Robust Design and Control of Extractive Distillation Processes under

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Robust design and control of extractive distillation processes under feed disturbances Hesam Ahmadian Behrooz Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b00004 • Publication Date (Web): 22 Mar 2017 Downloaded from http://pubs.acs.org on March 27, 2017

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Industrial & Engineering Chemistry Research

Robust design and control of extractive distillation processes under feed disturbances Hesam Ahmadian Behrooz 1

Chemical Engineering Faculty, Sahand University of Technology, Tabriz, Iran

Abstract An MINLP optimization framework based on the steady-state model of the extractive distillation process for benzene/acetonitrile azeotropic mixture separation with dimethyl sulfoxide solvent is developed. Two single-end control structures are evaluated for the designed plant for the nominal feed composition. Due to the fact that the adoption of inferential control strategies cannot guarantee the specification of the products, benzene feed composition is assumed to be a Gaussian variable. Then, the optimization problem is reformulated taking the setpoints of the controllers together with the design parameters as decision

1

Corresponding author. Tel.: (+9841)33459150, E-mail: [email protected]

1

ACS Paragon Plus Environment


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variables which lead to a closed-loop stochastic optimization problem in which unscented transform is used as the uncertainty propagation tool. Investigation of the steady-state properties, as well as the dynamic responses of the optimal solution under feed composition disturbances, reveals that the use of the control scheme with fixed reflux-to-ratio for the extractive and recovery columns handles feed composition disturbances more effectively while the desired purity of the products can be maintained. Keywords: Extractive distillation; Azeotrope; Optimization; Uncertainty; Unscented transform.

1. Introduction Benzene/acetonitrile mixture is formed in the production process of 2-Picoline as a pharmaceutical intermediate 1. Acetylene and acetonitrile are used as the reactants and the reaction is catalyzed by organic cobalt where pure benzene solution is used as the catalytic agent. Economic issues regarding the benzene/acetonitrile mixture separation make it necessary to effectively separate them where the formation of a minimum-boiling azeotrope makes the use of conventional distillation process impossible for the efficient separation of these two components. Mixtures forming azeotropes cannot be separated with the desired purity using conventional distillation process due to the existence of distillation boundaries in the residue curve map of these systems 2. In the case of azeotropic mixtures, the desired separation is performed using other methods such as extractive or azeotropic distillation 3

, reactive distillation 4, pressure swing distillation 5, membrane pervaporation

molecular sieves 3.

2


6

and

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Extractive distillation and azeotropic distillation are based on the addition of a third component to the original binary mixture in order to change the relative volatility of the components. In some cases, extractive distillation represents a more energy efficient solution than azeotropic distillation. For example, Knapp and Doherty

7

stated that

optimally designed extractive distillation is the most economical process for the production of anhydrous ethanol. However, sometimes it is difficult to obtain high purities with extractive distillation 8. In an extractive distillation process, a higher boiling point solvent is fed near the top of the extraction column above the binary feed mixture. The lighter component of the azeotropic feed is withdrawn at the top of the extractive distillation column and the mixture of the solvent and the heavier component of the azeotropic feed is sent to the recovery column. In the recovery column, the almost pure solvent leaves the bottom of the column ant it is recycled to the extractive distillation column while the other component leaves the recovery column as the distillate product. The extractive distillation process has been studied from various points of views, including solvent selection, synthesis of the separation sequence and controllability issues. The main variables affecting the economics of the extractive distillation process are the type of the solvent and its flow rate, reflux ratio of the columns, number of stages, azeotropic feed location, solvent tray and recovery column feed stage. This leads to a large number of degrees of freedom to cope with using the shortcut methods 8 and needs to be dealt with using systematic procedures. Kossack et al.

8

proposed a rigorous mixed-integer optimization of the entire

flowsheet to determine the best solvent of an extractive distillation process where the separation of benzene and methanol was studied. De Figueiredo et al.

9

presented a

systematic procedure to obtain the optimum condition for extractive distillation columns using a process simulator and the dehydration of aqueous mixtures of ethanol using 3



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ethylene glycol solvent was studied. García-Herreros et al.

10

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formulated a two-level

strategy that combines stochastic and deterministic algorithms to solve the mixed-integer nonlinear programming (MINLP) problem to find the optimal design and operating conditions of the extractive distillation of ethanol using glycerol as the entrainer. Shen et al.

11

presented a systematic approach for entrainer screening for separating maximum-

boiling azeotropes using thermodynamic feasibility analysis and a fuzzy logic approach. Control of maximum boiling systems such as acetone/chloroform boiling systems such as butanol/water Ramos et al.

14

13

12

and minimum

have been reported previously in the literature.

studied the effects of solvent content on the controllability of the

extractive distillation process for anhydrous ethanol production in the presence of feed disturbances. Zhang et al.

15

investigated the design and control of extractive dividing-

wall column for separating ethyl acetate/isopropyl alcohol mixture. Design and control of an extractive distillation system for tetrahydrofuran dehydration with ethylene glycol as the entrainer was the subject of the work by Fan et al.

16

and it was concluded that the

control structure with fixed reboiler heat duty-to-feed flow ratio shows a satisfactory performance under large disturbances in feed conditions. Wang et al.

17

studied the

effect of solvent flow rate on the controllability of the extractive distillation separation of n-heptane/isobutanol minimum-boiling azeotropic mixture. Application of extractive distillation in the separation of the benzene/acetonitrile mixture was studied by Yang et al.

1

from the viewpoint of the steady-state economic

design using a sequential iterative optimization procedure and dynamic controllability. Due to the impact of the dynamic controllability of the process on the economics of the process, the dynamic closed-loop performance of three different control structures including two “single temperature control” structures and a “dual temperature control” structure, were compared under feed composition and feed flow rate disturbances. The main objective was to maintain the purity of the products despite the imposed 4


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disturbances. It was concluded that under the same disturbances, the “dual temperature control” structure has better performance in maintaining the purity of the products compared to the “single temperature control” structures. It is worth mentioning that although a composition controller is capable of maintaining the desired purity, however, it introduces larger dead-times into the control loop compared to a temperature controller and slower responses are observed. On-line measurement of the concentration of the product is required to close the composition controller loop that is the other restriction of a composition controller as it is not always available. Accordingly, it is advantageous to use a tray temperature control loop instead, which makes it more appealing for industrial applications and also has faster control loop responses. However, adoption of an inferential control approach and using temperature controllers with set-points obtained using a deterministic cost optimization problem cannot necessarily guarantee the desired product purity. This issue is the main motivation of this work and we are seeking for optimal set-points for the whole control system along with the appropriate design parameters that can improve the robustness of the process against the imposed disturbances. Accordingly, the present paper addresses the optimization of extractive distillation separation plants under the azeotropic feed composition disturbances with emphasis on how to formulate the optimization problem to ensure the product quality and operational constraints as well. The steady-state simulation model of the process is used in the proposed optimization framework in order to obtain the design parameters as well as the set-points of the corresponding inferential control system whose effect is implemented in the steady-state model using the “Design spec/vary” utility in Aspen Plus. The resulting control system is capable of keeping the mass purity of the products in the distillate streams above the required values whilst the feed composition disturbances are imposed on the plant. 5



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The proposed method is illustrated by numerical examples on the separation of benzene/acetonitrile mixture by employing an extractive distillation system using DMSO as the solvent where the mass fraction of the benzene in the feed mixture is assumed to be a Gaussian random variable with known mean and variance. The unscented transform

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is used as a means for the propagation of the mean and variance of the

uncertain parameter through the simulation model of the plant to estimate the mean and variance of the objective function and constraints. A stochastic optimization model is formulated that can provide both the optimal configuration and operating conditions of the plant that can show robust performance under the unmeasured disturbances imposed by feed composition while the operational constraints are respected. For the final designs, the performance of two conventional “single-temperature control” schemes for an extractive distillation process are compared under feed composition disturbances against their robust counterparts in order to investigate the closed-loop performance of the solutions obtained based on the stochastic optimization of the steady-state model of the plant. In Section 2, the separation of benzene and acetonitrile using extractive distillation is discussed in detail. In Section 3, a deterministic optimization problem for the design and operation of the benzene/acetonitrile separation plant is formulated and the optimal design for various feed compositions are reported. The effects of the feed composition uncertainty are the subject of Section 4, where the optimization problem is reformulated in order to account for the uncertain feed compositions. In Section 5, the results are discussed for various control structures and their performance is evaluated under feed disturbances. Finally, the conclusions are drawn.

2. Problem statement

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Benzene and acetonitrile are two molecules which form a minimum-boiling azeotrope as repulsion forces are strong. Their normal boiling points (benzene = 353.24 K and acetonitrile = 354.8 K) are very close, however, they have considerably different molecular weights (benzene = 78.11 kg/kmol and acetonitrile = 41.05 kg/kmol). Figure 1 gives the Txy diagram for benzene/acetonitrile mixture at the pressure of 0.43 atm, where the azeotropic composition is 54.35 wt % of benzene and the temperature is 322.34 K. Among the various solvents including dimethyl sulfoxide (DMSO), phenol and sulfolane investigated by Yang et al.

1

to be used for the separation of

benzene/acetonitrile mixture using extractive distillation, DMSO was found to be the one with a higher relative volatility while no additional azeotrope is introduced in the system. The flowsheet of the benzene/acetonitrile separation plant using a conventional extractive distillation process with the aid of the DMSO as the solvent is shown in Figure 2. An azeotropic feed mixture composed of 65 wt % of benzene and 35 wt % of acetonitrile (that we subsequently refer to as the nominal case) is introduced in the lower part of the extractive distillation column C1 while the essentially pure DMSO solvent is recycled from the solvent recovery column C2 and is fed in the upper part of the column C1. The fresh feed flowrate is 3500 kg/hr with the temperature of 322 K. In the column C1, benzene with the mass purity of 99% is produced as the distillate stream while the mixture of acetonitrile and DMSO is sent to the column C2. In the column C2, distillate stream is a 99.9 wt % pure acetonitrile product while the 99.9 wt % DMSO solvent is recovered as the bottom product and after being cooled to the temperature of 322 K by the heat exchanger “Hx”, it is recycled back into the extractive column. The process is simulated using the Aspen Plus platform where the non-ideality of the liquid phase is modeled using the Wilson activity model

1

while the vapor phase is

assumed to be ideal. As discussed by Yang et al. 1, the operating pressures of both 7



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columns are determined in a manner that cooling water can be used in the condensers. Accordingly, the top pressure of C1 and C2 are 0.43 atm and 0.33 atm, respectively.

3. Deterministic optimization problem formulation After defining the structure of the extractive distillation process, the optimal value of the design and operating variables are determined based on a deterministic steady-state optimization problem whose details will be discussed as follows. The temperature, pressure and mass flow rate of the azeotropic feed stream are assumed to be constant, however, it can vary in composition. First, the optimization problem is solved for various values of the benzene mass purity in order to investigate the effect of azeotropic feed composition on the final optimum design. Then, the optimization problem will be reformulated in Section 4 considering the benzene wt % as a Gaussian random variable with known mean and variance to obtain robust solutions under azeotropic feed composition disturbance.

3.1

Decision variables and constraints

The decision variables of an extractive distillation plant as summarized in Table 1 can be divided into two categories including design and operating decision variables. The design variables include the total stage number of each column, feed and solvent stages of the extractive distillation column and feed plate of the recovery column. However, it is easier to assume that the extractive distillation column is composed of three sections separated by the feed stages. These sections include rectification, mid-section or extractive and stripping sections where the number of stages in each section as shown in Figure 2 is considered as a decision variable. Accordingly, the azeotropic feed and the

recycled solvent are fed to the stages ( + + 3) and ( + 2) of the extractive column

(using Aspen Plus tray numbering with the reflux drum as stage 1), respectively. For the 8


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solvent recovery column, the number of stages above ( ) and below ( ) the feed tray are considered as decision variables.

The choice of the solvent requires a careful screening as it can strongly influence the economics of the process

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. After deciding about the appropriate solvent (i.e., DMSO),

the solvent mass flow rate is one of the critical operating variables and the solvent-tofeed ratio ( /) is chosen as a decision variable. The mass flow rate of acetonitrile in the bottom stream of C1 ( ) is considered as

an operating decision variable and reboiler heat duty is manipulated to achieve the

desired value using a “Design spec/Vary” functionality in Aspen Plus. The reflux ratio of C1 is also considered to be a decision variable.

On the other hand, wt % of DMSO in the bottom product of C2 is fixed at the constant value of 99.9% as the recycled solvent should be essentially pure one and the reflux ratio is varied to reach this targeted value, which eliminates one operating degree of freedom regarding the C2 while reboiler duty is the other decision variable.

The selection of the aforementioned operating decision variables is made based on

the desire to decrease the challenges regarding the convergence issue as the two columns in the plant under study are coupled through a recycle stream, which makes obtaining converged solutions for the simulations a very challenging task. Accordingly, , and are considered as the decision variables.

The diameter of the two distillation columns can be calculated based on the maximum

flooding velocity criterion

20

designed at 80% of the flooding velocity using the “Tray

Sizing” utility in Aspen Plus considering sieve trays. However, the assumption of a constant pressure drop on each tray can lead to a discrepancy between the steady-state and dynamic models. Accordingly, column diameters are assumed to be decision variables and the column pressure profiles are updated using the Aspen Plus “Tray

9



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Rating” utility with fixed top pressure. Accordingly, minimum column diameter requirement due to flooding restriction should be considered as a constraint stated as: − ≤ 0

(1)

where is the minimum column diameter required to operate at 80% of the

flooding velocity in the column.

The above discussed decision variables which are summarized in Table 1, define together the design and operating decision variables of the extractive distillation process for the separation of benzene and acetonitrile. It is also noted that the variables to

are discreet while all other variables are continuous. Accordingly, the resulting optimization problem is an MINLP problem and the “NOMAD” optimization package

21

is

used to solve the formulated MINLP problems. In order to make sure that the final products are reusable, two constraints regarding the minimum purity of the products are considered. The wt % of the benzene product in the distillate of the column C1 and the wt % of the produced acetonitrile as the distillate of the column C2 should be equal or greater than the desired product purities of 99 wt % and 99.9 wt %, respectively. The reboilers are designed to operate below the approximate critical heat flux

( ⁄" )%&' of 32 kW/m2 to prevent film boiling 20. A very small amount of DMSO solvent

is lost at the extractive and recovery columns top streams and the “Calculator” tool provided in Aspen Plus is used to calculate the flow rate of the makeup DMSO added to recycle solvent before it is fed to the column C1.

3.2

Objective function

For

the

simultaneous

optimization

of

the

design

and

operation

of

the

benzene/acetonitrile extractive distillation process, the objective is the minimization of

10


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the total annual cost (TAC) of the plant. TAC is calculated as the sum of the operating costs (i.e., utility costs) and annual capital investment (i.e., investment for column shell and heat exchangers) as shown in Eq. (2), where the annual capital investment is defined as the capital cost divided by a three year payback period with an operating time of 8000 hr per year. (")($⁄+,) =

)/012/3 4562 + " _ T = U> V = X F − _ X > X _ − ^ W

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(6)

It is also worth mentioning that the optimization frameowrk is implemented in Matlab and the required communications between the optimizer (i.e. Matlab) and the simulator (i.e., Aspen Plus) are provided through Microsoft Excel.

3.4

Optimization results for various feed compositions

The main target of this work is to find a control structure which can maintain the specified purity levels in both product streams when changes in feed composition occur. As the first step, the temperature, pressure and mass flow rate of the azeotropic feed stream are assumed to be constant and through the proposed MINLP deterministic optimization framework in Section 3.3, different plants are designed for a range of feed compositions including: 55, 60, 65, 70, and 75 wt % benzene. The optimization problem is solved for deterministic cases and both design and operating conditions are summarized in Table S2 for various values of the benzene wt % in the feed mixture. It is worth mentioning that the run time of each deterministic optimization problem on an Intel Core i7-4770 PC, 3.40 GHz and 8 GB RAM is about 20 minutes corresponding to approximately 3000 to 3500 simulation runs. The operating pressure of the columns leads to approximate reflux drum temperatures of 328 K and 323 K in C1 and C2, respectively, and cooling water can be used in the condensers. In column C1, as the bottom temperature is in the range of 385405 K, low-pressure steam is used in the reboiler while medium-pressure steam is required in the reboiler of the C2 due to a higher temperature (~428 K).

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Optimization of the steady-state economics of the plants while satisfying the desired purities, requires extractive columns with different numbers of trays, different reflux ratios and also the variations of the feed composition alter the composition profile of the columns leading to the change of the optimal feed trays. The investigation of the results obtained using the steady-state model of the plant can reveal some guidelines in the design of the control structure. The results of the feed sensitivity analysis

22

are given in

Table 2 where the reflux ratio (), reflux-to-feed ratio (/) and % change from their corresponding nominal values are compared for each feed composition. In a “single-temperature control” structure, the most effective alternative among the two commonly used control structures

23

including: (1) a constant reflux ratio or (2) a

constant reflux-to-feed ratio, should be selected. This selection can be made considering the variations of the reflux ratio and reflux-to-feed ratio with feed composition. Accordingly, as the changes in the reflux-to-feed ratio are smaller than those of the reflux ratio, maintaining a constant reflux-to-feed ratio is the preferred choice to bring the product purities closer to their specified values under both feed composition and feed flow rate disturbances. However, both reflux ratio and reflux-to-feed ratio show significant changes which indicates the inefficiency of the “single-end control” and that a “two-end control” scheme is required for efficient disturbance handling 22. In conclusion, it can be stated that a “single-temperature” control scheme with fixed reflux-to-feed ratio can generate better regulatory control performances compared to the fixed reflux ratio while a “two-temperature control” scheme outperforms both of the “single-temperature” structures. Despite the fact that these conclusions are made using a steady-state feed sensitivity analysis, however, they are validated in the work of Yang et al. 1 where the performance of the three mentioned control structures are compared in the face of feed composition and feed flow rate disturbances.

13



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4. Azeotropic feed composition uncertainty 4.1

Introduction and formulation

Column temperature profile stabilization using tray(s) temperature measurement(s) with the aid of either a single or dual temperature control scheme, can not necessarily guarantee the product quality. As will be shown in Section 5, the inferential control of the composition of the products using a single temperature control structure is not efficient under feed composition disturbances and the degradation observed in its performance should be compensated in order to ensure robust operation. Due to the impacts and importance of the feed composition disturbances in the economic operation of the plant, a systematic procedure that can explicitly consider the feed composition as a major disturbance a priori at the design step can be helpful. Accordingly, as a first step, the feed composition is assumed to be an uncertain parameter and needs to be quantified. The assumption of uncertain parameters as Gaussian random variables characterized by a known means and variances is a common and adequate assumption in process engineering (Li et al., 2008a). Accordingly, benzene wt % of the feed stream is assumed to be a Gaussian variable that transforms the formulated deterministic optimization problem in Section 3.3 into a stochastic one (Li et al., 2004), where stochastic programming methods

24

can be

applied to solve the problem. The recourse approach

25

and the chance-constrained programming

26

are the two

approaches to the stochastic programming. Due to the requirement of a model for the penalty function in the recourse approach, it has limited applications. On the other hand, chance-constrained programming technique

26

is a viable solution to incorporate the

uncertainties into the optimization framework where a user-defined probability level of

14


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inequality constraint satisfaction is used to convert the original stochastic optimization problem into deterministic nonlinear programming counterpart. Two approaches can be distinguished in the formulation of the stochastic optimization problems using the chance-constrained methodology 27: open-loop and closed-loop. In the open-loop approach, the decision variables are the control (or manipulated) variables which are implemented to the open-loop system. The drawback of this approach is that the disturbances not included in the formulation can violate the process constraints. On the other hand, in closed-loop approach, set-points of the closed-loop system are considered as the decision variables and the characteristics of the major unmeasured disturbances are explicitly included in the optimization problem formulation while minor ones can be handled effectively with the regulatory control system. In the present work, the mass percent of the benzene in the feed mixture is considered to be a Gaussian variable whose mean and variance are both known. Then, the resulting stochastic MINLP problem based on the closed-loop approach is converted to its deterministic counterpart using chance-constrained programming technique. The details of formulating the optimization problem will be discussed as follows.

Considering b as the uncertain parameter corresponding to the benzene wt % of the

azeotropic feed, the stochastic optimization problem for the synthesis of the extractive distillation plant can be formulated as: min MND, bR

H7 I 6. 2. SND, bR = 0 TND, bR ≤ 0

15


(7) (8) (9)


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where the chance-constrained programming technique can be adopted to transform

this stochastic optimization problem (denoted as H7 ) into its deterministic MINLP equivalent.

A weighted sum of the mean and standard deviation of the stochastic objective function (7) results in the deterministic equivalent objective function as: min MND, bR = min(cdff + (1 − g)hi/,dMf)

(10)

where g ≥ 0 can be used to indicate relative importance between the mean and

standard deviation of the stochastic objective function (7) in minimization. According to

the chance-constrained programming framework, each inequality constraint 1 (denoted

as > ), due to its functionality from the random variable b, should be satisfied with the probability of p. In other words:

7d> ≤ 0f ≥ p

(11)

which implies that the probability of realizing > less than or equal to zero must be

greater than or equal to p. For nonlinear models and continuous (probability density function) PDF-based characterization of random effects

28

, the stochastic constraint (11)

can be restated as the deterministic constraint (12): cd> f + l(0)hi/,d> f ≤ 0

(12)

where l is the distribution function of the standard normal variable.

4.2

Control structure of the process

Due to the adoption of the closed-loop approach as discussed in Section 4.1, an appropriate control structure should be chosen. The set-point of the controllers are included in the optimization decision variables along with the design parameters. It is

16


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also worth mentioning that as we are dealing with the steady-state model of the plant, the effect of controllers is implemented by an equivalent “Design spec/Vary” utility in Aspen Plus. As an example, the reboiler duty of a column is manipulated to set the temperature at a specified stage in order to implement a tray temperature controller whose manipulating variable is the reboiler heat input. Accordingly, the dynamic effects of the controllers are not included in the optimization, however, the dynamic performances are investigated for the final optimal solutions found by the proposed optimization framework. Two conventional control structures recommended in the practical distillation control literature 1,29 including: 1. a “single-temperature control” scheme with fixed reflux ratio (CS1), and 2. a “single-temperature control” scheme with fixed reflux-to-feed ratio (CS2) are used for the process under study including the following pairings, and their performance is compared against their counterparts who have the same structures with set-points obtained using the proposed optimization framework in order to evaluate their robustness in handling feed composition disturbances. It will be shown that despite the inefficiency of the either CS1 or CS2 “single-temperature control” scheme with the setpoints and design parameters obtained using the deterministic optimization approach, their robust counterparts which have the same structure and pairings with set-points and design parameters obtained using the stochastic optimization formulation are capable of handling feed composition disturbances properly. Conventional PI controllers are used for all flows, pressures, and temperatures and

proportional controllers are used for all level controllers with m = 2. Flow rate controllers

are PI with m = 0.5 and no = 0.3 min. Reflux drums and sumps are sized considering a height-to-diameter ratio of 2 and a residence time of 5 min when the vessels are at 50% 17



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liquid level. A 1-min dead-time element is inserted into each temperature control loop and relay-feedback tests are run on the temperature controllers to determine ultimate gains and periods to tune the controllers using the tunings proposed by Tyreus and Luyben

30

. Figure 3 and Figure 4 show the control structures CS1 and CS2 where their

details are outlined as follows: 1. Feed is flow controlled (reverse acting). 2. The total solvent flow rate is ratioed to feed flow rate. 3. The operating pressure of each column is controlled by manipulating the heat removal rate of its condenser (reverse acting). The PI settings of the top pressure

control loops for both columns are given by m = 20 and no = 12 min.

4. The liquid level in the reflux drum in each column is controlled by manipulating the flow of distillate (direct acting). 5. Sump level in C1 is controlled by manipulating the bottom flow rate (direct acting). 6. Sump level in C2 is controlled by manipulating the makeup DMSO flow rate (reverse acting). 7. Solvent feed temperature is controlled by manipulating the heat removal in the cooler “Hx” (reverse acting). 8. The reflux ratio is held constant in each column for the control structure CS1, and the reflux flow rate in each column is ratioed to its feed flowrate for the control structure CS2. In C1 and C2, the reflux-to-feed ratios are Rq / and Rq / , respectively.

9. A temperature in C1 is controlled by manipulating the reboiler heat input using the controller TC1 (reverse acting). 10. A temperature in C2 is controlled by manipulating the reboiler heat input using the controller TC2 (reverse acting).

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For the column C1, sensitivity criterion

22

is used to find which tray has the largest

change in temperature when a small perturbation (+0.1%) is made in the reboiler heat duty as the manipulated variable. The temperature profiles and the open-loop gains between C1 tray temperatures and reboiler heat input are shown in Figure 5 for various feed compositions. The optimum number of total stages is different for different feed compositions, however, the location of the tray with the highest sensitivity to the manipulated variable is constant relative to the column bottom. It has also very small temperature variations in the range of 354.62 K to 355.59 K for different compositions. The selection of the temperature control tray location for C2 is based on the slope criterion

22

where temperature shows a large change from stage to stage. Temperature

profiles and the corresponding slope values within the recovery column C2 are plotted in Figure 6 against the number of stages for various feed compositions. Temperature profiles of the column C2 show a sharp break at stage 4 for different compositions which can be used to close the temperature controller loop at the corresponding tray. However, there is another tray (stage 9) with smaller temperature variations for different compositions which also has a temperature slope that is large enough regarding the slope criterion. It has also the added benefit of being closer to the bottom of the column and can be quickly affected by changes in the heat input. Accordingly, the 2nd stage from the bottom in C2 is selected as the most appropriate control stage in the “single temperature control” loop structure to close the temperature controller TC2. In conclusion and reasoning based on the sensitivity criterion and slope criterion as shown in Figure 5 and Figure 6, 9th and 2nd stages from the bottom in C1 and C2 are selected as the most appropriate control stages in the temperature control loops TC1 and TC2, respectively.

4.3

19

Decision variables and constraints



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The total condensers of column C1 and C2 located at the top of the columns operate at constant pressures of 0.43 and 0.33 atm, respectively. Feed flow rate is controlled at the steady value of the 3500 kg/hr and the set-point of the recycle solvent temperature controller is assumed to be constant at the value of 322 K. If the controlled variables that have no steady-state effect are neglected (i.e., the bottom holdup and condenser holdup), then we are left with five control degrees of freedom for each control structure which are considered as the operating decision variables. Accordingly, the decision variables of the stochastic formulation as shown in Table 1 are: 1. Number of stages in each section ( , , F , , ).

2. Diameter of the columns ( and ).

3. Heat exchange area of the reboiler and condenser of the columns (As , As , Aq q

and Aq ). q

q

q

4. Reflux ratio of the columns in the control structure CS1 and the reflux-to-feed ratios in the control structure CS2. 5. Solvent-to-feed ratio ( /).

6. Set-point of the temperature controller of each column ((tu , (tu ). For the v

v

temperature controllers of C1 and C2, 9th and 2nd stages from the bottom stage

are selected as the control stages, respectively. The value of the decision variables of the stochastic optimization problem can be initiated using the corresponding values obtained as the solution of the deterministic optimization problem H7 for nominal condition of benzene feed composition at 65 wt %

w in Table 3. Accordingly, set-points of the temperature controllers are denoted as D initialized with the values of 355.59 K and 414.96 K for TC1 and TC2, respectively.

A feasible solution in stochastic optimization iterations should satisfy the following constraints:

20


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1. Product quality specification: the wt % of the benzene and acetonitrile products should be equal or greater than the desired purity of 99% and 99.9%, respectively. 2. Minimum column diameter requirement due to flooding restriction stated as: − ≤ 0

(13)

3. The heat duty of each reboiler should be lower than the maximum value of %&' = x " ∆( .

4. The heat duty of each condenser should be lower than the maximum value of %&' = x " ∆( .

5. The limit on the heat flux in the reboilers: reboilers should be designed to operate below the approximate critical heat flux ( ⁄" )%&' of 32 kW/m2.

Accordingly, the vector of inequality constraints can be stated as: 99 [2 % − P] Y ` 99.9 [2 % − P X _ ] > _ Y > ` X − _ X >F _ X _ − X> _ X _ X > _ X − x " ∆( _ g= X > _ = X X z_ X _ − x " ∆( X >{ _ X _ − x " ∆( X >| _ X _ X >} _ X _ − x " ∆( W>~ ^ X _ ( /" ) ≤ ( ⁄" )%&' X _ W ( /" ) ≤ ( ⁄" )%&' ^

4.4

(14)

The unscented transform as the uncertainty propagation tool

The formulation of the stochastic optimization problems using the chance-constrained technique, as it can be seen in Eqs. (10) and (12), requires the mean and variance of

21



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Page 22 of 52

both the objective function and constraints. Accordingly, a methodology is required to propagate the mean and variance of the random variables through a nonlinear function. Monte Carlo method can be used to propagate a random variable through a nonlinear function, however, it requires a large number of sample points. As an alternative solution, unscented transform

18,31

resembles a Monte Carlo method and sigma points

are used instead of random sampling that will be discussed in details as follows.

If ∈ is a vector of Gaussian random variables with known mean and covariance

matrix (i.e., (g ∈ , ∈ × )) and + = Ψ() is a nonlinear function of several random variables where + ∈ and Ψ: → , then the mean and the covariance matrix of + (i.e., Ng , R) can be estimated using the unscented transform as

discussed in the following steps 18:

1. 2f) and variance (i/,dMf, i/,d>f) of the objective function and constraints using Eqs. (18) and (19).

The stochastic MINLP formulation of the optimization problem was discussed and its deterministic counterpart was obtained using the chance-constrained programming. Then, unscented transform was discussed as the uncertainty propagation tool which can be used in the calculation of the mean and standard deviation of the objective function and constraints. Now, the proposed optimization algorithm whose scheme is demonstrated in Figure 7 can be summarized as the following steps. 1. Initialize the value of the decision variables using the corresponding values

obtained as the solution of the deterministic optimization problem H7 for the

nominal condition.

2. The value of the decision variables is sent to the steady-state simulation model of the plant where solvent flow rate and the number of stages, feed location, reflux ratio or reflux-to-fed ratio, diameter and temperature of the desired stage as the “Spec” of the “Designs Spec/vary” utility are set in each column. 3. Run the simulation model for each sigma point and estimate the mean and variance of the objective function and constraints using steps P0 to P3 discussed previously as the unscented transform framework.

24


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4. The weighted sum of the mean and standard deviation of the objective function (Eq.(10)) as well as the left-hand side of Eq. (12) are passed to the optimizer. 5. Optimizer checks the convergence criterion of the optimization method in order to finish the optimization iterations or start a new iteration if the convergence is not achieved. Accordingly, if converged, then stop. Otherwise, go to step 2 and begin a new iteration with the updated values for the decision variables.

5. Results and discussion In Section 0, the optimized flowsheets for separation of azeotropic mixtures containing 55 to 75 wt % of benzene were obtained via the steady-state optimization of the extractive distillation process. Then, two control structures (CS1 and CS2) were chosen to handle the feed composition disturbances while CS2 has a better performance in feed composition disturbance handling. Now, the set-points of these control schemes along with the design parameters will be obtained in a manner that they are capable of efficient disturbance handling. Accordingly, feed composition is considered to be uncertain where benzene composition of the feed stream is assumed to be a Gaussian

random variable whose mean is 65 wt % and its standard deviation is 3.5 wt % (i. e. , b = (75, 3.5 ) [2 %). The stochastic optimization problem formulated in Section 4.1 is

solved using the proposed method in Section 4.5 assuming p = 0.999 and α = 1, β = 2

and κ = 1 in the unscented transform formulation.

The optimal plant designed for the nominal condition with control structures CS1 and

CS2 will be denoted as P1 and P2, respectively, while their robust counterparts which have the same structure and pairings with set-points and design parameters obtained using the stochastic optimization formulation will be denoted as P1* and P2*, respectively. Also, the control systems of the P1* and P2* are denoted as CS1* and CS2*, respectively.

25



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An important assumption regarding the proposed method is the assumption of the normality of the probability distributions of the objective function and more importantly the constraints, which should be validated. This can be done through Monte-Carlo sampling and obtaining the empirical PDFs by setting the value of the decision variables equal to the solution found for each designed plant and imposing the uncertain benzene feed wt %. However, the main aim of this work is the improvement of the dynamic performance of the aforementioned control structures under the feed composition disturbances which is obtained in a stochastic optimization framework using the steadystate simulation of the plant. Accordingly, it is sufficient to impose the feed composition disturbances on the designed plants having the control structures whose set-points are obtained using the proposed framework and investigate the responses in order to validate the results. The optimal design for the nominal condition is a 45-stage extractive column where the fresh feed is introduced on the stage 29, and the solvent on stage 4 while a 11-stage solvent recovery column is used that is fed on stage 6 with the solvent circulating at the rate of 8952.9 kg/hr. The reflux ratio of the columns are 1.362 and 0.433 with the control tray temperatures of 355.59 k and 414.96 k which are the 36th and 9th stages of C1 and C2, respectively.

w∗ and D w∗ for P1* and Now, the results of the stochastic cases shown in Table 3 as D

P2*, respectively, are compared to the deterministic cases P1 and P2 designed at the nominal feed composition. As shown in Table S2 for all deterministic cases, the purities of the benzene and acetonitrile products are at the minimum values of 99 and 99.9 wt %. However, this is not the case for uncertain feed composition situations and products with higher purities

are obtained as shown in Table 3 in order to make the process robust enough to tolerate the imposed disturbances by feed composition variations. 26


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Using the appropriate design and operating conditions provides a safety margin (i.e., a distance between the calculated purity and the minimum required value) which enables us to keep up with the minimum required purities despite the feed composition disturbances while all operational constraints (such as column flooding constraint) are respected and TAC is minimized. This safety margin can be obtained by increasing the reflux ratio or the number of stages. However, the solution of the stochastic optimization problem determines the best choices where both for P1* and P2*, simultaneously increasing the reflux ratio and also the number of stages of the recovery column are the preferred choices considering TAC minimization. Accordingly, two and one extra plates are required in C2 for plants P1* and P2*, respectively. In P1*, four extra plates are included in the column C1 while no additional plates are required in P2*. The safety margin can be increased by means of increasing the probability of the constraint satisfaction (p) or the standard deviation of the uncertain parameter which can result in control systems which can tolerate larger disturbances. The design parameters including the area required for heat transfer in the reboiler and condenser of the columns and columns diameter increase for stochastic cases. This results in a slight increase in the capital costs as well. The expected values of the TACs are also compared in Table 4 where the expected value of the TAC for the stochastic cases P1* and P2* are 9.65% and 6.94% higher compared to the nominal condition. This extra price is what we pay to have a robust plant under the unmeasured disturbances of the feed composition and gain greater operational flexibility in handling feed composition uncertainties. It is also noted that the robust counterpart of the control structure with a fixed reflux ratio has capital and operational costs 21.62% and 1.21% higher than the nominal case, respectively, while a 14.42% and 1.68% increase are observed in the capital and operational costs regarding the control structure with fixed reflux-to-feed ratio. Accordingly, capital cost is mostly 27



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affected in P1* while for P2*, greater energy consumption is expected. It is worth mentioning that these increased costs mostly originates from the modifications made in the recovery column. It is also interesting to know that stochastic formulation mostly affects the capital investment of the plant compared to operational costs while higher set-point for the temperature controller of the recovery column is required in the obtained solutions based on the stochastic formulation. Investigation of the results of the stochastic optimization of the benzene/acetonitrile extractive distillation separation problem showed its benefits over the deterministic approach from the steady-state point of view and considering TAC as the representative of the economic concerns, control structure with a constant reflux-to-feed ratio (i.e., CS2*) is the preferred choice. However, the dynamic infeasibility despite the steady-state feasibility can lead to the inapplicability of the results which makes the evaluation of the dynamic performance of the two approaches necessary. In order to investigate this issue, the performance of the control structures under feed composition disturbances for the plants designed based on expected condition and stochastic formulation are compared. Accordingly, ±5 wt % benzene feed composition disturbances relative to the nominal condition are introduced into the systems at 1 hr with design parameters and controller set-points obtained using deterministic and stochastic optimization formulations to evaluate the dynamic performance of the corresponding control structure. Table 5 lists the temperature controllers tuning parameters. The dynamic responses of the benzene and acetonitrile wt % in the product streams to disturbances imposed by 5% increase and decrease in benzene feed composition are shown in Figure 8 and Figure 9, respectively. It is noticed that the behavior of the system differs depending on whether a positive or negative disturbance is imposed.

28


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After the establishment of the new stable operating condition due to the actions taken by the control system of the P1 and P2, the mass purity of the benzene product is held quite close to the desired value of 99% while P1 and P2 perform better under positive and negative disturbances, respectively. However, this is not the case for acetonitrile product, as a large deviation from the desired value of 99.9 wt % is observed under positive benzene feed disturbances and the acetonitrile product purities are 96.35 wt % and 99.09 wt % for CS1 and CS2, respectively, showing the more favorable response of the constant reflux-to-feed ratio scheme compared to the fixed reflux ratio scheme. This conclusion was also drawn from the feed sensitivity analysis in Section 0. Despite the better performance of the constant reflux-to-feed scheme, results are still not satisfactory if the acetonitrile product specification had to meet strict requirements. Now we turn to the analysis of the control structures of P1* and P2* whose control structures have the same pairings as CS1 and CS2, however, their set-points and design parameters are obtained using the proposed stochastic optimization formulation. As shown in Figure 8 and Figure 9, the responses of the systems synthesized with the proposed stochastic approach confirm that the purities of both the benzene and acetonitrile products are held above the desired values under the positive and negative feed composition disturbances during transitions (except for the acetonitrile product under positive disturbance imposed on P1*) and also after the new steady state condition is reached. The control structure CS2* provides the faster transient responses than CS1*. Also, CS1* shows larger peak transients for the recovery column distillate composition while the violation of the acetonitrile specification occurs during transitions. Accordingly, CS2* achieves a better dynamic response than that of the CS1* with the proposed controller set-points.

29



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The dynamic responses of the concentration of the products show that considering the set-points of the control structure of the plant as well as the design parameters in the stochastic optimization problem can lead to improved performance of the controllers under the feed composition disturbances. This approach can also control the column flooding in both C1 and C2 properly which is shown in what follows. The diameter of the columns shows approximately a 9% and 20% increase for C1 and C2 compared to the nominal condition, respectively, which can provide the required safety margin to avoid flooding. The dynamic performance of the closed-loop plants can be also investigated from the viewpoint of the possibility of column flooding. Although for deterministic cases, column diameter is designed at 80% of the flooding velocity that fixes the upper limit of vapor velocity, however, under imposed disturbances this criterion may be violated and proximity to flooding should be checked throughout the columns. Investigation of the flooding approach profile of the columns at the nominal condition reveals that the recycle solvent tray of C1 and the last tray of the C2 operate at the maximum value of 80% of flooding velocity. However, for P1* and P2*, C1 has maximum values of 67.95% and 69.15% of flooding and C2 has maximum values of 57.70% and 61.10%, respectively. These safety margins can prevent the column flooding under feed composition disturbances as keeping the specified trays at their target temperatures by the control system requires the manipulation of the reboilers heat input, which can alter the vapor/liquid flows in the columns and lead to column flooding. To show this issue, the temporal behavior of the flooding approach of the recycle solvent tray of C1 and the bottom-most tray of C2 are shown in Figure 10 as they represent the maximum flooding approach of each column at any instance. In P1 and P2, as can be seen in this figure, increasing and decreasing the benzene throughput causes flooding in columns C1 and C2, respectively, which is not the case for the system obtained using the proposed stochastic optimization approach. 30


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The effectiveness of the discussed control structures in handling feed flow rate disturbances should be evaluated too. Figure 11 gives the extractive and recovery columns results for ±10% disturbances in the feed flow rate made at time equal to 1 hr imposed on P1* and P2* and as it can be seen, both product compositions are well controlled.

6. Conclusion The objective of this work was to design the extractive distillation process of benzene/acetonitrile separation using DMSO solvent with a high level of robustness against a range of feed composition unmeasured disturbances. Accordingly, a stochastic MINLP optimization problem was formulated whose solution provided the design parameters (such as the number of stages and the diameter of the extractive and recovery columns) as well as the set-points of the inferential composition control of the process. This included the solvent flowrate controller, tray temperature controllers and the reflux ratio or the reflux-to-feed ratio of the extractive and recovery columns, which can determine the optimal operating point of the plant. The optimization problem was formulated based on the steady-state model of the process and the effects of the controllers were included using the “Design Spec/Vary” feature in Aspen Plus. The benzene mass purity of the azeotropic feed was considered to be a major disturbance and it was assumed to be a Gaussian variable to explicitly incorporate the feed composition uncertainties. Feed sensitivity analysis showed that among the two conventional control schemes with fixed reflux ratio and fixed reflux-to-feed ratio, the latter has a better performance. However, it is not acceptable if we are obligated to meet strict requirements. Using the proposed framework, it was shown that it is possible to obtain robust solutions based on the “single end control” schemes which can tolerate feed composition disturbances while their deterministic counterparts are not efficient in maintaining product quality. 31



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The stochastic formulation of the design problem provides plants with a safety margin which enables us to keep up with the minimum required purities despite the feed composition disturbances which can mostly affect the capital investment of the plant compared to operational costs. Finally, it was concluded that considering TAC as the representative of the economic concerns, a “single end control” structure with a constant reflux-to-feed ratio is the preferred choice for the extractive distillation separation of benzene/acetonitrile mixture using DMSO solvent.

7. Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: ***.

Basis of cost estimation and results of the optimization for deterministic cases (PDF). 8. Nomenclature

N , N , NF , N, N , , , , q q , Q q , Q q

, Q s , Q s ( ⁄" )%&' , , q

q

,

number of trays in each column section reflux ratio of the columns reflux flow rate of the columns [kg/hr] condenser heat duty [kW]

? / / x , x " , " " ," , " , "

reboiler heat duty [kW]

,

critical heat flux in the reboiler internal diameter of column [m] minimum column diameter required to operate at 80% of flooding [m] column height [m] number of trays in a column solvent-to-feed ratio reflux-to-feed ratio heat transfer coefficient of condenser/reboiler [kW.K-1.m-2] heat exchange area of the condenser and reboilers of the column [m2]

TRD

reflux drum temperature [K]

TStm

utility steam temperature [K]

cw Tincw , Tout

cooling water inlet and outlet temperatures [K]

32


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TR

reboiler temperature [K]

PQQ , , D M S T P] P] m , no b = Ng , R cdbf i/,dbf Ng , R ∆( , ∆( 7d∙f > l p

capital cost of equipment ($)

c1 ,c2 ,c3

α, β , κ w D w∗ , D w∗ D ( , (

( ' Q' v v (tu - (tu

mass flow rate of acetonitrile in the bottom stream of C1 benzene wt % in feed stream distillate flow rate of C1 and C2 [kg/hr] bottom flow rate from C1 and C2 [kg/hr] vector of decision variables objective function simulation model of the plant inequality constraint vector wt % of benzene in wt % of acetonitrile in PID controller parameters random parameter - benzene wt % in the azeotropic feed mathematical expectation of the random variable b variance of the random variable b normal distribution with mean g and standard deviation condenser/ reboiler temperature difference [K] operator of probability computation ith inequality constraint distribution function of the standard normal variable inequality constraint satisfaction probability defined by the user (0 ≤ p ≤ 1 design parameter of unscented transform azeotropic feed flow rate [kg/hr] feed flow rate to column C2 [kg/hr] solution of the problem H7 for the nominal condition solution of the problem H7 for CS1 and CS2 temperature for stage n in column C1, C2 using Aspen plus notation [K] outlet temperature of the cooler “hx” [K] Heat duty of the cooler “hx” [kW ] set point of the temperature controller TC1-TC2 [K]

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Luyben, W. L. Distillation design and control using Aspen simulation; John Wiley & Sons, 2013. Luyben, W. L. Effect of feed composition on the selection of control structures for high-purity binary distillation. Ind. Eng. Chem. Res. 2005, 44 (20), 7800–7813. Sahinidis, N. V. Optimization under uncertainty: state-of-the-art and opportunities. Comput. Chem. Eng. 2004, 28 (6), 971–983. Khor, C. S.; Elkamel, A.; Ponnambalam, K.; Douglas, P. L. Two-stage stochastic programming with fixed recourse via scenario planning with economic and operational risk management for petroleum refinery planning under uncertainty. Chem. Eng. Process. Process Intensif. 2008, 47 (9), 1744–1764. Li, P.; Arellano-Garcia, H.; Wozny, G. Chance constrained programming approach to process optimization under uncertainty. Comput. Chem. Eng. 2008, 32 (1), 25– 45. Flemming, T.; Bartl, M.; Li, P. Set-point optimization for closed-loop control systems under uncertainty. Ind. Eng. Chem. Res. 2007, 46 (14), 4930–4942. Darlington, J.; Pantelides, C. C.; Rustem, B.; Tanyi, B. An algorithm for constrained nonlinear optimization under uncertainty. Automatica 1999, 35 (2), 217–228. Luyben, W. L.; Chien, I.-L. Design and control of distillation systems for separating azeotropes; John Wiley & Sons, 2011. Tyreus, B. D.; Luyben, W. L. Tuning PI controllers for integrator/dead time processes. Ind. Eng. Chem. Res. 1992, 31 (11), 2625–2628. Ahmadian Behrooz, H. Managing demand uncertainty in natural gas transmission networks. J. Nat. Gas Sci. Eng. 2016, 34, 100–111 DOI: 10.1016/j.jngse.2016.06.051. Van Der Merwe, R. Sigma-point Kalman filters for probabilistic inference in dynamic state-space models, Oregon Health & Science University, 2004. Kandepu, R.; Imsland, L.; Foss, B. A. Constrained state estimation using the unscented Kalman filter. In Control and Automation, 2008 16th Mediterranean Conference on; Citeseer, 2008; pp 1453–1458.



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

Figure 1. Txy diagram for benzene/acetonitrile at 0.43 atm

36


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Page 37 of 52

Hx

D2

D1 Makeup DMSO

Section 1

Benzene 99 wt %

Section 2 Azeotropic Feed 3500 kg/hr Benzene 65 wt % Acetonitrile 35 wt %

Section 4

Acetonitrile 99.9 wt %

C2

DMSO (99.9 wt %)

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


C1 Section 5 Section 3

B1

B2

Figure 2. Process flowsheet for benzene/acetonitrile/DMSO extractive distillation plant

37



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

Page 38 of 52

SP S/F

X

FC

∆T

TC

Hx

LC

PC LC

PC

X RRC2

FC

X RRC1

FC

FC

SP

C2

SP

∆T

C1

∆T

LC TC1

LC

Figure 3. Control structure CS1

38


TC2

Page 39 of 52

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


SP S/F

X

FC

∆T

TC

Hx

LC

PC LC

PC

FC FC

C1

F1

SP

C2

X RC1/F1

SP

F2

∆T

∆T

LC FC

TC1

LC

Figure 4. Control structure CS2

39

X RC2/F2


TC2


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

Page 40 of 52

Figure 5. Temperature profiles and temperature sensitivity profiles obtained using 0.1% increase in reboiler heat input for C1 for different benzene feed compositions

40


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


Figure 6. Temperature profiles and temperature slope value plots of C2 for different benzene feed compositions

41



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

Figure 7. Scheme of the proposed stochastic optimization procedure

42


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Page 43 of 52

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


Figure 8. Dynamic responses of the benzene and acetonitrile products wt % resulting from +5 wt % disturbance in benzene feed composition.

43



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

Page 44 of 52

Figure 9. Dynamic responses of the benzene and acetonitrile products wt % resulting from -5 wt % disturbance in benzene feed composition.

44


Page 45 of 52

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


Figure 10. Dynamic responses of the % of the flooding velocity in C1 and C2 resulting from ±5 wt % disturbances in benzene feed composition.

45



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Page 46 of 52

Figure 11. Dynamic responses of the benzene and acetonitrile products wt % resulting from ±10% disturbances in feed flowrate imposed on P1* and P2*. (+) positive disturbance and (-) negative disturbance.

46


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Table 1

Operating

Design

Design and operating decision variables for deterministic (H7) and stochastic (H7 ) formulation Decision variables

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


47

, , F , ,

, , F , ,

,

" , " , " , "

/

,

C S1


/

v v (tu , (tu

C S2

Rq /

Rq /


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

Page 48 of 52

Table 2. Feed composition sensitivity results

PQQ

*

% Change

55

1.752

28.64

60

1.684

23.67

65*

1.362

-

70

1.229

-9.78

75

1.208

-11.28

)

/ 0.97 3 1.02 0 0.89 4 0.86 9 0.91 5

% Change

Change

8.82

0.263

-39.29

14.14

0.303

-30.15

-

0.433

-

-2.83

0.563

29.88

2.39

0.713

64.65

Nominal condition

48

%


)

/ 0.04 5 0.04 7 0.05 1 0.05 6 0.06 0

% Change -13.13 -8.17 9.47 17.42

Page 49 of 52

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


Table 3. Results of the optimization for nominal condition and stochastic cases w ( 5, ) D

2

F

25

15

17

14

4

4

4

4

6

5

%]

99.00

99.32

99.38

%]

99.90

99.96

99.93

1.362

1.411

1.447

0.433

0.490

0.445

0.8940

0.9231

0.9459

0.0514

0.0595

0.0531

0.974

1.059

1.060

0.761

0.929

0.901

2.558

2.498

2.554

355.59

356.09

354.21

414.96

424.59

424.04

62.51

71.70

69.05

82.88

113.67

104.41

47.27

68.41

63.20

29.75

43.12

39.86

7.24

8.80

8.28

3.43

3.47

3.48

5.84

6.40

6.24

R / q

R / [m] q

[m]

v (tu v (tu " [m2] " [m2] " [m2] " [m2]

Solvent-to-feed ratio

5

Capital cost [10 $] Operating costs [105 $/yr] TAC [10 $/yr]

49

2

25

5

3

w∗ (∗ ) D

24

P] [wt P] [wt

w∗ (∗ ) D



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

Page 50 of 52

Table 4. Capital and operating costs of stochastic cases compared to the nominal condition Nominal

% P1*

Plant

50

4.691 2.279 2.548 1.147 7.240 3.426 5.839

% Change

5.207 2.291 3.076 1.193 8.283 3.484 6.245

10.99 0.52 20.74 3.99 14.42 1.68 6.94

Change

(P1 or P2) C1 capital cost [105 $] C1 operating costs [105 $/yr] C2 capital cost [105 $] C2 operating costs [105 $/yr] Capital cost [105 $] Operating costs [105 $/yr] TAC [105 $/yr]

P2*

5.511 2.259 3.294 1.209 8.805 3.468 6.402


17.47 -0.88 29.26 5.37 21.62 1.21 9.65

Page 51 of 52

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


Table 5. Temperature controllers tuning parameters for different process P1

Plant Controlled variable Manipulated variable

Gain, m Integral time, no [min] Set-point [K] Plant Controlled variable Manipulated variable

Gain, m Integral time, no [min] Set-point [K]

51

(Fz

Q s 2.637 11.88 355.59 q

(Fz

Q s 2.465 11.88 355.59 q

(}

( '

Q s 1.040 9.24 414.96 P2 q

Q'

(}

Q s 0.867 7.92 414.96

( '

Q'

q

0.170 3.96 322

0.170 3.96 322


(~

Q s 3.205 3.204891 356.09 q

(Fz

Q s 2.614 9.24 354.21 q

P1*

(

( '

Q s 1.996 11.88 424.59 P2* q

Q'

(~

Q s 3.871 10.56 424.04

( '

Q'

q

0.169 5.28 322

0.169 5.28 322


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

Page 52 of 52

Table of Contents graphic

S/F

SP

X

FC

∆T

TC

Hx LC

PC LC

PC

SP FC

Benzene wt % = 65%±5

C1

F2

X RC1/F1

TC1

LC

52

SP

X RC2/F2 ∆ T

∆T

LC FC

FC

C2


TC2

Robust Design and Control of Extractive Distillation Processes under

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