methanol system

Accepted Manuscript Title: Robust synthesis of the pressure-swing distillation process under azeotropic feed composition disturbance—Study of the Tetrahydrofuran/Methanol system Author: Hesam Ahmadian Behrooz PII: DOI: Reference:

S0098-1354(17)30185-0 http://dx.doi.org/doi:10.1016/j.compchemeng.2017.04.022 CACE 5800

To appear in:

Computers and Chemical Engineering

Received date: Revised date: Accepted date:

18-11-2016 25-4-2017 27-4-2017

Please cite this article as: & Ahmadian Behrooz, Hesam., Robust synthesis of the pressure-swing distillation process under azeotropic feed composition disturbance—Study of the Tetrahydrofuran/Methanol system.Computers and Chemical Engineering http://dx.doi.org/10.1016/j.compchemeng.2017.04.022 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Robust synthesis of the pressure-swing distillation process under azeotropic feed composition disturbance – Study of the Tetrahydrofuran/Methanol system Hesam Ahmadian Behrooz 1

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

1

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

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

Separation of tetrahydrofuran/methanol azeotropic mixture using pressure swing distillation



A stochastic MINLP formulation for the robust synthesis of the plant



A set-point optimization problem on the basis of the steady state model



Improved inferential control system under feed composition disturbances

Abstract Inferential control strategies on the basis of the temperature measurements in the separation process of the tetrahydrofuran/methanol azeotropic mixture using the pressure-swing distillation cannot guarantee the specification of the products. Accordingly, tetrahydrofuran mol% in the feed stream is assumed to be a Gaussian variable with known mean and standard deviation, and a stochastic mixed-integer nonlinear programming optimization framework is developed that is on the basis of the steady-state model of the plant. Proportional-integral controllers are implemented using the “Design Spec/Vary” utility in Aspen Plus and the optimization problem is formulated taking the set-points of the controllers together with the design parameters as decision variables. This leads to a closed-loop stochastic optimization problem in

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which unscented transform is used as the uncertainty propagation tool. The optimal solution shows more robustness against the imposed feed composition disturbances and handles them more effectively while the desired purity of the products can be maintained.

Keywords: Azeotrope; Pressure swing distillation; Stochastic Optimization; Uncertainty; Unscented transform; Nomenclature

DHPC , DLPC

distillate flow rate of HPC and LPC [kmol/hr]

BHPC , BLPC

bottom flow rate from HPC and LPC [kmol/hr]

F

azeotropic feed flow rate [kmol/hr] THF mol % in streams BHPC , BLPC , DHPC , DLPC and F

xBHPC , x

LPC B

, xDHPC , xDLPC , xF

T T N , N HPC , N LPC

number of trays in a column

N1 , N2 , N3 , N4 , N5

number of trays in each section

xazLPC , xazHPC

azeotropic compositions of the THF/methanol mixture at the pressure of the HPC and LPC [THF mol %] composition difference between the distillate composition and the corresponding azeotropic composition [THF mol %] critical heat flux in the reboiler

 HPC ,  LPC

QR

AR max

x

vector of decision variables objective function, simulation model of the plant inequality constraint capital cost of equipment ($)

𝑓 𝒉 𝒈

C1 ,C2 ,C3

ID , ID HPC , ID LPC LPC

HPC

IDFlooding , IDFlooding , IDFlooding

AC , AR , ACHPC , ARHPC , ACLPC , ARLPC

internal diameter for column [m] minimum column diameter required to operate at 80% of flooding [m] heat exchange area of the reboilers and condenser of the column [m2]

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QC , QR

condenser/reboiler heat duty [kW]

L

column height [m] condenser/ reboiler temperature difference [K]

TRD

reflux drum temperature [K]

TStm

steam temperature [K]

cw Tincw , Tout

cooling water inlet and outlet temperatures [K]

TR

Reboiler temperature [K]

U R ,UC

heat transfer coefficient of condenser/reboiler [kW.K-1.m-2]

Kc , I

PID controller parameters

 E  

random parameter - THF mol% in the azeotropic feed

Var  

variance of the random variable 

P 

operator of probability computation

gi ˆp

ith inequality constraint

TC , TR

mathematical expectation of the random variable 

R HPC , R LPC 𝒩(𝜇𝜃 , 𝜎𝜃2 )

inequality constraint satisfaction probability defined by the user ( 0  ˆp  1 ) distribution function of the standard normal variable. reflux ratio of the columns normal distribution with mean 𝜇𝜃 and standard deviation 𝜎𝜃

x*

Solution of the problem OP1

 x0.999

P HPC , P LPC T26HPC , T21HPC , T28HPC , T26LPC

Solution of the problem OP2 with ˆp =0.999 design parameter of unscented transform column pressure [atm] temperature of a tray using Aspen plus notation

TspHPC , TspLPC

set point of a temperature controller



 ,  ,

1. Introduction

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Tetrahydrofuran (THF) is an excellent solvent for the dissolution of polar and nonpolar chemical compounds in chemical and pharmaceutical industries. For example, THF and methanol separation is encountered in the production of steroid drugs (Wang et al., 2014), where the formation of a minimum-boiling azeotrope makes the use of conventional distillation process impossible for the efficient separation of these two components. This originates from the fact that distillation boundary exists in the residue curve map of this type of azeotropic systems (De Figueiredo et al., 2011) and other effective methods such as extractive or azeotropic distillation (Figueirêdo et al., 2015), reactive distillation (Maier et al., 2000), pressure swing distillation (PSD) (Wang et al., 2016) and membrane pervaporation (Hamad and Dunn, 2002) should be considered. Application of extractive distillation for the separation of THF/methanol mixture was discussed by ZHENG et al. (ZHENG et al., 2010). However, PSD can be an interesting alternative considering the fact that unlike extractive distillation, additional components are not involved in PSD as the solvent which can increase the chance of the product contamination (Li et al., 2016). Whatever method is decided to be the most appropriate approach for the separation of an azeotropic mixture, optimal synthesis of the corresponding plant considering both design parameters and operating conditions is of great importance. However, the final design changes with feed composition variations and robust designs that can handle feed composition unexpected disturbances can lead to economic benefits, since the

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quality of the product and satisfaction of the operational constraints can be guaranteed under the imposed disturbances. Here, we are interested in an optimization framework that systematically solves the problem of the optimal synthesis of the PSD process under the uncertainties originating from the azeotropic feed composition, and the separation of THF/methanol mixture is considered as the numerical case study to evaluate the performance of the proposed method. PSD is on the basis of the change of the system’s pressure in order to shift the azeotropic composition. This can restrict the application of the PSD to the azeotropic mixtures with azeotropic composition sensitive to pressure changes (Kossack et al., 2008). PSD process employs two columns, including a high pressure column (HPC) and a low pressure column (LPC) where their arrangement is dependent on the feed composition. In the case of the minimum azeotrope, the distillate of the first column is sent to the second column while the distillate of the second column is recycled back to the first column and high-purity products are obtained as the bottom streams. The optimal synthesis of the PSD plant for the separation of THF/methanol has been discussed by Wang et al. (Wang et al., 2014). They have also presented the optimal design for two THF feed mole fractions of 25 mol% and 75 mol% using a sequential iterative optimization procedure. It was also concluded that the feed composition can alter the sequence of the HPC and LPC. Minimization of the total annual cost (TAC) showed that the optimal column sequence should be LPC/HPC and HPC/LPC for the compositions of 25 and 75 mol% of THF, respectively. They have also investigated the

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possibility of fully/partially heat-integrated schemes to reduce the energy consumption of the plant. To be able to maintain the purity of the products despite the disturbances imposed by feed composition, dynamic controllability of the PSD process (Li et al., 2013; Wei et al., 2013; Yu et al., 2012) can be extremely crucial for the economics of the process. The control of the PSD process for separation of THF/methanol azeotropic mixture was discussed by Wang et al. (Wang et al., 2015b) considering the partially/fully heat integrated cases as well. In their work, despite the fast response of the temperature control, basic temperature control structure cannot guarantee the desired product purity. On the other hand, composition control can provide us with the desired purity while having larger dead-times and showing slower responses compared to temperature control. Accordingly, a cascade combination of the composition and temperature control was used by Wang et al. (Wang et al., 2015b) to maintain the product purity while implementing a fast control. However, the measurement required by the composition controller has larger deadtime than the temperature controller and more importantly, on-line measurement of the purity of the products is not always possible. In these situations, direct feedback control cannot be applied and inferential control strategies should be adopted. In this case, determination of the optimal set-points for the controllers that can provide robustness against the imposed disturbances and ensure the product quality and safety parameters is required.

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Development of a framework for the integrated process and control system design to identify an economically optimal solution that can guarantee a safe operation under a wide range of disturbances was the subject of many investigations (Mehta and Ricardez-Sandoval, 2016). A review of the subject of integration of design and control can be found in (Mehta and Ricardez-Sandoval, 2016). Different formulations of the optimization under uncertainty have been proposed in which uncertainty is explicitly included (Sahinidis, 2004). Grossmann et al. (Grossmann et al., 2016) have discussed the major challenges and recent advances in mathematical programming techniques for the optimization of the process systems with MILP formulation under uncertainty. The appearance of uncertain parameters in the model of the process system makes the objective function and the non-equality constraints uncertain as well. For bounded uncertainty, min-max algorithm can be applied, resulting in an overly conservative solution (Barz et al., 2010). If the probability density function (PDF) of the uncertain parameters are known, accurate calculations of the PDF corresponding to the objective function and other dependent variables require the calculation of multidimensional integrals (Li et al., 2008) mostly using numerical methods where an efficient method such as orthogonal collocation (on finite elements) method can be used. However, as the number of the uncertain parameters is increased, these methods become computationally intractable and curse of the dimensionality is the main drawback of these methods. To overcome these issues, mean-variance optimization models (Arellano-Garcia and Wozny, 2009) are commonly adopted where

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the calculation of the expectation of the objective function and feasibility of the nonequality constraints are the main concerns. Considering PDF-based characterization of random effects (Darlington et al., 1999) for general nonlinear models and considering the fact that normal or Gaussian distribution is believed in process engineering practice to be a sufficient assumption (Li et al., 2008), some uncertain parameters with known mean and variance are used to characterize the user demand. However, the PDF of the dependent variables of the process model might not be normal in general. In these cases, Monte Carlo methods, power series expansion (PSE) based uncertainty propagation methods (Bahakim et al., 2014; Chaffart et al., 2016) and polynomial chaos expansion (PCE) method (Crestaux et al., 2009; Nagy and Braatz, 2007) can be used. PCE is a probabilistic method that uses orthogonal stochastic polynomials in the random inputs in order to project model outputs. Barz et al. (Barz et al., 2010) experimentally verified the application of a two-layer chance-constrained optimization approach in the optimal operation of a high-pressure distillation pilot plant for the separation of the acetonitrile/water azeotropic mixture in the presence of various uncertainties. Mesfin and Shuhaimi (Mesfin and Shuhaimi, 2010) proposed an optimization model for a gas processing plant to provide optimal decisions for the operation of the plant on how to handle uncertain feed flow rate and composition to ensure the desired product with a known confidence level. Ricardez-Sandoval (Ricardez-Sandoval, 2012) studied the simultaneous design and control of dynamic systems under random perturbations using a distribution analysis of

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the worst-case variability expected during the normal operation of the system. Paramasivan and Kienle (Paramasivan and Kienle, 2012) formulated a mixed-integer dynamic optimization problem under uncertainty to simultaneously optimize the design of the control structure and the controller parameters where unscented transform (Julier and Uhlmann, 2004) was used to approximate the expectation and the variance of the outputs. The methodology was demonstrated using the design of an inferential control system in a reactive distillation column case study. Pintarič et al. (Pintarič et al., 2013) used sensitivity analysis to evaluate the influence of uncertain parameters on the first-stage design variables and the objective function in order to reduce the scenarios for the deterministic equivalent problem. Ostrovsky et al. (Ostrovsky et al., 2013) developed an approximate transformation to convert chance constraints into deterministic ones to avoid calculation of multiple integrals. Rasoulian and Ricardez-Sandoval (Rasoulian and Ricardez-Sandoval, 2015, 2014) applied PSE method to quantify the effect of parametric uncertainty in an epitaxial thin film growth process. Nagy and Braatz (Nagy and Braatz, 2007) evaluated the efficiency and accuracy of PSE and PCE in a simulated batch cooling crystallization process. Chaffart et al. (Chaffart et al., 2016) applied the PSE method in the analysis of the parametric uncertainty of a heterogeneous multi-scale catalytic reactor model. Zhang et al. (Zhang et al., 2016) proposed a chance-constrained optimization approach for optimization problems with continuous matrix uncertainty where the unknown true PDFs are estimated using Kernel Density Estimation. Uncertain parameters are

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assumed as intuitionistic fuzzy numbers in the fuzzy chance constrained programming technique proposed by Virivinti and Mitra (Virivinti and Mitra, 2015) to model different degrees of risk. Shen and Braatz (Shen and Braatz, 2016) proposed a PCEs based optimization framework for the design of nonlinear dynamical systems, where the dependence of the system outputs on the design parameters are parameterized by Legendre polynomials. The design of a hydroformylation process in a thermomorphic solvent system with uncertain parameters is formulated as a two-stage stochastic optimization problem (Steimel and Engell, 2016), where the first and second stage variables are the design and the operational parameters, respectively. A PCE based surrogate model relating the critical process parameters to deposit morphology was used in a hierarchical control system by Icten et al. (Icten et al., 2015) in the dropwise additive manufacturing process for pharmaceutical products. Unscented transform is an alternative to these methods which requires a small number of sample points compared to its computationally expensive counterpart. This makes the unscented transform a powerful tool that can be used to propagate a random variable through a nonlinear function. The application of the unscented transform is demonstrated in the steady-state stochastic optimization of the gas transmission networks (Behrooz and Boozarjomehry, 2015a, 2015b) under customer demand uncertainties (Behrooz, 2016). Ramos et al. (Ramos et al., 2014) studied the simultaneous design and control problem of an extractive distillation system for the production of fuel grade ethanol under composition disturbances where the mixed-

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integer dynamic optimization formulation of the problem was converted into a mathematical program with complementarity constraints problem. Bahakim and Ricardez-Sandoval (Bahakim and Ricardez-Sandoval, 2015) studied the optimal design of a CO2 capture pilot-scale plant for coal-based power plants under uncertainty. A PSE approximation (Bahakim et al., 2014) of the actual nonlinear process was used in computing the PDF of the process outputs due to uncertainty. Vallerio et al. (Vallerio et al., 2016) proposed a robust multi-objective dynamic optimization framework using unscented transform to demonstrate the trade-off between process safety and performance function in a chemical vapor deposition reactor case study with multiple uncertainties.

PSE based approximation was used in the framework proposed by Mehta and Ricardez-Sandoval (Mehta and Ricardez-Sandoval, 2016) in the simultaneous design of the process and control system considering a back-off from the optimal steady-state design. Ahmadian Behrooz (Ahmadian Behrooz, 2017) proposed a robust design and control framework for the extractive distillation process for benzene/acetonitrile azeotropic mixture separation with dimethyl sulfoxide solvent. In this work, the steady-state model of the PSD process is used to obtain the design parameters as well as the set-points of the corresponding inferential control system in a manner that under the unmeasured disturbances imposed by feed composition, the specification of the HPC and LPC products are above the required purity while the operational constraints are satisfied.

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Accordingly, a stochastic optimization model is formulated that can provide both the optimal configuration and operating conditions of pressure-swing distillation separation plants capable of robust operation under variable feed compositions. The applicability of the proposed method is illustrated for the separation of THF/methanol mixture using a PSD system where the mole % of THF in the feed mixture is assumed to be a Gaussian random variable with known mean and variance. The main idea of the work is incorporating the set-points of the regulatory controllers into the decision variables vector and using the unscented transform as a powerful uncertainty propagation tool to solve the formulated stochastic optimization problem. The main features of the proposed method are: 1. A closed-loop approach is adopted. 2. Steady state model is used to improve the dynamic performance which can facilitate the solution process of the resulting optimization problem. 3. The unscented transform is used which can reduce noticeably the computational load of the stochastic optimization problem. 4. The robustness of the final solution can be tuned using the standard deviation of the uncertain parameter. The solution provided by the stochastic formulation enables us to design and operate the plant with minimum TAC while respecting the operational constraints and achieving the desired purity of the products. Also, the performance of a conventional control structure for a PSD process is evaluated for the final design under feed composition disturbances to show the effects of the solution obtained on the basis of

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the stochastic optimization of the steady-state model, on the dynamic performance of the plant. In Section 2, the separation of THF and methanol using PSD is discussed in detail. In Section 3, a deterministic optimization problem for the design and operation of the THF/methanol 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 to account for the uncertain feed composition. In Section 5, the results are discussed and the performance of the control structure is evaluated. Finally, the conclusions are drawn. 2. Problem statement The mixture of the THF/methanol is found in the treatment of pharmaceutical wastewater and the significance of environmental and economic issues regarding their separation is high. However, THF and methanol are two molecules which are dissimilar and repulsion forces are strong, and a minimum-boiling azeotrope is formed which makes their separation more difficult. An azeotropic feed mixture of THF and methanol with the flowrate of the 100 kmol/hr and temperature of 320 K having a nominal composition of 75 mol% THF is to be separated into 99.9 mol% pure THF and methanol products. The applicability of the PSD process for obtaining high-purity THF and methanol products can be verified by investigating the effect of pressure on azeotrope composition as shown in Fig. 1. Accordingly, the azeotropic composition is 50.79

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mol% of THF at 1 atm while it becomes 15.37 mol% at 10 atm, making the PSD economically attractive for THF/methanol separation. The flowsheet of the THF/methanol separation plant using a PSD process is shown in Fig. 2, where an azeotropic feed mixture of THF and methanol is introduced in the first column (HPC). The bottom stream of the HPC is the 99.9 mol% THF product while the distillate stream is sent to the second column (LPC). In LPC, methanol with a molar purity of 99.9 mol% is separated as the bottom stream while the distillate stream is recycled back to the HPC. In the simulations, Aspen plus is used as the process simulator where the non-ideality of the liquid phase in the vapor-liquid equilibrium of the THF/methanol system can be well predicted using the NRTL model (Wang et al., 2014) and the fugacity of the vapor phase can be calculated using the Redlich-Kwong equation of state (Redlich and Kwong, 1949). Both columns are simulated using RadFrac blocks with “Azeotropic” convergence method and 50 “Maximum iterations”. Also, using “Broyden” as the flowsheet convergence method for the “Tears” has much better convergence behavior. The issues regarding the initialization of the recycle stream are discussed in Appendix A. 2.1

Columns pressure

The operating pressures of the LPC and HPC are chosen on the basis of the applicability of the cooling water in their condensers and the available utility steam levels that can be used in reboilers. Also, the boiling points of the two components at

15

different pressures are shown in Table 1. The pressure of the LPC is set to the value of 1 atm as the top temperature corresponding to the azeotropic composition at 1 atm is about 332.94 K and cooling water with an inlet temperature of 90°F (305.4 K) can be used in the condenser. On the other hand, the bottom product is (almost) pure methanol whose normal boiling point is 337.68 K which makes the utilization of lowpressure steam (LP steam with T=433 K) possible. As discussed by Wang et al. (Wang et al., 2015a), TAC is minimized when the HPC operates at the highest feasible pressure. Accordingly, to be able to use mediumpressure steam (MP steam with T=457 K) in the reboiler of the HPC while having a mean overall temperature driving force of 25°C (Seider et al., 2008), and considering the fact that the bottom product of the HPC is (almost) pure THF and the pressure drop of each tray is approximately 0.0068 atm, Table 1 suggests the operation of HPC at 10 atm. 3. Deterministic optimization problem After having defined the structure and parameters of the PSD process, two optimization problems are formulated, deterministic and stochastic ones, and their details will be discussed as follows. The feed composition is fixed in the deterministic cases while in the stochastic cases, a random feed composition is assumed. The solution of each optimization problem determines the optimal values of the design and operating variables. 3.1

Cost model

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The simultaneous optimization of the design and operating parameters of the PSD process is done by minimization of the TAC of the plant considered as the objective function whose details can be found in Appendix B. 3.2

Decision variables

Decision variables are specified by the decision maker (i.e., optimizer) to set the degrees of freedom to zero and can be divided into design and operating decision variables. Here, the design variables include the total stage number of each column, fresh feed and recycle stream trays of the HPC and feed plate of the LPC. However, it is easier to assume that the HPC and LPC are composed of three and two sections, respectively, separated by the feed stages where the number of stages in each section as shown in Fig. 2 is considered as a decision variable (i.e., 𝑁1 , 𝑁2 , 𝑁3 , 𝑁4 and 𝑁5 ). Accordingly, the azeotropic feed 𝐹 and the recycled stream “Recy” are fed to the trays (𝑁1 + 𝑁2 + 2) and (𝑁1 + 1) of the HPC, respectively, while in the LPC the feed stream DHPC enters on tray (𝑁4 +1). The number of operating degrees of freedom of the model of a simple distillation column having one reboiler and one total condenser, fixed pressure and feed stream with known design parameters is equal to two. However, if the required purities of the HPC

final products (i.e., xB

LPC

and xB ) are known, we can obtain the bottom product flow

rates of the HPC and LPC using an overall mole balance as Eq. (A.1) as discussed in

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the initialization of the recycle stream in Appendix A. Accordingly, one degree of freedom for each column is eliminated. If the composition of the distillates are defined HPC

as xD

 xazHPC   HPC and xDLPC  xazLPC  LPC , where xazLPC and xazHPC are the azeotropic

compositions of the THF/methanol mixture at the pressure of the HPC and LPC, respectively, then

 HPC

and

 LPC

as the composition difference between the distillate

compositions in columns and their corresponding azeotropic compositions are considered as the decision variables in the optimization problem. It is also noted that on the basis of the xy diagram of the THF/methanol mixture at 10 and 1 atm pressures,

 HPC

and

 LPC

should be positive and negative, respectively, making it

easier to rewrite them as:

xDHPC  xazHPC  HPC , HPC  0

(1)

xDLPC  xazLPC  LPC , LPC  0

(2)

Accordingly, the operating and design decision variables of the THF/methanol separation plant using PSD are summarized in Table 2. The resulting optimization problem is formulated as a mixed-integer nonlinear programming (MINLP) optimization problem since some of the decision variables (i.e., 𝑁1 to 𝑁5 ) are discrete and the others are continuous. 3.3

Constraints

The minimum purity of the final products is considered as a constraint where the mol% of the THF product in the bottom stream of the HPC and the produced methanol as the

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bottom product of the LPC should be equal or greater than the desired purity of 99.9%. Also, from the practical point of view to prevent film boiling in a reboiler, it should be designed to operate below the approximate critical heat flux  QR AR max of 32 kW m2 (Doherty and Malone, 2001). 3.4

Mathematical formulation

Defining 𝑥 = [𝑁1

𝑁2

𝑁3 𝑁4

𝑁5

𝜉𝐻𝑃𝐶

𝜉𝐿𝑃𝐶 ] as the vector of the decision variables,

the mathematical formulation of the MINLP problem for the optimal synthesis of the configuration and operation of the PSD process can be stated as the optimization problem OP1 as: min 𝑓(𝑥, 𝑥𝐹 )

(3)

𝑂𝑃1 { 𝑠. 𝑡. 𝒉(𝑥, 𝑥𝐹 ) = 0

(4)

𝒈(𝑥, 𝑥𝐹 ) ≤ 0

(5)

Equality constraint 𝒉 represents the simulation model of the plant and should be satisfied for all values of 𝑥. Inequality constraint 𝒈 represents the minimum purity of the products as discussed in Section 3.3 and is defined as follows: 𝑔1 99 𝑚𝑜𝑙 % − 𝑥𝐵𝐻𝑃𝐶 𝒈 = [𝑔 ] = [ 𝐿𝑃𝐶 ] 2 𝑥𝐵 − 0.01 𝑚𝑜𝑙 % 3.5

(6)

Optimization procedure

All the calculations including the objective function, constraints, (sigma-point calculations and uncertainty propagation required in mean and standard deviation calculations as will be discussed in Section 4.4) are implemented in Matlab software

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and communications required between the optimizer (i.e. Matlab) and simulator (i.e., Aspen Plus) are implemented through Microsoft Excel as shown schematically in Fig. 3. It is worth mentioning that all the optimization problems are solved using the optimization package NOMAD (Audet and Dennis Jr, 2006) that implements the “Mesh Adaptive Direct Search” algorithm for optimization under general nonlinear constraints. This algorithm has extended the “Generalized Pattern Search” algorithm (Audet and Dennis Jr, 2002) using local exploration. In deterministic cases, the operating specifications of the both columns are “Reflux ratio” and “Bottoms rate”. A “Design Spec/Vary” is defined for each column where reflux ratio is manipulated to set the THF mole purity in the distillate stream to the specified value. The optimization algorithm for deterministic case of the constant feed composition is shown schematically in Fig. 3 and its details are discussed as follows: HPC

1. Decide about the required purities of the methanol and THF products (i.e., xB LPC

and xB ). 2. Calculate the bottom flow rates (i.e., BHPC and BLPC ) using Eq. (A.1) and set their values in the simulation model. 3. The optimizer decides about the values of the decision variables (i.e., x ). 4. Set the number of stages and feed plate locations in the simulation file. 5. Calculate the distillate compositions using Eqs. (1) and (2). This also determines the composition of the recycle stream Recy.

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6. Calculate the molar flow rate of the recycle stream ( DLPC ) using the Eq. (A.2) to initialize this stream. LPC HPC 7. Set the distillate composition of the LPC and HPC (i.e., xD and xD ) and use

the corresponding reflux ratio as the manipulating variable to set the distillate composition using Aspen Plus “Design Spec/Vary” utility. 8. Run the simulation model to obtain the value of the dependent variables (such as the column diameter, reboiler/condenser duties and etc.) to calculate the numerical values of the objective function (i.e., Eq. (B.1)) and inequality constraints (Eq. (6)). These values are returned to the optimizer. 9. Optimizer checks the convergence criterion of the optimization method used to finish the optimization iterations or return to step 3 if the convergence is not achieved. 3.6

Optimization results

The temperature, pressure and molar flow rate of the azeotropic feed stream are assumed to be constant, however, it can vary in the composition. The optimization problem is solved for various values of the THF mole fraction around the design feed composition of 75 mol% THF to investigate the effect of azeotropic feed composition on the final optimum design. Design and operating conditions obtained through the proposed MINLP optimization problem are summarized in Table 3 for various values of the THF mol% in the feed mixture.

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It is noted that the condenser and reboiler temperatures of both columns remain almost unchanged with the feed composition variations. Operating pressure of the HPC and LPC are 10 atm and 1 atm leading to approximate reflux drum temperatures of 410 K and 333 K, respectively, and cooling water can be used in the condensers. In the HPC, as the bottom temperature is about 436 K, medium-pressure steam is used in the reboiler while low-pressure steam is required in the reboiler of the LPC due to a lower temperature (351 K). As shown in Table 3, fewer trays and reflux ratios are required in the HPC to achieve the given THF product purity specification, as feed stream with higher concentrations of THF are supplied into the PSD plant. This originates from the fact that we are getting further away from the azeotropic composition at 10 atm as the composition of THF increases in HPC feed stream which can facilitate the separation in the HPC and also decrease the heat loads of HPC condenser and reboiler. The same trend is observed in the LPC where high purity methanol is produced. The combined effect of all these issues results in a process which can achieve the desired purities with lower TAC for feed streams with higher THF content. Also, the variations of feed composition alter the composition profile of the columns leading to the change of the optimal feed stages. 4. Uncertain azeotropic feed composition 4.1

Introduction

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In Section 3, the synthesis of the PSD plant was formulated as a deterministic optimization problem where fixed values for the THF feed mole% were assumed. The optimal results, including the optimal values of the design variables as well as the operating variables for various values of the feed composition were discussed and it was shown that satisfaction of the product quality constraints as well as the operational constraints require different strategies for different feed compositions. On the other hand, the inferential control of the composition of the products as will be discussed in the Section 4.2 on the basis of the column temperature profile stabilization using tray temperature measurement, is not efficient under feed composition disturbances. Improvements in the performance of the control system can be achieved by assuming THF mol% in feed stream to be a Gaussian random variable characterized by a known mean and variance (Li et al., 2008), and a new synthesis method is proposed which can explicitly consider feed composition as a major uncertainty a priori at decision making step. Accordingly, it can provide optimal and robust solutions. Due to the appearance of this uncertain parameter in the simulation model of the PSD plant, the objective function and constraints will be uncertain which can lead to a stochastic programming (Li et al., 2004) optimization problem. Uncertainty can appear in various forms in process engineering such as modeling uncertainties, measurement uncertainties and uncertainties resulting from external disturbances (Kandepu et al., 2008a) where stochastic programming methods (Sahinidis, 2004) are the systematic solution of the problem.

23

Chance-constrained programming technique (Li et al., 2008) is a viable solution to incorporate the uncertainties into the optimization framework where a user-defined probability level of constraint satisfaction is used to convert the original stochastic optimization problem into deterministic nonlinear programming counterpart. In the formulation of stochastic optimization problem using the chance constrained methodology, two approaches can be adopted (Flemming et al., 2007) : 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 mole % of the THF in the feed mixture is assumed as a Gaussian random variable, then, chance-constrained programming technique on the basis of the closed-loop approach is applied to reformulate the stochastic MINLP optimization problem to its deterministic counterpart as follows. 4.2

Control structure of the PSD process

24

Adoption of the closed-loop approach as discussed in the Section 4.1 in the synthesis of the PSD plant capable of handling feed composition disturbances, needs the structure of the control system to be known. Accordingly, the conventional control structure recommended in the practical distillation control literature (Luyben and Chien, 2011; Wang et al., 2015b) is used for the PSD process including the following pairings. Fig. 4 shows the control structure of this system (Wang et al., 2015b) where conventional PI controllers are used for all flows, pressures, and temperatures and proportional controllers are used for all liquid levels with K c = 2. 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% liquid level. The details of the control structures are outlined as follows: 1. Feed is flow controlled (reverse acting). 2. 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 K c = 20 and  I = 12 min. 3. The liquid level in the reflux drum in each column is controlled by manipulating the distillate flow rate (direct acting). 4. Sump level in each column is controlled by manipulating the bottom flow rate (direct acting). 5. Reflux flow rate in each column is ratioed to its distillate flowrate to have a constant reflux ratio.

25

6. The temperature on fifth tray from the bottom in the HPC is controlled by manipulating the reboiler heat input into the HPC (reverse acting). 7. The temperature on fifteenth tray from the bottom in the LPC is controlled by manipulating the reboiler heat input into the LPC (reverse acting). A 1-min dead-time element is inserted into each temperature control loop and relayfeedback tests are run on the two temperature controllers to determine ultimate gains and periods to tune the controllers using the tunings proposed by Tyreus and Luyben (Tyreus and Luyben, 1992). The selection of the temperature control tray location is on the basis of the slope criterion (Luyben, 2013) where temperature shows a large change from stage to stage. Temperature profiles and their corresponding slope values are plotted in Fig. 5 and Fig. 6 against the number of stages for the HPC and LPC. Temperature profiles of the HPC for different compositions show a sharp break which can be used to close the temperature controller loops at the corresponding stages. However, this is not the case for LPC where no break is observed in its temperature slope profiles as the slopes of the temperature variations are almost the same along the column. Accordingly, sensitivity criterion (Luyben, 2013) 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 open-loop gains between LPC tray temperatures and reboiler heat input in shown in Fig. 7 for various feed compositions.

26

Despite the fact that the optimum number of total stages are different for different feed compositions, however, the location of the trays with the largest temperature change in HPC or the trays with the highest sensitivity to the manipulated variable in the LPC are constant relative to the columns bottom. On the basis of the slope criterion and sensitivity criterion as shown in Fig. 5 and Fig. 7, fifth and fifteenth trays from the bottom in the HPC and LPC are selected as the most appropriate control stages in the temperature control loop structures (Wang et al., 2015b), respectively. 4.3

Optimization problem revisited

If  is defined as the random variable corresponding to the THF mole % of the azeotropic feed, the stochastic optimization problem for the synthesis of the PSD plant can be formulated as: min 𝑓(𝑥, 𝜃)

(7)

𝑂𝑃2 { 𝑠. 𝑡. 𝒉(𝑥, 𝜃) = 0

(8)

𝒈(𝑥, 𝜃) ≤ 0

(9)

This stochastic optimization problem (denoted as OP2) can be transformed into its deterministic MINLP equivalent using the chance-constrained programming technique, as follows. The deterministic equivalent of the stochastic objective function (7) is constructed using a weighted sum of the mean and standard deviation of the stochastic objective function as:

min x

f  x ,   min x

 E  f   1   

Var  f 

27



(10)

where   0 is a parameter indicating the relative importance between the mean and standard deviation of the stochastic objective function (7) in minimization. Because of the functionality of the ith inequality constraint (i.e., g i ) from the random variable  , it should be satisfied with the pre-specified probability of ˆp in the chanceconstrained programming framework. In other words:

P  gi  0   ˆp

(11)

where the stochastic constraint (11) can be restated as the deterministic constraint (12): E  gi     ˆp  Var  gi   0

(12)

where  is the distribution function of the standard normal variable. As discussed previously, the optimal pressure for the LPC and HPC correspond to the minimum and maximum allowable pressure in the system and as a result, the total condensers located at the top of the columns operate at constant pressures of 1 and 10 atm, respectively. Also, feed flow rate must be controlled at the steady value of the 100 kmol/hr to ensure constant flow. Then, assuming fixed set-points for feed flow and condenser pressure controllers, and neglecting the controlled variables that have no steady-state effect (i.e., the bottom holdup and condenser holdup), we are left with four control degrees of freedom which are considered as the operating decision variables. Accordingly, the decision variables  x  of the stochastic formulation are:

28

1. Number of stages in each section ( N1 , N2 , N3 , N4 , N5 ). 2. Diameter of the columns ( ID HPC and ID LPC ). 3. Heat exchange area of the reboiler and condenser of the columns ( ACHPC , ARHPC , ACLPC and ARLPC ).

4. Reflux ratio of the columns ( R HPC and R LPC ). 5. Set-point of the temperature controller of each column. For the temperature controller of the HPC and LPC, fifth and fifteenth trays from the bottom stage are selected as the control stages, respectively. It is worth mentioning that as we are dealing with the steady-state model of the plant, the effect of these controllers are implemented using the “Design spec/Vary” utility of the Aspen Plus where for each column, the reboiler duty is manipulated to set the temperature of the specified stage of the column. It is worth mentioning that in this control structure, it is assumed that the composition of THF and methanol at the bottom stream of the HPC and LPC cannot be measured online and hence are not available for control. Accordingly, the reflux ratio and the heat duty in the reboiler are considered to be manipulated variables that can be used to ensure both the quality of the products and the operability of the process within its feasible domain. The initial point of the optimization procedure for stochastic cases can be chosen using the corresponding values form the solution of the deterministic optimization problem OP1 for nominal condition (i.e., THF feed composition of 75 mol%) as shown

29

* in Table 4 denoted as x where in the HPC and LPC, set-points of the temperature

controllers are initialized with the values of 426.83 K and 344.39 K. The objective is to design the distillation columns and determine the set-points of their control scheme that can be operated at minimum total annualized cost. For the stochastic optimization cases, a feasible solution should satisfy the following constraints: 1. Product quality specification: the mol% of the methanol product in the bottom stream of the LPC and the produced THF as the bottom product of the HPC should be equal or greater than the desired purity of 99.9 mol%. 2. Minimum column diameter requirement due to flooding restriction stated as: IDFlooding  IDC  0

(13)

where IDFlooding is the minimum column diameter required to operate at 80% of flooding velocity in the column. 3. The heat duty of each reboiler should be lower than the maximum value of

QRmax  U R AR TR 4. The heat duty of each condenser should be lower than the maximum value of

QCmax  UC AC TC 5. Limit on the heat flux in the reboilers: reboilers should be designed to operate below the approximate critical heat flux  QR AR max of 32 kW m2 . Accordingly, the vector of inequality constraints can be stated as:

30

99.9 mol%  xBHPC  g   1   LPC   g   xB  0.01 mol%   2   ID HPC  ID HPC Flooding   g3       ID LPC  ID LPC   g 4   Flooding HPC HPC HPC   g  QC  U C AC TC 5  g      LPC LPC LPC   g6  QC  U C AC TC   g   HPC HPC HPC   7  QR  U R AR TR   g 8  Q LPC  U ALPC T LPC R R R R      g 9   QRHPC ARHPC    QR AR   max  g    10   LPC LPC   QR AR    QR AR max  4.4

(14)

The unscented transform

Using the chance-constrained technique in the formulation of the stochastic optimization problem requires the mean and variance of both the objective function and constraints (Eqs. (10) and (12)). Propagation of the mean and variance of random variables through a nonlinear function can be done using the Monte Carlo methods, PSE based uncertainty propagation methods and PCE method. Computational load and curse of the dimensionality are the main drawbacks of these methods, especially when used in optimization loops. Unscented transform is an reasonable alternative to these methods which requires a small number of sample points that makes it a powerful tool in random variable propagation through a nonlinear function (Behrooz, 2016; Julier and Uhlmann, 2004).

31

For example, if 𝑛𝜔 is the number of uncertain parameters, uncertainty propagation on the basis of the evaluation of the multidimensional integrals using Gaussian quadrature (Paramasivan and Kienle, 2012), a third order PSE approximation (Bahakim et al., 2014) with forth-order accuracy and unscented transform requires 3 n , 6 n and 2 n +1 grid points, respectively. Feed composition is considered as the only uncertain parameter in this work, however, unscented transform framework can handle the cases with multiple uncertain parameters as well. This requires the mean vector and the associated covariance matrix of the uncertain parameters where the cross covariance between the parameters can be included too. Accordingly, the unscented transform is used in this work as a means of propagating the mean and covariance of the uncertain parameters through the simulation file as a nonlinear function that will be discussed in details as follows. If 𝜔 ∈ 𝑅𝑛𝜔 is a vector of Gaussian random variables with known mean and covariance matrix (i.e., 𝒩(𝜇𝜔 ∈ 𝑅𝑛𝜔 , 𝜎𝜔2 ∈ 𝑅𝑛𝜔×𝑛𝜔 )) and 𝑦 = Ψ(𝜔) is a nonlinear function of several random variables where 𝑦 ∈ 𝑅𝑛𝑦 and Ψ: 𝑅𝑛𝜔 → 𝑅 𝑛𝑦 , then the mean and the covariance matrix of y (i.e., 𝒩(𝜇𝑦 , 𝜎𝑦2 )) can be estimated using the unscented transform as discussed in the following steps (Julier and Uhlmann, 2004): 1) 2𝑛𝜔 + 1 sigma-points are calculated on the basis of the mean and covariance of :

32

i0   ,  i       Si , i  1, , n     S , i  n  1, , 2n i    

(15)

where 𝑆𝑖 is the 𝑖 th column of the matrix square root of 𝜎𝜔2 and 𝛾 is a scaling parameter (Van Der Merwe, 2004):   n   ,    2  n     n

(16)

where 𝑛𝜔 is the total number of Gaussian variables. 2) Each of the 2𝑛𝜔 + 1 sigma points is propagated through the function Ψ to form 𝑦(𝑖) : 𝑦(𝑖) = Ψ(𝜔(𝑖) ), 𝑦(𝑖) ∈ 𝑅𝑛𝑦

(17)

3) A weighted average of the transformed points is used to predict the mean: 2𝑛𝜔

𝜇𝑦 = 𝐸[𝑦] = ∑ 𝑤𝑖𝑚 𝑦(𝑖)

(18)

𝑖=0

4) The covariance is approximated using the weighted outer product of the transformed points: 2𝑛𝜔

𝜎𝑦2

= 𝑉𝑎𝑟[𝑦] = 𝑑𝑖𝑎𝑔(∑ 𝑤𝑖𝑐 (𝑦(𝑖) − 𝐸[𝑦])(𝑦(𝑖) − 𝐸[𝑦])𝑡 )

(19)

𝑖=0

where diag 



returns the main diagonal of its input matrix.

5) Each sigma point is associated with a weight, where the weights are defined as:

   m c w  , w   1   2    , i  0 i i  n   n     1  wim  wic  , i  1,  2  n   

33

(20)

, 2n

α, β and κ are the design parameters of unscented transform where choosing κ ≥ 0 (usually set to 0 (Kandepu et al., 2008b)) can guarantee the semi-positive definiteness of the covariance matrix. The spread of sigma points around 𝜇𝜔 is determined using α (0 ≤ α ≤ 1) (Van Der Merwe, 2004) and for a Gaussian distribution, β = 2 is the optimal choice (Van Der Merwe, 2004). However, the assumption of the Gaussian distribution of the objective function and constraints should be validated. The details of the application of the unscented transform in calculating mean and variance of the objective function and constraints are described in the following steps and it is demonstrated schematically is Fig. 8 as a part of the proposed optimization algorithm: P0. The values of the decision variables ( x ) and parameters α, β and κ should be known. P1. As THF mole% in feed stream is the only uncertain parameter ( n  1 ), the



mean and variance of  is used to calculated the sigma-points i  ,i  1,2,3



as illustrated by formula (15). P2. The simulation model of the plant is simulated for each of the sigma-points

i  . P3. The values of the objective function ( fi  ) and constraints ( gi  ) are calculated for each converged simulation.

34

P4. The unscented transform is used to obtain the mean ( E  f  , E  g  ) and variance ( Var  f  ,Var  g  ) of the objective function and constraints using Eqs. (18) and (19). The statistical linearization used in this transform eliminates the need for function linearization and it can also be applied for non-differentiable functions without needing a linearizing Jacobian matrix. The unscented transformation captures the first and second-order moments of the random variables propagation through a nonlinear transformation to second-order accuracy (Rezaie and Eidsvik, 2016). 4.5

The proposed optimization framework

In the stochastic formulation of the MINLP optimization problem, 𝜃 is assumed to be the THF mol% in the azeotropic feed which is a Gaussian random variable (𝒩(𝜇𝜃 , 𝜎𝜃2 )). The rationale behind this assumption is as follows. The original problem we are dealing with is the inefficiency of the inferential control system of the separation plant under the disturbances imposed by the feed composition which can be reasonably assumed to be stepwise in practical cases. Accordingly, the proposed method is aimed at the improvement of the control system performance. To do so, maximum of ±8.5% changes in the THF nominal mole % in the feed stream is assumed. A Gaussian random variable with its mean equal to the nominal value of the THF nominal mole % and a standard deviation of 2 mol% cover the range of (75±8.5)% with a probability close enough to unity and can represent the effect of the unknown feed composition discussed.

35

The resulting stochastic optimization problem is transformed to its deterministic counterpart with the aid of the chance-constrained programming approach where the mean and variance of the objective function and constraints can be estimated using the unscented transform. The proposed optimization algorithm can be summarized as the following steps and it is demonstrated schematically is Fig. 8. 1. Calculate the initial point of the stochastic optimization procedure using the corresponding values from the solution of the deterministic optimization problem * for the nominal condition denoted as x (i.e., solve OP1 with xF    75% ).

2. The optimizer decides about the value of the decision variables which are sent to the steady-state simulation model of the plant where the number of stages, feed location, reflux ratio, temperature of the desired stage as the “Spec” of the “Designs Spec/vary” utility and diameter of the columns in the “Tray rating” utility are set in each column. 3. For each sigma point, run the simulation model and estimate the mean and variance of the objective function and constraints using steps P0 to P4 discussed in Section 4.4 as the unscented transform framework. 4. The weighted sum of the mean and variance 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 used to finish the optimization iterations or start a new iteration if the convergence is not achieved. Accordingly, if converged, then stop. Otherwise continue on to the next step.

36

6. Go to step 2 and begin a new iteration with the updated values for the decision variables. 5. Results and discussion In Section 3.6, the optimized flowsheets for separation of azeotropic mixtures containing 65 to 85 mol% THF were obtained via the steady-state optimization of the PSD process. An important issue that can be understood from Table 3, Fig. 5 and Fig. 6 is that the effect of feed composition variations is less notable on reflux ratios and temperature profiles in comparison with that for the reboiler or condenser duties which makes the adoption of the closed-loop approach feasible. 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. However, stochastic cases took about three times as long as the deterministic cases due to computational load imposed by the unscented transform required in the calculation of the mean and standard deviation of the objective function and constraints as well. It is worth mentioning that as THF mol % in the azeotropic feed stream is the only uncertain parameter, three sigma-points are encountered in the unscented transform calculations. Now, we are interested in the case of the uncertain feed composition where THF composition of the feed stream is assumed to be a Gaussian random variable whose mean is 75 mol% and its standard deviation is 2 mol% (𝜃 = 𝒩(75, 22 ) mol%).

37

The stochastic optimization problem formulated in Section 4.3 is solved using the proposed method in Section 4.5 with ˆp =0.999 and assuming α = 1, β = 2 and κ = 0 in the unscented transform formulation and the resulting solution is shown in Table 4  denoted as x0.999 . However, the assumption of the normality of the distributions of the

objective function and more importantly the constraints should be evaluated. It is worth mentioning that the violation of normality assumption for the constraints can lead to solutions which are not compatible with the chosen ˆp as the probability of the constraint satisfaction. Accordingly, the distributions of the objective function and all the constraints ( g1 to g10 ) under the imposed uncertain THF feed mol% are obtained through Monte-Carlo sampling with 1e5 simulations where the value of the decision  variables are set as x  x0.999 and the results are shown as histograms in Fig. 9 for the

objective function and constraints g1 to g 3 . It is worth mentioning that ( g 4 to g10 ) have the same behavior of g 3 as will be explained in what follows. The estimated mean and standard deviation obtained for the distribution of the objective function and each constraint are shown in Table 5, and these values are compared to their corresponding values found using the unscented transform. For example, if the propagated PDF of the constraint corresponding to the minimum HPC HPC column diameter requirement of the HPC ( g3  IDFlooding  ID ) is assumed to be a

Gaussian random variable, it has the distribution of 𝒩(-0.0802, 0.02602) and 𝒩(0.0801, 0.02592) for the Monte Carlo simulations and unscented transform,

38

respectively. The comparison of the mean and standard deviation obtained using the unscented transform and Monte-Carlo simulations shows satisfactory agreement. The same trend is observed for the objective function and all the constraints. Also, the skew and kurtosis of the PDFs obtained using Monte-Carlo simulations are given where the non-zero value of the skew indicates the asymmetry of the corresponding distribution while for a Gaussian distribution the value of the kurtosis is 3. To highlight the importance of the issue, the probabilities of the constraint satisfaction under the imposed feed composition uncertainty are reported in Table 5 as well. In Table 5,  gi represents the value of each constraint corresponding to ˆp =0.999

  where F

which is obtained through solving the equation ˆp  Fgi  gi

gi

is obtained

using the experimental cumulative probability function of the distribution. On the other hand, the counterpart of  gi is calculated using the unscented transform denoted as  g whose value is calculated as: i

 g  E  gi     ˆp  Var  gi 

(21)

i

Despite the fact that the mean and standard deviation of the constraints can be estimated with enough accuracy using the unscented transform, however, due to the violation of the normality assumption of the constraints, the difference between the calculated values of  gi using the experimental PDF ( gi  Fgi 1  ˆp  ) and unscented transform ( g  E  gi     ˆp  Var  gi  ) can lead to wrong solutions. For example, for i

constraint g 2 , the value of the constraint corresponding to ˆp =0.999 is  g2  Fg21  ˆp  

39

0.0074 while the calculated value using the Eq. (21) is  g i  -6.42e-7. Accordingly, on the basis of the Monet-Carlo sampling, constraint g 2 is not satisfied while it is satisfied on the basis of the unscented transform estimated values. It is clear in Table 5 that the estimated probabilities of the constraint satisfaction ( p  ) using Monet-Carlo sampling for constraint g1 and g 2 are lower than the pre specified value of ˆp =0.999 which originates from the asymmetry of their PDFs as shown in Fig. 9. To resolve the issue raised by the non-zero skewness of the distributions of g1 and g 2 , other alternative forms of the unscented transform (Julier and Uhlmann, 2004, 1997) can be used to capture the higher moments of the distributions. However, we have proposed an alternative procedure as follows. The experimental PDF of the constraint i obtained using the Monte-Carlo approach called i is used to calculate the value of the left-hand side of Eq. (12) after  is replaced with i for i=1,2. After constructing the new PDFs for asymmetric  constraints, the stochastic optimization problem OP2 is solved again with x0.999 as the

 initial point and the new results are summarized in Table 4 denoted as x0.999 .

The obtained new results using the modified PDFs should be validated as well to make sure about the satisfaction probability of the constraints. Accordingly, for the  optimum values of the decision variables obtained for ˆp =0.999 (i.e., x0.999 ), 1e5

Monte-Carlo simulations are run and among all the realized simulations, the probability

40

of the constraint satisfaction is calculated counting the number of simulations not violating the imposed constraints. Accordingly, the constraint satisfaction probability of the constraints are summarized in Table 6 where putting aside the constraints which are inactive for all the Monte-Carlo realizations and have constraint satisfaction probability of 100%, satisfactory results are observed for active constraints as well (i.e., p  0.999).  Now, x0.999 can be regarded as an acceptable solution of the stochastic optimization

problem OP2 and we can be sure that the solution provided with stochastic approach can enable us to keep up with the minimum purity with a pre-specified probability despite the uncertainties imposed by the feed composition. Also, different values for constraint satisfaction probability ( ˆp ) are assumed and the results of the stochastic cases are compared to the deterministic one with nominal feed composition in Table 4. As shown in Table 3, for all deterministic cases, the purities of the THF and methanol products are at the minimum value of 99.9 mol%. However, this is not the case for uncertain feed composition situations and products with higher purities are obtained to make the process robust enough to tolerate the imposed uncertainties by THF feed composition variations. This safety margin, i.e., the distance between the calculated purities and the minimum value of 99.9 mol% increases as the probability of the constraint satisfaction ( ˆp ) is increased, which can result in the violation of the product quality constraint with lower probabilities.

41

The design parameters including the area required for heat transfer in the reboiler and condenser of the columns, columns diameter and number of stages increase as ˆp increases. This results in a slight increase in the TACs as well. Also, higher set-points for the temperature controllers are needed, if the required confidence level is higher.  Now, the results of the stochastic case x0.999 are compared with its corresponding

* deterministic expected condition (i.e., xF = 75 mol%) denoted as x in Table 4. The

safety margin required to meet the specified minimum purities enables us to compensate the variations of the feed composition. This safety margin can be obtained by increasing the reflux ratio or the number of stages. However, the solution of the optimization determines the best choices where both for HPC and LPC, simultaneously increasing the reflux ratio and also the number of stages were the preferred choices considering TAC minimization. Accordingly, two extra plates are required in HPC and five extra plates are included in the LPC. Diameter of the columns shows a 6.7% and 12.3% increase for HPC and LPC, respectively, which can provide the required safety margin to avoid flooding. The expected values of the TACs are also compared in Table 4 where the expected value of the TAC for the stochastic case with ˆp  0.999 is 5.74% 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 interesting to know that

42

stochastic formulation mostly affects the capital investment of the plant compared to operational costs. The results of the stochastic optimization of the THF/methanol PSD separation problem show its benefits over the deterministic approach, however, the dynamic infeasibility despite the steady-state feasibility can lead to inapplicability of the results which makes the evaluation of the dynamic performance of the two approaches necessary. The assumption of a constant pressure drop on each tray in steady state simulations 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 Rating” utility with fixed top pressure. Accordingly, the new steady operating condition achieved after the occurrence of a feed composition disturbance should be feasible. However, the quality of the transient behavior between the old and the new steady state conditions strongly depends on the tuning of the controllers and dynamic infeasibilities or big overshoots in dynamic responses can be avoided using an appropriate tuning for the controllers. The speed of the responses can be affected in the same way. To investigate this issue, the Aspen Plus steady state simulation file is pressure checked and export into Aspen Dynamics and the performance of the control structure under feed composition disturbances for the plants designed on the basis of expected condition and stochastic formulation with ˆp  0.999 are compared. Accordingly, ±8.5 mol% THF feed composition disturbances are introduced into the systems at 1 hr with

43

design parameters and controller set-points obtained using deterministic and stochastic optimization formulations to evaluate the dynamic performance of the corresponding control structure. Table 7 lists the temperature controllers tuning parameters. The dynamic responses of the THF and methanol mol% in product streams to disturbances imposed by 8.5% increase and decrease in THF feed composition are shown in Fig. 10. It is noticed that the behavior of the system differs depending on whether a positive or negative disturbance is imposed. After the establishment of the new stable operating condition due to the actions taken by the control system, the purity of the methanol and THF product fall below the desired value of 99.90 mol % for the plant designed on the basis of the deterministic approach when the control system encounters -8.5% and +8.5% feed composition disturbances, respectively, while the deviation is more notable for the methanol product as it settles around the purity of 99.5 mol%. The responses of the system synthesized with the stochastic approach show that the purities of products are held fairly near the desired value of 99.90 mol % after the new steady state condition is reached. 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 control the column flooding in both LPC and HPC properly which is shown in what follows.

44

The dynamic performance of the closed-loop plant can be investigated form the viewpoint of the possibility of column flooding. Although in deterministic cases, the column diameter are 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 design condition reveals that the bottom and top of the HPC and LPC operate at the maximum value of 80% of flooding velocity, respectively. However, for their stochastic counterparts, HPC and LPC have maximum values of 69.5% and 63.75% of flooding at the bottom and top, 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 bottom and top trays of the HPC and LPC is shown in Fig. 11 as they represent the maximum flooding approach of each column at any instance. As can be seen in this figure, increasing the THF throughput causes flooding in both columns which is not the case for the system obtained using the proposed stochastic optimization approach. 6. Conclusion In the design step of a pressure swing distillation (PSD) process, parameters such as the number of stages of the low and high pressure columns, reboilers and condensers

45

heat exchange areas, diameters and the feed stage of these columns should be decided by the designer. On the other hand, the values of the process variables such as the reflux ratio of the columns as well as its control system should be determined for keeping the operation at optimum. In this work, a stochastic optimization problem is proposed for THF/methanol separation using PSD whose solution provides the design parameters as well as set points of the indirect purity control system of the process which shows robustness against feed composition unmeasured disturbances. The steady-state model of the PSD process is used in the optimizations and the effect of the temperature controllers are included using the “Design Spec/Vary” feature in Aspen Plus. In the proposed optimization framework, the uncertainty effects of feed azeotropic composition on the optimal design of a PSD plant are discussed. It is shown that it is possible to obtain a robust solution which can tolerate feed composition disturbances. The results are validated using a Monte-Carlo simulation. The optimum structure obtained on the basis of the steady-state model can be dynamically infeasible. Accordingly, the dynamic performance of the plant is studied where the concentration of the products are indirectly controlled by the stabilization of the temperature profile in the column. It is concluded that the use of appropriate design and operation condition enables us to meet the minimum required purity under feed composition disturbances while all operational constraints (such as column flooding) are respected. It is also noted that the proposed stochastic optimization

46

framework mostly affects the capital investment of the plant compared to the operational costs. Appendix A. The initialization of the recycle stream The recycle stream of the low-pressure distillate back to the HPC (i.e., “Recy”) is initialized (i.e., its flow rate and composition are estimated) using a total and component mole balance equations for the binary system of methanol and THF in the following manner: 1. The required purity of the products specifies the mole fractions of the bottom HPC

streams (i.e., xB

LPC

= 99.9 mol% and xB

= 0.01 mol%).

2. Azeotrope composition is a function of the pressure and the composition of the distillates whose values are limited to the corresponding azeotropic compositions, have great effects on the operating and capital cost of the LPC and HPC (Luyben and Chien, 2011). As the difference between the distillate composition and azeotropic composition becomes smaller, the required number of theoretical stages increases because more difficult separations are involved in the columns, while the flow rates of the stream DHPC and recycle stream

DLPC decrease resulting in lower energy consumption in the reboilers. The composition of the distillate stream of the LPC is limited to a maximum value corresponding to the azeotropic composition at 1 atm. The same is true for the HPC, considering the azeotropic composition at 10 atm as the minimum

47

reachable value. One can adopt a compromise between the energy consumption and capital cost of the process by setting the distillate compositions of the LPC and HPC slightly lower and higher than the corresponding azeotropic compositions, respectively. Despite the fact that the two distillate compositions are design optimization variables which are determined by the results of the economic optimization of the PSD process, however, a 2 mol% difference between the distillate compositions and their corresponding azeotropic values are used (Wang et al., 2015a) for the initialization purposes. Accordingly, the compositions of the HPC

distillate streams (i.e., xD

LPC

and xD ) are specified as well.

3. The bottom product flow rates of the LPC and HPC are obtained using an overall mole balance:  F  xBLPC  xF  F  B  B   B   HPC LPC   HPC xBLPC  xBHPC  HPC LPC   xF F  xB BHPC  xB BLPC   BLPC  F  BHPC

(A.1)

4. Finally, distillate flow rates are obtained using a mole balance around the HPC:



 BLPC xDLPC  xBLPC D  D  B  D   HPC LPC LPC   HPC  HPC xDLPC  xDHPC LPC LPC x D  x D  x B  HPC D LPC B LPC  D   DLPC  DHPC  BLPC

 (A.2)

Appendix B. Details of the cost model TAC is considered as the sum of the annual operating costs (i.e., utility steam cost) and annual capital cost (i.e., investment for the major devices in the plant including two

48

column shells and four heat exchangers) as shown in Eq. (B.1), where a three-year payback period and operating time of 8000 h per year is considered. 𝑇𝐴𝐶 (

$ 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 𝑐𝑜𝑠𝑡 )= + 𝐴𝑛𝑛𝑢𝑎𝑙 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑛𝑔 𝑐𝑜𝑠𝑡 𝑦𝑟 𝑃𝑎𝑦𝑏𝑎𝑐𝑘 𝑝𝑒𝑟𝑖𝑜𝑑

(B.1)

The details of the sizing relationships, parameters, equations required to calculate the capital cost of the equipment and utility costs are summarized on the basis of the cost models of Douglas (Douglas, 1988; Luyben and Chien, 2011) in Table B.1 where Marshall & Swift (M&S) index is 1431.7 (Zhu et al., 2015). The height of a distillation column with N trays is calculated considering 0.61 m spacing between the trays plus 20% extra length. The diameters of the two distillation columns are calculated by the maximum flooding velocity criterion (Doherty and Malone, 2001) designed at 80% of flooding velocity using the “Tray sizing” utility in Aspen Plus considering sieve trays.

49

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53

Fig. 1. xy curve for THF/methanol mixture at 1 atm and 10 atm

Fig. 2. Schematic of the THF/methanol pressure swing distillation plant Fig. 3. Scheme of the proposed deterministic optimization procedure Fig. 4. Control structure of the THF/methanol PSD system. Fig. 5. Temperature profiles and temperature slope value plots of HPC Fig. 6. Temperature profiles and temperature slope value plots of LPC Fig. 7. Temperature sensitivity profiles obtained using 0.1% increase in reboiler heat input for LPC Fig. 8. Scheme of the proposed stochastic optimization procedure Fig. 9. Distribution of the objective function and constraints g1 to g 3 obtained through  Monte-Carlo sampling for x  x0.999 .

Fig. 10. Comparison of the dynamic responses of the THF and methanol mol% in streams BHPC and BLPC , respectively, resulting from ±8.5 mol% disturbances in THF feed composition imposed on the plants with parameters obtained using the proposed stochastic framework and deterministic approach for nominal condition. Fig. 11. Comparison of the dynamic responses of the % of the flooding velocity in HPC and LPC resulting from ±8.5 mol% disturbances in THF feed composition imposed on the plants with parameters obtained using the proposed stochastic framework and deterministic approach for nominal condition.

54

Fig. 1

1

10 atm

Mole fraction of THF in vapor

0.8

0.6

0.4

Azeotropic Composition

Azeotropic Composition 0.2

1 atm 10 atm

1 atm

0 0

0.2

0.4

0.6

0.8

Mole fraction of THF in liquid

Fig. 1. xy curve for THF/methanol mixture at 1 atm and 10 atm

1

Fig. 2

Fig. 2. Schematic of the THF/methanol pressure swing distillation plant

Fig. 3

Updated Decision variables

Simulation Model

NOMAD Optimizer

No

Converged?

Yes

Objective function & Constraints Calculation Stop

Fig. 3. Scheme of the proposed deterministic optimization procedure

Fig. 4

Fig. 4. Control structure of the THF/methanol PSD system.

Fig. 5

439 85 mol%

434

Temperature [K]

80 mol%

429

75 mol% 70 mol%

424

65 mol%

419 414 409 1

6

11

16 Stage

11

16

21

26

31

6 85 mol%

Temperature slope [K]

5

80 mol%

4

75 mol%

3

70 mol% 65 mol%

2 1 0 1

6

21

26

Stage Fig. 5. Temperature profiles and temperature slope value plots of HPC

31

Fig. 6

352

Temperature [K]

347 85 mol%

342

80 mol% 75 mol% 70 mol%

337

65 mol%

332 0

5

10

15

20

25

30

35

Stage 0.7 85 mol% 80 mol%

Temperature slope [K]

0.6

75 mol% 70 mol%

0.5

65 mol%

0.4

0.3 1

6

11

16

21

26

Stage Fig. 6. Temperature profiles and temperature slope value plots of LPC

31

36

Fig. 7

0.04 85 mol%

0.03

80 mol% 75 mol%

[K/kW]

70 mol%

0.02

65 mol%

0.01

0 1

6

11

16

21

26

31

36

Stage Fig. 7. Temperature sensitivity profiles obtained using 0.1% increase in reboiler heat input for LPC

Fig. 8

Unscented Transform

Updated

Decision

variables

i  , i  1, 2,3

NOMAD Optimizer

Objective function & Constraints calculation

Simulation Model No

for each sigma point ( f i  ,g i  )

Converged?

Yes Stop

E  f   1    Var  f 

E  f  ,Var  f 

E  gi     p j  Var  gi 

E  gi  ,Var  gi 

Fig. 8. Scheme of the proposed stochastic optimization procedure

Fig. 9

Fig. 9. Distribution of the objective function and constraints g 1 to g 3 obtained through Monte-Carlo  sampling for x  x0.999 .

Fig. 10

99.96

THF mol%

99.92

99.88 Nominal +8.5% Nominal -8.5%

99.84

Stochastic +8.5% Stochastic -8.5%

99.8

0

1

2

3

4

5

6

7

Time [hr] 100

Methanol mol%

99.9 Nominal +8.5%

99.8

Nominal -8.5%

99.7

Stochastic +8.5% Stochastic -8.5%

99.6 99.5 0

1

2

3

4

5

6

7

Time [hr] Fig. 10. Comparison of the dynamic responses of the THF and methanol mol% in streams BHPC and BLPC , respectively, resulting from ±8.5 mol% disturbances in THF feed composition imposed on the plants with parameters obtained using the proposed stochastic framework and deterministic approach for nominal condition.

Fig. 11

HPC bottom tray % flooding velocity

95 90 85 80 Nominal +8.5%

75

Nominal -8.5%

70

Stochastic +8.5%

65

Stochastic -8.5%

60 55 0

1

2

3

4

5

6

7

Time [hr]

LPC top tray % flooding velocity

100 90 80 70

Nominal +8.5%

Nominal -8.5%

60

Stochastic +8.5%

Stochastic -8.5%

50 40 0

1

2

3

4

5

6

7

Time [hr]

Fig. 11. Comparison of the dynamic responses of the % of the flooding velocity in HPC and LPC resulting from ±8.5 mol% disturbances in THF feed composition imposed on the plants with parameters obtained using the proposed stochastic framework and deterministic approach for nominal condition.

Table 1 Boiling point of the pure THF and methanol at different pressures Boiling point [K] Pressure [atm] THF

Methanol

1

339.12

337.68

2

362.18

356.41

3

377.47

368.51

4

389.26

377.66

5

399.01

385.12

6

407.40

391.45

7

414.80

396.99

8

421.46

401.93

9

427.53

406.39

10

433.12

410.48

11

438.31

414.25

55

Table 2 Design and operating decision variables for deterministic cases Number of stages in section 1 ( N 1 ) Number of stages in section 2 ( N 2 ) Design decision variables

Number of stages in section 3 ( N 3 ) Number of stages in section 4 ( N 4 ) Number of stages in section 5 ( N 5 ) “Design Spec/Vary” functionality in Aspen Plus with:

 HPC

Controlled variable:

xDHPC  xazHPC   HPC

Manipulated variable: HPC reflux ratio Operating decision variables “Design Spec/Vary” functionality in Aspen Plus with:

 LPC

Controlled variable:

xDLPC  xazLPC   LPC

Manipulated variable: LPC reflux ratio

56

Table 3 Results of the optimization for deterministic cases 85 80 THF mol% in Feed

75

70

65

P HPC [atm]

10

P LPC [atm]

1

 HPC [mol%]

8.000

7.370

7.220

6.780

6.673

 LPC [mol%]

5.231

4.696

4.695

4.212

4.370

N1

11

14

14

17

18

N2

7

6

6

5

4

N3

7

7

7

7

7

N4

9

10

10

11

11

N5

21

22

23

23

23

T N HPC

27

29

29

31

31

T N LPC

31

33

34

35

35

R HPC

1.656

1.701

1.711

1.740

1.746

R LPC

2.762

2.929

2.932

3.095

3.079

THF mol% in DHPC

99.901

99.901

99.900

99.900

99.901

Methanol mol% in DLPC

99.900

99.900

99.900

99.900

99.900

ID HPC [m]

0.967

1.035

1.102

1.159

1.219

ID LPC [m]

0.599

0.683

0.759

0.826

0.889

57

DHPC [kmol/hr]

30.6

39.3

48.8

57.0

66.5

DLPC [kmol/hr]

15.7

19.4

23.9

27.1

31.5

QCHPC [kW]

-633.7

-828.2

-1033.6

-1220.7

-1426.5

QCLPC [kW]

-535.0

-689.0

-851.4

-1003.6

-1164.1

QRHPC [kW]

1172.4

1366.7

1574.0

1759.9

1967.9

QRLPC [kW]

451.8

583.3

720.5

851.9

987.4

Capital cost [105 $]

11.70

13.49

14.99

16.58

17.88

Operating costs [105 $/yr]

3.79

4.54

5.34

6.08

6.87

TAC [105 $/yr]

7.69

9.04

10.34

11.60

12.83

58

Table 4 Results of the optimization for stochastic cases and nominal condition

x*

 x0.999

 x0.999

 x0.99

 x0.9

ˆp

-

0.999

0.999

0.99

0.9

N1

14

16

17

16

15

N2

6

6

5

5

5

N3

7

6

7

7

7

N4

10

15

14

14

13

N5

23

23

24

24

23

T N HPC

29

30

31

30

29

T N LPC

34

39

39

39

37

R HPC

1.711

1.865

1.865

1.845

1.827

R LPC

2.932

3.478

3.484

3.463

3.274

THF mol% in DHPC

99.900

99.923

99.928

99.919

99.912

Methanol mol% in DLPC

99.900

99.932

99.942

99.929

99.921

ID HPC [m]

1.102

1.177

1.176

1.155

1.133

ID LPC [m]

0.759

0.855

0.852

0.837

0.814

TspHPC

426.83

428.22

428.64

427.99

427.45

59

TspLPC

344.39

346.39

346.54

346.42

345.60

ACHPC [m2]

12.62

15.60

15.55

14.76

13.90

ACLPC [m2]

55.53

68.29

67.71

64.98

60.46

ARHPC [m2]

132.56

153.52

153.83

147.64

141.06

ARLPC [m2]

22.52

27.95

27.71

26.59

24.64

Capital cost [105 $]

14.99

16.88

16.96

16.45

15.69

Operating costs [105 $/yr]

5.34

5.31

5.28

5.29

5.32

TAC [105 $/yr]

10.34

10.93

10.93

10.77

10.55

60

Table 5  Analysis of the solution of the optimization problem OP2 with ˆp =0.999 ( x0.999 )

Monte-Carlo sampling std.

ske

Mean deviation w

f

1093028.

31879.6

Unscented transform kurtos p  P  gi  0  gi  Fgi 1  ˆp  is

0.00

9

g1

-

-

0

0.0074

g2 -0.0300

i

deviation

31785.26

-

21 -6.42E-

0.57 -0.0225

 g

1093152. 3.01

04

std. Mean

3.67

0.0074

0.994

-0.0226

0.0073

6

07

1.12

-5.82E-

0.0102

5.42

0.0198

0.988

-0.0301

0.0097 07

5 -

g3

-2.65E-0.0802

0.0260

0.07

3.03

-0.0026

0.999

-0.0801

0.0259 06

1 -

g4

-8.61E-0.0918

0.0298

0.12

3.05

-0.0050

1.000

-0.0916

0.0297 06

7

g5

0.00 -0.2515

0.0815

-6.56E3.01

-0.0003

0.999

4

-0.2512

0.0813 07

61

-

g6

-5.51E-0.2022

0.0655

0.00

3.01

-0.0012

0.999

-0.2019

0.0653 07

7

g7

0.00 -0.2537

0.0822

-3.76E3.01

0.0002

0.999

-0.2533

0.0820

7

g8

07

0.01 -0.5581

0.0574

3.01

-0.3805

1.000

-0.5579

0.0572

-0.3810

1

g9

0.00 -21.8539

0.5322

3.01

-20.2128

1.000

-21.8518

0.5307

5

20.2118

-

g10 -6.1757

2.0020

0.00

3.01

-0.0361

0.999

7

62

-6.1678

1.9959

-0.0001

Table 6 Monte-Carlo sampling of the solution of the optimization problem OP2 with modified  PDFs and ˆp =0.999 ( x0.999 )

p  P  gi  0 

f

-

g1

0.9989

g2

0.9988

g3

0.9993

g4

0.9996

g5

0.9990

g6

0.9991

g7

0.9990

g8

1.0000

g9

1.0000

g10

0.9991

63

Table 7 Temperature controllers tuning parameters for different PSD process

Nominal case ( x )

*

 Stochastic case ( x0.999 )

TCHPC

TCLPC

TCHPC

TCLPC

Controlled variable

T26HPC

T21HPC

T28HPC

T26LPC

Manipulated variable

QRHPC

QRLPC

QRHPC

QRLPC

Gain, K c

1.042

3.650

1.434

6.790

Integral time,  I [min]

10.56

5.28

9.24

6.6

Set-point [K]

426.83

344.39

428.64

346.54

64

Table B.1 Basis of cost estimation Equipment Cost Model 1.066 0.802  M &S  C1  $     L   2.18  3.67 FP   937.636  ID   280 

Column shell

Pressure [atm] Up to 3.4 6.8

FP

L  N  0.61 1.2

1.00



0.65  M &S  C2  $    2.29  3.75  0.8  Fp   474.668  AC   280 

AC  Condenser

TC

1.05 1.15



QC , U C = 0.852 kW.K-1.m-2 U C  TC

T 

RD

cw  Tout   TRD  Tincw 

log

T T

RD RD

cw  Tout 

 Tincw 



0.65  M &S  C3  $    2.29  3.75 1.35  Fp   474.668  AR   280 

Reboiler

AR 



QR

min U R  TR , QR AR max





,  QR AR max ~ 32kW m2

U R = 0.852 kW.K-1.m-2, TR  TStm  TR Utility cost

65

13.6

Pressure [atm]

FP

Up to 10.2

0.00

20.4

0.10

27.2

0.25

LP steam (433 K, 6 bar) = $7.78/GJ MP steam (457 K, 11 bar) = $8.22/GJ

66