Residue curve maps of reactive membrane separation

Chemical Engineering Science 59 (2004) 2863 – 2879

www.elsevier.com/locate/ces

Residue curve maps of reactive membrane separation Yuan-Sheng Huanga , Kai Sundmachera; b;∗ , Zhiwen Qia , Ernst-Ulrich Schl/undera a Max-Planck-Institute

for Dynamics of Complex Technical Systems, Sandtorstrasse 1, D-39106 Magdeburg, Germany Magdeburg, Process Systems Engineering, Universit)atsplatz 2, D-39106 Magdeburg, Germany

b Otto-von-Guericke-University

Received 15 October 2003; received in revised form 5 April 2004; accepted 15 April 2004

Abstract A batch reactive membrane separation process is analysed and compared with a batch reactive distillation process by means of residue curve maps. In both processes, the chemical reaction takes place (quasi-) homogeneously in the liquid bulk phase and vapour–liquid equilibrium is assumed to be established. Additionally, in the reactive membrane separation process, selective vapour phase permeation through a membrane is incorporated. A model is formulated which describes the autonomous dynamic behaviour of reactive membrane separation at non-reactive and reactive conditions when vacuum is applied on the permeate side. The kinetic e9ect of the chemical reaction is characterized by the Damk/ohler number Da, while the kinetic e9ect of multicomponent mass transfer through the membrane is characterized by the matrix of e9ective mass transfer coe=cients. The process model is used to elucidate the e9ect of selective mass transfer on the singular points of reactive membrane separation for non-reactive conditions (Da = 0), for kinetically controlled reaction (0 ¡ Da ¡ ∞), and for equilibrium controlled reaction (Da → ∞). Scalar, diagonal and non-diagonal mass transfer matrices are considered. As examples, the simple reaction A ⇔ B + C in ideal liquid phase, and the cyclization of 1,4-butanediol to tetrahydrofurane in non-ideal liquid phase are investigated. ? 2004 Elsevier Ltd. All rights reserved. Keywords: Membrane separation; Reactive distillation; Azeotrope; Vapour permeation; Residue curve map; Bifurcation analysis

1. Introduction During the last decade Reactive Distillation (RD) has emerged as one of the most important innovative separation technologies. As advantages of RD, chemical equilibrium limitations can be overcome, higher selectivities can be achieved, the heat of reaction can be recovered in situ for distillation, auxiliary solvents can be avoided, and azeotropic or closely boiling mixtures can be more easily separated than in non-reactive distillation. Increased process e=ciency and reduction of investment and operational costs are the direct results of the integration of chemical reaction and distillation. Some of these advantages are realized by using reactions to improve separation, others are realized by using separation steps to improve the chemical reaction (Taylor and Krishna, 2000; Sundmacher and Kienle, 2003). ∗ Corresponding author. Max-Planck-Institute for Dynamics of Complex Technical Systems, Sandtorstrasse 1, D-39106 Magdeburg, Germany. Tel.: +49-391-6110351; fax: +49-391-6110353. E-mail address: [email protected] (K. Sundmacher).

0009-2509/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2004.04.018

For the conceptual design of RD processes, Residue Curve Maps (RCM) were introduced as a very important and powerful tool and were used by many researchers (e.g. Venimadhavan et al., 1994; Ung and Doherty, 1995a; Thiel et al., 1997; Qi et al., 2002). RCMs represent the dynamic behaviour of the liquid phase composition in a simple batch reactive distillation process, as depicted in Fig. 1a. The analysis of the location and stability of the singular points in RCMs, i.e. the steady states of this simple process, yields valuable information on the attainable products of a RD process. The existence of these stationary points was also proven experimentally (Song et al., 1997, 1998). For the stationary points in a distillation system undergoing equilibrium-controlled chemical reactions the term reactive azeotrope was introduced by Doherty and coworkers (Barbosa and Doherty, 1988a,b; Ung and Doherty, 1995b). In RD systems with kinetically controlled chemical reactions, the singular points are called kinetic azeotropes according to Rev (1994) who investigated the general situation of the simultaneous occurrence of a chemical reaction and a separation process. Several groups studied the bifurcations of kinetic azeotropes in homogeneous

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Y.-S. Huang et al. / Chemical Engineering Science 59 (2004) 2863 – 2879

Fig. 1. Considered reactive separation processes (a) reactive distillation, (b) reactive membrane separation, (c) equivalent continuous reactive membrane separation process.

systems (Venimadhavan et al., 1994; Thiel et al., 1997) and also in mixtures undergoing liquid phase splitting (Qi and Sundmacher, 2002). Most of these singularity analyses were carried out using the Damk/ohler number of Krst kind, Da, as most important bifurcation parameter. As important limiting cases, at Da = 0 the classical azeotropes of non-reactive distillation are recovered, and at Da = ∞ the (chemical equilibrium controlled) reactive azeotropes are found.

As has been shown in the cited works, new types of azeotropes will be formed by combining distillation and chemical reactions, even in ideal reaction mixtures without any non-reactive azeotrope. Due to this, despite the convincing success of RD in many applications, such as esteriKcations and etheriKcations, RD is not always advantageous. In some cases RD does not yield the desired products (Sundmacher and Kienle, 2003). In non-reactive azeotropic distillation processes, Castillo and Towler (1998) and Springer et al. (2002) investigated the importance of vapour–liquid interface mass transfer with respect to the feasible products. The latter authors found that it is possible to cross distillation boundaries by mass transfer e9ects. Schl/under (1977, 1979) and Fullarton and Schl/under (1986) demonstrated that separational limitations imposed by azeotropes can be overcome by application of an entraining medium. Due to di9erent di9usion rates of the components in the entrainer, the azeotropic points are shifted. Since these points are formed under mass transfer control, the term pseudo-azeotrope was proposed. Analogously, Nguyen and Clement (1991) observed and analysed pseudo-azeotropic points which appear in the separation of water–ethanol mixtures–water at pervaporation membranes. From the analysis of non-reactive systems one can expect that selective vapour–liquid mass transfer will also have a signiKcant e9ect on the attainable products of reactive separation processes. In particular, the stationary points of reactive distillation processes might be inMuenced in a desired manner by selective membranes, as has been shown recently by Aiouache and Goto (2003) who integrated a pervaporation membrane into a reactive distillation process. In the present work, in order to get a deeper understanding of the role of membranes on the feasible products of RD processes, the batch reactive membrane separation process in Fig. 1b is considered. The considered batch process is the equivalent of a continuous membrane process, as depicted in Fig. 1c. There, the time coordinate is replaced by the axial special coordinate. Therefore, the RCM acquired from the batch process can be directly used for continuous process design. The main objective of this work is to analyse the RCM and the stationary operating points of the proposed reactive membrane separation process and to elucidate the role of the membrane mass transport properties for the feasible products. For this purpose, a process model is formulated which is applied to an ideal reaction system with constant relative volatilities and to a strongly non-ideal reaction system, namely the cyclization of 1,4-butanediol.

2. Model of reactive membrane separation The process model is based on the following assumptions (see also Fig. 1b): • The liquid phase and the retentate vapour bulk phase are assumed to be in phase equilibrium based on the

Y.-S. Huang et al. / Chemical Engineering Science 59 (2004) 2863 – 2879

• •

•

•

fact that the membrane holds major mass transfer resistance. Vacuum is applied on the permeate side of the membrane (pp → 0). The chemical reaction takes place at boiling temperature in the reactive holdup Hr of the liquid phase, i.e. either as a homogeneous reaction in the liquid bulk (Hr = H ), or at a heterogeneous catalyst which is fully immersed in the liquid bulk (Hr = Hcat ). A vapour permeation process is considered, i.e. the membrane is placed above the (retentate) vapour–liquid interface. The reason for choosing a vapour permeation process is to make an illustrative comparison with RD. Accumulations of components within the retentate vapour phase and within the membrane are neglected, i.e. process dynamics are considered to be dominated by the liquid phase.

2.1. Reaction kinetics and mass balances The chemical reaction taking place in the liquid phase is represented by Cr
r=1

|r |Ar ⇔

Cp


|p |Ap ;

(1)

p=1

where i is the stoichiometric coe=cient of component i. By convention, i ¡ 0 if component i is a reactant, i ¿ 0 if component i is a product. Ar and Ap are the reacting species, Cr is the number of reactants and Cp is the number of products. Thus, the total number of components is NC = Cr + Cp . The rate expression of this reaction is given by r = kf (T ) · R(a);

(2)

where kf is the temperature-dependent rate constant of the forward reaction, and R is the dimensionless reaction rate depending on the liquid phase composition, expressed in terms of activities ai = xi · i . The component and the total mass balances in the liquid phase are formulated in molar units as follows: d(H · xi ) dt = − n i +  i · H r · kf · R

(3)

dH (4) = −nT + T · Hr · kf · R; dt where H represents the total molar liquid holdup, Hr the holdup in which the reaction proceeds, ni the component NC mass Mux through a certain membrane area S, nT = i=1 ni NC the total mass Mux, and T = i=1 i the total mole change of the chemical reaction. Expanding the time derivative in Eq. (3) and combining the result with Eq. (4) yields   Hr d xi ni + · (i − T xi ) · kf R = xi − d nT nT with i = 1 : : : NC − 1;

with i = 1 : : : NC − 1;

(5)

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where  stands for the dimensionless time with d=(nT =H )· dt. Now, a Damk/ohler number Da is introduced which is deKned as the ratio of the characteristic reaction rate (Hr; 0 · kf; ref ) and the characteristic escaping total Mux nT; 0 : Da =

Hr; 0 · kf; ref ; nT; 0

(6)

where Hr; 0 and nT; 0 are the initial reactive holdup and the initially escaping total Mux through the membrane, respectively; kf; ref is the forward reaction rate constant at a reference temperature. Using Eq. (6), the mass balances, Eqs. (5), are rewritten as   d xi ni Hr nT; 0 kf = xi − +Da · (i −T xi ) · · ·R d nT Hr; 0 nT kf; ref with i = 1 · · · NC − 1:

(7)

Eq. (7) cannot be solved without specifying policies for Hr and nT; 0 , which a9ect the relative weighting of the inMuences of mass transfer and chemical reaction. To study the autonomous dynamic behaviour, two suitable strategies are assumed in this paper: (i) nT =nT; 0 = Hr =Hr; 0 , if the chemical reaction takes place as a homogeneous reaction in the whole liquid bulk phase, (ii) nT =nT; 0 =const:, if the chemical reaction takes place in a solid catalyst phase whose molar holdup is constant during the process (Hr = Hr; 0 = const.). At both strategies the reaction kinetic e9ect is characterized by a single parameter, the Damk/ohler number Da, and Eq. (7) become   kf d xi ni + Da(i − T xi ) R = xi − d nT kf; ref with i = 1 : : : NC − 1:

(8)

2.2. Vapour–liquid equilibria The composition of the retentate vapour phase below the membrane is quantiKed by the mole fractions yi . The liquid bulk is assumed to be in the boiling state such that it is in phase equilibrium with the retentate vapour phase at any time. Assuming ideal gas phase behaviour at moderate pressures, yi can be calculated from yi =

xi · i (xj ; T )pisat (T ) p

with i = 1 : : : NC;

(9)

where p is the system pressure, pisat the saturated vapour pressure of component i, T the boiling temperature at the given pressure and liquid phase composition, i the liquid phase activity coe=cient. The corresponding boiling point

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Y.-S. Huang et al. / Chemical Engineering Science 59 (2004) 2863 – 2879

T is calculated from the vapour phase summation equation NC
yi = 1: (10) i=1

2.3. Kinetics of vapour permeation In a partial pressure-driven vapour permeation process, the NC-dimensional vector of mass Muxes through the porous membrane (n) can be expressed as (e.g. Krishna and Wesselingh, 1997) S (n) = (11) · [k] · (y · p − yp · pp ); RT where [k] is the NC × NC-dimensional matrix of e9ective binary mass transfer coe=cients kij , S the e9ective permeation area, yp and pp denote the vector of mole fractions and the total pressure on the membrane permeate side, respectively. If vacuum is applied on the permeate side, the partial pressures of the di9using components on the permeate side are negligible (pp → 0). For the subsequent process analysis, it is useful to formulate Eq. (11) in terms of dimensionless mass transfer coe=cients ij which are the ratios of the e9ective mass transfer coe=cients kij related to a reference coe=cient. Here, the Krst main diagonal element k11 is taken as reference coe=cient: kij S ·p · k11 · [] · (y) with kij ≡ : (12a,b) (n) = RT k11 In general, all elements of the mass transfer matrix depend on the process variables, in particular on the vapour phase composition (y). The mass transfer mechanisms in membranes can be rather complicated and diverse. E.g. in porous membranes, transport mechanisms such as Knudsen di9usion, molecular di9usion, surface di9usion and viscous Mow are coupled according to the Dusty Gas Model (Krishna and Wesselingh, 1997). In case of systems close to the boiling point, capillary e9ects will play a role, too (Yeh et al., 1991). However, for the conceptual analysis of the considered membrane process, it is not useful to go into the mechanistic details of membrane mass transport. Therefore, in the following the e9ective binary mass transfer coe=cients kij are assumed to be constants. Three cases can be distinguished concerning the mathematical structure of the [k]-matrix: (i) scalar [k]-matrix if the membrane is non-selective, i.e. it has no separation e9ect on the components. In this case the membrane process coincides with the distillation process (Fig. 1a) and the []-matrix in Eq. (12a) is equal to the identity matrix. (ii) diagonal [k]-matrix if the permeating components are only driven by their own partial pressure di9erences. This case is valid, e.g., for a Knudsen-membrane. (iii) non-diagonal [k]-matrix if the permeating components interact with each other within the membrane. This will be the case if, e.g., bulk di9usion and/or competitive adsorption e9ects are involved.

Despite the fact that the matrix elements can be positive or negative, the [k]-matrix has to be positive deKnite because of the constraints imposed by the second law of thermodynamics (Taylor and Krishna, 1993). 2.4. Singular points Stationary operating points of batch separation processes, as the here considered batch reactive membrane process, are of special interest for the conceptual design of continuous separation processes. The stationary operating points are the singular points of Eq. (8) and can be classiKed as stable nodes, unstable nodes and saddle points. Stable nodes represent feasible product mixtures of a continuously operated reactive or non-reactive separation process. The locations of the singular points in the phase space depend on the type of the separation applied (distillation/membrane separation) and on the presence of chemical reactions (non-reactive/reactive operation). Four cases can be distinguished (see Table 1): (i) Non-reactive distillation. For non-reactive membrane separation, i.e. Da = 0, and for the special case that [k] is a scalar matrix, i.e. no separation e9ect by the membrane, the stationary points correspond to those obtained for phase equilibrium controlled distillation. The condition is x i = yi

with i = 1 : : : NC − 1:

(13)

Eq. (13) are fulKlled by pure components and by non-reactive azeotropic mixtures. (ii) Non-reactive membrane separation. If Knudsen diffusion and/or bulk di9usion within a porous membrane are involved, [k] is a non-scalar matrix and the following singularity condition results from Eq. (8): xi =

ni nT

with i = 1 : : : NC − 1:

(14)

Eq. (14) are valid for mass transfer controlled non-reactive separation and corresponds to the azeotropic condition for equilibrium controlled non-reactive distillation, Eq. (13). In the literature, these singular points were termed as pseudo-azeotropes (Schl/under, 1977; Nguyen and Clement, 1991) because they behave as if they were azeotropes. Since the meaning of the preKx “pseudo-” is quite unspeciKc, a more precise terminology is proposed here. Considering the Greek origin of the word “azeotrope”, it is composed of three fragments (Atkins, 1990): “a” means “not”, “zeo” means “boiling” and “trope” means “changing”. The complete meaning of “azeotrope” is therefore “the liquid composition is not changing with boiling”. Following the same nomenclature logic, we propose the term “arheotrope” representing the stationary points in mass transfer controlled processes, where “rheo” means “Mux”. The full meaning of “arheotrope” can be stated as “the liquid composition is not changing with Mux”.

Y.-S. Huang et al. / Chemical Engineering Science 59 (2004) 2863 – 2879

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Table 1 ClassiKcation of singular points in (non-)reactive distillation and membrane separation processes Reaction

Separation Phase equilibrium controlled separation: distillation   1 0 0    [] =  0 1 0

Mass transfer controlled separation: membrane permeation   1 AB AC    [] =   BA BB BC  CA CB CC

0 0 1 No reaction: Da = 0 Kinetically controlled reaction: Da ∈ (0; ∞) Chemical equilibrium controlled reaction: Da → ∞

Azeotrope, Eq. (13) First introduced by Wade and Merriman (1911) Kinetic azeotrope, Eq. (15) First studied by Venimadhavan et al. (1994) Reactive azeotrope, limiting case of Eq. (15) First introduced by Barbosa and Doherty (1988a,b)

(iii) Reactive distillation. Formally, a conventional reactive distillation process can be interpreted as a special reactive membrane separation process with a scalar [k]-matrix, i.e. no separation e9ect is exerted by the membrane, Eq. (8) yield the following singularity condition: kf 0 = (xi − yi ) + Da(i − T xi ) R kf; ref with i = 1 : : : NC − 1:

(15)

The stationary points deKned by Eq. (15) are termed as kinetic azeotropes for kinetically controlled reactive distillation (Qi et al., 2004) and as “reactive azeotropes” for chemical equilibrium-controlled reactive distillation (Barbosa and Doherty, 1988a,b). It is important to note that the term kinetic refers to the chemical reaction kinetics rather than to the mass transfer kinetics. (iv) Reactive membrane separation. The most general case is a reactive membrane separation process whose [k]-matrix is a non-scalar matrix, i.e. a diagonal or a non-diagonal matrix. From Eq. (8) the following conditions for stationary points are obtained:   kf ni 0 = xi − + Da(i − T xi ) R nT kf; ref with i = 1 : : : NC − 1:

(16)

According to the terminology proposed above, the stationary points can be denoted as kinetic arheotropes (0 ¡ Da ¡ ∞) or as reactive arheotropes (Da → ∞). In the latter case, the reaction approaches the chemical equilibrium. The mixture composition of kinetic arheotropes is determined by both, the chemical reaction kinetics and the mass transfer kinetics through the membrane. 2.5. Potential singular point curve (PSPC) As has been demonstrated for the singularity analysis of reactive distillation systems (Barbosa and Doherty, 1988a, b; Qi et al., 2004), it is useful to separate the inMu-

Arheotrope, Eq. (14) First introduced “pseudo-azeotrope” by Schl/under (1977) Kinetic arheotrope, Eq. (16) This work

as

Reactive arheotrope, limiting case of Eq. (16) This work

ences of the chemical reaction kinetics and of the vapour –liquid equilibria. The same is true for the analysis of reactive membrane separation processes. By elimination of the reaction term from the (NC − 1) singularity conditions, Eq. (16), it is possible to Kx the potential singular point curve (PSPC) on which all possible singular points will be located. The PSPC is deKned by a set of equations which can be formulated analogously to the azeotropic conditions for non-reactive distillation, Eq. (13), by introducing transformed composition variables for the liquid phase, Xi , and for the vapour phase, Yi (see derivation in Appendix A). By means of these transformed variables, the singularity conditions of the here discussed separation processes can be expressed in compact manner: Xi = Yi

with i = 1 : : : NC − 2;

i = k;

(17a)

where Xi =

(k xi − i xk ) ; (k − T xk )

(17b)

Yi =

(k ni − i nk ) : (k nT − T nk )

(17c)

The transformed vapour composition variable, Yi , can be successively expanded for the di9erent possible structures of the mass transfer matrix: NC NC (k j=1 ij yj − i j=1 kj yj ) Yi = NC NC NC (k i=1 j=1 ij yj − T j=1 kj yj ) for a non-diagonal []-matrix; =

(18a)

(k ii yi − i kk yk ) NC (k i=1 (ii yi ) − T kk yk )

=

for a diagonal[]-matrix;

(18b)

(k yi − i yk ) (k − T yk )

(18c)

for a scalar []-matrix:

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Y.-S. Huang et al. / Chemical Engineering Science 59 (2004) 2863 – 2879

The deKnition in Eq. (18c) is identical to the Yi -deKnition which was proposed Krst by Barbosa and Doherty (1988a,b) for reactive distillation processes.

C 1

Da=0 0.8

3. Results and discussion 0.6

xC

3.1. Example I: Ideal reaction system As Krst example, the e9ect of a selective membrane on the ternary mixture A=B=C undergoing a single reversible chemical reaction in ideal liquid phase is considered: B: main product; C: byproduct:

0.2

(19)

The dimensionless rate of the chemical reaction obeys the kinetic law x B xC R = xA − (20) K with K as chemical equilibrium constant. For the sake of a simpliKed analysis, the following assumptions are made: • The chemical equilibrium constant is independent of temperature: K = 0:2. • The rate constant kf is independent of temperature: kf =kf; ref = 1. • The relative volatilities of the product B and the byproduct C with respect to the reactant A, !BA and !CA , are assumed to be constants. Then, the vapour phase equilibrium composition can be calculated as !iA xi yi = C : (21) m=A !mA xm • Three []-matrix structures are considered here: identity matrix, diagonal matrix and non-diagonal matrix.

0 0

0.2

0.4

0.6

0.8

xB

A

(a)

1

B

C 1

Da=1 0.8

0.6

Kinetic Azeotrope

xC

A⇔B+C

0.4

0.4

Chemical Equilibrium Curve

0.2

0 0

0.2

0.4

A

(b)

xB

0.6

0.8

1

B

C 1

Da=∞ 0.8

0.6

Reactive Azeotrope

xC

3.1.1. Residue curve maps (RCM) First, the e9ect of a membrane was studied with the help of residue curve maps (RCM), i.e. phase portraits of the liquid phase composition. They were determined for the reactive distillation process (Fig. 1a) as well as for the reactive membrane separation process (Fig. 1b). The selected volatility values are !BA = 5:0 and !CA = 3:0, i.e. the reactant A is the high boiler, the main product B is the low boiler and the byproduct C is the intermediate boiler. Fig. 2 shows residue curve maps for the classical reactive distillation process at three di9erent Damk/ohler numbers. In the non-reactive case, i.e. Da=0 (Fig. 2a), the map topology is structured by one unstable node (pure B), one saddle point (pure C) and one stable node (pure A). The arrows along the residue curves indicate increasing time. Since pure A is the only stable node of non-reactive distillation, this is the feasible bottom product to be expected in a continuous distillation process. At kinetically controlled chemical reaction (Da = 1, Fig. 2b), the stable node moves from the pure A vertex into the composition triangle, i.e. the feasible product is a ternary

0.4

0.2

0 0

(c)

A

0.2

0.4

0.6

xB

0.8

1

B

Fig. 2. Residue curve maps for reactive distillation; A ⇔ B + C; K = 0:2; constant relative volatilities: !BA = 5:0; !CA = 3:0. Legend: (o) unstable node, ( ) saddle point, (•) stable node, (a) no chemical reaction (i.e. non-reactive distillation), (b) kinetically controlled chemical reaction, (c) equilibrium controlled chemical reaction.

Y.-S. Huang et al. / Chemical Engineering Science 59 (2004) 2863 – 2879

mixture instead of pure A. According to the classiKcation in Table 1, this stable node is a kinetic azeotrope. At equilibrium controlled chemical reaction (Da → ∞, Fig. 2c), the reaction approaches its chemical equilibrium, and therefore the stable node is exactly located on the equilibrium curve. In this case, the residue curves are Krst dominated by the reaction stoichiometry and then move along the equilibrium curve towards the stable node which is a reactive azeotrope. In analogous manner, RCMs of the reactive membrane separation process are determined from Eq. (8). First, a diagonal []-matrix is considered with CC ¿ 1 and BB = 1, i.e. the undesired by-product C permeates preferentially through the membrane (CC ¿ 1), while A and B are assumed to have the same mass transfer coe=cients. The RCMs are depicted for two di9erent CC -values (Figs. 3a–c: CC = 5; Figs. 3d– f: CC = 10) at three di9erent Damk/ohler numbers. Figs. 3a and d reveal the e9ect of the membrane at non-reactive conditions. The trajectories move from pure C to pure A, while in non-reactive distillation (Fig. 2a) they move from pure B to pure A. Thus, by application of a C-selective membrane, the C vertex becomes an unstable node, while the B vertex becomes a saddle point. This is due to the fact that the membrane changes the e9ective volatilities (i.e. the products ii · !iA ) of the reaction system such that CC !CA ¿ BB !BA . At kinetically controlled reactive conditions (Da = 1), Figs. 3b and e show that the stable node (kinetic arheotrope) moves into the composition triangle as in reactive distillation (Fig. 2b). But, this node moves towards the B vertex with increasing C-selectivity of the membrane. At inKnite Damk/ohler number the system is equilibrium controlled (Figs. 3c and f), and therefore the stable node (reactive arheotrope) is located exactly on the chemical equilibrium curve. The di9erent stable node locations in Fig. 3c (CC = 5) and in Fig. 3f (CC = 10) reveal that pure B is a feasible product if the membrane selectivity with respect to C exceeds a certain critical value. 3.1.2. Singular points For practical applications, the mixture composition at the singular points are of major interest. As discussed above, all potential singular points of a certain system are located on a unique curve (PSPC) whose location is deKned by Eq. (17). In the here considered case, the PSPC is given by XB = YB ;

(22)

where the reactant A is chosen as the reference component because T ¿ 0 (see Appendix A). For conventional reactive distillation, Eq. (18c) deKnes the transformed vapour phase composition YB and this yields in combination with Eq. (22): (−xB − xA ) (−yB − yA ) = (for RD): (23) (−1 − xA ) (−1 − yA ) At the given set of relative volatilities (!BA =5:0, !CA =3:0), the PSPC deKned by Eq. (23) is a hyperbola (see Fig. 4). For the sake of completeness, the PSPC is also shown outside the physically relevant composition space. The left part of

2869

the hyperbola intersects the chemical equilibrium curve at the reactive azeotrope. Between this point (Da → ∞) and the origin of the triangle (Da = 0), the branch of stable nodes is located (solid line). Each point along this branch corresponds to a certain value of the Damk/ohler number. Formally, the reactive distillation process can be seen as a special case of the reactive membrane separation process, namely a membrane process with a scalar [k]-matrix. However, now the question arises how the PSPC and the branch of stable singular points are inMuenced by a membrane having a diagonal mass transfer matrix. In Fig. 5 the PSPC of the reactive distillation process is depicted along with the PSPCs of the reactive membrane separation process for three di9erent CC -values. Only the physically relevant composition space is considered in this diagram. The solid lines represent the di9erent branches of stable nodes. Obviously, there exists a critical value, CC; crit = 5=3, above which the PSPC is turned from the C vertex towards the B vertex. At the intersections of the PCPCs with the chemical equilibrium curve (Da → ∞), reactive arheotropes are located. They are moved towards the B vertex with increasing C-selectivity of the membrane, i.e. increasing CC -value. Above a certain value the reactive arheotrope coincides with the pure B vertex and at this point the PSPC is tangential to the chemical equilibrium curve. 3.1.3. Generalization For a more generalized analysis of the qualitative inMuence of membranes on the singular points, now the reactive membrane separation process is considered with a non-diagonal []-matrix. Then, Eq. (17) in combination with Eq. (18a) yield the condition for kinetic arheotropy: (−xB − xA ) (−nB − nA ) = (for RMS): X B = YB ⇒ (−1 − xA ) (−nT − nA ) (24) It is easy to show that Eq. (24) can be casted into the following quadratic form: (x)T · [A] · (x) + (b)T · (x) + c = 0

(25)

with (x)T = (xB ; xC ) and the following matrix–vector deKnitions (note that !AA = 1):  a2 =2 a1 ; (26a) [A] = a2 =2 a3 (b)T = (a4 ; a5 ); (26b) c = a6 ; a1 = −1 − BA + AB !BA + BB !BA ;

(26c) (27a)

a2 = −BA + CA − AB !BA − CB !BA + AC !CA +BC !CA ;

(27b)

a3 = 1 + CA − AC !CA − CC !CA ;

(27c)

a4 = 1 + 2 · BA − CA − BB !BA + CB !BA ;

(27d)

a5 = −1 + BA − 2 · CA − BC !CA + CC !CA ;

(27e)

a6 = −BA + CA :

(27f)

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Y.-S. Huang et al. / Chemical Engineering Science 59 (2004) 2863 – 2879

C

C

1

1

Da=0

Da=0 0.8

0.6

0.6

xC

xC

0.8

0.4

0.4

0.2

0.2

0 0

(a)

0.2

0.4

0.6

0.8

xB

A

0 0

1

B

(d)

0.2

0.4

0.6

1

B

C

C 1

1

Da=1

Da=1

0.8

0.8

0.6

0.6

0.4

Kinetic Arheotrope

0.2

0 0

0.2

0.4

0.6

Kinetic Arheotrope

0.2

0.8

xB

A

Chemical Equilibrium Curve

xC

xC

Chemical Equilibrium Curve

0.4

(b)

0.8

xB

A

0 0

1

B

(e)

C 1

0.2

0.4

A 1

xB

0.6

0.8

1

B

C

Da=∞

Da=∞

0.6

0.6

xC

0.8

xC

0.8

0.4

0.4

Reactive Arheotrope

Reactive Arheotrope

0.2

0 0

A

(c)

0.2

0.2

0.4

0.6

0.8

xB

0 0

1

B

(f)

A

0.2

0.4

0.6

xB

0.8

1

B

Fig. 3. Residue curve maps for reactive membrane separation; A ⇔ B + C, K = 0:2; constant relative volatilities: !BA = 5:0, !CA = 3:0; membrane: diagonal []-matrix. Legend: (o) unstable node, ( ) saddle point, (•) stable node     1 0 0 1 0 0     (3a–c) : [] =  1 0 (3d–f ): [] =  1 0  0 ; 0 . 0

0

5

0

0

10

Y.-S. Huang et al. / Chemical Engineering Science 59 (2004) 2863 – 2879

The following cases can be distinguished:

1.2

C

1.0

%1 · %2 ¡ 0 ⇒ hyperbolic system;

Potential Singular Point Curve (PSPC)

%1 · %2 ¿ 0 ⇒ elliptic system: %1 = 0

0.8 Da= ∞ (Reactive Azeotrope)

1.0

0.4

0.5 0.2

0.2 Da=0

Stable Node Branch (Kinetic Azeotropes)

0.0

B

A -0.2 0.2

0.0

0.2

0.4

0.6

0.8

1.0

(28)

a1 = −1 + BB !BA ;

(29a)

a2 = 0;

(29b)

a3 = 1 − CC !CA ;

(29c)

a4 = −a1 ;

(29d)

a5 = −a3 ;

(29e)

a6 = 0:

(29f)

(−1 + BB !BA ) · (1 − CC !CA ) ¡ 0 ⇒

Fig. 4. Potential singular point curve and stable node bifurcation behaviour for reactive distillation; A ⇔ B + C; K = 0:2; constant relative volatilities: !BA = 5:0, !CA = 3:0. Legend: (•) branch of stable nodes.

hyperbolic system; (−1 + BB !BA ) · (1 − CC !CA ) ¿ 0 ⇒ elliptic system;

In the composition triangle, Eq. (25) describes a second-order curve whose shape is Kxed by the signs of the eigenvalues %1 and %2 of the symmetric matrix [A].

(−1 + BB !BA ) = 0

or

(1 − CC !CA ) = 0 ⇒

parabolic system:


CC

PSPC 1 2 3 4

C

%2 = 0 ⇒ parabolic system:

In this case, the shape of the PSPC can be classiKed as follows:

1.2

xB

1

or

For the special case of a membrane being characterized by a diagonal []-matrix, Eqs. (27a–f) reduce to

Chemical Equilibrium Curve

xC

0.6

2871

(30)

Process Reactive distillation Reactive membrane separation Reactive membrane separation Reactive membrane separation

1.0 2.0 5.0 10.0

0.8

PSPC 1 0.6

Chemical Equilibrium Curve

xC

Reactive Azeotrope 0.4

PSPC 2

Reactive Arheotrope

0.2

Da=0

PSPC 3 0 0

A

0.2

PSPC 4

0.4

0.6

xB

0.8

1

B

Fig. 5. Potential singular point curves (PSPCs) and stable node bifurcation behaviour of reactive membrane separation at di9erent mass transfer conditions; A ⇔ B + C, K = 0:2; constant relative volatilities: !BA = 5:0, !CA = 3:0; membrane with diagonal []-matrix   1 0 0   ⇒ [] =  1 0  0 . Legend: (•–•) Branch of stable nodes (Damk/ohler numbers: Da ∈ (0; ∞)). 0

0

CC

2872

Y.-S. Huang et al. / Chemical Engineering Science 59 (2004) 2863 – 2879 PSPC

1.4

1.4 1.2

1.2

C

C

1.0

1.0 0.8

0.8

Da=∞

xC

xC

0.6 0.4 0.2

Da=0

0.6 0.4 0.2

Da=∞

Da=0

0.0

0.0

A

-0.2 -0.4 -0.4 -0.2

0.0

0.2

0.4

0.6

-0.2 -0.4

0.8

1.0

1.2

-0.4 -0.2

1.4

xB

(a)

B

A

B

Chemical Equilibrium Curve

0.0

0.2

0.4

0.6

0.8

1.0

1.2

v

1.4 1.4 1.2

1.2

C

C

1.0 0.8

1.0 0.8

Da=0~∞

0.6

xC

0.6

xC

1.4

xB

(b)

0.4 0.2

0.4

Da=∞

0.2

0.0

A

A

0.0

B

-0.2

B -0.2

-0.4

Da=0

-0.4 -0.4 -0.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

(c)

xB

-0.4 -0.2

(d)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

xB

Fig. 6. Potential singular point curves (PSPC) and stable node bifurcation behaviour intrinsically hyperbolic system; A ⇔ B + C; K = 0:2 (For parameters see Table 2). Legend: (•) Branch of stable nodes.

For the simple case of a scalar mass transfer matrix, the criterion for the shape of the PSPC is (−1 + !BA ) · (1 − !CA ) ¡ 0 ⇒ intrinsically hyperbolic system; (−1 + !BA ) · (1 − !CA ) ¿ 0 ⇒ intrinsically elliptic system; (−1 + !BA ) = 0

or

(1 − !CA ) = 0 ⇒

intrinsically parabolic system:

(31)

In Eq. (31), the term “intrinsically” reMects the fact that, if no membrane is applied, the shape of the PSPC only depends on the relative volatilities of the reaction components. Eq. (31) is in full agreement with the conclusion stated in the literature (Barbosa and Doherty, 1988a,b) that the ranking of the volatilities of the reaction components determines whether a reactive distillation system exhibits reactive azeotropy or not. If the reactant A is highest boiler or lowest boiler of the reaction system (as assumed in Fig. 4: !BA =5:0, !CA =3:0), one has an intrinsically hyperbolic system in which the PSPC intersects with the chemical equilibrium line within the physically relevant composition range, i.e. a reactive azeotrope will appear.

Due to the fact that, in reactive membrane separation, the eigenvalues %1 and %2 of [A] are dependent on the binary mass transfer coe=cients, ij , the shape and the orientation of the PSPC of a reactive distillation process (i.e. the attainable products) can be changed by application of membranes. This is illustrated in the next subsection. 3.1.4. In>uence of membranes on singular points An intrinsically hyperbolic system is studied under the inMuence of membrane permeation (Fig. 6). The applied parameters (volatilities !iA , mass transfer coe=cients ij ) and the corresponding eigenvalues of the [A]-matrix are summarized in Table 2. For comparison, again the PSPC for the reactive distillation process is given in Fig. 6a. This situation is equivalent to a reactive membrane separation process with a scalar mass transfer matrix, i.e. [] = [I ]. The branch of stable nodes is indicated by a solid line for all positive Damk/ohler numbers Da ∈ [0; ∞). In Fig. 6b, the e9ect of a selective membrane having a diagonal []-matrix is illustrated. It is assumed that the membrane preferentially removes the undesired byproduct C from the liquid phase (CC = 5, see Table 2). The membrane changes the vertically oriented hyperbola to a horizontally oriented hyperbola. Based on Eqs. (26) and (29a, c), one can show that the

Y.-S. Huang et al. / Chemical Engineering Science 59 (2004) 2863 – 2879

2873

Table 2 Parameters used for the potential singular point curves presented in Fig. 6 Fig.

Parameters Volatilities

(6a)

(6b) (6c)

(6d)

!BA = 3:0 !CA = 2:0

!BA = 3:0 !CA = 2:0 !BA = 3:0 !CA = 2:0

!BA = 3:0 !CA = 2:0

Eigenvalues of [A] []-matrix Scalar  1 0  0 1  0

0 1

Diagonal  1 0  1 0

0

0

0

%2

2

−1

Vertical hyperbola

Intrinsic hyperbolic system (membrane without any separation e9ect, corresponds to reactive distillation)

2

−9

Horizontal hyperbola

Selective membrane which preferentially transfers C

2

0.6

Ellipse

(Improper) membrane creates a stable node at the undesired by-product C, membrane causes change to elliptic system

2.05

−9:55

Rotated and deformed hyperbola Interaction of A and C in selective membrane creates non-reactive binary arheotrope at A − B edge



 0

0



0

 0  0:2

Non-diagonal   1 0 0:5   1 0   0 0:5

%1

5

Diagonal  1 0  1 0 0

Physical meaning



 0

0

PSPC shape

0

5

coe=cients a1 and a3 Kx the orientation of the hyperbola:

3.2. Example II: Non-ideal reaction system

|a1 | ¿ |a3 |; i:e: | − 1 + BB !BA | ¿ |1 − CC !CA | ⇒

Now, after the analysis of an ideal ternary reaction system, a non-ideal example of technical relevance is considered. This is the heterogeneously catalysed cyclization of 1,4-butanediol (1,4-BD) to tetrahydrofurane (THF) and water:

vertical hyperbola;

(32a)

|a1 | ¡ |a3 |; i:e: | − 1 + BB !BA | ¡ |1 − CC !CA | ⇒ horizontal hyperbola:

H+

1; 4-butanediol−→tetrahydrofurane + H2 O; (32b)

The PSPC has an elliptic shape if an (improper) membrane, which preferentially retains the byproduct C, is applied (Fig. 6c). In this case, the pure C vertex is the only stable node of the system. That means the only feasible product is the pure byproduct C at reactive and also at non-reactive conditions. The e9ect of a non-diagonal []-matrix is illustrated in Fig. 6d. O9-diagonal elements will appear if molecular interactions between the permeating species are involved. As an example, here the interaction of A and C is embodied in the []-matrix by assuming AC =0:5 and CA =0:5. All other parameters are equal to the case in Fig. 6b. As can be seen the horizontal hyperbola is slightly rotated and deformed. As a consequence, a binary mixture on the A–B edge, instead of pure A, is a stable node at non-reactive conditions (arheotrope). Moreover, pure B is achievable at Da → ∞ which cannot be attained in the absence of A–C interactions at the given parameter settings (compare to diagonal case in Fig. 6b).

SR H 0 = −13:4 kJ=mol:

(33)

Limbeck et al. (2001) have studied the kinetics of this irreversible reaction at macroporous acidic ion exchange resins as catalysts. In this reaction, the byproduct water has a signiKcant a=nity with the catalyst and this inhibits the catalytic reaction severely. Therefore, the authors proposed the rate expression in Eq. (34): r = '(aH2 O ) · =

kf · KBD aBD 1 + KBD aBD

kf · KBD aBD 1 · ; √ 1 + KH2 O aH2 O 1 + KBD aBD

(34)

where kf represents the rate constant, KBD and KH2 O the sorption constants of 1,4-BD and water, and ' the water inhibition factor. The expressions of kf , KBD and KH2 O as functions of temperature are as follows:     kf −18282:42 −18282:42 exp ; (35) = exp kf; ref T=[K] 404:2

Y.-S. Huang et al. / Chemical Engineering Science 59 (2004) 2863 – 2879

1335:1 KBD = 6:29 × 10 × exp T=[K]   764:97 KH2 O = 2:43 × exp : T=[K]

 ;

(36b)

The boiling point of THF at the operating pressure p=5 atm is chosen as the reference temperature. The liquid phase activity coe=cients were calculated from the Wilson equation using the interaction parameters Aij and molar volumes Vi L as listed below (Gmehling and Onken, 1977):   ABB ABT ABH    ATB ATT ATH    AHB AHT AHH   0 1030:1867 410:4832   0 1140:7179  =  105:6295  (cal=mol); 675:7784 1819:4033 0 (40) L VTHF

 = 8:835

VHL2 O

0.8

0.4

0

Thus, water will pass preferentially through the Knudsen membrane.

0.2

549.3K

0.4

0.6

0.8

1

xTHF

THF 404.2K

water (0.0143, 0.8196) 428.4K

1

Da=0.4

0.8

(0.0328, 0.6935) 429.0K 0.6

(0.6709, 0.3291)

0.4

(0.9760, 0.0091) 404.3K

0.2

0

0

0.2

(b) 1,4 BD

0.4

0.6

0.8

1

THF

xTHF

water Da=∞

(m3 =mol): (41)

Since water is the byproduct and it has an additional undesired inhibition e9ect at the catalyst, it has to be separated e=ciently from the reaction mixture. To achieve this, both conventional reactive distillation (Fig. 1a) and reactive membrane separation (Fig. 1b) are considered as process alternatives. In the second process, a Knudsen membrane is applied. Consequently, the mass transfer matrix [] has a diagonal structure and the diagonal elements are the Knudsen selectivities, i.e. the square roots of the ratios of the molecular weights Mi (1,4-BD as reference component):   1 0 0     MBD =MTHF 0 [] =  0   0 0 MBD =MH2 O   1 0 0   0  = (42)  0 1:1 : 0 0 2:5

0

(a) 1,4 BD

1

 1:807 × 10−5

(0.6709, 0.3291) 396.3K

0.2



8:155

0.6

0.8

0.6

xwater

L VBD

1

Da=0

The saturated vapour pressures of the reaction component are calculated from the following Antoine equations: 3905:1 sat log10 (pBD ; (37) =[bar]) = 8:0236 − T=[K] − 15:96 1442:3 sat log10 (pTHF =[bar]) = 4:27 − ; (38) T=[K] 1985:0 sat : (39) =[bar]) = 5:33 − log10 (pH 2O T=[K]



water 429.4K

(36a)

xwater



−2

xwater

2874

(0.6709, 0.3291)

0.4

0.2

0

(c)

0

1,4 BD

0.2

0.4

xTHF

0.6

0.8

1

THF

Fig. 7. Residue curve maps for reactive distillation; 1,4-butanediol → THF + water; p = 5 atm. Legend: (o) unstable node, ( ) saddle point, (•) stable node, (a) no chemical reaction (i.e. non-reactive distillation), (b) kinetically controlled chemical reaction, (c) equilibrium controlled chemical reaction.

3.2.1. Residue curve maps (RCM) First, RCMs were determined for non-reactive distillation (Fig. 7a), for reactive distillation at kinetically controlled reaction (Fig. 7b) and for reactive distillation at equilibrium

Y.-S. Huang et al. / Chemical Engineering Science 59 (2004) 2863 – 2879

3.2.2. Singular points Fig. 9 shows the PSPC and the bifurcation behavior of simple reactive distillation. Qualitatively, the curve of potential singular points is shaped like a hyperbola due to the boiling sequence of the involved components. Along the left-hand part of the PSPC, the stable node branch and the saddle point branch 1 coming from the water vertex, meet each other at the kinetic tangent pinch point (x)T =

1

water 429.4K Da=0

xwater

0.8

(0.4093, 0.5907) 397.2K

0.6

0.4

0.2

0 0

(a)

0.2

1,4 BD 549.3K 1

0.4

0.6

0.8

xTHF

1

THF 404.2K

water (0.0073, 0.9307) 425.7K

Da=0.4

0.8

xwater

(0.4093, 0.5907) 0.6

0.4

(0.2313, 0.2520) 425.9K

(0.9540, 0.0081) 405.3K

0.2

0 0

0.2

1

0.4

0.6

0.8

xTHF

(b) 1,4 BD

1

THF

water Da=∞

0.8

(0.4093, 0.5907) 0.6

xwater

controlled reaction (Fig. 7c). The topology of the maps at non-reactive conditions (Da = 0) is structured by a binary azeotrope (unstable node) between water and THF. Pure water and pure THF are saddle nodes while the 1,4-BD vertex is a stable node. It has the highest boiling point in this system. At Da = 0:4 (Fig. 7b), the two saddle points are moved from the pure vertices into the composition triangle. The same is true for the stable node which is moved from the 1,4-BD vertex to the kinetic azeotrope at (x)T = (0:0328; 0:6935). Pure water and pure THF are still singular points, but they become stable nodes under reactive conditions. The unstable node on the water-THF edge remains unmoved. It forms two separatrices with the two saddle points. Therefore, the whole composition space is divided into three subspaces which have each a stable node, namely pure water, pure THF and the kinetic azeotrope. At Da → ∞ (Fig. 7c), only pure water and pure THF remain stable nodes. The residue curves Krst are dominated by the reaction stoichiometry and approach the water-THF edge, which is actually the chemical equilibrium line of this irreversible reaction, and then converge to the water vertex or to the THF vertex. When starting from pure 1,4-BD, the undesired by-product water will be the Knal product in the distillation still. The e9ect of a Knudsen membrane on the process behaviour is illustrated in Fig. 8a which is valid at non-reactive conditions. Comparing Fig. 7a and Fig. 8a, the unstable node on the THF-water edge is moved closer to water vertex by application of the Knudsen membrane, while the two saddle points and stable node are not a9ected. At Da = 0:4 (Fig. 8b), the RCM is structured by one unstable node, two saddle points and three stable nodes (as in Fig. 7b). But the locations of the two saddle points and the stable node are changed. In particular, the kinetic arheotrope at (x)T =(0:2313; 0:2520) is located closer to the THF vertex than the kinetic azeotrope at (x)T = (0:0328; 0:6935). Similar to the reactive distillation process, at Da → ∞ the composition space is again divided into two subspaces which have either pure THF or pure water as attractors (Fig. 8c). However, as very important feature of the reactive membrane separation process, when starting with pure 1,4-BD it is possible to attain pure THF as the desired product. This is not possible with conventional reactive distillation. The reason for this behaviour is the fact that the saddle point on the THF-water edge is shifted towards the water vertex.

2875

0.4

0.2

0 0

(c)

1,4 BD

0.2

0.4

xTHF

0.6

0.8

1

THF

Fig. 8. Residue curve maps for reactive membrane separation; 1,4-butanediol → THF + water; p =5 atm; Knudsen membrane. Legend: (o) unstable node, ( ) saddle point, (•) stable node, (a) no chemical reaction (i.e. non-reactive membrane separation), (b) kinetically controlled chemical reaction, (c) equilibrium controlled chemical reaction.

(0:0246; 0:7462) where the critical Damk/ohler number is Da = 0:414. The right hand part of the PSPC is the saddle point branch 2 which runs from pure THF to the binary azeotrope between THF and water.

2876

Y.-S. Huang et al. / Chemical Engineering Science 59 (2004) 2863 – 2879

Da=0

1

water

Da=0 Saddle Point Branch 1

0.3

0.8

1

water 0.4

Saddle Point Branch 2

0.4

0.8

Kinetic

0.414

Tangent

0.4

1.0 1.5

Pinch

2.0

0.6

0.3

0.4

Stable Node Branch

0.1

3.0 4.0

Da=0

4.0

1,4 BD

0.6

xTHF

0.8

1.8

THF

0 0

Fig. 9. Potential singular point curve and bifurcation behaviour for reactive distillation; 1,4-butanediol → THF + water; p = 5 atm. Legend: (o) unstable node, ( ) saddle point, (•) stable node.

(a)

Da=0

1.0

0.2

0.4

0.4

1,4 BD

1

0.6

0.8

1

THF

xTHF

water

1.0 3.0

Saddle Point Branch 2

7.2

0.8

xwater

In a similar manner, Fig. 10 illustrates the PSPC for the reactive membrane separation process. In Fig. 10a, the effect of a Knudsen membrane on the PSPC can be seen by comparison with the corresponding diagram for reactive distillation (i.e. []-matrix equals identity matrix) depicted in Fig. 9. Generally, the membrane turns the vertical hyperbola into an horizontal hyperbola. In particular, the membrane shifts the stable node branch towards the THF vertex such that THF-rich products can be achieved in the considered Knudsen membrane reactor. Because the boiling temperature of 1,4-BD is much higher than of the two products and the cyclization reaction is irreversible, the bifurcation behaviour is only a9ected by the ratio of water =THF if BD is not extremely high or low. There exists a critical value of water =THF = 2:1, above which the stable node branch approaches the THF vertex. Fig. 10b shows the e9ect of a membrane which is assumed to have with a higher selectivity of water with respect to 1,4-BD than the Knudsen membrane, namely water = 11 instead of 2.5 in the Knudsen case. At this higher selectivity the location of stable node branch is closer to BD-THF edge, i.e. a higher THF content can be achieved in the Knal liquid residue. In order to illustrate the di9erence between the two considered membranes on a quantitative basis, it is helpful to simultaneously integrate Eqs. (2) and (3) to get the two diagrams given in Fig. 11. With the initial condition xBD; 0 = 1:0 at Da = 100 (i.e. equilibrium controlled chemical reaction), Fig. 11a shows the time needed for insitu puriKcation. A Knudsen membrane needs about four times longer than the more selective membrane process to achieve a reasonable product purity. In Fig. 11b, the relative liquid holdup (H=H0 ) is plotted versus the liquid mole fraction of the desired prod-

Da=0 1.5

Pinch

0.001 0.4 1

1.87

Tangent

1.0

0.4

1.8

Kinetic

Da=0

2.0

0.2

Saddle Point Branch 1

0.1

0.2 0.01

Da=0 0 0

6.0

0.4

Saddle Point Branch 2

0.01

8.0

0.4 Stable Node Branch 1.0 1.5

7.29

0.2

9.47

xwater

xwater

0.6

0.6

0.4 Kinetic

Stable Node Branch

Tangent

Saddle Point Branch 1

Pinch

Da=0

0.2

Da=0 0 0

(b)

0.1

0.001 0.01

0.2

0.2

0.3

0.4 0.438

0.4

0.6

0.4

0.8

xTHF

1,4 BD

0.3

1

THF

Fig. 10. Potential singular point curves (PSPC) and bifurcation behavior of reactive membrane separation at di9erent mass transfer conditions; 1,4-butanediol → THF + water; p = 5 atm. 

1  (a) Knudsen membrane [] =  0 

0

 (b) membrane with [] =  0

0 0

1:1



 0  ,

1:1

0 1

0

0

2:5 

 0  .

0 0 11 Legend: (o) unstable node, ( ) saddle point, (•) stable node.

uct THF. Although pure THF is a feasible product for a Knudsen membrane reactor, the Knal product amount is almost negligible, while the more selective membrane produces an acceptable amount of pure THF (ca. 70% of max. value).

Y.-S. Huang et al. / Chemical Engineering Science 59 (2004) 2863 – 2879

More Selective Membrane 1

xTHF

Knudsen Membrane 0.5

Reactive Distillation

0 1

2

3

4

5

6

Characteristic residence time (H0/nT0)

(a) 2

More Selective Membrane

Liquid holdup (H/H0)

1.5

1

Knudsen Membrane 0.5

0 0

0.2

0.4

0.6

0.8

1

xTHF

(b)

Fig. 11. Transient behaviour of a reactive membrane separation process based on di9erent membranes (Knudsen membrane/more selective membrane); 1,4-butanediol → THF + water; p = 5 atm; initial condition: xBD; 0 = 1:0, Da = 100;   1 0 0   Knudsen membrane [] =  1:1 0  0 , 0



0 1

 more selective membrane [] =  0 0

2:5 0 1:1 0

0



 0  . 11

(a) THF liquid mole fraction versus characteristic residence time, (b) liquid holdup versus THF liquid mole fraction.

2877

tive liquid phase. However, the proposed methodology can be also applied to a reactive liquid permeation process (pervaporation). The integration of membrane separation into a reactive distillation process inMuences the e9ective volatilities of the reaction components. Above certain critical values of the membrane selectivities, the highest boilers are preferentially transferred to the permeate side while the light boilers are kept back on the retentate side. This in turn inMuences the rate of the chemical reaction taking place in the liquid phase. As a consequence, the singular points are shifted and thereby the topology of the residue curve maps is changed signiKcantly. Depending on the structure of the matrix of e9ective membrane mass transfer coe=cients (scalar/diagonal/non-diagonal), the attainable product compositions are shifted to a desired direction or to an undesired direction. This has been demonstrated for a simple ideal reaction system (A ⇔ B +C) and also for a selected non-ideal reaction example (cyclization of 1,4-butanediol to THF). A scalar mass transfer matrix corresponds to the conventional (non-)reactive distillation process. A diagonal mass transfer matrix can be realized in practice e.g. by application of a Knudsen membrane. However, due to interactions of the permeating components within the membrane, the o9-diagonal elements of the mass transfer matrix can play an important role. These elements lead to the appearance of (non-reactive) singular points along the edges of the phase triangle. These points correspond to the pseudo-azeotropic points which were discussed in the Keld of pervaporation processes (Nguyen and Clement, 1991) and also in di9usion distillation (Fullarton and Schl/under, 1986). The appearance of singular points in (non-)reactive membrane separation is a process phenomenon, and is not a pure thermodynamic phenomenon. Therefore, one should denote these singular points as reactive arheotropes which means in translation that the liquid composition is not changing with the Muxes. The condition of arheotropy can be formulated in terms of a set of transformed composition variables in analogous manner to reactive distillation (Appendix A). This condition yields the potential singular point curve (PSPC) whose shape depends on the reaction stoichiometry, the vapour–liquid equilibria, and on the mass transfer properties of the membrane applied.

Notation

4. Conclusions

ai a1 –a6

As demonstrated by means of residue curve analysis, selective mass transfer through a membrane has a signiKcant e9ect on the location of the singular points of the considered batch reactive separation process. The analysis was carried out for a vapour permeation process, i.e. the membrane is considered to be placed in the vapour phase above the reac-

A,B,C Da H H0 Hr kf

liquid phase activity of component i coe=cients in the quadratic equation, Eq. (27a–f) chemical species Damk/ohler number, Eq. (6) molar liquid holdup, mol initial molar liquid holdup, mol holdup in which the reaction proceeds, mol forward reaction rate constant, 1/s

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kf; ref [k] kij K KBD KH2 O Mi ni nT nT 0 NC pisat p r R R S t T (x) xi Xi yi Yi

Y.-S. Huang et al. / Chemical Engineering Science 59 (2004) 2863 – 2879

forward reaction rate constant at reference temperature, 1/s matrix of binary e9ective mass transfer coe=cients, Eq. (11) e9ective binary mass transfer coe=cients of pair i–j, m/s chemical equilibrium constant sorption constant of 1,4-butanediol sorption constant of water molecular weight of component i mass Mux of component i through membrane, mol/s total mass Mux through membrane, mol/s initial total mass Mux, mol/s number of reacting species saturated vapour pressure of component i, bar system pressure, bar reaction rate, mol=(mol s) universal gas constant, 8:314 J=(mol K) dimensionless reaction rate, Eq. (2) e9ective permeation area, m2 time, s boiling temperature, K vector of liquid phase mole fractions liquid phase mole fraction of component i transformed liquid phase mole fraction of component i, Eq. (17b) mole fraction of component i in vapour phase transformed vapour phase mole fraction of component i, Eq. (17c)

Greek letters !ij i ij %1; 2  i T '

relative volatility of i with respect to j, Eq. (21) liquid phase activity coe=cient of component i dimensionless binary mass transfer coe=cient, Eq. (12b) eigenvalues of matrix [A] deKned by Eq. (26a) dimensionless time, d = (nT =H ) dt stoichiometric coe=cient of component i total mole change of reaction water inhibition factor, Eq. (34)

components A, B and C, respectively 1,4-butanediol components i, j permeation side tetrahydrofurane

Abbreviations PSPC RCM RD

The authors like to thank Dr.-Ing. Michael Mangold from the Max Planck Institute in Magdeburg for his linguistic advice when creating the word “arheotrope”.

Appendix A. Potential singular point curves (PSPC) (Qi et al., 2004) allow more insight into the intrinsic process types and the e9ect of incorporating a membrane. We start with the component and total material balances, Eqs. (3) and (4). To eliminate the reaction term (Hr · kf · R) in both equations, so that the potential singular point curve covers all extents of reaction, it is convenient to utilize the material balance for a reference component k: d(H · xk ) = −nk + k · Hr · kf R dt

(A.1)

and substituting this result into Eq. (3) yields     d xi xi xk xk dH H + − − dt i k i k dt  =−

nk ni − i k



i = 1 : : : NC − 1; i = k:

(A.2)

Applying Eq. (A.2) to the overall material balance Eq. (4) gives    k n T −  T nk d xk dH T =− : (A.3) +H dt k −  T x k (k − T xk ) dt Eqs. (A.2) and (A.3) can be     d xi xi xk H + − − dt i k i    ni xi nk =− − − i k i

combined to    T d xk xk k k −  T x k dt   k nT − T nk xk ; − k  k −  T xk

i = 1 : : : NC − 1;

(A.4)

i = k:

This set of equations can be written in a compact form if one deKnes transformed composition variables as follows:

Subscripts A; B; C BD i; j p THF

Acknowledgements

potential singular point curve residue curve map reactive distillation

Xi =

(k xi − i xk ) ; (k − T xk )

Yi =

(k ni − i nk ) : (k nT − T nk )

(A.5)

With the transformed variables, Eq. (A.4) become dXi (k nT − T nk ) =− (Yi − Xi ); dt H (k − T xk )

i = 1 : : : NC − 1; (A.6) i = k

and by deKning the time variable, Eq. (A.7): d+ =

(k nT − T nk ) dt H (k − T xk )

(A.7)

Y.-S. Huang et al. / Chemical Engineering Science 59 (2004) 2863 – 2879

one gets the Knal (NC − 2) linearly independent reduced equations, in analogous manner to distillation and reactive distillation (Barbosa and Doherty, 1988a,b): dXi = Xi − Y i ; d+

i = 1 : : : NC − 1; i = k:

(A.8)

It is worth noticing that, if all Muxes ni are positive (i.e. directed from reacting phase to permeate phase) as the usual condition of membrane reactors, the dimensionless time + can be guaranteed to increase monotonically with t, by selecting the reference component k as (i) a reactant, if T ¿ 0, (ii) a product, if T ¡ 0, (iii) arbitrarily, if T = 0. Additionally, Xi and Yi obey the following summation equations: NC


Xi = 1;

i=1 i=k

NC


Yi = 1:

(A.9)

i=1 i=k

As a consequence of Eq. (A.8), at the singular points in a reactive membrane separation process, the transformed vapour and liquid phase mole fractions are equal: Xi = Y i ;

i = 1 : : : NC − 1; i = k:

(A.10)

References Aiouache, F., Goto, S., 2003. Reactive distillation—pervaporation hybrid column for tert.-amyl alcohol etheriKcation with ethanol. Chemical Engineering Science 58, 2465–2477. Atkins, P.W., 1990. Physikalische Chemie. VCH, Weinheim, Germany. Barbosa, D., Doherty, M.F., 1988a. The inMuence of equilibrium chemical reactions on vapour–liquid phase diagrams. Chemical Engineering Science 43, 529–540. Barbosa, D., Doherty, M.F., 1988b. The simple distillation of homogeneous reactive mixtures. Chemical Engineering Science 43, 541–550. Castillo, F.J.L., Towler, G.P., 1998. InMuence of multicomponent mass transfer on homogeneous azeotropic distillation. Chemical Engineering Science 53, 963–976. Fullarton, D., Schl/under, E.-U., 1986. Di9usion distillation—a new separation process for azeotropic mixtures. Chemical Engineering and Processing 20, 255–263. Gmehling, J., Onken, U., 1977. Vapour–liquid equilibrium data collection, chemistry data series. Frankfurt am Main, DECHEMA.

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Krishna, R., Wesselingh, J.A., 1997. The Maxwell-Stefan approach to mass transfer. Chemical Engineering Science 52, 861–911. Limbeck, U., Altwicker, C., Kunz, U., Ho9mann, U., 2001. Rate expression for THF synthesis on acidic ion exchange resin. Chemical Engineering Science 56, 2171–2178. Nguyen, Q.T., Clement, R., 1991. Analysis of some cases of pseudo-azeotropes in pervaporation. Journal of Membrane Science 55, 1–19. Qi, Z., Sundmacher, K., 2002. Bifurcation analysis of reactive distillation systems with liquid-phase splitting. Computers in Chemical Engineering 26, 1459–1471. Qi, Z., Flockerzi, D., Sundmacher, K., 2004. Singular points in reactive distillation systems. A.I.Ch.E. Journal, in press. Qi, Z., Kolah, A., Sundmacher, K., 2002. Residue curve maps for reactive distillation systems with liquid phase splitting. Chemical Engineering Science 57, 163. Rev, E., 1994. Reactive distillation and kinetic azeotropy. Industrial and Engineering Chemical Research 33, 2174. / Schl/under, E.-U., 1977. Uber den EinMuW der Sto9/ubertragung auf die Selektivit/at der Thermischen Trennverfahren - dargestellt am Beispiel der Schleppmitteldestillation. Verfahrenstechnik 11, 682–686. Schl/under, E.-U., 1979. The e9ect of di9usion on the selectivity of entraining distillation. International Chemical Engineering 19, 373–379. Song, W., Huss, R.S., Doherty, M.F., Malone, M.F., 1997. Discovery of a reactive azeotrope. Nature 388, 561. Song, W., Venimadhavan, G., Manning, J.M., Malone, M.F., Doherty, M.F., 1998. Measurement of residue curve maps and heterogeneous kinetics in methyl acetate synthesis. Industrial and Engineering Chemical Research 37, 1917. Springer, P.A.M., Baur, R., Krishna, R., 2002. InMuence of interface mass transfer on the composition trajectories and crossing of boundaries in ternary azeotropic distillation. Separation and PuriKcation Technology 29, 1–13. Sundmacher, K., Kienle, A., 2003. Reactive Distillation—Current Status and Future Directions. Wiley-VCH, Weinheim, Germany. Taylor, R., Krishna, R., 2000. Modelling reactive distillation. Chemical Engineering Science 55, 5183–5229. Thiel, C., Sundmacher, K., Ho9mann, U., 1997. Residue curve maps for hetero-geneously catalysed reactive distillation of fuel ethers MTBE and TAME. Chemical Engineering Science 52, 993–1005. Ung, S., Doherty, M.F., 1995a. Calculation of residue curve maps for mixtures with multiple equilibrium chemical reactions. Industrial Engineering & Chemical Research 34, 3195. Ung, S., Doherty, M.F., 1995b. Vapour–liquid phase equilibrium in systems with multiple chemical reactions. Chemical Engineering Science 50, 23–48. Venimadhavan, G., Buzad, G., Doherty, M.F., Malone, M.F., 1994. E9ect of kinetics on residue curve maps for reactive distillation. A.I.Ch.E. Journal 40, 1814–1824. Wade, J., Merriman, R.W., 1911. InMuence of water on the boiling point of ethyl alcohol at pressures above and below the atmospheric pressure. Journal of the Chemical Society Transactions 99, 997–1011. Yeh, G.C., Yeh, B.V., Schmidt, S.T., Yeh, M.S., McCarthy, A.M., Celenza, W.J., 1991. Vapour–liquid equilibrium in capillary distillation. Desalination 81, 161–187.