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Numerical Determination of Distillation Boundaries for Multicomponent Homogeneous and Heterogeneous Azeotropic Systems Juan A. Reyes-Labarta, Jose A. Caballero, Antonio Marcilla Department of Chemical Engineering University of Alicante. Ap. Correos 99, Alicante E-03080, Spain, E-mail: [email protected]

Abstract As it is very well known, especially when dealing with the planning and operation of distillation processes and equipment, it is essential to characterise the intrinsic limitations due to the VL equilibrium of the system under consideration. This paper addresses a new approach to directly and numerically calculate the distillation boundaries present in ternary and quaternary azeotropic systems, simply involving the concept that such boundaries are distillation trajectories passing through specific singular points. To generate the tested distillation trajectories, that after the optimization process will be the searched boundary, we use cubic splines and polynomial equations (although any other adequate equation could be used). The commercial software package Simulis Thermodynamics (ProSim) has been used to calculate the VLE using the available thermodynamic models such as NRTL and UNIFAQ. Keywords: Distillation boundary, azeotropic distillation, liquid-vapour equilibrium, VLE.

1. Introduction Many strategies have been searched in the past to define the distillation boundaries present in a multicomponent system with azeotropic compositions. In this sense, basic theory and different necessary but not sufficient properties have been published [1-4]. In addition, several topological and approximated methods have been developed to predict the general location of these boundaries [5-8], having in mind their importance when dealing with azeotropic distillation processes. As very well known [9-11], distillate and residue curve maps present a set of singular or critical points related to the distillation trajectories that can be classified (by checking the boiling temperatures or the signs of the derivatives of the function (yi-xi) in the singular points), as follows: x stable node: critical point with all paths approaching x unstable node: critical point with all paths departing x saddle: singular point with finitely many paths both approaching and departing. In the present work, we suggest a new algorithm to directly calculate the distillation boundaries present in ternary and quaternary azeotropic systems, simply applying the concept that define such boundaries as distillation trajectories passing through specific singular points, previously classified.

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2. Mathematical treatment As commented before, the proposed algorithm is based on the concept that distillation boundaries are distillation trajectories that pass through specific singular points such as points that represent azeotropic mixtures (binary or ternary) or pure components. Additionally, the concept the tie lines are chords of the distillation curves is also used (Fig. 1). x0 L-V tie lines x1

y0

distillation curve

Figure 1. Relationship between distillation curves and L-V tie lines.

y1 x2 y2 x3

y3 x4

To generate the probe distillation trajectories, that after the optimization process will be the searched boundary, we use cubic splines or nth-order polynomial equations (although any other adequate equation could be used). Fig. 2 shows the general scheme of the proposed calculation algorithm. This proposed procedure consists in the following steps: 1. Classify the different singular points of the composition diagram, i.e. the azeotropic compositions and pure components present in the system, as stable, unstable or saddle points. 2. Define the origin and the end of the boundary trajectory searched, its number of intermediate points to be calculated (nipt) and the number of intermediate points to be used for the cubic splines (nipcs) or the number of parameters (n) in the case of using a nth-order polynomial function or any other algebraic function. 3. Define the composition of the component that will be the independent variable (e.g. x2) 4. Select the independent variable values for each intermediate point homogeneously distributed, i.e.:

x2, k (k= 1,2,…,nipt) and x 2,k ' (k’= 1,2,…,nipcs).

5. Guess initial values for

x1,k ' (k’= 1,2,…,nipcs), to be used in the cubic spline

calculations or the initial values for the parameters Aj (j= 1,2,…,n) of the algebraic function used.

Numerical determination of distillation boundaries for multicomponents homogeneous 645 and heterogeneous azeotropic systems Singular points

Trajectory extremes, nipt and nipcs (or n)

Independent variable (e.g. x2)

Compositions: x2,k ( k 1,2,..., nipt )

Initial values for the dependent variable: x1,k’ (or Aj)

. Compositions: x1cal 1,2,..., nipt ) ,k ( k

Compositions: y ieq, k. ( k 1,2,..., nipt )

. Compositions: y1cal 1,2,..., nipt ) ,k ( k

yes . ¿ y1cal ,k

y1eq, k. ?

End

no New values for the dependent variable x1,k’ ( k ' 1,2,..., nipcs ) Figure 2. General scheme of the proposed method to calculate distillation boundaries using cubic splines (or algebraic equations) to generate distillation trajectories.

6. Calculate

. th x1cal , k , using the cubic spline (cs), the n -order polynomial or any other

adequate algebraic function (af) and k= 1,2,…,nipt and k’= 1,2,…,nipcs or

. x 2,k , i.e.: x1cal ,k . x1cal ,k

f

af

( x 2,k , A j ) with j= 1,2,…,n.

7. Calculate, the LV equilibrium of each liquid point thermodynamic model:

yieq,k.

f cs ( x 2,k , x1,k ' , x 2,k ' ) with

xi , k , using an adequate

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8. Calculate

. th y1cal , k , using the cubic spline (cs), the n -order polynomial or any other . y2eq, k. , i.e.: y1cal ,k

adequate algebraic function (af) and k= 1,2,…,nipt and k’= 1,2,…,nipcs or

. y1cal ,k

f

af

f cs ( y 2eq,k. , x1,k ' , x 2,k ' ) with

( y 2eq,k. , A j ) with j= 1,2,…,n.

9. Using an optimization solver over the cubic spline or the algebraic function, calculate the trajectory ( x1,k ' with k’= 1,2,…,nipcs, passing through the selected singular points of origin and ending) that satisfies the following objective function: nipt

¦ y

eq. 1, k

. y1cal ,k

2

=0.

k 1

In the present work, the commercial software package Simulis Thermodynamics (ProSim) has been used to calculate the VLE using the available thermodynamic models such as NRTL and UNIFAQ, although any other empirical equation could be used [12].

3. Results and discussion To validate the proposed algorithm, the following systems have been studied using the suggested procedure: System A: Chloroform + Methanol + Acetone, 760 mmHg with 3 binary azeotropes and 1 homogeneous ternary azeotrope. System B: Benzene + Isopropanol + Water, 760 mmHg with 3 binary azeotropes and 1 heterogeneous ternary azeotrope. System C: Acetone + Methanol + Propanol + Water, 760 mmHg with 2 binary azeotropes. 1 0.9 0.8

Methanol

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Chloroform

Figure 3. Results for the studied system A: Chloroform + Methanol + Acetone at 760 mmHg. a) distillation boundaries ( cubic spline nodes, * distillation boundary). Figure 3 shows the results for the calculated distillation boundaries of the homogeneous system A, following the procedure previously described. As we can observe, this system

Numerical determination of distillation boundaries for multicomponents homogeneous 647 and heterogeneous azeotropic systems shows 3 different distillations boundaries, starting all of them at the ternary azeotrope and ending at each of the binary azeotropes present in the system. Figure 4 and 5 also show the results of the distillation boundaries and surfaces for the heterogeneous ternary and homogeneous quaternary studied systems, respectively. 1 0.9 0.8

Isopropanol

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Benzene Figure 4. Results for the studied heterogeneous ternary system: Benzene + Isopropanol + Water at 760 mmHg, including the liquid-liquid equilibrium at 298 K ( cubic spline nodes, * distillation boundary, O azeotropic mixtures, * LL tie-lines). Acetone

Methanol i-propanol

Water Figure 5. Results for the studied homogeneous quaternary system: Acetone + Methanol +Propanol + Water at 760 mmHg showing the corresponding distillation boundary surfaces.

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4. Conclusions The algorithm proposed to calculate distillation boundaries, using thermodynamic models in the VLE calculations, produces very goods results in the homogeneous and heterogeneous azeotropic ternary and quaternary studied systems, having in mind that if we increased the number of intermediate points for the cubic spline (nipcs) or in same cases the type of algebraic function used, the results would be improved.

5. Acknowledgements Vicepresidency of Research (University of Alicante), Generalitat Valenciana (GV/2007/125) and Ministry of Science & Innovation (PHB2008-0090-PC).

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