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Rigorous Modeling, Simulation and Optimization of a Dividing Wall Batch Reactive Distillation Column: a comparative study Edna Soraya Lopez-Saucedoa , Juan Gabriel Segovia-Hernandeza , Ignacio E. Grossmannb and Salvador Hernandez-Castroa a Department

of Chemical Engineering; Universidad de Guanajuato; Guanajuato, Mexico of Chemical Engineering; CMU, Pittsburgh, USA [email protected]

b Department

Abstract A model and solution strategies are investigated for the optimization of a Dividing Wall Batch Reactive Distillation Column (DWBRC). In order to accomplish this objective, we describe a dynamic model that involves tray-by-tray calculations for the time varying column proﬁles. In order to compare the simultaneous solution of the system of differential and algebraic equations, two different approaches are used: equation oriented based on orthogonal collocation implemented in GAMS, and control vector parameterization (CVP) as implemented in gPROMS. Keywords: Reactive batch distillation, dynamic optimization, Dividing Wall Columns

1. Introduction Reactive Batch Distillation Columns (RBC) have been studied as a promising technology due to its dual functionality: separation and reaction. Modeling, simulation, and optimization of batch distillation processes rely on dynamic models. A number of different solution approaches for this kind of systems, described by a set of differential and algebraic equations (DAEs) have been proposed in the literature. One of these approaches has been developed by Biegler (1984), in which the dynamic optimal control problem is approximated by a ﬁnite dimensional nonlinear program (NLP) through the discretization of all variables using ﬁnite elements with orthogonal collocation points. This equation oriented approach can then be solved with GAMS (General Algebraic Modeling System 24.2.2) as an NLP problem to simultaneously perform the optimization while converging the DAEs. The other solution method is the Control Vector Parametization (CVP) proposed by Vassiliadis et al. (1994) which relies on the iterative solution of DAEs in the space of the control variables in order to perform the optimization with a Successive Quadratic Programming (SQP) method. In this study we propose the optimal design and operation of a Dividing Wall Batch Reactive Distillation Column (DWBRC). We propose an esteriﬁcation reaction for the production of ethyl acetate (as the distillate product). This study investigates with the two solution approaches how the parameters such as vapor ﬂow rate (V , kmol/hr) and optimal control variable reﬂux ratio (RR = R/V ) are to be adjusted to maximize the productivity in the column for a given product speciﬁcation. First, we provide the problem statement followed by the proposed mathematical model in section 3. The model was solved using two different approaches presented in section 4 for the production of ethyl acetate. The column conﬁguration and operational conditions are

Edna Soraya Lopez-Saucedo et al.

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presented in section 5. The results are shown in section 6 followed by the conclusions in section 7.

2. Problem statement In a general form the problem can be stated as follows:

Given a feed consisting of a mixture of NC components, the column conﬁguration, and product purity speciﬁcation for a key distillate component. The goal is to maximize an objective function by manipulating the column reﬂux ratio RR(t) and vapor ﬂowrate V to purify a given mixture until an NC pure component is obtained (inside some pre-speciﬁed tolerance).

V

R

Condenser stage NT Reflux Drum

D

1

V

2

Dividing Wall

V

2

1

R

R Liquid distributor

2

R x j+1,i 2

j

Internal Trays j 2

V y j-1,i 2

The speciﬁc dynamic optimization problem can then be described as: Given:

Determine: To maximize: Subject to:

Vapor distributor

1

1

V

R

2

R

2

V

Reboiler Column conﬁguration, feed mixture, stage 1 V R vapor ﬂow rate, product purity and batch time. Optimal reﬂux ratio. Figure 1: Batch Reactive Distillation The amount of distillate product. Column (BRC) Equality and inequality constraints.

The reﬂux ratio RR(t) is considered the control variable in the optimization problem. A general proﬁt function P that combines the minimum time and the maximum distillate problem is given by Mujtaba (2004) and is used in this study: P=

C1 D −C2 MB0 −C3 tB + tS

(1)

where P is the proﬁt ($/hr), D is the amount of distillate product (kmol), C1 is the sales value of the distillate product ($/kmol), MB0 is the initial raw material charge (kmol), C2 is the cost of raw material ($/kmol), C3 is the ﬁxed operating cost (energy, wages, depreciation, etc., $/hr), tB is the batch time (hrs) and tS is the set up time (charging and cleaning time between batches, hrs). In mathematical terms, the optimization problem can be represented as: maxRR(t) s.t.

D Dynamic process equations x product (t)≥xdesired Bounds on QREB and D product

(equality constraints) (inequality constraint) (inequality constraint)

3. Dynamic Process Equations In order to solve the optimization problem shown in the previous section, it is necessary to develop a rigorous model to successfully predict the behavior of the variables with respect to time. Two basic assumptions are applied in the formulation of the model: 1. The vapor phase holdup is assumed to be negligible compared to the liquid phase holdup on each plate. 2. Chemical reactions in the vapor phase are neglected.

Rigorous Modeling, Simulation and Optimization of a Batch Reactive Distillation Column: a comparative study

655

The proposed set of differential and algebraic equations (DAEs), can be decomposed into different equations: mass balances, energy balances, equilibrium equations (chemical, physical and thermodynamic) and other equations such as reaction rate, summation of compositions, etc. The set of equations that constitutes the proposed model is presented in the set of Equations 2-19, which are derived from the distillation column on Figure 1. The heat of reaction in the energy balance equations is omitted because heat of formation at the standard conditions is used as a base for enthalpy calculations. The notation for the variables is given in Figure 2. The stages are numbered from bottoms to top of the batch column, stage 1 being the reboiler and condenser the stage 10. More details about the column are given in Section 5. Total mass balances Reboiler: j = 1n dMB = −D + Δn1 MB dt

(2)

Distribuitors: j = NT − 1 and Tray 2 dM j = R1j+1 + R2j − R j +V j+1 −V j + Δn j M j dt

(3)

Internal Trays: j = 2 and NT − 2 dM 1j dt

1 −V j1 + Δn1j M 1j = R1j+1 − R1j +V j−1

(4)

Component mass balances Reboiler: j = 1 MB

dxB,i = R2 x2,i − RB xB,i −VB (xB,i − yB,i ) + rB,i MB − Δni MB dt

(5)

Distribuitors: j = 2and j = NT − 1 d(M j x j,i ) = R1j+1 x1j+1,i + R2j+1 x2j+1,i − R j x j,i +V j−1 y j−1,i −V j y j,i + r ji M j dt

(6)

Internal Trays (section 1 and 2): j = Distributor + 1, ..., NT − 2 d(M 1j x1j,i )

1 y j−1,i −V j1 y1j,i + r1ji M 1j = R1j+1 x1j+1,i − R1j x1j,i +V j−1,i

(7)

2 y j−1,i −V j2 y2j,i + r2ji M 2j = R2j+1 x2j+1,i − R2j x2j,i +V j−1,i dt Condenser: j = NT

(8)

dt d(M 2j x2j,i )

d(MNT xNT,i ) = VNT −1 yNT −1,i − (V j + ΔnNT MNT )xNT,i + rNT,i MNT dt

(9)

Energy balance Reboiler: j = 1 0 = QREB + R2 hL2 − RB hLB +VB (hLB − hVB )

(10)

Distribuitors: j = NT − 1 and Tray 2 1 2 R1j+1 hLj+1 − R2j+1 hLj+1 − R j hLj +V j−1 hVj−1 −V j hVj = 0

(11)

Internal Trays: j = Distributor + 1, ..., NT − 2 1 1 1 R1j+1 hLj+1 − R1j hLj 1 +V j−1 hVj−1 −V j1 hVj 1 = 0

(12)

Condenser: j = NT QCOND −VNT −1 hVNT −1 + (VNT −1 + ΔnNT MNT )hLNT = 0

(13)

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Equilibrium relationship and vapor and liquid summations 1,2 y j,i = K j,i x j,i and y1,2 j,i = K j,i x j,i

,

Σy j,i = 1 and x j,i = 1

(14)

Vapor-Liquid Equilibrium constant 1,2 1,2 1,2 K j,i = K j,i (x j,i , T j , Pj ) and K 1,2 j,i = K j,i (x j,i , T j , Pj )

(15)

Enthalpy L

L

V

V

1,2 hLj,i = hLj,i (x j,i , T j , Pj ) and h j,i1,2 = h j,i1,2 (x1,2 j,i , T j , Pj )

(16)

1,2 hVj,i = hVj,i (y j,i , T j , Pj ) and h j,i1,2 = h j,i1,2 (y1,2 j,i , T j , Pj )

(17)

Reaction and Reaction Terms AcOH (acetic acid)

+

EtOH (ethanol)

↔

EtOAc (ethyl acetate)

+

H2 O (water)

Δn j = Σr j,i when r j,i = r j,i (k j,i , x j,i )

j and i are the number of trays and components, respectively x j,i is the liquid mole fraction for tray j and component i y j,i is the vapor mole fraction for tray j and component i D is the distillate ﬂowrate R j is the liquid ﬂowrate in tray j V j is the vapor ﬂowrate in tray j MB and MNT are the reboiler and distillate holdup, respectively

(18) (19)

QREB is the energy consumed in the reboiler QCOND is the energy consumed in the condenser hLj is the liquid enthalpy in tray j hVj is the vapor enthalpy in tray j superscript 1 and 2 represent left side (section 1) and right side (section 2) of the dividing wall batch column, respectively

Figure 2: Notation used on the dynamic model

4. Solution approaches In order to determine the optimal solution of the dynamic model presented in the previous section, we summarize below some of the issues that arise in the two approaches used in this study. 4.1. Solving the optimization problem by an Equation Oriented Approach In this approach the set of DAEs (Equations 2 - 19) is discretized into a set of algebraic equations by applying ﬁnite elements and the orthogonal collocation method developed by Cuthrell and Biegler (1987). These equations are then used in a large-scale NLP model. The use of ﬁnite elements and collocation points provides more ﬂexibility but the error in the discretization cannot be easily controlled. The proposed DAE system involves a complex set of equations that leads to an index 2 problem. To be solved, the index should be reduced to 1 by differentiating the equation ∑NC i=1 x j,i = 1. This leads to a new algebraic equation that substitutes the internal trays mass balances given in Equation 7 for one component. 4.2. Solving the optimization problem by Control Vector Parameterization Approach To formulate the optimal control problem as a reduced NLP problem, the control variable RR(t) is approximated by a ﬁnite dimensionally equation. The time interval is divided into a ﬁnite number of subintervals, each involving a ﬁnite number of parameters for the control variables to be optimized. This new problem is subjected to the constraints of the model and can be solved using a Succesive Quadratic Programming (SQP) algorithm. This approach has the advantage of providing a direct control of the discretization error by adjusting the size and order of the integration steps using integration techniques.

Rigorous Modeling, Simulation and Optimization of a Batch Reactive Distillation Column: a comparative study

657

5. Case study The reaction-separation is carried out using a 10 tray Dividing Wall Batch Reactive Column (DWBRC). The operational conditions are given in Table 1. An Table 1: Operational conditions for the amount of 100 kmol is fed into the reboiler at the batch reactive column: acetic acid esteristart of the operation with the next composition in ﬁcation Number of Trays mole fraction: 0.45 acetic acid, 0.45 ethanol and (including reboiler and condenser) 10 0.10 water. The distillate product must achieve a pu- Feed, MB0 , kmol 100 Variable rity of at least 0.70 in mole fraction of ethyl acetate Vapor, V, kmol/h β1 = 0.70 in the distillate with a ﬁxed batch time of 1 hour. The Vapor and Liquid Distributors F1 = 0.70 performance is modeled assuming a constant molar Composition Feed, x0 , mole fraction i holdup for each tray (a total column holdup of 4% of Acetic Acid 0.45 0.45 the initial feed where 50% is taken as the condenser Ethanol 0 hold up and the rest is equally divided in the plates). Ethyl Acetate Water 0.10 Constant relative volatilities, calculated using NRTL Activity coefﬁcients (NRTL), αi are used for the various components for modeling the Acetic Acid 0.98 0.99 phase equilibrium. The initial values for the system Ethanol 2.3 variables are calculated at total reﬂux in the steady Ethyl Acetate Water 2.5 state. The reaction zone extends from tray 1 to 9. The production of ethyl acetate by the esteriﬁcation Column Holdup, M j , kmol 0.283 of acetic acid with ethanol is accomplished by the Condenser Internal trays 0.071 stoichiometric equation 18.

6. Results

Column Pressure, P, bar Condenser Internal Trays Reboiler

1.05 1.12-1.08 1.2

The optimization problem is solved by discretizing the differential equations using the two approaches Reaction equations given in Mujtaba (2004) presented in section 4 with the next speciﬁcations: r = k1C1C2 - k2C3C4 −4 the CVP approach implemented in gPROMS and the k1 = 4.76x10−4 k1 = 1.63x10 equation oriented approach with 10 ﬁnite elements and 3 collocation points implemented in GAMS (24.2.2) using CONOPT as the NLP solver. The objective is to maximize the productivity by converting the maximum proﬁt problem in a maximum productivity problem when C1 = 1 and C2 = C3 = tS = 0 in Equation 1. The reﬂux ratio is used as the control variable. Five different cases with different vapor ﬂowrates were studied. All examples were solved on a Workstation with 8 GB RAM memory and IntelCoreTM i7 CPU (2.20 GHz).

The CVP approach results in a system of 530 equations and 600 variables. The optimization problem is solved in 40 seconds. From the results presented in Table 2(a). The ﬁxed batch time is 1.0 hour, while the optimal reﬂux ratio is piecewise as shown in Figure 3(c) for all vapor ﬂowrates. It is clear that the maximum amount of distillate product Dethylacetate (kmol) is achieved for the maximum vapor ﬂowrate V = 90 kmol/hr. The equation oriented approach results in a system of 532 equations and 698 variables. The optimization problem is solved in 318 seconds. The results are presented in Table 2(b). As in the CVP approach, the batch time is 1.0 hour. The maximum amount of distillate product Dethylacetate (kmol) is achieved for the maximum vapor ﬂowrate V = 80 kmol/hr. In both approaches, the amount of distillate and the energy consumption were directly proportional to increases in vapor ﬂowrate. Therefore, any higher purity would require higher energy consumption. A comparison between the calculated reboiler energy shows that the EOA needs a better initial point for the temperatures in the trays. The accumulated distillate composition proﬁles for the maximum distillate product for the two approaches are shown in Figures 3(a) and 3(b).

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Edna Soraya Lopez-Saucedo et al.

Table 2: Results for the productivity maximization (a) Results for the CVP approach

Distillate composition profiles in mole fraction

1.0 0.9 0.8 0.7 0.6

acetic acid ethanol ethyl acetate water

0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.2

0.4

0.6

0.8

Time (hrs)

(a) CVP approach

1.0

(b) Results for the EOA approach Vapor (kmol/hr) 50 60 70 80

DEtOAc (kmol) 10.00 12.32 14.16 15.83 16.85

Composition profiles in the distillate (mole fraction)

QREB (kJ) 47.40 56.88 66.48 76.13 86.13

QREB (kJ) 78.43 94.14 109.78 125.18

0.8

0.6 0.5 0.4 0.3 0.2 0.1

DEtOAc (kmol) 10.32 12.09 13.27 12.33

1.0

Acetic Acid Ethanol Ethyl acetate Water

0.7

0.9

Reflux ratio (R/V)

Vapor (kmol/hr) 50 60 70 80 90

Vapor flowrate (kmol/hr) 50 60 70 80 90

0.8

0.7

0.6

0.5

0.0 0.0

0.5

Time (hrs)

(b) EOA

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Time (hrs)

(c) Optimal reﬂux ratio for the CVP approach

Figure 3: Distillate composition proﬁles for the production of ethyl acetate in a Dividing Wall Batch Reactive Distillation Column for the CVP and EOA approaches and optimal piecewise reﬂux ratio for the production of ethyl acetate in a Dividing Wall Batch Reactive Distillation Column for the CVP approach

7. Conclusion In this work, a model for the optimization of a dividing wall batch reactive distillation column when the esteriﬁcation of ethanol using acetic acid to produce ethyl acetate is studied. A maximum productivity optimization problem is solved under ﬁxed distillate product purity (ethyl acetate concentration higher than 0.70 in mole fraction). The results show that the problem is solved using the two different approaches: the ﬁnite elements with collocation points implemented in GAMS (24.2.2), and the control vector parametization implemented in gPROMS (2004) with no major differences on the calculated variables values. These differences are due to the discretization error carried out during the discretization. Also, the results show that signiﬁcantly more CPU time is required with the EOA approach.

References Biegler, L. T., 1984. Solution of dynamic optimization problems by succesive quadratic programming and orthogonal collocation. Comput. Chem. Eng. 8, 243–248. Cuthrell, J. E., Biegler, L. T., 1987. On the optimization of differential-algebraic process systems. AIChE J 33, 257. Mujtaba, I. M., 2004. Batch Distillation Design and Operation. Series on Chemical Engineering Vol. 3, University of Bradford, UK. Vassiliadis, V. S., Sargent, R. W. H., Pantelides, C. C., 1994. Solution of a class of multistage dynamic optimization problems. part i-algorithmic framework. Ind. Eng. Chem. Res. 33, 2115–2123.

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