Equilibrium Theory Analysis of a Binary Chromatographic System

Article pubs.acs.org/IECR

Equilibrium Theory Analysis of a Binary Chromatographic System Subject to a Mixed Generalized Bi-Langmuir Isotherm Franziska Ortner, Simon Jermann, Lisa Joss, and Marco Mazzotti* Institute of Process Engineering, ETH Zurich, Zurich, Switzerland ABSTRACT: In this study, an equilibrium theory solution for a binary system subject to a specific subclass of generalized biLangmuir isotherm is derived. This solution provides a thorough understanding for the type of isotherm investigated, which exhibits several interesting features. Depending on initial and feed compositions, the order of breakthrough and desorption changes at frontal analysis conditions, and a changing impact of component 1 (from competitive to cooperative) on the adsorption behavior of component 2 can be observed at displacement conditions. Furthermore, the equilibrium theory solution includes some rare transitions, such as semishock transitions with multiple shock and wave parts. Due to its interesting behavior, this isotherm is able to accurately describe an experimental system investigated in the context of the delta-shock phenomenon. With the aid of the derived equilibrium theory solution, it was possible to identify several types of transitions and intermediate states, and to fit isotherm parameters to experimental elution profiles. The interesting behavior accounted for by this type of isotherm, and its ability to describe a specific and peculiar experimental system, which had previously been misinterpreted, may motivate the future investigation of the entire class of generalized bi-Langmuir isotherms. Hxi and Kxi denote the Henry’s and equilibrium constant of component i in term x, ci and ni are the liquid and adsorbed phase concentration, respectively, and the variables pxi can take the values ±1. In this study, we present the equilibrium theory solution valid for a subclass of generalized bi-Langmuir isotherms (including the isotherm describing the experimental system), which provides a thorough understanding of the column dynamics arising with this isotherm and illustrates the ability of this isotherm to describe the experimental profiles. The paper is structured as follows: section 2 introduces the main equilibrium theory equations for binary systems and explains different types of transitions (complex semishock transitions are illustrated in detail with the aid of an exemplary single-component system in the Appendix, section A). The presented equations are applied to a system subject to a generalized bi-Langmuir isotherm in section 2.2. In the following, the equilibrium theory solution for an exemplary binary system subject to a L-M1 bi-Langmuir isotherm is presented. General properties are explained through the analysis of the behavior of characteristics in the hodograph plane and their associated composition fronts (section 3), and selected chromatographic cycles illustrate the adsorption/ desorption (section 4) and displacement behavior (section 5) of the system (a rigorous derivation of the equilibrium theory solution is also provided in the Appendix, sections B and C). Elution profiles simulated by an equilibrium dispersive model16 matching the corresponding equilibrium theory predictions verify the derived solution. After evaluating the general validity of the solution with respect to a variation of isotherm parameter values (section 6.1), the obtained behavior is correlated in

1. INTRODUCTION The local equilibrium theory, developed in the early second half of the 20th century,1−6 plays an important role for the understanding, description and design of chromatographic processes. The conservation laws that equilibrium theory is based on only account for convection and exchange between the mobile and adsorbed phase, which are assumed to be at local equilibrium. A solution, which is analytical for several adsorption isotherms, can be obtained through the method of characteristics,5,6 offering accurate predictions at a low computational effort. Until today, equilibrium theory has been further developed for different applications such as multicolumn (simulated) moving bed processes,7−9 and for different types of thermodynamic adsorption equilibria, such as the generalized Langmuir isotherm.10−12 Recently, in the context of investigating the delta-shock phenomenon,13 an interesting binary adsorption behavior was observed for the system phenetole (PNT) and 4-tertbutylphenol (TBP) in methanol:H2O 63:37 (v:v) and with the adsorbent Zorbax 300SB-C18.14 Exhibiting a change in the order of breakthrough and desorption in frontal analysis experiments and a changing impact of PNT on the adsorption behavior of TBP from competitive to cooperative during displacement experiments, this system could not be described with a mixed generalized Langmuir isotherm (M2 type), as was initially assumed.15 A new class of isotherms named generalized bi-Langmuir isotherms of the form ni =

Hiaci Hibci + δa δb

(i = 1, 2)

(1)

with δ x = 1 + p1x K1xc1 + p2x K 2xc 2

(x = a , b)

Received: Revised: Accepted: Published:

(2)

was established, and an accurate description of the experimental system was achieved with a subclass of these isotherms.14 Here, © 2015 American Chemical Society

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Industrial & Engineering Chemistry Research

slopes ξj = dc1/dc2 of the characteristics Γj in the hodograph plane are given by

section 6.2 to experimental data reported in a companion paper.14 Finally, section 6.3 discusses opportunities offered by the interesting behavior of the new class of isotherms, as well as challenges arising in their study.

Γj:

2. THEORY 2.1. Equilibrium Theory for Binary Systems. Since the derivation of an equilibrium theory solution for a binary system in liquid chromatography has been widely discussed in the literature,6,10,17 we will provide here only the basic equations and refer to the literature for detailed derivations. Under the assumptions of isothermal conditions and of constant velocities (dilute liquid phase concentrations), as well as neglecting dispersive effects and mass transport limitations, the mass balance equations for fixed bed chromatography can be written as ∂c ∂ (ci + νni) + v i = 0 (i = 1, 2) ∂t ∂z

(3)

(4)

the mass balances can be recast to form a system of first-order, homogeneous and reducible partial differential equations (PDEs): (I + ν N )

∂c ∂c +v =0 ∂t ∂z

θj̃ =

(5)

where I is the identity matrix and N = [nij] with nij = ∂ni/∂cj. c is the vector of concentrations of components 1 and 2. This set of PDEs is completed by defining the initial conditions, which are typically the values of the solution along a line in the physical plane (plane with time as vertical coordinate and space as horizontal coordinate). Here only Riemann problems are considered, with a constant initial state in the column (at t = 0) and a constant inlet state being fed to the column (at z = 0). A solution of the problem can in general be provided via the method of characteristics,6 by defining the slopes of the characteristics Cj (lines of constant liquid phase composition in the physical plane) as Cj:

(1 + νθj) dt = = σj dz v

(j = 1, 2)

n11 + n22 ±

(6)

(n11 − n22)2 + 4n21n12 2

(8)

[n1] [n ] = 2 [c1] [c 2]

(j = 1, 2)

(9)

In the hodograph plane, states that can be reached by a shock starting from a state B′ belong to a shock path Σj. This path is tangent to the Γj characteristic in the initial state B′. Generally, at every point on a Γj characteristic, there exists a corresponding tangent Σj path. Simple waves and shock transitions are the most common, but not the only existing transitions. If two states, connected by a discontinuity, propagate with the same speed 1/σj as the discontinuity itself, the transition is called a contact discontinuity. In contrast to shock transitions, contact discontinuities are not self-sharpening. If the directional derivative of σj along the corresponding path changes sign, transitions can be composed of multiple simple wave, contact discontinuity and shock parts. An example for such “multipart transitions” are semishocks. It is worth noting that these kind of transitions do not exist in the cases of the generalized Langmuir isotherm,10 whereas they are a key feature in the generalized biLangmuir isotherms dealt with in this work. One single change in sign of the directional derivative (from negative to positive or from positive to negative), can result in semishocks consisting of a shock and a wave part, which in the following will be denoted as shock-waves or wave-shocks, respectively. Transitions consisting of three parts, such as shock-wave-shocks and wave-c.d.-waves (c.d. denotes contact discontinuity) can be observed upon two changes in sign of the directional derivative. Properties and necessary conditions of these “multipart transitions” are explained in detail in the Appendix, section A. 2.2. Application of Equilibrium Theory to a System subject to a Generalized Bi-Langmuir Isotherm. In this

Here θj are the eigenvalues of the matrix N, which for a binary system are calculated as θ1,2 =

(j = 1, 2)

Given an initial state B and an inlet state A, the transition path connecting these two states is uniquely described by segments of a Γ1 and a Γ2 characteristic, which intersect at an intermediate state I. Every state on a Γ1 characteristic is related to a simple wave characteristic C1 with slope σ1 in the physical plane, and accordingly every state on a Γ2 characteristic is related to a simple wave characteristic C2. Since by definition σ1 < σ2, the unique (physically correct) path connecting state B with state A (moving from bottom up in the physical plane) always starts along a Γ1 characteristic, before continuing along a Γ2 characteristic. The type of transition connecting two states located on the same characteristic Γj (e.g., state B and state I, or state I and state A) is decided by the evolution of σj along Γj. In the following, we will denote these two states located on the same characteristic as B′ and A′. Simple waves can only be observed if σj increases while moving along a Γj characteristic on the path connecting state B′ with state A′, which is the case for increasing θj. On the contrary, decreasing slopes σj (corresponding to decreasing θj) result in an unphysical situation (regions in the physical plane defined by multiple states) and thus require that a ’weak solution’ be introduced. In that case, the continuous transition is replaced by a discontinuity (shock), described by a shock path Sj in the physical plane with a slope σ̃j given by eq 6, where, however, θj is replaced by θ̃j:

where t and z are the time and space coordinates, ν = (1 − ϵ)/ϵ is the phase ratio with total porosity ϵ, and v = u/ϵ with u being the superficial velocity, i.e., the ratio fluid flow rate to column cross section. Since ni is given through the adsorption isotherm as a function of the liquid phase composition

ni = f (c1 , c 2)

θj − n22 dc1 = = ξ3 − j dc 2 n21

(7)

In the case where the two eigenvalues are distinct and real (i.e., the discriminant of eq 7 is greater than zero), the system is strictly hyperbolic and can be solved by equilibrium theory. Note that θ1 < θ2 implies that σ1 < σ2. It is further convenient to determine the characteristics in the hodograph plane (state plane with coordinates (c1, c2)), which, given the adsorption isotherm of the system, can be laid down once and for all, independently of the initial and feed state. The 11421

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Industrial & Engineering Chemistry Research Table 1. Sets of Isotherm Parameters Used in This Studya Ha1 set α (derivation of ET solution) set β (fitted to experimental system) set γ (further illustration) a

Hb1

Ha2

Hb2

Ka1 −1

Kb1

Ka2

−1

−1

Kb2

[−]

[−]

[−]

[−]

[L g ]

[L g ]

[L g ]

[L g−1]

0.80 0.03 0.60

1.66 2.00 var.b

2.82 2.27 2.40

0.35 0.40 0.80

0.060 0.028 0.010

0.028 0.022 0.020

0.080 0.052 0.024

0.008 0.012 0.008

Note that in all cases pa1 = pa2 = pb2 = 1 and pb1 = −1. bThis parameter is varied for illustration purposes, see Figure 7

3. GENERAL PROPERTIES OF THE L-M1 BI-LANGMUIR SYSTEM IN THE HODOGRAPH PLANE In the following, the equilibrium theory solution for a system subject to a L-M1 bi-Langmuir isotherm with the specific parameters reported in Table 1 (parameter set α) shall be presented. As one can see in the subsequent sections, the equilibrium theory solution for the considered isotherm is complex and bears many different interesting features, such that a complete and detailed derivation of the solution is rather lengthy. Therefore, we will here only highlight and explain the most important phenomena, and provide a rigorous derivation in the Appendix. The single component isotherm of component 1 corresponding to the parameter set α exhibits an inflection point at a concentration c1 = 0.186 g/L, whereas that of component 2 has no inflection points. The network of simple wave characteristics in the hodograph plane for the system under consideration is shown in Figure 1, namely for a broad concentration range in Figure 1a, as well as for a low concentration region in Figure 1b. The physically meaningful region of the hodograph plane is the first quadrant, with nonnegative liquid phase concentrations of both components. A further constraint to that region is that adsorbed phase concentrations, and thus δa and δb (see eq 2), be nonnegative. In the investigated system and for nonnegative liquid phase concentrations, only δb can become negative, and the corresponding boundary of the physically meaningful region is indicated by a thick dashed line in Figure 1a. Furthermore, the discriminant in eq 7 is always nonnegative within the first quadrant, so that an equilibrium theory solution is possible for the entire physically meaningful region. For a large part of the physically meaningful region (at rather high concentrations), the two families of characteristics exhibit typical features: rather flat Γ1 characteristics intersect with the vertical axis, whereas rather steep Γ2 characteristics intersect with the horizontal axis. While Γ1 characteristics always exhibit a positive slope (but for a partly horizontal characteristic along the horizontal axis, where the slope equals zero), slopes of Γ2 characteristics change from negative at low concentrations of component 1 to positive at high concentrations of component 1. The locus of points where the slopes of the characteristics change from negative to positive is indicated in Figure 1a by a thin dashed line. The features of the two families of characteristics change when moving to low liquid phase concentrations (see Figure 1b), namely when crossing the Γ1 and Γ2 characteristics marked with a thicker (red/blue) line in Figure 1b, which are in the following designated as separatrices. To the left and below the Γ2 separatrix, Γ2 characteristics intersect the vertical axis below WS1 and the horizontal axis between WS2 and X1. Below the Γ1 separatrix, Γ1 characteristics no longer reach the vertical axis but bend to intersect with the horizontal axis left of WS2. The point at which the Γ2 separatrix leaves the vertical axis (at c1 = 4.285 g/L) is a watershed-point (WS1), i.e., a point at

study, equilibrium theory is applied to a system subject to a generalized bi-Langmuir isotherm (see eqs 1 and 2). Component 1 is defined to be the component with the smaller value of the sum of Henry’s constants, i.e., of Hai + Hbi . This class of isotherms can be divided into several subclasses, depending on the type of the two terms a and b, which in turn is determined by the sign of the parameters pxi (based on the nomenclature of a previous study10). If px1 = px2 = 1, term x is a classic Langmuir term (L), if px1 = px2 = −1, it is an antiLangmuir term (A). If the two parameters pxi differ in sign, we call the term mixed. Depending on the signs of px1 and px2 and the values of Hx1 and Hx2 the term can be either a M1 or a M2 term. The term is denoted as M1, if Hxi > Hxj and pxi = −1, while pxj = 1. On the contrary, the term is called M2, if Hxi < Hxj , and pxi = −1, pxj = 1. Note that this nomenclature, which is able to clearly classify generalized Langmuir isotherms of different behavior,10 is not rigorous for the generalized Bi-Langmuir isotherms investigated in this study, but admits several subtypes in one subclass. As an example the L-M1 subclass admits systems with pa1 = pa2 = pb2 = 1, pb1 = −1, Ha1 < Ha2 and Hb1 > Hb2 (subtype A), but also contains systems with the parameter combination pa1 = pb1 = pa2 = 1, pb2 = −1, with Hb1 < Hb2 and for all (Ha1 + Hb1) < (Ha2 + Hb2) (subtype B). This study focuses on L-M1 Bi-Langmuir isotherms of subtype A. The partial derivatives for a generalized bi-Langmuir isotherm, required for the calculation of the eigenvalues θi, are given by n11 =

n22 =

H1a(1 + p2a K 2ac 2) δa2

H2a(1 + p1a K1ac1)

n12 = −

n21 = −

δa2

p2a K 2aH1ac1 δa2 p1a K1aH2ac 2 δa2

−

−

+

+

H1b(1 + p2b K 2bc 2) δb2

(10a)

H2b(1 + p1b K1bc1) δb2

(10b)

p2b K 2bH1bc1 δb2

(10c)

p1b K1bH2bc 2 δb2

(10d)

For the examined L-M1 bi-Langmuir isotherm, characteristics as well as shock paths in the hodograph plane cannot be solved analytically as in the case of a generalized Langmuir isotherm,10 but must be solved numerically. Simple wave characteristics given by the ordinary differential eq 8 were calculated using a built-in Matlab ode45 routine, whereas shock-paths were obtained by minimizing the difference between right- and left-hand side of eq 9 by a built-in Matlab trust-region-dogleg algorithm. 11422

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the directional derivative of σj along Γj characteristics changes sign. The locations at which the sign of the directional derivative changes are connected by continuous black lines, which for both σ1 and σ2 meet at WS1. This change in sign of the directional derivatives of both σ1 and σ2 indicates the possible occurrence of semishock transitions in the solution. From the hodograph planes presented in Figure 1, it is possible to infer several interesting effects in the adsorption, desorption, and displacement behavior. Sections 4 and 5 shall explain and illustrate the most relevant of these effects, which are • A change in the order of breakthrough of the two components, from the breakthrough of component 1 occurring first at low feed concentrations to component 2 breaking through earlier at high feed concentrations. Similarly, also the order of desorption of the two components changes. • A changing impact of component 1 on the adsorption behavior of component 2, exhibited at displacement conditions (displacing component 1 by component 2) with different initial and feed concentrations. This effect is associated with an intermediate concentration level of component 2 changing from lower to higher than its feed concentration. • Exceeding certain initial and feed concentration limits, the redisplacement of component 2 by component 1 always occurs along the same path in the hodograph plane, which approaches the origin along a horizontal Γ1 characteristic, then leaves the horizontal axis at the state X1, before reaching the origin, to follow the Γ2 separatrix. For a detailed explanation of the complete adsorption/ desorption and displacement behavior, please refer to the Appendix, sections B and C.

4. RELEVANT FEATURES OF FRONTAL ANALYSIS PROFILES In the following, the main features of the equilibrium theory solutions for the investigated L-M1 bi-Langmuir system at frontal analysis conditions with the initial state B located in the origin (i.e., starting with a completely regenerated column) shall be discussed. Solutions for the entire chromatographic cycle, i.e., adsorption from initial state B to feed state A, as well as desorption from the feed state A back to the initial state B, are determined within the hodograph plane (Figures 2 and 3), and are illustrated by exemplary elution profiles and mappings in the hodograph plane for different prototypical compositions (Figure 4). Feed states of the examples given in Figure 4 are also indicated in Figures 2 and 3. The exemplary concentration profiles in Figure 4 are compared to detailed simulations by the equilibrium dispersive model, which was implemented using a high resolution semidiscrete flux limiting finite volume scheme as described elsewhere.16 For the detailed simulations, we assumed a high number of theoretical plates (NT = 106), corresponding to a very low dispersion coefficient, and a high number of finite volumes used for discretization (1000). With intermediate states and elution times being consistent for detailed model and equilibrium theory predictions in all considered cases, the accuracy of the developed equilibrium theory solutions could be confirmed. 4.1. Adsorption. As discussed in section 2.1, the path in the hodograph plane connecting an initial state B to a feed state A always starts along a Γ1 characteristic (or Σ1 shock path) to

Figure 1. Hodograph plane with simple wave characteristics for a L-M1 bi-Langmuir system, (a) broad concentration range: Thick dashed line indicates the border at which the denominator δb becomes zero, thin dashed line indicates states at which the slope of Γ2 characteristics (ξ1) changes from negative to positive (b) zoom to low concentrations: Arrows indicate the direction along which the respective slope of the characteristics in the physical plane, σj, increases. Black continuous lines indicate states at which the sign of σj changes. The Γ1 and Γ2 separatrices (partly vertical Γ1 and Γ2 characteristic) are marked by a thicker red/blue line. Significant points: WS1 and WS2: Watershed points, X1: intersection of the Γ2 separatrix with the horizontal axis.

which the discriminant of eq 7 equals zero, and thus the two simple wave characteristics in this point have the same slope (ξ1 = ξ2). A second watershed-point (WS2) is located on the horizontal axis at a concentration c2 = 2.505 g/L. The intersection of the Γ2 separatrix with the horizontal axis is in the following termed X1. Arrows on the characteristics in Figure 1b indicate the direction along which the slope σj of the corresponding characteristics in the physical plane increases. While at high concentrations of both component 2 and component 1, the slopes σj exhibit a strictly monotonic behavior along the characteristics Γj, there are regions at low concentrations where 11423

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characteristics and shock paths coincide). Thus, starting from the origin, the Γ1 separatrix (see Figure 1b) has to be considered. Along this characteristic, the directional derivative of the slope in the physical plane changes sign twice, from negative to positive at the inflection point of the single component isotherm of component 1, and back to negative at the watershed point WS1. Accordingly, different types of transition can connect the initial state to intermediate states located in different regions of the hodograph plane. Intermediate states on the vertical axis (located between the origin and WS1) are connected to the origin by shocks and shock-waves. Intermediate states within the first quadrant of the hodograph plane, which are all located on the dashed black line indicated in Figure 2 (detailed explanation see Appendix, section B), can be reached via shock-wave-shocks or shocks. Finally, it can be shown that intermediate states located on the horizontal axis (beyond the intersection of the black dashed line with the horizontal axis) can be reached through shock transitions. The type of the first transition is determined by the location of the corresponding feed state in the hodograph plane. The areas in Figure 2 colored differently identify regions of feed states which result in different types of first transition, i.e., shocks/shock-waves along the vertical axis (yellow), shockwave-shocks (blue), shocks to intermediate states in the first quadrant (red) and shocks along the horizontal axis (green). Adsorption profiles in the left column of Figure 4 illustrate examples representative for each of these colored areas, with the corresponding feed states indicated in Figure 2. These profiles reveal a change in the order of breakthrough of the two components, with component 1 breaking through first at low concentrations, a simultaneous breakthrough of the two components occurring at intermediate feed concentrations, and component 2 breaking through first at high concentration. This is in accordance with the paths of the first transitions in the hodograph plane, mapping on the vertical axis at low feed concentrations, but being located on the horizontal axis at high feed concentrations. 4.2. Desorption. The development of desorption profiles (from the feed state A back to the initial state B) for frontal analysis conditions with the initial state located in the origin of the hodograph plane is illustrated in Figure 3. It can be shown that intermediate states can be located either on the horizontal axis between the origin and WS2 (connected to the initial state B in the origin through a simple wave) or on the vertical axis above WS1 (connected to the initial state through a shock). The location of the intermediate state (as in the adsorption case) depends on the corresponding feed state. Since all Γ1 characteristics above the Γ1 separatrix intersect with the vertical axis, all feed states located above this separatrix (violet region in Figure 3) lead to an intermediate state located on the vertical axis. On the contrary, feed states below this separatrix (nonviolet regions in Figure 3), located on Γ1 characteristics intersecting the horizontal axis, result in intermediate states located on the horizontal axis. As for the adsorption case, colored areas in Figure 3 define regions of feed states A resulting in different types of first transition, as specified in the Appendix, section B. Desorption profiles corresponding to the feed states indicated in Figure 3 are shown in the center column of Figure 4. Since second transitions map on the horizontal axis for the corresponding feed states located below the Γ1 separatrix, but map on the vertical axis for feed states above

Figure 2. Derivation of adsorption paths in the hodograph plane for frontal analysis conditions with an initial state located in the origin. Colored regions indicate feed states leading to the different types of first transitions. Second transitions occur along blue simple wave (continuous) and shock (dashed) paths. Feed states Aa to Ad correspond to the feed states of exemplary concentration profiles a to d in Figure 4. The black dashed line indicates all possible intermediate states located within the first quadrant, which can be reached from the origin.

Figure 3. Derivation of desorption paths in the hodograph plane for frontal analysis conditions with an initial state located in the origin. Intermediate states are located on the vertical axis above WS1 and on the horizontal axis between the origin and WS2 (indicated by black dashed lines). Colored regions indicate feed states leading to different types of first transitions: yellow and violet: simple waves, green: waveshock, red: wave-c.d.-wave, dark blue: shock, light blue: shock-wave. Second transitions in the desorption profiles are shocks for intermediate states on the vertical axis, and simple waves for intermediate states on the horizontal axis. Feed states Aa to Ad correspond to the feed states of exemplary concentration profiles a to d in Figure 4.

reach an intermediate state, before continuing along a Γ2 characteristic (or Σ2 shock path) that connects the intermediate state to the feed state (note that along the axes, simple wave 11424

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Figure 4. Concentration profiles (left: adsorption, center: desorption) for a L-M1 bi-Langmuir system at exemplary frontal analysis conditions predicted by equilibrium theory, and compared to predictions by the numerical equilibrium dispersive model (thin black lines). The right column presents the corresponding paths (thick black lines) in the hodograph plane, with the direction indicated by arrows. For reasons of clarity, the dashed arrow in (c) does not represent the shock path, but simply connects the initial state to the intermediate state. Initial state: c1 = c2 = 0 g/L, feed states: (a) c1 = 2 g/L, c2 = 1 g/L, (b) c1 = 3 g/L, c2 = 1 g/L, (c) c1 = 6 g/L, c2 = 4 g/L, and (d) c1 = 9 g/L, c2 = 9 g/L. Simulations by the equilibrium dispersive model were performed at a number of discretization cells k = 1000 and NT = 106.

conditions−displacement of a pure solution of component 1 as the initial state by a pure solution of component 2 as the feed state, as well as the redisplacement to reach the initial state. For the detailed derivation, the interested reader is referred to the Appendix, section C. Here, we focus on displacement conditions at rather high initial and feed concentrations, i.e., initial states above WS1 and feed states beyond X1. Figure 5 presents the displacement paths of component 1 by component 2 for two exemplary cases e and f, on the background of shock paths emanating from various initial and feed states (thin gray

the separatrix, the desorption profiles exhibit a change in the order of desorption, with component 1 desorbing first in Figures 3a to c (feed states below the separatrix), and component 2 desorbing first in Figure 3d (feed state above the separatrix).

5. RELEVANT FEATURES OF DISPLACEMENT PROFILES In the same way as for frontal analysis conditions, we have derived the equilibrium theory solution for displacement 11425

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Figure 5. Paths for the displacement of component 1 by component 2 corresponding to the concentration profiles of examples e and f illustrated in Figure 6, on the background of shock paths (gray lines) emanating from different initial and feed states located on the vertical and horizontal axis, respectively. Black continuous lines indicate boundary shock paths for the states WS1 and X1. The black dashed line indicates intermediate states for which the intermediate concentration of component 2 equals the feed concentration.

lines). Black continuous lines in Figure 5 indicate the border shock paths corresponding to the initial state WS1 and the feed state X1, respectively. The corresponding elution profiles of examples e and f are reported in Figure 6. The good agreement between profiles derived by equilibrium theory and profiles simulated with the detailed model confirms the accuracy of the equilibrium theory solution. For the displacement of component 1 by component 2, it can be shown that both the first and the second transition are always shocks. Since Σ1 shock paths for initial states above WS1 indicated as gray lines in Figure 5 (flatter set of shock paths intersecting the vertical axis), always have a positive slope, intermediate states connected to an initial state through the corresponding shock path always exhibit an intermediate concentration of component 1 which is higher than the initial concentration. This is reflected in the elution profiles in Figure 6, where component 1 is enriched during the intermediate state in both examples e and f. In contrast, slopes of Σ2 shock paths (steeper paths intersecting the horizontal axis) change in sign, i.e., from negative at low initial and feed concentrations to positive at high initial and feed concentrations. Accordingly, the concentration level of component 2 of intermediate states connected to the feed state through a corresponding Σ2 path is lower than the feed concentration at low initial and feed conditions but higher at high initial and feed conditions. As a boundary, the dashed line in Figure 5 indicates all the intermediate states with a concentration of component 2 equaling the corresponding feed concentration. With intermediate states of examples e and f being located below and above this boundary, respectively, the corresponding elution profiles indicate an intermediate concentration level of component 2 which is lower than the feed concentration in example e but higher in example f. These changing intermediate concentration levels of component 2 indicate a changing impact of component 1 on the adsorption behavior of component 2,

Figure 6. Concentration profiles (whole chromatographic cycle) for a L-M1 system at exemplary displacement conditions predicted by equilibrium theory and compared to predictions by the numerical equilibrium dispersive model (thin black lines). Initial (component 1) and feed (component 2) concentrations: (e) c1 = 15 g/L, c2 = 20 g/L and (f) c1 = 15 g/L, c2 = 60 g/L. Simulations by the equilibrium dispersive model were performed at a number of discretization cells k = 1000 and NT = 106.

from competitive at low initial and feed concentrations to cooperative at high initial and feed concentrations. For the redisplacement of component 2 by component 1, we can see from Figure 1b that all feed states beyond X1 are connected to initial states above WS1 via the same path, which is a simple wave along the Γ1 characteristic located on the horizontal axis until reaching X1, and then continues along the Γ2 separatrix as a wave-c.d.-wave. Accordingly, all redisplacement profiles for initial and feed states located beyond the indicated limits (e.g., examples e and f) coincide.

6. DISCUSSION 6.1. General Validity of the Solution. As already mentioned in section 2.2, the derived equilibrium theory solution is not valid for all types of L-M1 bi-Langmuir isotherms. First of all, one has to discriminate between isotherms with parameters fulfilling the conditions pa1 = pa2 = pb2 = 1, pb1 = −1, Ha1 < Ha2 and Hb1 > Hb2 (subtype A) and those systems fulfilling the conditions pa1 = pb1 = pa2 = 1, pb2 = −1, Hb1 < Hb2 for all (Ha1 + Hb1) < (Ha2 + Hb2) (subtype B). The derived solution is not valid for systems belonging to subtype B. However, also within subtype A, the derived behavior is not exhibited by all isotherms. This becomes clear when examining the envelopes for various sets of parameters belonging to subtype A. These envelopes connect all states at which the 11426

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Industrial & Engineering Chemistry Research discriminant D of eq 7 is zero, thus representing the boundaries between regions in which the system of partial differential equations is hyperbolic (D > 0) and regions where it is elliptic (D < 0). Figure 7a shows the envelopes for parameter set α

quadrant. These systems behave according to the behavior predicted and discussed in sections 3 to 5. A limit is reached at Hb1 = 2.0, where elliptic regions are detected neither in the first, nor in the second quadrant (however, still two points on the positive vertical axis and one on the positive horizontal axis with D = 0 can be identified). Systems with Hb1 = 1.8 and Hb1 = 1.6 in Figure 7b exhibit elliptic regions within the first quadrant, for which the equilibrium theory solution cannot apply. Accordingly, these systems do not exhibit the predicted behavior. In this context, it should be noted that, while the considered subtype A of L-M1 bi-Langmuir isotherm is in general not consistent with the Gibbs isotherm (in the framework of the ideal adsorbed solution theory, IAS), those representatives with D > 0, and thus with two distinct and real eigenvalues, in the first quadrant can still be called thermodynamically consistent in a broader sense.18,19 In contrast, those representatives, for which our solution fails, do not fulfill the necessary thermodynamic condition of two distinct and real eigenvalues, since D < 0 in a physically meaningful region (first quadrant and δb > 0). A solution for these cases might however be obtained in a similar way as for the generalized Langmuir isotherm of the M2 type, where certain conditions, resulting in paths crossing the border of the hyperbolic region, lead to the nonclassical transition of a delta-shock.13,20 Unfortunately, the derivation of an exact mathematical condition discriminating between systems, which exhibit or not the derived behavior is not straightforward, due to the rather complex expressions of partial derivatives (see eq 10), which enter into the expression giving the discriminant D. Thus, when applying the derived equilibrium theory solution, attention has to be paid whether the underlying isotherm parameters fulfill the necessary condition (D > 0 within the first quadrant) or not. 6.2. Experimental Results. By performing frontal analysis and displacement experiments for a rather broad range of experimental conditions, it was found that the system PNT (component 1) and TBP (component 2) in Methanol:H2O 63:37 (v:v) with the adsorbent Zorbax 300SB-C18 can be very well described by a L-M1 bi-Langmuir isotherm.14 Exemplary elution profiles, adapted from a companion paper,14 are compared to profiles simulated by an equilibrium dispersive model16 in Figures 8 (binary frontal analysis) and 9 (displacement). Figures 8 and 9 also include hodograph planes, in which the experimental data (concentrations of PNT and TBP) corresponding to the presented frontal analysis and displacement experiments is depicted. Comparing the equilibrium theory solution to this experimental data, it is possible to correlate characteristic features of the equilibrium theory solution to phenomena observed in the experimental profiles. 6.2.1. Identification of Characteristic Features. The changing features of Γ1 and Γ2 characteristics when moving from low to high concentrations in the hodograph plane yield different types of transition from and into the origin. As a consequence the isotherm is able to account for a change in the order of breakthrough and of desorption between the two components. This effect is in fact observed in the elution profiles of binary breakthrough experiments. With the breakthrough of PNT occurring before the breakthrough of TBP in experiment BT7 (Figure 8), the first transition maps on the vertical axis (compare yellow region in Figure 2). In contrast, with TBP clearly breaking through before PNT in experiment BT6, the first transition corresponds to a shock mapping on the

Figure 7. Envelopes (i.e., lines on which the discriminant D of eq 7 equals 0) for different sets of isotherm parameters given in Table 1. All parameter sets belong to the subclass of L-M1 isotherms, subtype A. (a) Envelopes for parameter sets α and β used for the derivation of the equilibrium theory solution and fitted to the experimental system, respectively. (b) Envelopes of a further exemplary parameter set γ with varying Henry constants Hb1. Besides the designated areas, regions completely enclosed by the envelope are characterized by D < 0.

employed in the derivation of the equilibrium theory solution, as well as for a set β (see Table 1) fitted to the experimental profiles (see section 6.2 below). Figure 7b shows the envelopes for a different exemplary set of parameters (set γ in Table 1), for varying values of Hb1. Note that all parameter sets in Figure 7 belong to subtype A. For all systems belonging to subtype A, envelopes touch the axes delimiting the first quadrant in three points, i.e. two on the vertical axis, and one on the horizontal axis. While the upper of the two points on the vertical axis is always located in the region where δb < 0, and is thus physically not meaningful, the other two points on the axes correspond to the two watershed points WS1 and WS2. The different systems belonging to subtype A differ in the location of elliptic regions (with D < 0). Besides those in Figure 7 designated with D < 0, all regions that are completely enclosed by the envelope exhibit negative values of D. For the systems with envelopes shown in Figure 7a, as well as for systems belonging to the parameter set γ with Hb1 = 2.4 and Hb1 = 2.2 (Figure 7b), all elliptic areas are located outside the first 11427

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Figure 8. Elution profiles (concentration of PNT and TBP) of binary breakthrough experiments (adapted from the companion paper14), as well as the corresponding liquid phase concentration data plotted in the hodograph plane. Feed states: BT7: cPNT = 3.0 g/L, cTBP = 1.0 g/L, BT9: cPNT = 14.4 g/L, cTBP = 4.8 g/L, BT6: cPNT = 24.0 g/L, cTBP = 90.0 g/L. The experimental profiles are compared to detailed simulations (black lines) based on a L-M1 bi-Langmuir isotherm and performed with the equilibrium dispersive model (NT = 106, number of discretization cells k = 1000). Note that for reasons of visibility, concentrations of TBP of BT6 are scaled with a factor of 0.1 in the hodograph plane. The experimental data in the hodograph plane is connected through dashed lines to guide the eye. Arrows along the paths in the hodograph plane indicate the direction in which the liquid phase compositions evolve with increasing elution times.

experiments presented in Figure 9 exhibit an enrichment in PNT during the interaction zone (intermediate state), whereas the intermediate concentration level of TBP is lower than the feed concentration at rather low feed concentrations (experiment DS1), but higher at high feed concentrations (experiment DS3). This behavior indicates a changing impact of PNT on the adsorption behavior of TBP, from competitive at low initial and feed concentrations to cooperative at high initial and feed concentrations. All transitions observed when component 1 is displaced by component 2 can be clearly identified as shocks. Plotting the concentration data of displacement experiments DS1 to DS3 in the hodograph plane (see Figure 9), it can be noted that the corresponding redisplacement paths (connecting feed state A to initial state B) of all three experiments overlap (they also coincide with the paths of experiments DS4 to DS6 at a different initial concentration of PNT, presented elsewhere14). Also this finding is in accordance with the theory (see Appendix, section C.2), which predicts the same paths for the displacement of a high concentration of TBP (higher than X1) by a high concentration of PNT (higher than WS1). It should be noted that displacement experiments with a high initial concentration of PNT (near the solubility limit) exhibited a behavior which could not be easily described in terms of simple wave, shock, or semishock transitions.14 As a consequence, this experimental behavior could not be in agreement with the equilibrium theory predictions, neither quantitatively nor qualitatively. We suspect that such behavior,

horizontal axis of the hodograph plane, followed by a simple wave adsorption profile of PNT (compare the green region in Figure 2). At intermediate concentrations and a rather high ratio of cPNT to cTBP in the feed, initial and feed states are connected through transitions enabling a simultaneous breakthrough of TBP and PNT (blue and red region in Figure 2), as observed experimentally in BT9. Similar correspondences between experiments and simulations can be established for the desorption profiles. With TBP clearly desorbing before PNT in experiment BT9, the second transition maps onto the vertical axis; this behavior corresponds to the violet region in Figure 3. However, at low feed concentrations such as in experiment BT7, PNT is desorbed before TBP, with a second transition mapping on the horizontal axis. Due to the change in the order of breakthrough and desorption, the corresponding experimental paths in the hodograph plane are followed in different directions (indicated as arrows in Figure 8), i.e., clockwise in BT7 and counterclockwise in BT9 and BT6. Concerning displacement conditions, due to changing slopes of Γ2 (and corresponding Σ2) characteristics in the hodograph plane, the investigated isotherm is able to account for the changing intermediate concentration levels of component 2 (higher or lower than the feed concentration). In contrast, component 1 is always enriched at the intermediate state, due to the always positive slopes of Γ1 (and Σ1) characteristics (see examples e and f in Figures 5 and 6). This effect can indeed be observed for the experimental system: the displacement 11428

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Figure 9. Elution profiles (concentration of PNT and TBP) of displacement experiments (adapted from the companion paper14), as well as the corresponding liquid phase concentration data plotted in the hodograph plane. Initial state: cPNT = 12.0 g/L, Feed states: DS1: cTBP = 20.2 g/L, DS2: cTBP = 40.0 g/L, DS3: cTBP = 79.9 g/L. The experimental profiles are compared to detailed simulations (black lines) based on a L-M1 bi-Langmuir isotherm and performed with the equilibrium dispersive model (NT = 106, number of discretization cells k = 1000). In the hodograph plane, the experimental data is connected through dashed lines to guide the eye.

parameters examined in the context of the derivation of the equilibrium theory solution, the single component isotherm of PNT based on the fitted set of parameters does not exhibit an inflection point for positive values of its concentration, cPNT. Thus, adjusting the equilibrium theory solution to the fitted set of parameters would require minor alterations, in fact simplifications, since the monotonic behavior of σ1 along the vertical Γ1 characteristic would not change below WS1, and transitions emanating from the origin along this characteristic would start as a wave (and not as a shock or as a shock-wave). Experimental profiles in Figures 8 and 9 agree very well with simulations (indicated as black lines) using an equilibrium dispersive model16 and assuming a L-M1 bi-Langmuir isotherm with the fitted set of parameters given in Table 1 (parameter set β). Note that the peak predicted by the equilibrium dispersive model in BT7 corresponds to a short intermediate state in the equilibrium theory solution, and that dispersive effects are the very likely reason for its absence in the experimental data. 6.3. Summary and Conclusions. In this study, an equilibrium theory solution for systems subject to a subclass of generalized bi-Langmuir isotherms was derived. Since the structure of this isotherm is rather complex (as compared to isotherms investigated previously by equilibrium theory, such as the generalized Langmuir isotherms10), the equilibrium theory equations can no longer be solved analytically, but require numerical computations. Furthermore, as shown in section 6.1, the validity of the derived solution is limited not only to qualitative features (such as signs of the values of pxi ) but also to quantitative features of the set of isotherm parameters. Explicit

which is so far unexplained, is due to considerable nonidealities in the liquid and the adsorbed phase, which are not accounted for by the current model. 6.2.2. Parameter Fitting and Mathematical Description. Having identified, with the aid of the derived equilibrium theory solution, characteristic features and transitions of the experimental elution profiles, it was possible to estimate the isotherm parameters so as to describe qualitatively and quantitatively these characteristic phenomena. Among others, clearly identified shock transitions and intermediate state concentrations of displacement profiles, as well as shock and wave transitions of experimental frontal analysis profiles with feed states located in the green region in Figure 2 were included in the fitting. The fitting was performed by minimizing the error between values of the composition of specific, characteristic states calculated by equilibrium theory and the corresponding experimental values. This fitting procedure offers a computationally inexpensive alternative to the fitting of isotherm parameters by an inverse method, i.e. employing a detailed model and a numerical algorithm to calculate composition profiles during column experiments.21 The latter approach requires a high computational effort and in our application did not achieve satisfactory results, due to the high number of experiments to be included into the estimation procedure of the large number of adsorption isotherm parameters in order to account for the various different effects observed experimentally. The obtained set of parameters is reported in Table 1 (parameter set β). Note that, in contrast to the set of 11429

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changing sign at least once. An exemplary isotherm with two inflection points J1 and J2 is illustrated in Figure 10.

mathematical constraints on the isotherm parameters explicitly defining the conditions for the applicability of the derived equilibrium theory solution has yet to be established. Such applicability can be assessed during the solution procedure based on the considerations discussed in section 6.1. Despite these drawbacks, it is indeed the complexity of this adsorption isotherm and of its equilibrium theory solution that leads to its ability of accounting for several interesting phenomena. Several of these phenomena have been explained in detail, such as a change in the order of breakthrough and desorption at frontal analysis conditions, and a changing impact of component 1 on the adsorption behavior of component 2 (competitive/cooperative) at displacement conditions, which is manifested in changing intermediate concentration levels (lower/higher than the feed concentration). Apart from common shock and simple wave transitions, the derived solution exhibits also semishocks with multiple shock and wave parts. Due to these remarkable features, this type of isotherm was able to accurately describe the experimental system PNT and TBP in Methanol:H2O with the adsorbent Zorbax 300SB-C18, which has previously been studied in the context of the deltashock phenomenon.14 The derived equilibrium theory solution did not only provide a thorough understanding of the experimental behavior, which had previously been misinterpreted, but it also enabled the fitting of isotherm parameters in a targeted and efficient (computationally inexpensive) manner (see section 6.2.2). After several unsuccessful approaches of fitting isotherm parameters using an inverse method and employing an equilibrium dispersive model, this new and efficient approach yielded a set of parameters accurately describing the experimental behavior (see section 6.2 and our experimental study14). It can thus be concluded that, despite its lack of generality, the presented equilibrium theory solution is indeed very powerful and useful and motivates a thorough investigation of other types of generalized bi-Langmuir isotherms by equilibrium theory, in order to assess the existence of similarly novel and interesting features.

■

Figure 10. Exemplary single component isotherm with two inflection points J1 and J2, illustrating conditions resulting in different types of semishock transitions. (a) Exemplary adsorption conditions leading to shock-waves, wave-shocks and shock-wave-shocks, (b) Exemplary desorption conditions leading to wave-c.d.-waves (possible shock-wave and wave-shock conditions not indicated).

Let us first consider the adsorption case (cA > cB), with an initial state cB < cJ1 (e.g., state d in Figure 10a). With n″(c) < 0 at the initial state, the transition starts as a shock. In the case where

APPENDIX

A. SEMI-SHOCKS AND CONTACT DISCONTINUITIES This section aims at explaining properties and necessary conditions of transitions consisting of multiple simple wave, contact discontinuity and shock parts (such as semi-shocks), which were derived in general5,6 as well as specifically for a single-component system.22 Properties of these transitions, which form the necessary basis for the detailed derivation of the equilibrium theory solution in sections B and C, can be very well illustrated with an exemplary single-component system, for which eq 6 reduces to σ=

(1 + νn′(c)) v

σ(B) > σ (B, k ) > σ (k ) ̃

(12)

with k being any state on the path connecting state B to state A, the transition is a shock, with simple wave characteristics of initial and feed state in the physical plane impinging on the shock path. If σ̃(B,A) < σ(A), as, e.g., for an initial state located in d and a feed state located in f in Figure 10a, the transition is a semi-shock, namely a shock-wave (n″(c) changes from negative to positive, and thus slopes of simple wave characteristics in the physical plane first decrease and then increase). The state T1 (state e in Figure 10a) at which the transition changes from a shock to a wave fulfills the condition

(11)

σ(B) > σ (B, T) ̃ 1 = σ (T) 1

Thus, the type of transition in a single-component system, whether a simple wave or a shock, directly depends on the monotonic behavior of n′(c) or on the sign of n″(c). Accordingly, semi-shocks and other “multipart transitions” in a single-component system can be observed for isotherms containing one or more inflection points, i.e., with n″(c)

(13)

It should be noted that, in contrast to shock transitions, the shock parts in semi-shocks only exhibit a partially selfsharpening tendency on one side (here from state B), while the state on the other side (here state T1) moves with the same propagation velocity as the shock part itself. 11430

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respectively, correspond to the exemplary elution profiles given in Figure 4.

A transition from an initial state located between the two inflection points (e.g., state e) to a feed state beyond inflection point J2 (e.g., state g) is a wave-shock (n″(c) changes from positive to negative). The state T2 (state f in Figure 10a) connecting the wave to the shock part fulfills the requirement σ(T2) = σ (T ̃ 2 , A) > σ(A)

(14)

Initial state d and feed state g embracing both inflection points can be connected through a shock-wave-shock, with the different shock and wave parts being connected by states e (T1) and f (T2), fulfilling the requirements given in eqs 13 and 14, as well as the inequality: σ(T) 1 ≤ σ (T2)

(15)

The border of possible shock-wave-shock transitions with an initial state in the origin is illustrated in Figure 10a, connecting the origin to a feed state x2. The two states T1 and T2 connecting shock and wave parts merge into one state T12 (x1 in Figure 10a), with σ(B) > σ (B, T12) = σ(T12) = σ (T ̃ ̃ 12 , A) > σ(A)

(16)

Feed states beyond x2 cannot be connected to the initial state in the origin through a shock-wave-shock, since eq 15 cannot be fulfilled. These feed states are reached through a shock, for which again simple wave characteristics of initial and feed state in the physical plane impinge on the shock path. Let us finally consider a desorption case (cB > cA) with an initial state exceeding state z1 and a feed state below state z2 (see Figure 10b). In this case, initial and feed state are connected through a transition which consists of a simple wave, a contact discontinuity and another simple wave, in our study designated as wave-c.d.-wave transition. Again, the two states T2 and T1 connecting the different wave and discontinuity parts (corresponding to z1 and z2 in the exemplary system, respectively) have to fulfill requirements which are a combination of eqs 13 and 14: σ(T2) = σ (T ̃ 2 , T) 1 = σ (T) 1

Figure 11. Line of possible intermediate states in the adsorption profile, resulting in different types of first transitions. red part: shock or shock-wave transition, gray part (dashed): shock-wave-shock transition, black part (dashed): shock transition. Intermediate states Ia to Id correspond to those of examples a to d in Figure 4. The point at which the transition changes from shock to wave is designated as T1. T2 indicates the point at which the transition changes from a wave to a shock for the conditions of example b.

B.1. Adsorption

As discussed in section 4.1, the solution for the adsorption case is derived starting along the Γ1 separatrix. Figure 11 deals with this first transition connecting the initial state B to an intermediate state I1. The red (dashed and continuous), gray dashed, and black dashed lines indicate all possible intermediate states I1 which can be reached from the origin through the first transition. Arrows in Figure 11 indicate two states connected by a shock transition. B.1.1. First Transition. Following the Γ1 characteristic from the origin, the slope σ1 of the corresponding characteristics in the physical plane first decreases until reaching the inflection point of the single component isotherm of component 1 at c1 = 0.186 g/L, and then increases until reaching WS1, where the characteristic leaves the vertical axis and at the same time the behavior of σ1 changes again (decreases again along the characteristic). Thus, transitions from the initial state always start as a shock. The state T1 at c1 = 0.279 g/L on the vertical axis fulfills eq 13 and thus connects the shock part at lower concentrations to a wave part at higher concentrations. Intermediate states I1 on the vertical axis below this state are connected to the origin through a shock, whereas intermediate states on the vertical axis at a higher concentration of component 1 (but below WS1) are connected to the origin through a shock-wave. The latter is also the case for intermediate state Ia corresponding to example a in Figure 4. Since the entire transition takes place along the vertical axis, this first transition of example a describes a breakthrough of component 1 only. The second change in the behavior of σ1 at WS1 indicates the existence of shock-wave-shock transitions for intermediate

(17)

As in this case, the states T2 and T1, connected by the discontinuity, travel with the same propagation velocity as the discontinuity itself, the discontinuity in this case is not selfsharpening. It is by definition a contact discontinuity. The requirements for the different types of “multipart transitions” derived in eqs 13 to 17 in the illustrative context of a single component system are valid for binary and multicomponent systems with neither restrictions nor the necessity of further requirements. In the context of binary and multicomponent systems, the single component liquid phase concentration c is replaced by the vector c.

B. EQUILIBRIUM THEORY SOLUTION FOR FRONTAL ANALYSIS CONDITIONS In the following, we provide a detailed and complete derivation of equilibrium theory solutions for the investigated L-M1 biLangmuir system at frontal analysis conditions with the initial state B located in the origin. Solutions for the entire cycle are described and illustrated in the hodograph plane given in Figures 2 and 3, which have already been shortly discussed in section 4. Further illustration for the first transition of the adsorption process is provided by Figure 11. Intermediate and feed states indicated in Figure 11, and Figures 2 and 3, 11431

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it are reached through a shock (as in examples a and b). Accordingly, Figure 2 indicates simple wave characteristics (continuous blue lines) connecting intermediate states to the region above, and shock paths (dashed blue lines) emanating from the line of intermediate states to the region below. In a small region close to the first watershed point (mentioned at the beginning of this section), a change in the directional derivative of σ2 along Γ2 characteristics (compare Figure 1b) can result in different types of semishocks: Intermediate states located on the vertical axis below and very close to WS1 are connected to feed states located below and very close to the Γ2 separatrix by wave-shocks. In turn, intermediate states located in the blue region (on the dashed line) in Figure 2 very close to WS1 are connected to feed states slightly above the Γ2 separatrix by shock-wave-shocks or shockwaves. Apart from the occurrence of semishocks instead of shock transitions in its surrounding, this region is neither very interesting from a phenomenological point of view, nor does it play an important role for the system behavior, due to its limited size in the case of the adsorption isotherms considered here.

states being located on the gray dashed line in Figure 11 (e.g., Ib). These transitions start as a shock-wave along the vertical axis. Every intermediate state along the gray dashed line is then connected through a shock to a state T2 on the vertical axis between T1 and WS1, which fulfills eqs 14 and 15 and which is depicted in Figure 11 for the conditions of example b in Figure 4. In contrast to the shock-wave transition of example a, which maps on the vertical axis of the hodograph plane, the shockwave-shock transition of example b leaves the vertical axis of the hodograph plane during the second shock part. Accordingly, both components 1 and 2 break through within the first transition, however, the breakthrough of component 2 (during the second shock part) is still delayed with respect to the breakthrough of component 1 (during the first shock part). The last intermediate state on the gray dashed line (Figure 11), which is the point in common with the black dashed line, is connected through a shock to the first switching state (T1 = T2), and thus reduces the wave part to one single state, fulfilling eq 16. This transition is comparable to the B-x1-x2 transition in the single component example of section A, and, fulfilling the two conditions σ̃1(B,T1) = σ̃1(T1,I1) = σ̃1(B,I1) it can be regarded as a shock-wave-shock that collapses into a pure shock. Intermediate states on the black dashed lines (in the first quadrant, such as Ic corresponding to example c, and along the horizontal axis, such as Id corresponding to example d in Figure 4) do not fulfill eq 15, and are therefore connected to the origin by a shock (compare states beyond x2 in the single component example). While the first transition in the case of example c − not mapping on either of the axes−describes a simultaneous breakthrough of components 1 and 2, the first transition of example d, mapping on the horizontal axes, describes a breakthrough of component 2 only. In principle, every point on the horizontal axis can be connected to the origin through a shock (fulfilling the Rankine-Hugoniot condition given in eq 9). However, σ̃1(B,I1) > σ2(I1) for states on the horizontal axis left of the horizontal black dashed line, such that there is no physically valid solution for a Γ2 path connecting a feed state to these states. The location of the first intermediate state depends on the feed state. Areas of feed states leading to the different types of first transition are identified in Figure 2 using different colors, namely shocks or shock-waves along the vertical axis (yellow), shock-wave-shocks (blue), shocks to an intermediate state in the first quadrant (red), or shocks to an intermediate state on the horizontal axis (green). Each of the colored areas contain one of the feed states of the examples illustrated in Figure 4. By comparing the first transitions of the prototypical examples a to d, it can be concluded that the different types of possible first transitions when starting from the origin enable a change in the order of breakthrough, with component 1 being eluted first at low feed concentrations, a simultaneous breakthrough occurring at intermediate feed concentrations, and component 2 breaking through first at high feed concentrations. B.1.2. Second Transition. The second transition along a Γ2 characteristic connects the intermediate state I1 to the feed state A. Let us first leave aside the small region near WS1, in which the directional derivative of σ2 along Γ2 changes in sign (this region will be discussed at the end of this section). Apart from that region, the slope σ2 behaves strictly monotonically along all Γ2 characteristics. Thus, feed states located above the line of intermediate states are always reached through a simple wave (as in examples c and d in Figure 4), whereas feed states below

B.2. Desorption

Figure 3 in the main part illustrates in the hodograph plane the development of desorption profiles (from the feed state A back to the initial state B) for frontal analysis conditions with the initial state located in the origin of the hodograph plane. Exemplary desorption profiles for the feed states depicted in Figure 3 are provided in the center column of Figure 4. B.2.1. Second Transition. As discussed in section 3, Γ1 characteristics (determining the path of the first transition) intersect the vertical axis for rather high feed concentrations of component 1 (above the Γ1 separatrix, i.e. in the violet region in Figure 3), but intersect the horizontal axis for low feed concentrations of component 1 (below the Γ1 separatrix, i.e. all regions but the violet one in Figure 3). Thus, possible intermediate states on the path from a feed state to the origin (initial state) are located on the vertical Γ2 characteristic above WS1, or on the horizontal Γ2 characteristic left of WS2. With σ2 decreasing when moving along the partly vertical Γ2 characteristic in the direction toward the origin, the second transition of the desorption profile from an intermediate state on the vertical axis to the origin is always a shock. This behavior is illustrated by example d in Figure 4 (corresponding feed state located in the violet region of Figure 3). Mapping on the vertical axis, this transition describes a complete desorption of component 1 in the absence of component 2, which indicates that complete desorption of component 1 in the second transition takes place after complete desorption of component 2 in the first transition. In contrast, with σ2 increasing along the horizontal Γ2 characteristic toward the origin, intermediate states on the horizontal axis (as in examples a to c in Figure 4) are connected to the origin through a simple wave. These simple waves, mapping on the horizontal axis, describe the desorption of component 2 in the absence of component 1, which accordingly has to take place after the complete desorption of component 1 during the first transition. Since for increasing feed concentrations, the intersection of the corresponding Γ1 characteristics with the horizonal axis, which identifies the intermediate state, approaches the origin, intermediate concentration levels decrease from example a to c. Without detailed knowledge of the behavior of the corresponding first transitions, we can see that due to the 11432

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Industrial & Engineering Chemistry Research changing features of Γ1 characteristics when moving from low to high concentrations, the isotherm is able to account for a change in the order of desorption. B.2.2. First Transition. A higher complexity is encountered when considering the first transition of the desorption profile, since the behavior of σ1 along some Γ1 characteristics is not strictly monotonic (states at which the directional derivative of σ1 changes in sign were discussed in section 3 and are indicated in Figure 3 by a thin continuous black line). For feed states being located in the yellow and violet region (examples a and d in Figure 4), σ1 increases strictly monotonically until reaching the horizontal or vertical axis, respectively. In these cases, the first transition in the desorption profile is a simple wave. For feed states located in the red region, the sign of the directional derivative of σ1 along Γ1 characteristics connecting the corresponding feed state to an intermediate state on the horizontal axis changes twice (positive−negative-positive), thus yielding a wave-c.d.-wave transition. This is the case of example b in Figure 4 (however, the second wave part is too steep to be clearly distinguished from the preceding contact discontinuity part). The border separating the red and the light blue region corresponds to T2 states fulfilling eq 17 with cT1 1 ≥ 0 g/L. Thus, feed states above this border (red region) are connected to the intermediate state I2 by a wave-c.d.-wave, whereas feed states below this border (light blue region) result in a shock-wave. At the border separating the red and the green region, eq 17 can still be fulfilled, however, state T1 merges with the intermediate state I2 on the horizontal axis (cT1 1 = 0 g/L), such that the second wave part is reduced to one single point coinciding with the intermediate state; this transition can also be considered as a wave-contact discontinuity. States in the green region beyond this border can no longer fulfill eq 17, since σ1(T2) = σ̃1(T2,I2) > σ1(I2). Thus, feed states in the green region are connected to the intermediate state by a wave-shock (compare example c in Figure 4), with the switching state T2 fulfilling eq 14. States of T2 form the border between the green region exhibiting waveshock transitions and the dark blue region exhibiting shock transitions. The dark blue shock region and light blue shockwave region are separated by states fulfilling the requirement σ̃1(A,I2) = σ1(I2).

Figure 12. Derivation of an equilibrium theory solution for the displacement of component 1 by component 2 in the hodograph plane. (a) Note that red and blue lines are not simple wave characteristics, but indicate states which can be reached from a specific initial state on the vertical axis or can be connected to a specific feed state on the horizontal axis, respectively. The black dashed line corresponds to states which fulfill the Rankine-Hugoniot condition (eq 9) from an initial state in the origin, the gray dashed line corresponds to states reached from the origin through a shock-wave-shock (compare Figure 2). Continuous black borderlines indicate states which can be reached from WS1 and from X1 through a shock. The black borderline corresponding to X1 is extended until reaching X1 by a dotted line. (b) Paths corresponding to exemplary displacement conditions illustrated in Figure 14. The dashed arrow corresponding to the first transition of example d indicates the connection of the initial to the intermediate state through a shock, but does not reflect the course of the shock path.

C. EQUILIBRIUM THEORY SOLUTION FOR DISPLACEMENT CONDITIONS This section provides a detailed derivation of an equilibrium theory solution for displacement conditions, i.e. displacing a pure solution of component 1 by a pure solution of component 2. The whole cycle, i.e., displacement of component 1 by component 2 (Figure 12), as well as redisplacement of component 2 by component 1 (Figure 13), was investigated. Exemplary elution profiles are provided in Figure 14; their corresponding paths in the hodograph plane are presented in Figures 12b and 13b. As for the investigations concerning frontal analysis conditions, elution profiles derived by equilibrium theory and presented in this section are compared in Figures 14 to numerical simulations based on the equilibrium dispersive model. The agreement between the two solutions confirms the validity of the equilibrium theory approach.

correspond to the Γ1 and Γ2 characteristics given in Figure 1. The investigated displacement conditions include a range of initial states and feed states (in contrast to frontal analysis conditions with a fixed initial state located in the origin). Therefore, the red and blue lines given in Figure 12a each indicate states which can be reached from a specific initial state on the vertical axis or can be connected to a specific feed state

C.1. Displacement of Component 1 by Component 2

Possible transitions and intermediate states for the displacement of component 1 by component 2 are shown in Figure 12a. Note that red and blue lines in this figure do not 11433

DOI: 10.1021/acs.iecr.5b03089 Ind. Eng. Chem. Res. 2015, 54, 11420−11437

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shock-wave-shock, respectively, as already discussed in the context of frontal analysis. However, in Figure 12a, the black dashed line does not stop at the point where the shock-waveshock line starts, but is extended until reaching the vertical axis. States on this extension fulfill the Rankine-Hugoniot condition (eq 9), but in reality cannot be reached from the origin, since the correct path is a shock-wave-shock reaching the states on the gray dashed line (see section B). C.1.1. First Transition. Let us first consider the red lines connecting the initial to an intermediate state. For initial states above WS1, red lines are shock paths (since σ1 decreases along Γ1 characteristics when moving away from the vertical axis), which emanate from the corresponding initial state. This is the case for examples b and c, with their paths in the hodograph plane and their elution profiles given in Figures 12b and 14, respectively. In principle, also the initial state of example a is located above WS1, however, the corresponding first transition does not follow the shock path as indicated for examples b and c. The reason for the different behavior will be given in the context of the analysis of the second transition path. Lines for initial states below WS1 are derived on the basis of the partly vertical Γ1 characteristic (already discussed in the context of frontal analysis), along which the sign of the directional derivative of σ1 changes twice. Thus, initial states should reach intermediate states through a wave-shock (or shock-wave-shock, depending on the position of the initial state) fulfilling eq 14. These intermediate states are located on the dashed gray line derived in the previous section. For σ1(B) > σ̃1(B,k), the requirement for a wave-shock transition can no longer be fulfilled, and all further states k along the red lines, deviating from the gray dashed line, are connected to the initial state through a shock. Thus, all red lines corresponding to initial states below WS1 coincide along a part of the gray dashed line (connected through a wave-shock), until they leave this line at different points to indicate states that can be reached from the corresponding initial state through a shock. Accordingly, also the first transition of example d with an initial state below WS1 is a shock; the corresponding initial and intermediate states are connected by a dashed arrow in Figure 12b. C.1.2. Second Transition. The second set of lines (blue) indicate possible intermediate states which can be connected to a specific feed state. The connecting transition is always a shock, since above and to the right of the Γ2 separatrix (and thus in the entire physically meaningful region in Figure 12a) σ2 decreases when moving along a Γ2 characteristic from a possible intermediate state toward a feed state on the horizontal axis (see main part, Figure 1b). States on the blue lines fulfill, assuming a specific feed state, the Rankine-Hugoniot condition (eq 9), and the requirement that σ̃1(B, k) ≤ σ̃2(k, A), where k is any state along a blue path. These lines are limited by the straight black dashed line and the vertical axis, fulfilling the border condition σ̃1 (B, k) = σ̃2 (k, A). In principle, paths fulfilling the Rankine-Hugoniot condition could be extended beyond these limits (as is exemplarily indicated by the black dotted line for the feed state being located in X1), however, violating the second requirement, states on these extensions do not provide physically meaningful intermediate states. As intermediate states correspond to intersections of red and blue lines for the corresponding initial and feed state, possible intermediate states are located in the first quadrant above the black dashed line. For a feed state beyond X1 (border path marked by a continuous black line), blue paths intersect with

Figure 13. Derivation of paths for the redisplacement of component 2 by component 1 in the hodograph plane. (a) Relevant Γ1 and Γ2 characteristics, arrows on the characteristics indicate the direction in which the slope σj of the corresponding characteristics in the physical plane increases. Thin black lines indicate states at which the directional derivative of σj changes in sign. (b) Paths in the hodograph plane corresponding to the redisplacement profiles illustrated in Figure 14.

on the horizontal axis, respectively. Intersections of blue and red lines identify the intermediate states which are reached for the initial and feed states belonging to the intersecting paths. States in the region below the black dashed line (spare white region and region between the gray dashed and the black dashed line) cannot be reached as intermediate states, as explained in detail below. Thus, a continuation of blue lines within this region is mathematically possible, but physically not meaningful. The continuous black borderlines indicate the respective lines corresponding to the initial state located in WS1 and to the feed state located in X1. For the sake of illustration, the line corresponding to the feed state located in X1 is continued as a (physically not meaningful) dotted line until reaching X1. The black dashed line and the gray dashed line indicate states connected to the origin through a shock and 11434

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Figure 14. Concentration profiles (left: displacement of component 1 by component 2, right: redisplacement of component 2 by component 1) for a L-M1 system at exemplary displacement conditions predicted by equilibrium theory and compared to predictions by the numerical equilibrium dispersive model (thin black lines). initial (component 1) and feed (component 2) concentrations: (a) c1 = 10 g/L, c2 = 1 g/L, (b) c1 = 10 g/L, c2 = 3 g/L, (c) c1 = 10 g/L, c2 = 6 g/L, and (d) c1 = 2 g/L, c2 = 10 g/L. Simulations by the equilibrium dispersive model were performed at a number of discretization cells k = 1000 and NT = 106.

region. For these combinations, the correct intermediate state, fulfilling the condition σ̃1(B,I1) ≤ σ̃2(I1,A), is located in the origin. This intermediate state is connected to initial states through a desorption shock (or shock-wave or wave for initial conditions slightly above or below the inflection point of the single component isotherm of component 1, respectively), and to feed states through an adsorption shock. This is the case for example a: The blue path corresponding to the feed state of example a (c2 = 1 g/L) in Figure 12b reaches the vertical axis

every possible red path. Thus, for these feed states, intermediate states exist within the marked region for every possible initial state, connected to both the initial and the feed state through shock transitions (as for examples c and d). On the left of the discussed border (feed state left of X1), blue paths do not intersect with every possible red path, due to their limitation by the vertical axis and the black dashed line. Thus, there are combinations of initial and feed states, which are not connected via an intermediate state in the discussed 11435

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It can be concluded that feed states beyond X1 and initial states above WS1 are always connected to each other by the same path in the hodograph plane (simple wave transition mapping on the horizontal Γ1 characteristic and wave-c.d.-wave transition, for which the second wave part maps onto the Γ2 separatrix).

without intersecting the red path corresponding to an initial state c1 = 10 g/L. Therefore, the initial and feed states in example a are connected via the origin, which corresponds to complete desorption of component 1 before adsorption of component 2. In contrast, blue and red paths corresponding to initial and feed states of example b intersect at an intermediate state located in the first quadrant, and accordingly the corresponding elution profile exhibits an interaction zone of components 1 and 2. It should further be noted that, since blue paths do not reach the gray dashed line, intermediate states are always connected to initial states through a shock (and not a wave-shock). Exceptions are (as already mentioned) low initial concentrations of component 1 near or below the inflection point of the single component isotherm, which are connected to an intermediate state located in the origin through a shock-wave or wave.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +41 44 632 24 56. Fax: +41 44 632 11 41. Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Glueckauf, E. The general theory of two solutes following nonlinear isotherms. Discuss. Faraday Soc. 1949, 7, 12−25. (2) Lax, P. D. Hyperbolic systems of conservation laws II. Commun. Pure. Appl. Math. 1957, 10, 537−566. (3) Helfferich, F.; Klein, G. Multicomponent Chromatography; Dekker: New York, 1970. (4) Rhee, H. K.; Aris, R.; Amundson, N. R. On the Theory of Multicomponent Chromatography. Philos. Trans. R. Soc., A 1970, 267, 419−455. (5) Rhee, H. K.; Aris, R.; Amundson, N. R. First-Order Partial Differential Equations, Vol. I: Theory and Applications of Single Equations; Prentice Hall: Englewood Cliffs, NJ, 1986. (6) Rhee, H. K.; Aris, R.; Amundson, N. R. First-Order Partial Differential Equations, Vol. II: Theory and Applications of Hyperbolic Systems of Quasilinear Equations; Prentice Hall: Englewood Cliffs, NJ, 1986. (7) Storti, G.; Mazzotti, M.; Morbidelli, M.; Carrá, S. Robust design of binary countercurrent adsorption separation processes. AIChE J. 1993, 39, 471−492. (8) Mazzotti, M.; Storti, G.; Morbidelli, M. Optimal operation of simulated moving bed units for nonlinear chromatographic separations. J. Chromatogr. A 1997, 769, 3−24. (9) Chiang, A. Complete separation conditions for a local equilibrium TCC adsorption unit. AIChE J. 1998, 44, 332−340. (10) Mazzotti, M. Local Equilibrium Theory for the Binary Chromatography of Spiecies Subject to a Generalized Langmuir Isotherm. Ind. Eng. Chem. Res. 2006, 45, 5332−5350. (11) Mazzotti, M. Design of Simulated Moving Bed Separations: Generalized Langmuir Isotherm. Ind. Eng. Chem. Res. 2006, 45, 6311− 6324. (12) Mazzotti, M. Equilibrium theory based design of simulated moving bed processes for a generalized Langmuir isotherm. J. Chromatogr. A 2006, 1126, 311−22. (13) Mazzotti, M. Nonclassical Composition Fronts in Nonlinear Chromatography: Delta-Shock. Ind. Eng. Chem. Res. 2009, 48, 7733− 7752. (14) Jermann, S.; Ortner, F.; Mazzotti, M. Absence of experimental evidence of a delta-shock in the system phenetole and 4-tertbutylphenol on Zorbax 300SB-C18. J. Chromatogr. A 2015, accepted. (15) Mazzotti, M.; Tarafder, A.; Cornel, J.; Gritti, F.; Guiochon, G. Experimental evidence of a delta-shock in nonlinear chromatography. J. Chromatogr. A 2010, 1217, 2002−12. (16) Javeed, S.; Qamar, S.; Seidel-Morgenstern, A.; Warnecke, G. Efficient and accurate numerical simulation of nonlinear chromatographic processes. Comput. Chem. Eng. 2011, 35, 2294−2305. (17) Mazzotti, M.; Rajendran, A. Equilibrium theory-based analysis of nonlinear waves in separation processes. Annu. Rev. Chem. Biomol. Eng. 2013, 4, 119−41. (18) Kvaalen, E.; Neel, L.; Tondeur, D. Directions of quasi-static mass and energy transfer between phases in multicomponent open systems. Chem. Eng. Sci. 1985, 40, 1191−1204.

C.2. Redisplacement of Component 2 by Component 1

Redisplacement of the feed state (pure solution of component 2) by the initial state (pure solution of component 1) is considered in Figure 13a. The derivation of the corresponding paths is straightforward considering simple wave characteristics along the axes and in the low concentration region of the hodograph plane, as well as the behavior of the slopes σ1 and σ2 along these characteristics (indicated by arrows). Again, paths in the hodograph plane for the exemplary concentration profiles in Figure 14 are shown in Figure 13b. C.2.1. First Transition. For a feed state of pure component 2 beyond WS2, the first transition maps onto the horizontal Γ1 characteristic, until reaching the corresponding intermediate state. If the feed state is located beyond X1 (as is the case for examples c and d in Figure 14), this transition is always a simple wave (the intermediate state is always located left of the feed state). If the feed state is positioned between WS2 and X1, the transition is a shock for intermediate states right of the feed state (as for example b), and a simple wave for intermediate states left of the feed state. For feed states on the left of WS2, the first transition occurs no longer along the horizontal axis. This transition is a shock for the characteristics exhibiting a strictly monotonic behavior of σ1 (compare example a), and a shock-wave or shock-wave-shock for the Γ1 characteristics exhibiting a directional derivative of σ1 changing in sign. The shock, shock-wave and shock-wave-shock paths deviate slightly from the simple-wave characteristics, but still intersect with all relevant Γ2 characteristics. For the sake of clarity, they are not shown in Figure 13a. C.2.2. Second Transition. Initial states above WS1 are connected to intermediate states via the Γ2 separatrix. Since the directional derivative of σ2 changes sign twice along this Γ2 characteristic, the corresponding transition is a wave-c.d.-wave (or shock-wave, if the intermediate state is located within the discontinuity part), whose path exhibits a marginal deviation from the Γ2 separatrix. The wave-c.d.-wave transition is exhibited by examples a to c. Since the second wave part of the transition maps onto the vertical axis, component 2 is completely desorbed within the discontinuity part of the transition. Initial states below WS1 are reached by simple waves along the indicated Γ2 characteristics, as illustrated by example d, except for initial states near WS1, which due to a change in sign of the directional derivative of σ2 along the characteristics are reached through a wave-shock. 11436

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