray holes (i.e. weeping) in the lower trays, i.e. the 9th stage. This situation is due to the fact that the design did not consider the dynamics of the CD column; in particular, the liquid and vapor holdups in the stages, which directly affect the system’s hydrodynamics. This is the key motivation to pursue a design that satisfies these constraints while considering the dynamic behavior of the system at once, as described in the next section. Simultaneous optimal design and control The simultaneous design and control of the ETBE production CD column is considered next. The

investment cost #"o is the extra term added to the objective function used for the optimal control

formulation. This term considers no weighting parameter as it is part of the economic objective function, whereas the remaining terms have the same weighting parameters as in the optimal control formulation problem with the nominal-case design scenario. In addition, the design parameters, i.e. column diameter, weir height and stages height, were explicitly defined as optimization variables in this formulation. The results for the SDCP under the nominal-case scenario using the different disturbances described in Equation 19 are discussed next. Each of this optimization problems resulted in an NLP formulation with 45,706 equations and 44,607 continuous variables.


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Figure 6. Over weir pressure and hole velocity profiles for the most likely stages to be affected by downcomer flooding and weeping for the disturbances considered and for the sequential and simultaneous ODCP solutions


28


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Sinusoidal disturbance

As in the previous section, the profiles for manipulated variables and the ETBE molar composition in the bottoms are depicted in Figure 7, Figure 8 and Figure 9 for each frequency considered in the sinusoidal disturbance cases; the reference values shown correspond to the solutions obtained from the steady-state problem. Also, Table 7 summarizes the results obtained for each disturbance considered. Table 7. Optimal solution results of the simultaneous design and control with sinusoidal and step disturbances

Column Diameter £¤ Stage height ¥¦ Downcomer height ¥§ Feed flow rate ¨©

Units [m] [m] [m] [mol/min]

Reference Reboiler Duty ª«¬¨ Molar reflux ratio «««¬½ Entrainment flooding? Downcomer flooding? Weeping?

«¬½

»

Sinusoidal disturbance · = 1/2 · = ¹ · = 3/2 0.1100 0.1106 0.1113 0.0500 0.05 0.0500 0.0166 0.0166 0.0166 5.337 5.277 5.234

Step disturbance 0.1295 0.0500 0.0166 6.257

[kJ/min]

276.72

276.86

279.43

309.70

[-]

2.925 No No No

2.958 No No No

3.020 No No No

2.792 No No No

[(mol/mol)2]

1.96E-8

6.46E-8

2.02E-8

1.18E-5

[(kJ/min)2]

393.949

327.25

220.202

983.24

[-]

0.383

0.1707

0.093

0.032

[$/hr] [$/hr] [$/year] [$/year]

13.262 21.412 10,034 102,109

13.089 21.263 10,034 100,712

13.004 21.163 10,034 100,000

13.479 23.728 10,040 103,271

ETBE bottoms composition tracking

Z^ , − ,, b l¼

»

º

Reboiler duty quadratic deviation

Z [ª«¬¨ − ª«¬¨, j l¼

»

«¬½

º

Reflux quadratic deviation

Z^«««¬½ − «« b l¼

º

Profit ETBE Operating cost Annualized Investment cost Annualized Profit

The dynamic response of the SDCP solution is not similar to the sequential design and control approach: in the simultaneous approach, the response in the manipulated variables oscillates with


29

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the same frequency considered in the disturbances. Since the reference values for these variables are calculated while taking into account design and process dynamics decisions, the variability observed in these variables is less when compared to the sequential design approach. For instance, the abrupt responses in the manipulated variables obtained in the sequential design and control approach were not observed in the responses from the simultaneous approach. Moreover, the geometrical parameters obtained for the simultaneous approach differ from the sequential solutions. The column’s specifications resulted in a smaller diameter and a lower feed flow rate compared to the sequential solutions. As shown in Table 4 and in Table 7, the reference reflux ratio is higher and the reboiler duty is lower in the simultaneous approach than that observed in the sequential approach. Investment costs for the former are lower than any of the cases presented on the latter, as reported in Table 7 and in Table 6, respectively. The profits from the ETBE sells are higher for the majority of cases of sequential design and control, but the operating costs are also higher. Thus, the annual total profit in the simultaneous approach is higher than the nominal-case and the conservative-case

sequential solutions except for the case where = 0.5, where it is slightly lower than the nominal-case (sequential). For the sinusoidal disturbances in the SDCP, it was obtained a 10.8%, 13.5% and 14.6% detriment in the total annual profit compared to the conservative-case

sequential design and control when solving the problems for the frequencies = 0.5,1,1.5, respectively. The difference with the nominal case was always below 2%.

As shown in Figures 2 through 4 and 7 through 9 for both the sequential and simultaneous approaches respectively, the molar reflux ratio and the reboiler duty change over time in order to meet the purity constraint, which is satisfied at all times. This constraint is active during most of the time horizon for two reasons: there is a penalty term for deviations from this minimal ETBE


30


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composition in the objective function, and, there are economic considerations in the objective function. Both of them are minimized and produce a design that maintains the composition at its lower limit during operation. Even though the control actions in the manipulated variables have a sinusoidal trend, they are not in phase with the disturbance. This may be due to the mass inertia of the system; which is the accumulation of mass in the column through time, and the nonlinearity of the reaction coupled with the separation.

Figure 7. ETBE in bottoms and manipulated variables profiles for the simultaneous ODCP solution to the sinusoidal disturbance and frequency ω=0.5

The reflux ratio and the reboiler duty adjustment over time imply changes on the flows and on the compositions throughout the column. These changes in flow rates have an effect on other key variables such as the stage pressure, the compressibility factor or the reaction rate. The resulting profiles are shown in the Supporting Information.


31

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The tray capacity constraints are satisfied for the simultaneous approach; however, some of them were violated for the sequential approach, as shown in Figure 6. This result is reasonable since the geometrical parameters in the sequential approach were obtained without prior consideration of the disturbances and process dynamics. On the other hand, the simultaneous approach results satisfy the tray capacity constraints and the dynamic equations, obtaining geometrical parameters that can accommodate the sinusoidal disturbances at different frequency contents.


32


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Regarding the process controllability, the tracking objectives were evaluated at the solution for each approach. The nominal-case in the sequential approach returned the best tracking objective for the controlled variables. The simultaneous approach returned the lowest values for the tracking in the manipulated variables. Note that the reference values were decision (optimization) variables in this approach whereas on the sequential approach those values were fixed according to the optimal steady-state design results. Step disturbance

A summary of the results obtained for the case of a step disturbance is presented in Table 7. The profiles in the manipulated variables and the ETBE molar composition in the bottoms for the simultaneous approach under the effect of a step disturbance are shown in Figure 10.


33

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Figure 10. ETBE in bottoms and manipulated variables profiles for the SDCP solution to the step disturbance.

The manipulated variables’ response to the step disturbance is affected by an abrupt change in the last time steps to benefit the economic objective in both the SDCP and the sequential design and control approach, where the disturbance highlights the dynamic interaction of the manipulated variables and the feed composition. As the amount of the inert n-butene in the feed increases, the reboiler duty has to increase to satisfy the purity constraint of the ETBE in the bottoms. At the same time, the reflux ratio increases to maintain most of the isobutene inside the column to produce ETBE and it slowly decreases to remain closer to its reference value. The simultaneous design and control solution to the step disturbance resulted in a column with a larger diameter than any of the sequential approach solutions considered in this study. This large diameter is required to hold the tray capacity constraints with the large reference reboiler duty and reflux ratio obtained. Both the profit for the ETBE and the operating cost obtained for the simultaneous design and control solution are higher than those of the sequential-case nominal


34


Page 36 of 63

scenarios but lower than those of the conservative-case scenario. As the investment cost is proportional to the column diameter, the simultaneous approach converged to a slightly higher investment cost than the two sequential case scenarios. The annualized profit for the present scenario is 23.9% higher than the nominal-case scenario. The same trend from the sinusoidal disturbance is obtained with the step disturbance in terms of the tracking objectives, i.e. the conservative-case sequential approach returned the largest value in the tracking objectives of the manipulated variables whereas the simultaneous returned the lowest. Note that the nominal case returned the lowest ETBE bottoms set-point tracking objective. The tray capacity constraints, hole velocity and over weir pressure for the critical stages are shown in Figure 9. The conservative-case sequential solution is beyond the tray capacity limits in terms of weeping. Weeping occurs on the last time steps in the ninth stage for the conservativecase and the nominal-case sequential design and control with all the disturbances considered in this work. On the other hand, none of the simultaneous design and control solutions violated the tray hydrodynamic constraints. Note that the overweir pressure in the second stage for the sequential cases is far from the upper limit given by the downcomer flooding. On the other hand, taking the hydrodynamic constraints into account in the optimal design and control problem allowed the design parameters to change following an economic criterion but still able to satisfy the dynamic operability constraints. The model hereby proposed is more comprehensive than the models used previously35; thus, there are differences between the models’ results. Without an experimental analysis, it is not possible to determine which of them describes the actual column’s behavior more accurately. Based on experimental data, a model could be validated and used in the future without the need for any further experiments.


35

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The latter condition derives from the fact that this comprehensive rigorous dynamic model requires more input parameters than a simplified one. Despite including a thoughtful description of the various phenomena occurring in the tower, the parameters’ adjustment may be linked to the incorporation of noise and errors propagation along the results.

Conclusions In this work, a comprehensive mechanistic model based on fundamental principles for a catalytic distillation column has been developed. Using large-scale NLP algorithms, the optimization of an ETBE production catalytic distillation column was performed. The rigorous dynamic model presented in this work included hydraulic and product specification constraints, as well as considerations of both liquid and vapor phases behaving in a non-ideal fashion. The inclusion of these phenomena allows for a more detailed modeling of the key characteristic of this process, which can eventually help to avoid malfunctioning of the column. An optimal design optimization formulation aimed at minimizing the process economics was considered. The results from this problem were compared to those reported in the literature that have used less detailed mechanistic models. The results proved the importance of considering key phenomena for the specification of more attractive design and control decisions. Results for the optimal control of the system subject to sinusoidal show that the reaction rate seems to be strongly affected by the disturbance in the feed compositions. The dynamic optimization control problems presented in this work considered tracking and economic objective functions. To optimize both contradictory objectives, a weighted sum of these terms was considered as the objective function. A methodology based on an offline utopia tracking multi-objective optimization was developed to determine the weighting parameters in the objective function.


36


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The simultaneous design and control of the column was also considered. The results showed that the sequential approaches were not able to satisfy the design specifications, even if the design was performed under the conservative-case scenario. This weakness may be overcome by using overdesign design factors at the expense of higher investment costs, which would represent a suboptimal solution for the optimal design problem. On the other hand, the simultaneous approach solution satisfied the design specifications for the entire time horizon without sacrificing economic profitability. Future work can head in different directions. The first is related to the simultaneous design and control formulation, where improvements and refinements on the optimization formulation can be made. Also, the simultaneous design and control formulation can be extended to consider integer structural decisions in the analysis. Moreover, future work can also be focused on the implementation of advanced control strategies, such as Nonlinear Model Predictive control (NMPC). Furthermore, other multi-objective optimization alternatives can be considered to improve the weighting parameter estimation proposed in this study.


37

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Appendix A – Conservation balance equations The following equations model the dynamic behavior of a CD column with one single reaction. The equations represent total mass balances, component molar balances (both, with or without the chemical reaction), energy balances phase equilibrium and summation equations (i.e. MESH equations).

Total mass balance

." 1 = >"¾: − =" ¿1 + À = 0, ∀ ∈

." = G" + ="9: + >"¾: − =" − >" , ∀ ∈

." = G" + ="9: + >"¾: − =" − >" + SD*+," ℛ" Z Â# , ∀ ∈ #∈7

(A.1.a)

(A.1.b)

(A.1.c)

." = ="9: − =" − >" = 0, ∀ ∈

(A.1.d)

.",# 1 = >"¾: !"¾:,# − =" %",# ¿1 + À , ∀ ∈ , ∀ ∈

(A.2.a)

Partial mole balance

.",# = G" ",# + ="9: %"9:,# + >"¾: !"¾:,# − =" %",# − >" !",# , ∀ ∈ , ∀ ∈

.",# = G" ",# + ="9: %"9:,# + >"¾: !"¾:,# − =" %",# − >" !",# + SD*+," ℛ" Â# , ∀ ∈ , ∀ ∈

(A.2.b)

(A.2.c)

.",# = ="9: %"9:,# − =" %",# − >" !",# , ∀ ∈ , ∀ ∈

(A.2.d)

Ã" 1 = >"¾: /,"¾: − =" ¿1 + À − h7E" , ∀ ∈ 2,"

(A.3.a)

Energy balance

Ã" = G" Ä," + ="9: 2,"9: + >"¾: /,"¾: − =" 2," − >" /," , ∀ ∈ ∪ Ã" = hcdi + ="9: 2,"9: − =" 2," − >" /," , ∀ ∈

(A.3.b)

(A.3.c)


38


Page 40 of 63

Note that the reaction does not add any term to the energy balance since the reference status of the enthalpies is 298K, which makes the heat of reaction equal to the difference of the formation enthalpy of the components 54. Internal energy

Ã" ≅ ./," /," −

Thermodynamic equilibrium

!",# = $",# %",# , ∈ , ∀ ∈

$",# =

Summation equations

(" + .2," 2," , ∀ ∈ 0̅/,"

(#)*+ ,",# , ∈ , ∀ ∈ (" -",#

Z^!",# − %",# b = 0, #∈7

∈

(A.4)

(A.5) (A.6)

(A.7)

Concerning the reaction term referred to in Equation A.2.c, it is based on the Langmuir, Hinshelwood, Hougen, and Watson (LHHW) pseudo-homogeneous kinetics model, which considers the kinetic factor, the driving force of the reaction and the adsorption over the catalyst. This mechanism assumes two active adsorption sites for the ethanol and one for the isobutene. Taking into account that the liquid is strongly non-ideal, the reaction rate is expressed in terms of the activity of the components ,% instead of the molar composition. The kinetic expression was

obtained by Jensen et al. 67 and described below. ℛ=

,%CJ\C À ÇCÉ ,%C+uÊ 1 + ÇË ,%C+uÊ

ÇK*+d ,%CJ\C ¿,%È\ −

(A.8)

where the reaction rate constants are given by the following expression. ln ÇCÉ = 10.387 +

4060.59 − 2.89055 ln − 0.01915144 + 5.28586 ∗ 109 − 5.32977

∗ 10

9Ð

(A.9)


39

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ÇK*+d = 2.0606 ∗ exp

−60.4 ∗ 10

ln ÇË = −1.0707 +

1323.1

(A.10)

(A.11)

where is the temperature in Kelvins, and is the ideal gas constant. The computation of this

reaction rate has to be made in each reactive stage independently.

Further information regarding the ETBE reaction mathematical model is available in the Supporting Information.


40


Page 42 of 63

Appendix B – Tray hydrodynamic constraints The tray hydrodynamic constraints are included to correlate geometrical parameters of the internal design of the column. These constraints are included such that we have a feasible design given the geometry of the trays and to make sure that the operation of the distillation column is done within hydraulic capacity limits. Being inside these limits avoids effects such as entrainment flooding, downflow flooding and weeping-dumping. Geometrical relations

The geometrical relations used in this work are based on the sieve tray design stated by Gomez et al.54 and mentioned in optimal steady-state design formulation section. It can be noted that the holes are located in the corner of equilateral triangles, with a distance

denoted as the Iℎ. For further considerations in this work, the hole diameter and the

Iℎ are constant and satisfy the design ratio between 2.5 and 5.

Based on the arrangement shown in Figure B.1, the following geometrical relationships can be

obtained. Some building issues like the stage height or the downcomer height are determined from sizing correlation obtained by Kister 55 and Douglas 68.

Figure B.1. Diagram of plain and elevated view of the sieve-tray 69.


41

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Hole area

Active area

Downcomer area

Ô = 0.907Ô* Iℎ

(B.1)

Ô* = Ô J − 2ÔÕ7

(B.2)

Õ À Ö − sinÖ 2 =Ø Ö = 2 arcsin ¿ À Õ

ÔÕ7 = 0.5 ¿

Weir Length

Stage height

Weir height

Tray Capacity Limits

(B.3) (B.4)

=Ø = 0.767

(B.5)

J ≥ 1.15 Z ℎ5

(B.6)

ℎ5 ℎ5 ≤ ℎÙ ≤ 20 3

(B.7)

"∈_

The tray capacity is limited by several hydraulic undesirable effects, which affect the efficiency of the trays and the successful operation of the distillation column. Since one of the assumptions of the models described in this work is that there is thermodynamic equilibrium in the column, these undesirable effects must be avoided. The undesirable effects analyzed here are the entrainment flooding, the downcomer flooding and the weeping. The vapor and liquid flowrates are related with the geometrical parameters of the column (e.g. the column diameter, the hole area) resulting in the phase velocities through several parts of the tray, which are the variables to be compared in order to prevent these undesirable effects.


42


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Entrainment flooding Flooding is the excessive accumulation of liquid in the column. The entrainment flooding occurs when the upward vapor velocity is high enough to suspend liquid droplets and carry them to the upper trays. The vapor velocity passing through the tray is defined as: H/," =

>" , ∈ + Ô J 0̅/,"

(B.8)

The maximum allowed vapor velocity through the plate to avoid this behavior is given by: 02," − 0/," `*Ú H/," = )ie Û , ∈ + 0/,"

(B.9)

where the capacity parameter is given by the Kister and Hass correlation 54 as: )ie = 0.37

Ü" 02,"

m.:

0/," 02,"

m.:

¿

ℎ5 m. À ℎDÝ

(B.10)

where ℎDÝ is the height of the clear liquid at the transition from froth to spray regimes, and is

given by:

ℎDÝ =

0.157m.Ð 9m.Þß

1 + 1.04 ∗ 109à ¿

996 9:.Þß: 02,"

=" m. ß À 0̅2," =Ø

m.ß: m. ¿:9 À n

(B.11)

The liquid velocity passing through the downcomer is defined as: H2," =

=" , ∈ + ÔÕ7 0̅2,"

(B.12)

The maximum allowed liquid velocity to prevent entrainment flooding is given by: `*Ú H2,"

Downflow flooding

02," − 0/," = Ü" O 02,"

:/à

, ∈ +

(B.13)

The downflow flooding or downcomer flooding is when the column cannot handle the large amount of liquid available. To prevent this issue, excessive backup should be avoided.


43

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The pressure balance equation, obtained by Cicile 68 is given by: ℎØ + ℎ5 ≥ ℎ2 +

∆( + ∆(Õ7

O^02," − 0/," b

(B.14)

where ℎÝ is the height of the liquid over the tray, determined as the sum of the stage weir

height and the weir height crest.

ℎ2 = ℎØ + ℎEÙ

The pressure drop across the downcomer, called ∆(Õ7 is given by ∆(Õ7 = 1.6202," á

=" >" + â 0̅2," É 0̅/," É]

(B.15)

(B.16)

where É and É] are the downcomer cross areas as depicted in Figure B.1.

Rearranging the terms in the pressure balance, the pressure drop of the liquid over the tray is

constrained by the following term.

Weeping

`*Ú (EÙ = ℎEÙ O^02," − 0/," b ≤ (EÙ = ℎ5 O^02," − 0/," b − ∆( + ∆(Õ7

(B.17)

When the vapor velocity through the tray holes is too low, the liquid starts draining through them. To avoid this phenomenon, the vapor velocity through the holes, defined as: >" , ∈ + Ô 0̅/,"

(B.18)

0.68 ± 0.12 , ∈ + 0/," ä0 Oℎ

Simultaneous design and control of catalytic distillation columns using

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