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Theoretical evaluation of two-phase flow in a chromatographic reactor Franziska Ortner, and Marco Mazzotti Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b00871 • Publication Date (Web): 09 Apr 2018 Downloaded from http://pubs.acs.org on April 9, 2018

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Industrial & Engineering Chemistry Research

Theoretical evaluation of two-phase ow in a chromatographic reactor Franziska Ortner and Marco Mazzotti∗ Institute of Process Engineering, ETH Zurich, 8092 Zurich, Switzerland E-mail: [email protected]

Abstract The performance of a chromatographic reactor in the presence of phase split and multiple convective phases is assessed theoretically. For this purpose, an equilibrium theory model is established, accounting for adsorption, two-phase ow, and a reversible reaction with negligible kinetic limitations. The model is solved for an exemplary esterication system, yielding n-hexyl-acetate and water as products. Evaluating the batch elution behavior under standard single phase and novel two-phase ow conditions, it is found that pure products can be achieved in both cases, and that the two-phase ow behavior can positively aect the separation performance.

1

Introduction

Reactive chromatography, a hybrid process coupling chemical reaction and chromatographic separation, has received great attention in the past decades (for an overview see Refs. 1,2). This process is particularly advantageous in the context of equilibrium limited reversible reactions, as the selective removal of reaction products from the reaction site allows to overcome the conversion limit imposed by chemical equilibrium. However, it can occur that products and/or reactants are only partially miscible. 3 In this 1

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case, enrichment of the partially immiscible components due to ongoing reaction within the chromatographic reactor can result in a liquid-liquid phase split - and consequently in two convective phases. A possible example of such a system consists of esterication reactions, which constitute the reaction class considered most frequently in the context of chromatographic reactors. 49 In these reversible, equilibrium limited reactions, an alcohol (D) and an acid (C) react to form water (B) and an ester (A) according to C + D − )− −* − A + B. In the presence of a polymeric ion exchange resin acting as catalyst and selective adsorbent, the two products water and ester are commonly the strongest and the weakest adsorbing components, respectively, while the two reactants adsorb with intermediate strength. 4,7,10 As a consequence, the two products are simultaneously separated and removed from the reaction site. In the case of long-chained, hydrophobic esters and reactants, miscibility of water and ester, and possibly of water and one or both reactants, can be limited. As a consequence, the enrichment of water due to ongoing reaction in the column can cause a phase split and possibly two-phase ow. The occurrence of multiple convective phases in the context of liquid chromatography has not been discussed in the literature until recently, when we presented an extensive experimental investigation and theoretical description of two-phase ow for a reversed phase (nonreactive) chromatographic system. 11,12 Due to a lack of understanding, a common approach is to adapt process conditions to prevent a phase split, which for some reactive chromatographic processes severely limits the applicable concentration ranges and the process performance, hence its attractiveness. 9 Yet it was shown experimentally that exceeding solubility limits in the exemplary case of methylformate hydrolysis in a packed-bed reactor (operated at steady state without chromatographic separation) can signicantly increase the yield. 13 In this study we build on the deep physical understanding of two-phase ow in non-reactive chromatography attained during our earlier work, 11,12 and we investigate theoretically the physical implications of a phase split and subsequent two-phase ow in a chromatographic re2


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actor, as well as its consequences for process performance. We evaluate an exemplary system consisting of n-hexanol (D) and acetic acid (C), reacting to water (B) and n-hexyl-acetate (A). The evaluation is based on an equilibrium theory model, assuming thermodynamic equilibrium between all (convective and adsorbed) phases, as well as reaction equilibrium. Furthermore, one of the reactants (in esterication reactions commonly the alcohol 4,8,9 ), is assumed to be present in excess (solvent), while all other components are diluted in the solvent. A simple analytical solution for such equilibrium theory model with a single convective phase has been presented previously. 1416 This model shall now be extended and solved, when accounting for the presence of multiple convective phases (in the spirit of our previous study 11,12 ). The paper is structured as follows: In section 2, we explain the underlying model assumptions and derive the model equations. Then, the exemplary esterication system is introduced, characterizing crucial system properties, such as thermodynamic equilibria, as well as reaction and two-phase ow behavior (section 3). The equilibrium theory model is used to derive elution proles for specic initial and feed conditions, located in the single-phase region and hence preserving a single-phase ow, or located in the two-phase region and thus resulting in phase split and two-phase ow (section 4). Qualitative properties of the elution proles are discussed (section 4.1), and the process performance in the presence of single-phase and two-phase ow is evaluated (section 4.2). For the sake of brevity and readability, the underlying mathematical derivation of the equilibrium theory solution, which is novel for the applied model and provides a deeper mathematical and physical understanding, is explained in appendix A. Conclusions are drawn in section 5.

3



2

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Equilibrium theory model

2.1 Derivation of model equations We set up the following conservation laws, accounting for adsorption, multi-phase ow and chemical reaction:

∂Ci ∂ni ∂Fi +ν + = κi R, ∂τ ∂τ ∂ξ

i = 1...3.

(1)

Note that, in the context of the mathematical model, components A to D are renamed to components 1 to 4. The independent dimensionless time and space variables are designated as ξ = x/Lc and τ = tu/Lc , where Lc is the column length and u is the supercial velocity. The phase ratio ν is dened as (1 − )/, with denoting the total porosity. The adsorbed phase concentration of component i (in mole per L volume of adsorbent) is denoted as ni , while R is the overall specic reaction rate of the equilibrium reaction (over all convective phases) and κi is the stoichiometric coecient of the reaction, equaling 1 for components 1 and 2 and -1 for component 3. Finally, variables Ci and Fi describe the overall liquid concentration and overall fractional ow of component i over all convective phases. It should be pointed out that only conservation laws for the rst three components are considered, as the fourth component can be determined through the stoichiometric correlation 4 X

Zi = 1,

(2)

i=1

where Zi is the mole fraction of component i over all convective phases. For the interconversion between Zi and Ci , see the supporting information. In this contribution, the fourth component (in our case the alcohol) is assumed to be present in excess, as a solvent. The overall liquid concentrations Ci of the other three components (one reactant and the two products) are assumed to be rather low, such that volumetric changes due to adsorption, reaction or mixing are neglected. As a consequence, constant volumes of adsorbed and convective phases (constant porosity ) and a constant overall ow rate (constant supercial 4


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velocity u) are assumed in equations 1. For convective phases in particular, we assume specic volumes to equal the specic volume of the bulk component (alcohol); details concerning this assumption are discussed in the supporting information. In addition, we neglect heat effects upon adsorption or reaction and consider a thermostated chromatographic setup, hence the process is assumed isothermal (in agreement with various literature studies of dierent reactive chromatographic systems 3,4,1719 ). The overall liquid concentrations Ci and overall fractional ows Fi (in mole per unit volume of convective phases) are dened as:

Ci =

NP X j=1

zij ρj Sj =

NP X

cij Sj ,

Fi =

j=1

NP X

zij ρj fj =

j=1

NP X

cij fj

(3)

j=1

Here, NP corresponds to the number of convective phases, zij and cij denote the molar fractions and concentrations of component i in phase j , respectively, and ρj is the molar density of phase j . The so-called phase saturation Sj indicates the volumetric fraction of phase j with respect to the overall volume of all convective phases, while the fractional ow

fj describes the volumetric ow fraction of phase j with respect to the overall volumetric PNP PNP ow. It is obvious that j=1 fj = 1. Dierent convective phases can j=1 Sj = 1 and move with dierent velocities (fj 6= Sj ). The fractional ows fj are often expressed as a rational function of Sj (a detailed discussion is provided in section 3). It follows in the case of multiple convective phases that Fi 6= Ci , but Fi = Fi (C). Only in the case of a single convective phase, the system simplies to S1 = f1 = 1 and Fi = Ci = ci . It is further assumed that all convective phases are in thermodynamic equilibrium (i.e., activities ai of each component are identical in all convective phases), and that convective phases and the adsorbed phase are in equilibrium (which is described by an adsorption isotherm, being a function of the liquid phase activities, as discussed in section 3). Under the additional assumption of chemical equilibrium, the equilibrium relationship, de-

5



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ned through the equilibrium constant, provides another constraint, namely:

K=

a1 a2 a3 a4

(4)

With this constraint, the degrees of freedom (number of unknown variables), can be further reduced from 3 to 2. This means that, in a quaternary system, the concentrations of only 2 components uniquely dene the system at chemical and phase equilibrium. Since conservation equations 1 are linear in the reaction rate R, this linear term can be eliminated by summation, as described in detail in the literature. 1416,20 Doing so, the system of conservation equations is reduced to two homogeneous partial dierential equations:

∂n ˜ i ∂ F˜i ∂ C˜i +ν + = 0, ∂τ ∂τ ∂ξ

i = 1, 2.

(5)

For the assumed type of reaction, x ˜i = xi + x3 , with x = n, c, C, F . Accordingly, one can also dene C˜i and F˜i as:

C˜i =

NP X

F˜i =

c˜ij Sj ,

j=1

NP X

c˜ij fj .

(6)

j=1

˜ and F˜i = F˜i (C) ˜ ) only Note that the reduction to the variables C˜1 and C˜2 (both n ˜i = n ˜ i (C) provides a sucient, unique description of the system under the assumption of chemical and phase equilibrium (where degrees of freedom are reduced to 2). Equations 5 constitute a system of rst order, homogeneous partial dierential equations, which can be solved analytically with the method of characteristics, 21,22 as described in greater detail in appendix A.

2.2 Critical assessment of the model assumptions We would like to comment on the model assumptions outlined above, in particular regarding their physical relevance. As stated above, one of the reactants (the alcohol) is assumed to be present in excess, such that overall liquid concentrations of the other three components 6


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are comparatively low. We would like to point out that this assumption does not entail thermodynamic ideality: Operating at elevated concentrations close to or even above the solubility limit clearly requires the consideration of thermodynamic nonideality of the convective phases, and of a non-linear adsorption equilibrium (based on liquid phase activities). The main purpose of assuming one component to be present in excess is to allow neglecting volumetric changes due to adsorption, mixing or reaction. It is obvious that the assumption of a constant specic volume of the convective phases (equaling the specic volume of the excess component), bears a certain error, which, as assessed in greater detail in the supporting information, is not expected to exceed 4 vol % for the system and conditions considered in this work. We are hence convinced that, under the conditions investigated, the eects of volumetric changes are minor as compared to the impact of adsorption and hydrodynamic properties on the elution behavior. It is also worth noting that relaxing the assumption of negligible volumetric changes results in a considerable complication of the model equations and its solution, since in that case a variable porosity and/or supercial velocity u would have to be considered. The second major assumption of the equilibrium theory model is that of negligible kinetic limitations of any kind, entailing thermodynamic equilibria between all convective and adsorbed phases, as well as chemical equilibrium of the reversible reaction at any time and position in the column. It has recently been shown for a dierent chromatographic system that the displacement of partially miscible states, involving adsorption and two-phase ow, could be quantitatively described by a similar equilibrium theory model as the one suggested in this work (also neglecting kinetic limitations). 11,12 In general, we expect thermodynamic equilibria between the dierent convective and adsorbed phases to be attained fast in chromatographic systems, due to the large interfacial areas between the phases. In contrast, a spontaneous phase split, i.e. the creation of a new phase within the column, as well as reversible chemical reactions, are prone to kinetic limitations. These aspects clearly require an in-depth examination for the system under investigation, to assess whether the assumption 7



of thermodynamic and chemical equilibrium is close enough to reality.

3

Hexyl-Acetate system

We consider an exemplary model system, namely the esterication of n-hexanol (component D/4) and acetic acid (C/3) to water (B/2) and n-hexyl-acetate (A/1), with the strongly acidic ion-exchange resin Amberlyst CSP2 as catalyst and adsorbent, operating at 298.15 K. The system has not been characterized experimentally by us in the scope of this study, but realistic model and parameter assumptions are made based on literature data.

3.1 Thermodynamic equilibria The thermodynamic behavior of convective (liquid) phases is estimated using the fully predictive modied UNIFAC model. 23 Three of the four ternary diagrams, resulting when one of the four components is absent (zi = 0), are presented in Figure 1. Blue tie-lines, connected through binodal curves, indicate the liquid-liquid phase regions, while white regions correspond to miscible regions of a single liquid phase. The fourth ternary diagram for the system of n-hexanol, n-hexyl-acetate and acetic acid is trivial, since this ternary system is completely miscible. To further illustrate the behavior of the quaternary system, the simulated tie-lines in the quaternary diagram are presented in Figure 2. While n-hexanol, n-hexyl-acetate and water are all completely miscible with acetic acid, a considerable miscibility gap exists for both n-hexanol and n-hexyl-acetate with water. As acetic acid is added to these mixtures, the miscibility increases. In the equilibrium theory model presented in section 2, thermodynamic equilibrium (as predicted using the modied UNIFAC model) between the convective phases is assumed at every position and time in the column, so as all species feature identical liquid phase activities in all convective phases. Furthermore, thermodynamic equilibrium is assumed between the convective phase(s) and 8


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the adsorbed phase, which is described through an adsorption isotherm, being a function of the liquid phase activities. It has been shown previously that the adsorption behavior of multicomponent systems in the nonideal range (which is certainly the case here, dealing with compositions close to and above the solubility limit), can often be described with high accuracy as a function of the liquid phase activities. 11,24 In accordance with the literature, 10 we use a competitive Langmuir isotherm for this system:

ni =

qisat Ki ai , 3 P 1+ K k ak

(7)

k=1

where qisat and Ki are the saturation capacity and adsorption equilibrium constant of component i, respectively. Adsorption of component 4 is neglected (q4sat = K4 = 0) because of the assumption that it behaves as a solvent, while parameters for components 1 to 3 are estimated from Ref. 10 and reported in Table 1. AcOH

20

AcOH

80

40

20

60

60

40

40

60

80

20

60

60

20

20

80

40

80

HexOH

HexOH

40

40

80

H 2O

HexOAc

40

60

80

60

60

20

20

80

40

80

H 2O

HexOAc

20

20

40

60

80

H 2O

Figure 1: Ternary diagrams (mole fractions) of the model system (resulting from one of the four components being absent), predicted by the modied UNIFAC model. Liquid-liquid equilibria are indicated by blue tie-lines. The fourth ternary system resulting when H2 O is absent is completely miscible. Table 1: Isotherm parameters for the exemplary system, estimated from Ref. 10. component 1 2 3

qisat [molL−1 ]

Ki [-]

1.83 14.65 4.40

0.50 8.10 1.70

9



AcOH

80

80 80 60

60 60

40

40 40

20

20

HexOH

60

40

20 80

20

80

20

HexOAc

40

60

60

40 20

80

H2 O

Figure 2: Thermodynamic behavior of the quaternary model system, predicted by the modied UNIFAC model. Light blue lines are tie-lines, connecting liquid phases in equilibrium, dark blue points indicate plait points.

3.2 Reaction equilibrium The chemical equilibrium between the esterication reaction and its reverse ester hydrolysis reaction is dened through its equilibrium constant K , see equation 4. From Ref. 10, K is estimated to be equal to 22.07 at 298.15 K. As discussed in section 2, considering chemical equilibrium for the quaternary system at a specied temperature, the degrees of freedom determining possible compositions reduce to two. Thus, compositions in the quaternary system in chemical equilibrium are uniquely dened through two variables only, which may be z1 and z2 , but they can also be chosen to be C˜1 and C˜2 , as in the equilibrium theory equations 5 (for an interconversion between the variables see the supporting information). Concerning the two-phase region, not all tie-lines fulll the condition for reaction equilibrium (equation 4). Tie-lines fullling such condition are all located on the same surface, which is indicated through exemplary tie-lines in Figure 3a. Converting the composition z of the phases in equilibrium (ends of the tie-lines) to the variables c˜1 and c˜2 , the surface of tie-lines fullling chemical equilibrium can be mapped onto the (C˜2 , C˜1 ) plane (compare Figure 3b). It can be noted that on the horizonal axis (C˜2 = 0), C˜1 corresponds to C1 , the concentration of 10


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n-hexyl-acetate, while on the vertical axis (C˜1 = 0), C˜2 corresponds to C2 , the concentration of water. The range of values spanned by the vertical axis is considerably larger than that spanned by the horizontal axis, which is due to the fact that water (component 2) has a lower molecular weight, hence a higher molecular density, than n-hexyl-acetate (component 1). (b)

(a)

50

AcOH 40 80

80

C˜2 [mol L−1 ]

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


80

60 60

60 40

40

30

20

40 20

10 20

HexAc

60

40

20

80

H2 O

20

20

40 60

80

0

80 60

40

20

0

HexOH

2

4

6

8

10

12

C˜1 [mol L−1 ]

Figure 3: Liquid phases in thermodynamic and reaction equilibrium, indicated by red tielines in the quaternary system in (a). In (b), these tie-lines are mapped on the (C˜2 , C˜1 ) plane.

3.3 Two-phase ow behavior In the case of two (or multiple) convective phases, interstitial velocities of these phases are commonly not identical. The macroscopic properties of two-phase ow in porous media have been studied extensively in the context of applications in natural reservoirs. 25,26 The twophase ow behavior can be derived from the ow rate vs. pressure relationship, described macroscopically for a multi-phase system through the extended Darcy's law:

uj = fj u = −

11

krj K dPj µj dx


(8)


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As one can deduce from equation 8, the supercial velocity uj of phase j is linked linearly to the pressure drop; the proportionality constant involves the permeability K (an intrinsic property of the porous medium), the dynamic viscosity µj of phase j , and the relative permeability krj of phase j . The relative permeability commonly ranges between 0 and 1 and accounts for the fact that, in the presence of multiple convective phases, not the entire void space, and thus not the entire void cross-sectional area, is available for the uid phase

j. Neglecting pressure dierences due to capillary pressures in the dierent phases, so as Pj = P , one can directly express the fractional ow fj as a function of kr and µ:

fj =

krj µj N PP i=1

(9)

kri µi

At this point, we would like to underline that capillary eects are not neglected in the twophase ow description. In fact, they are the key property determining the system-specic relationship of the relative permeabilities krj with respect to the phase saturations Sj , which we will establish in the following. However, for most systems, the pressure dierence of the dierent uid phases (corresponding to the capillary pressure) is negligible with respect to their absolute pressure levels. It has been observed that for a great number of experimental systems in natural reservoirs, 27,28 but recently also in the context of liquid chromatography, 11 the relative permeabilities krj can be described as a function of S through a power-law relationship, which for two convective phases R and L is expressed as:

krL = 0; krR = 1;

L S L ≤ Slim

(10a)

L L n R L m krL = kr,max (Seff ) ; krR = kr,max (1 − Seff ) ;

L R Slim ≤ S L ≤ (1 − Slim )

(10b)

krL = 1; krR = 0;

R (1 − Slim ) ≤ SL

(10c)

12


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with L = Seff

L S L − Slim . L R 1 − Slim − Slim

(11)

In our application, the superscripts R and L denote the water-rich and the water-lean phase, X indicates the respectively. The exponents m, n are commonly > 1. The parameter Slim

limiting volume fraction of phase X (X =R,L), at which this phase becomes trapped in the porous medium (hydraulically disconnected), such that f X = 0. The maximum relative X permeability kr,max determines the velocity of phase X at a certain pressure drop, in the Y presence of the other phase Y being completely trapped at a saturation Slim . In this paper,

lacking experimental data for the system under consideration, we assume the most standard X X form of this equation, with m = n = 2, Slim = 0 and kr,max = 1. Furthermore, for equation

9, we assume equal viscosities µj for both phases. The resulting relationship f R = f R (S R ) is illustrated in Figure 4 (blue continuous line), next to the relationship which would be obtained when assuming equal interstitial velocities of the convective phases, i.e. f R = S R (dashed black line). The typical S-shape of the relationship accounts for the common ow behavior that at low volume fractions S X (X =R,L), the interstitial velocity of phase X is very slow (slower than that of the other phase Y ). In turn, at high volume fractions S X (X =R,L), the interstitial velocity of phase X is higher than that of the other phase Y .

13



1

0.8

0.6

f R [-]

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

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

S R [-]

Figure 4: Fractional ow function assumed for the model system (blue continuous line), next to the relationship obtained when assuming equal interstitial velocities (black dashed line).

14


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4

Elution properties and process performance

In this section we present and discuss elution proles for dierent initial and feed states, resulting from the solution of the equilibrium theory model (section 2) using the method of characteristics. Important physical aspects of the elution proles are discussed, and their implications on the process performance are assessed. For the interested reader, appendix A oers a detailed explanation of the equilibrium theory solution, which is novel for the model established in section 2 and provides a thorough mathematical and physical insight on the chromatographic behavior.

4.1 Elution proles for single- and two-phase ow conditions In the following, we consider entire chromatographic cycles, consisting of an adsorption step, where an initial state A is displaced by a feed state B until the entire chromatographic column is equilibrated with state B, and a desorption step, where B is redisplaced by A until the column is reequilibrated with A. The column is initially equilibrated with pure n-hexanol, such that the initial state A is dened as C˜1A = C˜2A = 0. At time τ = 0, a certain composition of acetic acid and n-hexanol (solution with concentration C3 of acetic acid) is fed to the column at a constant supercial velocity u. The feed composition instantaneously achieves chemical equilibrium through reaction to n-hexyl-acetate and water, to fulll equation 4. Since equimolar amounts of n-hexyl-acetate and water are formed during the reaction, and none of the components was initially present in the mixture, it can be concluded that the feed ow into the column is characterized by F˜1B = F˜2B = C3 . In the case where the feed ow at chemical equilibrium consists of a single phase, the feed state within the column corresponds to C˜1B = C˜2B = F˜1B = F˜2B . In contrast, if the feed ow consists of two convective phases, they can move with dierent velocities, so as one phase accumulates in the column, while the other propagates faster. Then, the concentrations C˜1B and C˜2B of the corresponding feed state in the column dier (C˜1B 6= C˜2B ), and can be derived from F˜1B , F˜2B on the basis of

15



equations 3 with the inverted fractional ow function (equations 9 to 11). We consider four dierent feed states Bi (i = 1 − 4), with F˜iB corresponding to 1.0, 1.5, 2.0 and 3.0 mol/L. The feed states Bi and the initial state A are plotted in the (C˜2 , C˜1 ) plane in Figure 5. This plane has already been considered in Figure 3b, and Figure 5 constitutes a zoom to the lower left corner of Figure 3b, assuring relatively low concentrations of components 1 to 3 and hence an excess of component 4 (as assumed when developing the equilibrium theory model). The continuous black line in Figure 5 constitutes the binodal, separating the single phase from the two-phase region. Accordingly, feed states B1 and B2 are located in the single phase region, while feed states B3 and B4 are located in the two-phase region and lead to a phase-split and two-phase ow within the chromatographic column. Blue lines constitute the mapping of the equilibrium theory solution for the specic initial and feed states in the (C˜2 , C˜1 ) plane, i.e. the evolution of C˜1 and C˜2 throughout the chromatographic cycles, for the mathematical derivation see appendix A. In addition, grey lines and symbols belong to the general solution of the equilibrium theory model, independent of initial and feed states, and are explained in detail in appendix A. The dashed black line in Figure 5 denotes the conditions C˜2 = C˜1 , and only the rst two feed states B1 and B2 in the single phase region are located on that line, for reasons explained above. Elution proles (concentration C˜i and ow F˜i proles) are presented in Figure 6 (adsorption steps, i.e. displacement of A by Bi ) and in Figure 7 (desorption steps, i.e. displacement of Bi by A). Concentration proles represent the concentration of a component over all liquid phases at a specic location of the column (here at the end of the column, ξ = 1). Meanwhile, ow proles describe the overall amount of a component moving per volume of ow over all liquid phases (overall fractional ow), in this case again at ξ = 1. Experimentally, it is the overall fractional ows of the dierent components that can be easily measured by gathering the eluate in fractions and analyzing these. 11 With reference to Figures 6 and 7 it can be noted that, wherever the solution path in Figure 5 is located in the miscible region, concentration and ow proles are identical. This is the 16


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case for the entire proles of the rst two examples (located entirely in the miscible region), while it is only partly the case for examples 3 and 4. It can also be observed that, in any of the adsorption proles (Figure 6), there is a period (intermediate state Ia ) where C˜1 6= 0 and C˜2 = 0 (and accordingly F˜1 6= 0 and F˜2 = 0). In Figure 5, this corresponds to a part of the solution path, namely that between A and Ia , mapping on the horizontal axis. From the denition of C˜i and F˜i , one can deduce that during this period, only component 1, i.e. the main product

n -hexyl-acetate, is present in

the bulk component n -hexanol, while components 2 and 3 (water and acetic acid) are absent. Likewise, in any of the desorption proles (Figure 7), there is a period where C˜1 = 0 and

C˜2 6= 0, which corresponds to parts of the solution paths in Figure 5 (between Id and A) mapping on the vertical axis. During this period, only component 2 (water) is present in the solvent and components 1 and 3 are absent. We can hence conclude that a purication of both products can be achieved for any of the four feed states, regardless whether a phase split and two-phase ow occurs or not. Let us have a closer look at the adsorption proles reported in Figure 6. As we explain in greater detail in appendix A, an initial state A and a feed state Bi are commonly connected by two transitions and one intermediate state I. This is indeed the case for examples 1, 2 and 4, whereas example 3 exhibits an additional plateau at a state P3 . With reference to Figure 5, it can be noted that P3 is at the intersection of the solution path with the binodal curve, hence it corresponds to the solubility limit. From Figure 4, we have concluded that phases with a low saturation S (here the water-rich phase) propagate with a very low interstitial velocity. Accordingly, states in the two-phase region located close to the binodal curve feature a low propagation velocity, which is smaller than the propagation velocity of a state in the single-phase region close to the binodal curve. Crossing the binodal, i.e. connecting downstream single-phase states to upstream two-phase states close to the binodal curve, hence leads to the creation of a plateau at the solubility limit, as observed for example 3. We have further deduced from Figure 4 that the interstitial velocity of a phase 17



increases with increasing saturation S . Therefore, states located in the two-phase region at a greater distance from the binodal feature higher propagation velocities, which exceed the propagation velocities of states in the miscible region. Connecting such two-phase, upstream states to single-phase, downstream states no longer results in the formation of a plateau at the solubility limit, as observed for example 4. To conclude our discussion of the adsorption proles, we focus on the intermediate states Ia . The values of C˜1 or F˜1 of these intermediate states increase with increasing feed concentrations, i.e. from example 1 to example 4. Meanwhile, the duration of these states decreases from example 1 to 2, stagnates from example 2 to 3 and decreases again slightly from example 3 to 4. It is a well-known fact (see the relevant literature 21,22,29 and appendix A), that systems with a curved downward isotherm, as the Langmuir isotherm assumed here, feature a shock transition when a downstream state with lower concentration is connected to an upstream state with higher concentration, and that the propagation velocity of such shock increases with increasing concentration of the upstream state. Shocks with a higher propagation velocity are expected to appear at the column outlet at lower retention times. The retention time of the rst shock transition in the adsorption proles, being already close to the void time, only decreases slightly from example 1 to 4. The retention time of the second shock transition decreases considerably from example 1 to 2. Since the second transition in example 3 connects the intermediate state to the state P3 instead of the feed state B3 , this transition spans a similar concentration range as the second transition in example 2, connecting Ia2 to B2 . The second transitions of examples 2 and 3 thus feature similar retention times. This is the case for all feed states located in the two-phase region and suciently close to the binodal to create a second plateau in the adsorption proles at the solubility limit. Since feed state B4 is located in the two-phase region at a greater distance from the binodal and enables a direct shock transition from the intermediate state Ia4 to the feed state B4 , this transition spans a greater range of concentrations, hence it appears at a lower retention time than the second transition in example 3. 18


Page 18 of 42

Page 19 of 42

Finally, with reference to Figure 7, we observe that the desorption proles feature more complex transitions than the adsorption proles, as explained in detail in appendix A. It is however important to note that the last tailing (wave transition) occurs entirely in the single-phase region (along the vertical axis in Figure 5), and is identical for all four examples. Hence, the same time is required in all four examples to completely regenerate the column, i.e. to reach initial state A with C˜1 = C˜2 = 0. The physical reason for this behavior is that water, featuring a very high Henry constant H2 , exhibits a strong anity to the adsorbent, so as in all four examples low concentrations of water are desorbed very slowly during the mentioned wave transition. In this system, the desorption of small concentrations of water requires more time than the removal of low fractions of a water-rich phase forming during examples 3 and 4.

10 Id4

B4

8

C˜2 [mol L−1 ]

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


6

B3

Id3 4

2 Id 2

A0

Id1 0

B1

Ia1 1

B2

P3

Ia2

Ia3

Ia4

2

3

4

C˜1 [mol L ] −1

Figure 5: Solution paths (blue) in the (C˜2 , C˜1 ) plane for examples i = 1 - 4 with initial state A and feed states Bi . Entire chromatographic cycles (i.e. adsorption and desorption steps) are considered. Intermediate states are denoted as Ix , with x = a, d for the adsorption and the desorption step, respectively. The binodal curve is plotted as a continuous black curve, the dashed black line denotes states with C˜2 = C˜1 .

19



4

A-B1

4

Ia1

A

B1

2

1

C˜1

2

F˜1

1

C˜2

0

F˜2

0 0

2

4

6

8

10

12

0

2

4

τ [-] 4

A-B2

4

B2

8

10

Ia2

A

12

B2

3

F˜i [mol L−1 ]

3

C˜i [mol L−1 ]

6

τ [-]

Ia2

A

2 C˜1

C˜2

1

2 F˜1

F˜2

1

0

0 0

2

4

6

8

10

12

0

2

4

τ [-] 10

A-B3

B1

3

F˜i [mol L−1 ]

C˜i [mol L−1 ]

Ia1

A

3

6

8

10

12

τ [-] 4

Ia3

A

P3

Ia3

A

B3

P3

B3

8

F˜i [mol L−1 ]

C˜i [mol L−1 ]

3 C˜2

6

4 C˜1

2

F˜2

2

F˜1

1

0

0 0

2

4

6

8

10

12

0

2

4

τ [-] 10

A-B4

4

C˜2

F˜i [mol L−1 ]

3

6

4

Ia4

A

B4

8

6

8

10

12

10

12

τ [-]

Ia4

A

C˜i [mol L−1 ]

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

Page 20 of 42

C˜1

B4

F˜1

F˜2

2

1

2

0

0 0

2

4

6

τ [-]

8

10

12

0

2

4

6

8

τ [-]

Figure 6: Adsorption proles for examples i 20 = 1 - 4 with initial state A and feed states Bi . Left column: concentration proles, Right column: ow proles ACS Paragon Plus Environment

Page 21 of 42

4

A-B1

4

B1 Id1

→A

3

F˜i [mol L−1 ]

C˜i [mol L−1 ]

B1 Id1

→A

3

2

1

2

1

F˜2

C˜2

C˜1

F˜1

0

0 0

2

4

6

8

10

12

0

2

4

τ [-] 4

A-B2

4

B2 Id2

→A

8

10

12

B2 Id2

→A

3

F˜i [mol L−1 ]

C˜i [mol L−1 ]

6

τ [-]

3

2

1

2

F˜2

1

C˜2

C˜1

F˜1 0

0 0

2

4

6

8

10

12

0

2

4

τ [-] 10

A-B3

6

8

10

12

τ [-] 4

Id3

B3

Id3

B3

→A

→A

8

F˜i [mol L−1 ]

C˜i [mol L−1 ]

3

6

4

C˜2

2 F˜2

F˜1

1

C˜1

2

0

0 0

2

4

6

8

10

0

12

2

4

10

A-B4

6

8

10

12

τ [-]

τ [-] 4

B4 Id4

→A

B4 Id4

→A

8 3

F˜i [mol L−1 ]

C˜2

C˜i [mol L−1 ]

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


6

4 C˜1

2 F˜2

F˜1 1

2

0

0 0

2

4

6

τ [-]

8

10

12

0

2

4

6

8

10

12

τ [-]

Figure 7: Desorption proles for examples i 21 = 1 - 4 with initial state A and feed states Bi . Left column: concentration proles, Right column: ow proles ACS Paragon Plus Environment


Page 22 of 42

4.2 Process Evaluation As discussed in the previous section, pure products n-hexyl-acetate and water (diluted in the solvent) can be collected during the intermediate states of the adsorption and desorption steps, regardless whether the feed state is located in the miscible or in the immiscible region. Concentration levels of the corresponding pure product at these intermediate states increase with increasing concentrations of the feed state. The duration of the intermediate state during adsorption, featuring pure n-hexyl-acetate, remains virtually constant for immiscible feed states with a certain proximity to the binodal (compare examples 2 and 3), and is only reduced for immiscible feed states located at a greater distance from the binodal (see example 4). At the same time, complete regeneration (corresponding to the complete removal of the strongest adsorbing component water), requires the same time in all four examples. Based on the elution behavior, let us now evaluate the productivity of the process. We assume an identical time for the chromatographic cycle (which is legitimate, considering the fact that regeneration requires the same time in all cases), and an operation of the process on the same column. The area under the ow prole in the presence of pure n-hexyl-acetate (area under state Ia ) corresponds to the cyclic productivity P , i.e. to the amount of n-hexylacetate puried per chromatographic cycle and column void volume (not the total column volume, due to the denition of τ ):

P = ∆τ F˜1I

(12)

where F˜1I denotes the overall fractional ow of n-hexyl-acetate at the column outlet during state Ia , and ∆τ denotes the duration (dimensionless) of this state. Under the assumption of a constant duration of the chromatographic cycle, P can be directly related to the productivity in amount of puried product per time and column volume, and it is plotted as a function of the overall fractional feed ow F˜1B in Figure 8. The dashed line in Figure 8 indicates the solubility limit; values of F˜1B on the left of that line form one stable liquid phase (corresponding to a single-phase ow), whereas overall

22


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fractional feed ows exceeding this limit result in liquid-liquid phase split and two-phase ow. For feed states located in the miscible region, the productivity approaches a plateau value with increasing F˜1B . This can be explained by the trade-o between increasing concentration levels of Ia (which has a positive eect on productivity), and the decreasing duration of state Ia (negative eect on productivity). When crossing the binodal, the productivity increases almost linearly, since the concentration level of Ia increases, while the duration of Ia remains virtually constant. At a certain value of F˜1B in the immiscible region, the prole exhibits another kink, and subsequently attens, again approaching a plateau. At overall fractional feed ows beyond this kink, Ia is directly connected to the feed state B through a shock transition, without the formation of a plateau P at the binodal. The propagation velocity of this shock increases with increasing concentrations of B. Accordingly, the duration of the intermediate state Ia in the elution proles decreases with increasing values of F˜1B and limits the increase in productivity. To conclude, the eect of a spontaneous phase split and subsequent two-phase ow can considerably enhance the performance of the batch process. Physically, this can be explained by the fact that the water-rich phase propagates slowly at low saturations S R , thus enhancing the retention of water, on top of its strong interaction with the adsorbent. The water-rich phase accumulates to a certain extent within the column, and allows for a longer duration of the elution of component 1 pure. Due to the high Henry constant of water, low concentrations of water exhibit a very low propagation velocity, and thus complete desorption occurs more slowly than the removal of the water-rich phase (through continuous adjustment of thermodynamic equilibria). For this reason, complete regeneration of the adsorbent requires the same amount of time, regardless of the location of the feed state (single-phase or two-phase region). It should be pointed out that this is not necessarily the case, and that the removal of the second phase can be the time limiting factor over desorption (as observed for another chromatographic system in the literature 11 ), especially in the case of lower Henry constants, of upwards curved adsorption 23



R isotherms, or of trapped uid phases (Slim > 0).

15

single phase

P [mol L−1 ]

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

Page 24 of 42

two phases

10

5

0 0

0.5

1

1.5

2

2.5

3

F˜1B [mol L−1 ] Figure 8: Cyclic productivity P as a function of the overall fractional feed ow F˜1B . Since the overall process time until complete regeneration (complete elution of component 2) is constant, P can be correlated with the productivity (per cycle time and column volume) of the batch process. The dashed line indicates the solubility limit (assuming F˜1B = F˜2B ).

24


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5

Conclusion

In this contribution, we have assessed the separation behavior and performance of a chromatographic batch reactor in the presence of a phase split and two-phase ow. An equilibrium theory model was derived, accounting for adsorption, two-phase ow, and a reversible reaction in chemical equilibrium. This model and its solution is based on previous work in the eld of chromatographic reactors 1416 and of multiphase ow. 11,12 Elution proles were derived for feed states located in the miscible and in the two-phase region for an exemplary esterication system (with main product n-hexyl-acetate and side product water). It can be concluded that pure products (diluted in eluent n-hexanol) are achieved for miscible or immiscible feed states. Furthermore, the two-phase ow behavior exhibits an advantageous eect on the separation behavior, enhancing the amount of puried main product per chromatographic cycle. The equilibrium theory model applied in this study is based on several strong assumptions, in particular neglecting kinetic limitations concerning mass transfer, chemical reaction, and liquid-liquid phase split. It certainly has to be assessed for the specic application, whether these assumptions are consistent enough with reality. Further, it is clear that the batch process discussed in this work would not be considered for an industrial application, where more ecient simulated moving bed reactors (SMBRs) are preferred. However, the equilibrium theory solution for a single batch column provides a deep understanding of the physical implications of two-phase ow in a chromatographic reactor, which are expected to remain valid also for continuous reactive chromatography. In particular, it was found that the formation of a second uid phase rich in the strongest adsorbing component (and with low saturation Sj ) can increase the separation eciency hence the process productivity, but the increased retention of the strongest adsorbing component also bears the risk of making the column regeneration more dicult. We are convinced that these features (discussed in detail in section 4) would aect the performance of the SMBR process in a similar manner. 25



Page 26 of 42

The current work, together with refs. 11 and 12, clearly shows that two-phase ow in chromatography can be well described and understood, and that its eects can be tolerable for some applications and advantageous for others.

Acknowledgement This work was supported by ETH Research Grant ETH-44 14-1.

Supporting Information Available Supporting information to this work is available free of charge on the ACS publications website at DOI: . It includes a detailed explanation of the transformation between mole fractions, concentrations, and the transformed variables c˜i (C˜i ), as well as of the calculation of the Jacobian N = ∂ n ˜ /∂˜ c.

References (1) Sundmacher, K.; Kienle, A.; Seidel-Morgenstern, A.

Integrated Chemical Processes -

Synthesis, Operation, Analysis and Control ; Wiley-VCH: Weinheim, 2005. (2) Rodrigues, A. E.; Pereira, C. S.; Santos, J. C. Chromatographic Reactors.

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(5) Mazzotti, M.; Neri, B.; Gelosa, D.; Kruglov, A.; Morbidelli, M. Kinetics of Liquid-Phase Esterication Catalyzed by Acidic Resins.

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(6) Lode, F.; Houmard, M.; Migliorini, C.; Mazzotti, M.; Morbidelli, M. Continuous reactive chromatography.

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J. Chromatogr. A 2014, 1360, 196208.

(8) Kulkarni, A. A.; Zeyer, K. P.; Jacobs, T.; Kaspereit, M.; Kienle, A. Feasibility studies and dynamics of catalytic liquid phase esterication reactions in a micro plant.

Chem.

Eng. J. 2008, 135, 270275. (9) Mitra Ray, N.; Ray, A. K. Modelling, simulation, and experimental study of a simulated moving bed reactor for the synthesis of biodiesel.

Can. J. Chem. Eng. 2016, 94, 913

923. (10) Schmitt, M.; Hasse, H. Chemical equilibrium and reaction kinetics of heterogeneously catalyzed n-hexyl acetate esterication.

Ind. Eng. Chem. Res. 2006, 45, 41234132.

(11) Ortner, F.; Mazzotti, M. Two-phase ow in liquid chromatography, Part 1: Experimental investigation and theoretical description. Ind. Eng. Chem. Res. 2018, 57, 32743291. (12) Ortner, F.; Mazzotti, M. Two-phase ow in liquid chromatography, Part 2: Modeling.

Ind. Eng. Chem. Res. 2018, 57, 32923307. (13) Reymond, H.; Vitas, S.; Vernuccio, S.; Von Rohr, P. R. Reaction Process of ResinCatalyzed Methyl Formate Hydrolysis in Biphasic Continuous Flow.

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AIChE J. 2006, 52, 10101026.

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(19) Graça, N. S.; Pais, L. S.; Silva, V. M. T. M.; Rodrigues, A. E. Analysis of the synthesis of 1,1-dibutoxyethane in a simulated moving-bed adsorptive reactor. Chem. Eng. Process.:

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(20) Barbosa, D.; Doherty, M. F. A New Set of Composition Variables for the Representation of Reactive-Phase Diagrams.

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1. Theory and Application of Single Equations ; Prentice-Hall: Englewood Clis, N. J., 1986. (22) Rhee, H.-K.; Aris, R.; Amundson, N. R.

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Ind.


A

Page 30 of 42

Solution of the equilibrium theory model

In this section, we derive the solution of the equilibrium theory model introduced in section 2 by the method of characteristics. In sections A.1 and A.2, the general solution of the model is provided separately for the miscible and the immiscible region. The resulting equations are then applied to the chromatographic system under investigation, and the equilibrium theory solution for this system is illustrated, independently of initial and feed states, in the hodograph plane in section A.3. Finally, the elution proles for specic initial and feed states presented in section 4.1 are derived and explained mathematically in section A.4.

A.1 Solution in the miscible region In the miscible region, with only one convective phase, the overall fractional ows are identical to the overall liquid concentrations (which equals the single liquid phase concentration), and thus F˜i = C˜i = c˜i . In this case, equations 5 correspond to the model discussed in refs. 1416. With its mathematical structure being identical to the classical binary chromatographic model, the mathematical solution procedure has been discussed in great detail in the literature. 21,22,29,30 Here, we just provide the most important results, without explaining the mathematical solution procedure in detail. There are two types of characteristics Γ1 and Γ2 in the Hodograph (C˜1 , C˜2 ) plane, dened through their slopes ζ as:

Γj : ζ3−j =

θj − n ˜ 22 dC˜1 = ; ˜ n ˜ 21 dC2

j = 1, 2.

(13)

The variables n ˜ ij are dened as ∂ n ˜ i /∂˜ cj , whereas θj are the eigenvalues of the matrix N =

[˜ nij ]. We assign the condition that θ1 < θ2 . It is worth noting that the relationship of n ˜i in terms of c˜j , and thus the determination of the partial derivatives n ˜ ij , is not trivial. With adsorbed phase concentrations ni being described by the isotherms provided in equation 7, the variables n ˜ i are function of the liquid phase activities aj of all four components. In turn, 30


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the liquid phase activities ai are function of the mole fractions zi of the rst three components (the mole fraction z4 can be determined from the other three mole fractions). These three mole fractions can be expressed in terms of the three liquid phase concentrations ci , which can then be converted into the variables c˜i = C˜i . The determination of N is explained in detail in ref. 16, important equations are summarized in the supporting information. The Γi characteristics, resulting from the solution of the ODEs 13, describe the change in composition (C˜1 ,C˜2 ) that occurs when connecting an initial state to a feed state. Each composition located on a Γi characteristic travels along the column at a specic propagation velocity λi . The inverse of the propagation velocity, σi = λ−1 i , corresponds to the slope of the characteristics in the physical (ξ, τ ) plane, which is dened as

σj = 1 + νθj

(14)

With the denition θ1 < θ2 , it can be concluded that σ1 < σ2 hence λ1 > λ2 . Let us now consider a Riemann problem with an initial (downstream) state A and a feed (upstream) state B. The solution in the physical (ξ, τ ) plane consists of three states, namely the initial state A, the intermediate state I, and the feed state B, connected by two transitions, each of which maps either on a Γ1 or a Γ2 characteristic. The type of transition depends on the directional derivative of σi along the Γi characteristic; if σi decreases from an upstream state U to a downstream state D (i.e. the propagation velocity λi increases), then the transition is a simple wave. In the other case, when σi increases from U to D, a simple wave transition would cause a multivalued, unphysical solution. The physically meaningful solution is then a discontinuous shock transition, which directly connects U to D, and with a slope σ ˆ obtained from the integral form of the conservation laws, i.e.:

ˆ σ ˆ = 1 + ν θ,

31


(15)


Page 32 of 42

with

[˜ n1 ] [˜ n2 ] θˆ = = . [C˜1 ] [C˜2 ]

(16)

Here, [·] denotes the jump in the quantity enclosed across the discontinuity. Equation 16 is the Rankine-Hugoniot condition, dening the possible states which, assuming a certain downstream state D (a certain upstream state U), can be connected to D (U) through a shock. These states are located on a shock path Σi , which is tangent to the characteristic Γi in D (U). A further condition of the shock transition is that σiU < σ ˆ < σiD . The two transitions connecting A to B map on two dierent types of characteristics Γ1 and Γ2 , which intersect at the intermediate state I. For a physically meaningful solution, downstream states have to propagate faster than upstream states. With σ1 < σ2 (λ1 > λ2 ), the same state is reached faster through a Γ1 than through a Γ2 characteristic, such that intermediate states should always be connected to downstream states through a Γ1 characteristic, and to upstream states through a Γ2 characteristic. As a consequence, the physically correct path connecting the initial, downstream state A to the feed, upstream state B corresponds to the sequence A - Γ1 - I - Γ2 - B. In the case of shock transitions, Γi characteristics in this sequence are replaced by Σi paths, which can deviate slightly from the corresponding characteristics.

A.2 Solution in the immiscible region In the case of two convective phases, equations 5 are mathematically very similar to the model discussed and solved in a previous contribution, 12 with minor dierences: In the current model, both modied components 1 and 2 are adsorbing (n ˜ 2 6= 0), the porosity is assumed to be constant, and the fractional ow f R is a function of S R only (not of a tieline indicator η ). Although the model could also be solved based on the unknowns C˜1 and

C˜2 , it is mathematically simpler and provides more physical insight to perform a variable transformation to the new unknowns η and S R . While S R is the phase saturation of the

32


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water rich phase, η denotes a tie-line indicator, which uniquely identies the tie-line. In this contribution, we dene η as c˜L1 , i.e. the concentration of the modied component 1 in the water lean phase. Since the derivation is analogous to the derivation provided in ref. 12, only the main results are summarized here. The variable transformation results in a system of homogeneous PDEs in η and S R of the form

Ai

∂η ∂η ∂S R ∂S R + Bi + Ci + Di = 0; ∂ξ ∂τ ∂ξ ∂τ

i = 1, 2

(17)

with dc˜L dc˜R i + (1 − f R ) i dη dη R dc˜ dc˜L dn ˜i Bi = S R i + (1 − S R ) i + ν dη dη dη R df Ci = c˜R ˜Li i −c R dS Di = c˜R ˜Li i −c

Ai = f R

(18a) (18b) (18c) (18d)

Note that, with c˜X ˜ i depending on η only, and with f R depending on S R only, i and n derivatives in equations 18 are normal (not partial) derivatives. The necessary values for

c˜X ˜ i are determined from spline interpolations tted to the 60 predetermined tie-lines i and n (illustrated in Figure 3) as a function of the tie-line indicator η . Derivatives of these spline functions with respect to η also provide d˜ cX ni /dη . i /dη and d˜ Applying the method of characteristics based on the new unknowns provides two sets of characteristics Γx (x =t,nt), dened through their slopes ψx = dη/dS R : (19)

Γt : ψt = 0 Γnt : ψnt =

B2 C1 − B1 C2 + A1 D2 − A2 D1 A2 B1 − A1 B2

33


(20)


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From equation 19, it is clear that Γt characteristics map on tie-lines, while the second set of characteristics dened through equation 20 consists of non-tieline characteristics. The slopes σx of characteristics in the physical plane, corresponding to states located on Γx characteristics, are dened as:

−1 df R σt = dS R B1 ψnt + D1 σnt = A1 ψnt + C1

(21) (22)

With n ˜ i = f(a) and a =const. along tie-lines, no ad- or desorption occurs along Γt characteristics, and thus the propagation velocities λt = σt−1 are a function of the two-phase ow behavior only. As discussed for the miscible region, if the directional derivative of σx along a Γx characteristic, connecting a downstream state D to an upstream state U, is positive, a wave transition is the physically correct solution; if the directional derivative is negative, a weak shock solution is required. The slopes σ ˆ of the shock paths in the physical plane, derived from the integral material balance, read:

σ ˆ=

[C˜1 ] + ν[˜ n1 ] [C˜2 ] + ν[˜ n2 ] = . [F˜1 ] [F˜2 ]

(23)

As in the miscible region, the Rankine-Hugoniot condition given by equation 23 also determines the shock paths Σx , as it identies all states U (D), which can be connected to a downstream state D (upstream state U) through a shock, while satisfying the integral form of the material balances. Equation 23 is also valid for shock transitions crossing the binodal, i.e. connecting a miscible to an immiscible state. In this case, F˜i can be replaced by c˜i for the miscible state. In principle, tieline and non-tieline characteristics can be calculated for the entire two-phase region, as shown in Figure 9a. The characteristics in the (C˜1 , C˜2 ) plane are constructed by solving the ordinary dierential equations 19 and 20, and by transforming the resulting values for η and S R to C˜1 and C˜2 . We would like to emphasize that characteristics in the 34


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hodograph plane exhibit the same qualitative properties as in ref. 12. Furthermore, propagation velocities λx (x =t,nt) along dierent tie-line characteristics are illustrated in Figure 9b. With equation 21 being a function of S R only (and not of η ), proles of λt are identical along all tie-lines. This is not the case for the proles of λnt . Neither of the two propagation velocities λx is always greater than the other, as it is the case in the miscible region (λ1 > λ2 ). It is also worth noting that λt → 0 as the binodal is approached (S R → 0 or S R → 1). The physical interpretation of this behavior is that phases at a low saturation S X propagate very slowly (low f X , compare Figure 4), thus approaching a fractional ow of 0 at the limiting X , which according to our assumption equals 0. saturation Slim

Considering the entire immiscible region violates our assumption of low overall liquid concentrations (and thus negligible volumetric changes due to adsorption, reaction, or nonideal mixing). Hence, we constrain the applicable region to concentrations C˜1 < 4 molL−1 and

C˜2 < 10 molL−1 , corresponding to the grey box in Figure 9a. In this region (reaching a maximum saturation S R ≈ 0.2), one can deduce from Figure 9b that λnt > λt . Following the terminology of the miscible region, with λ1 > λ2 , we rename the Γnt and Γt characteristics to Γ1 and Γ2 characteristics, respectively, in the applicable region. It is worth noting that, with the same reasoning as applied for the miscible region (section A.1), one can deduce the correct solution, connecting an initial (downstream) state A to a feed (upstream) state B, to map on a sequence A-Γ1 -I-Γ2 -B (or, in the case of shock transitions, Σi paths instead of Γi characteristics). The state I is the intermediate state determined by the intersection of the

Γ1 and the Γ2 characteristic, passing through A and B, respectively.

A.3 Illustration of the solution in the Hodograph plane The characteristics in the applicable region of the hodograph plane for the miscible and the immiscible region are plotted in Figure 10. The miscible and the immiscible region are separated by a black continuous line, corresponding to the binodal curve. Γ1 and Γ2 charac35



(a)

50

Γnt Γt

C˜2 [mol L−1 ]

40

30

20

10

0 0

2

4

6

8

10

12

C˜1 [mol L−1 ]

(b)

2 λt

1.5 λnt

λ [-]

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

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1

0.5

0 0

0.2

0.4

0.6

0.8

1

R

S [-] Figure 9: (a) Γt (red) and Γnt (blue) in the entire immiscible region, mapped in the (C˜2 , C˜1 ) plane. (b) Propagation velocities λt (red) and λnt (blue) along the tie-line characteristics.

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teristics are illustrated in blue and red, respectively. Furthermore, arrows in the color of the respective characteristic indicate the direction in which σi increases. Following characteristics in this direction when connecting a downstream to an upstream state, wave transitions are physically possible, while in the opposite direction, shock transitions are required (note that shock paths Σi can deviate slightly from Γi characteristics). Blue lled circles indicate states at which the directional derivative of σ1 switches from negative to positive. Crossing these states can create semi-shock transitions, which are a combination of shock and wave transitions (for a detailed discussion of semi-shocks see the relevant literature 21,22,31,32 ). It can be noted that all Γ2 characteristics, but only a few Γ1 characteristics in the immiscible region, intersect with the binodal. These characteristics are continued by the corresponding

Γ2 and Γ1 characteristics in the miscible region. However, slopes of the Γi characteristics, as well as propagation velocities λi , or their reciprocals σi , exhibit a discontinuity at the binodal. If the jump in σi is positive when crossing the binodal from a downstream to an upstream state, the upstream limit at the binodal propagates more slowly than the downstream limit. As a consequence a plateau in the concentration prole at the intersection of Γi with the binodal forms. Note that this plateau does not correspond to a classical intermediate state, since the solution path at this plateau continues along the same type of characteristic. In the opposite case, where the jump in σi at the binodal is negative when connecting a downstream to an upstream state, the binodal has to be crossed through a shock (since the upstream limit would propagate faster than the downstream limit, thus creating a multivalued, non-physical situation). As the binodal is only crossed by few Γ1 characteristics, we neglect these cases and focus on the crossing of the binodal along Γ2 characteristics. According to Figure 9b, λ2 → 0, and thus σ2 → ∞, when approaching the binodal from the immiscible region. In the miscible region, σ2 never approaches ∞. Thus, for any Γ2 characteristic, the limiting value of σ2 in the immiscible region is greater than that in the miscible region. As a consequence, downstream miscible states connected to upstream immiscible states through a Γ2 characteristic create a plateau at the intersection with the binodal, while 37



upstream miscible states connected to downstream immiscible states yield a shock crossing the binodal. First, let us consider the scenario of a downstream miscible state D connected to an upstream immiscible state U through a Γ2 characteristic, i.e. the scenario which in principle creates a plateau at the intersection with the binodal with composition P. According to the decrease in σ2 along Γ2 , indicated by the arrows in Figure 10, D should be connected to P through a shock with slope σ ˆ DP , and P should further be connected to U through a shock with slope

σ ˆ PU . This is the case for states U in the two-phase region located relatively close to the binodal, where σ ˆ DP < σ ˆ PU , i.e. where the downstream shock propagates faster than the upstream shock. In the opposite case when σ ˆ DP > σ ˆ PU , which occurs for states U located at a greater distance from the binodal, the physically valid solution is a direct shock transition connecting D to U with a slope σ ˆ DU which travels faster than the shock connecting D to P. Second, we examine the case of a downstream immiscible state D connected to an upstream miscible state U through a Γ2 characteristic. In this case, increasing slopes σ2 along Γ2 (compare arrows in Figure 10) indicate that the connection of D to the binodal, and of the binodal to U, consist of simple wave transitions. However, the binodal itself has to be crossed by a shock. The physically correct solution of such cases is the formation of semishock transitions. As an illustrative example, we consider the Γ2 characteristic mapping on the vertical axis. The two empty circles in Figure 10 indicate two states S1 and S2 , which are connected across the binodal by a contact discontinuity, fullling the necessary condition

ˆ S1S2 = σ2S2 . Any downstream state D located above S1 , is connected to S1 through σ2S1 = σ a simple wave, and every upstream state U located below S2 is reached from S2 by a simple wave. In turn, downstream and upstream states located between S1 and S2 are accessed through a shock which crosses the binodal. As a consequence, there exists a spectrum of dierent possible transitions, reaching from (1) a wave - contact discontinuity - wave for states D and U located beyond S1 and S2 , (2) a semi-shock (wave-shock or shock-wave) if one of the two states is located between and the other one beyond S1 and S2 , to (3) a shock 38


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if both states D and U are located between S1 and S2 .

10 Γ1 Γ2

8

C˜2 [mol L−1 ]

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


S1 6

4

2 S2 0 0

1

2

3

4

C˜1 [mol L−1 ] Figure 10: Γ1 (blue) and Γ2 (red) characteristics of the model system in the Hodograph plane. The black line denotes the binodal (separating miscible and two-phase region). Arrows indicate the direction in which the slope of the physical plane σi increases. Blue lled circes denote the positions at which the directional derivative of σ1 chances from negative to positive. The red empty circles on the veritical axis indicate two states S1 and S2 which are connected through a contact discontinuity along a Γ2 characteristic (σ2S1 = σ˜2 = σ2S2 ).

39



A.4 Derivation of elution proles In the following, we derive mathematically the solutions (entire chromatographic cycles, i.e. adsorption and desorption steps) for the inital state A and feed states Bi (i = 1−4) discussed in section 4.1. We will refer to the solution paths in the hodograph plane presented in Figure 5, as well as to the adsorption and desorption proles in Figures 6 and 7, respectively. Let us rst examine the solution paths in the hodograph plane (Figure 5). The solutions for the adsorption and desorption steps of all four exemplary cases always map on a sequence of

Γ1 characteristic (emanating from the initial state) and Γ2 characteristics (passing through the feed state), which intersect at an intermediate state I. Superscripts a and d denote the aliation of a state to the adsorption or the desorption step, respectively. Due to the location of state A in the origin, solution paths of the adsorption steps involve the Γ1 characteristic located on the horizontal axis, such that all intermediate states Ia are located on this axis. With the same reasoning, solution paths of the desorption step involve the Γ2 characteristic located on the vertical axis, such that all intermediate states Id are located on the vertical axis. As discussed in section 4.1, the location of parts of the solution paths (connecting A and the intermediate states Ia or Id ) on the horizontal and the vertical axis corresponds to regions of pure component 1 (n -hexyl-acetate) during adsorption and pure component 2 (water) during desorption in the elution proles (compare Figures 6 and 7) for any of the four examples. As a consequence, pure products are achieved in all four examples, regardless whether a phase-split and two-phase ow occurs (examples 3 and 4) or not (examples 1 and 2). In addition, with the distance of intermediate states Iai and Idi from the origin increasing from example 1 to example 4 (compare Figure 5), concentration levels of the intermediate states in the corresponding elution proles increase. With regard to the adsorption steps, all transitions are shock transitions, as one can deduce from the directional derivatives of σi along Γi characteristics indicated qualitatively by arrows in Figure 10. For examples 1 and 2, located entirely in the single-phase regime, the solution consists of the usual sequence of A - transition - Ia - transition - B. In the case of example 3, 40


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where the binodal is crossed along a Γ2 characteristic, an additional concentration plateau P3 forms at the solubility limit, as explained in the previous section A.3. With the feed state B4 being located in the two-phase region at a greater distance from the binodal, this state features a higher propagation velocity, and is directly connected to Ia4 through a shock, without the formation of an additional plateau. Also this behavior was discussed in section A.3. Considering the desorption proles in Figure 7, a variety of dierent transitions, especially concerning the connection of Id to A, can be observed. However, it is important to note that the last part of these transitions, reaching state A, is always a simple wave mapping on the same Γ2 characteristic (vertical axis). Accordingly, this last part of the elution proles is identical for all four cases, such that complete desorption (state A) is reached at the same time. The detailed explanation for this behavior is as follows: In the miscible region, intermediate states on the vertical axis (such as Id1 and Id2 ) are connected to A through a simple wave. As in the adsorption case, the transition becomes more complex for intermediate states located in the immiscible region (examples 3 and 4), since the binodal has to be crossed to reach A. The intermediate state Id3 is located below the state S1 dened in section A.3, whereas the intermediate state Id4 is located above this state. As a consequence (see section A.3), Id3 is connected to A through a semi-shock (shock-wave), with the shock part crossing the binodal and changing into a wave at a state located above S2 (see Figure 10). From Id4 , state A is reached through a wave - contact discontinuity - wave (see the discussion in section A.3). Despite the more complex transition in cases 3 and 4, state A is still reached through a simple wave, overlapping with the solutions of cases 1 and 2.

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Theoretical evaluation of two-phase flow in a ... - ACS Publications

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