Local Equilibrium Theory for the Binary ... - ACS Publications

352

Ind. Eng. Chem. Res. 2011, 50, 352–377

Local Equilibrium Theory for the Binary Chromatography of Species Subject to a Generalized Langmuir Isotherm. 2. Wave Interactions and Chromatographic Cycle Arvind Rajendran*,† and Marco Mazzotti‡ Nanyang Technological UniVersity, School of Chemical and Biomedical Engineering, 62 Nanyang DriVe, Singapore, Singapore 637459, and Institute of Process Engineering, ETH Zurich, Sonneggstrasse 3, CH 8092 Zurich, Switzerland

The generalized Langmuir isotherm model was recently developed to describe the competitive adsorption equilibria of a binary mixture [Mazzotti, M. Ind. Eng. Chem. Res. 2006, 45, 5332-5350]. This formalism captures the competitive effect of solutes that show either Langmuirian or anti-Langmuirian behavior. This results in four isotherm types that show markedly different chromatographic behavior. The equilibrium theory of chromatography was developed in a previous work to study Riemann problems (saturation, elution, and the chromatographic cycle) that are of practical relevance [Mazzotti, M. Ind. Eng. Chem. Res. 2006, 45, 5332-5350]. This work focuses on the development of the equilibrium theory to account for wave interactions and is divided into two parts. In the first part, the theory of wave interactions is developed on the basis of the method of characteristics. It is shown that the rules developed for competitive Langmuir isotherms can be extended to describe possible wave interactions in the case of the generalized Langmuir isotherm [Rhee, H. K. et al. First-Order Partial Differential Equations; Dover Publications: Mineola, NY, 2001; Vol II]. The second part of the study deals with the development of the chromatographic cycle for binary pulse injections. The general case of short pulses in which the waves from the adsorption and desorption front interact within the column is considered. The solution of this problem results in a set of algebraic equations that describe the movement of the solutes in the column. The solution of the problem is unique to the type of adsorption isotherm considered owing to the different sequence in which wave interactions occur in the column. Finally, the elution profiles for the four isotherm types obtained from the equilibrium theory are shown to be fully consistent with those obtained through numerical simulations. 1. Introduction Chromatography is not only a common analytical tool but also a powerful and efficient tool for preparative separations in the pharmaceutical, food, and agrochemical industries. Both single-column and multicolumn operating modes of various degrees of complexity have been developed.3-5 Preparative chromatography is almost always operated at overloaded conditions, where the concentrations in the column correspond to the nonlinear portion of the adsorption isotherm. Under these conditions, the quantitative description of the solute movement provides the basis to understand and to predict the complex dynamics of chromatographic columns and to eventually design the separation process.6 The equilibrium theory of chromatography is a powerful tool for studying solute movement in chromatographic columns, as well as more in general in (real) fixed bed and (theoretical) moving bed adsorption columns.2,7-10 Within the frame of the theory, the solute movement is described only through the consideration of convection and of the equilibria between the fluid and the solid phases, while mass transfer resistance and dispersion effects are neglected. Although based on a simplification of the physical phenomena that occur in reality, the theory provides a remarkably accurate description of several important features, e.g., the formation and propagation of shocks, simple waves, and semishocks. The theory has been successfully * To whom correspondence should be addressed. E-mail: arvind@ ntu.edu.sg. † Nanyang Technological University. ‡ ETH Zurich.

used in the literature to design both fixed bed6 and simulated moving bed5,11,12 processes. When applied to a common though simple isotherm such as the Langmuir isotherm, equilibrium theory leads to results that are given in explicit form and are very easy to use, also, in the case of multicomponent systems.2 Recently, a new model for describing the competitive adsorption of a binary mixture, called “generalized Langmuir isotherm”, was introduced;1 this isotherm formulation allows for the description of Langmuirian and antiLangmuirian behavior and their possible competitive or cooperative behavior in a rather simple way. The fundamentals of solute movement for the generalized Langmuir isotherm, including the solution of Riemann problems, i.e., piecewise constant initial value problems, were developed using again equilibrium theory and showing that also in this more general case the same structure of explicit solutions as for the Langmuir isotherm applies.1 In this paper, we extend such development to wave interactions, i.e., an aspect that was not considered earlier. Wave interactions are important in studying several problems of practical interest such as the chromatographic cycle, where the aim is to describe the movement along the chromatographic column of a short overloaded pulse, which is large enough to be subjected to a nonlinear adsorption isotherm and so small as never to saturate the column. Herein, we briefly review the basics of equilibrium theory as applied to the description of the generalized Langmuir isotherm. Then, the theory of wave interactions is developed. Finally, we apply the theory to describe the chromatographic cycle and compare the results obtained in this way with numerical simulations.

10.1021/ie1015798  2011 American Chemical Society Published on Web 11/24/2010

Ind. Eng. Chem. Res., Vol. 50, No. 1, 2011

[H1(d - p1K1c1) + H2(d - p2K2c2)] - 4H1H2d > 0

2. Background on the Equilibrium Theory for the Generalized Langmuir Isotherm

(5)

2.1. Model Equations and Generalized Langmuir Isotherm. The fundamentals of the local equilibrium theory are described in detail in several publications.1,2,10 To introduce the concepts and notation needed to study wave interactions, we summarize in the following the most relevant aspects of the theory applied to binary systems subject to the generalized Langmuir isotherm. Under assumptions of negligible mass transfer resistance, negligible axial dispersion, uniform void volume, and constant velocity along the column, the mass balance equation for solute i is given by ∂ci ∂ ) 0 (i ) 1, 2) (c + νni) + ∂τ i ∂x

(1)

(2)

with the denominator defined as d ) 1 + p1K1c1 + p2K2c2

Such a condition defines a region of the hodograph plane (c1, c2) where the system of PDEs is hyperbolic, i.e., it is associated to two distinct characteristics, Cj, in the physical plane (x, τ), whose slope σj is given by

In the above equation, Ni and Ki are the saturation capacity and the adsorption equilibrium constant of component i. In the discussions that follow, we assume that component 2 is the strongly adsorbed component, i.e., H2 > H1, where Hi ) NiKi is its Henry’s constant. The coefficients p1 and p2 are allowed to take the values +1 or -1. A value of pi ) +1 indicates that component i exhibits Langmuirian behavior, whereas pi ) -1 corresponds to anti-Langmuirian behavior. Hence, for a binary mixture, there are four possible combinations. When p1 ) p2 ) 1, the classical competitive Langmuir isotherm is obtained; p1 ) p2 ) -1 yields the anti-Langmuir isotherm. Then, there are two mixed cases: the M1 mixed Langmuir isotherm where p1 ) -p2 ) 1 and the M2 mixed Langmuir isotherm where -p1 ) p2 ) 1. It is worth noting that only compositions yielding d > 0 are physically meaningful; hence the domain of interest in the (c1,c2) plane is defined by c1 g 0, c2 g 0, and d > 0. 2.2. Method of Characteristics. Equations 1 and 2 represent a set of homogeneous reducible partial differential equations (PDE), which can be solved after properly defining the initial value problem. The method of characteristics can be used when the system is hyperbolic, as defined below. In this context, it can be proved that there exists a one-to-one mapping between the concentrations (c1, c2) and the two characteristic parameters (ω1, ω2) calculated as roots of the following equation (with ω1 < ω2): dω2 - [H1(d - p1K1c1) + H2(d - p2K2c2)]ω + H1H2 ) 0 (4) Two such values are distinct when the discriminant of eq 4 is positive, i.e., when the following condition applies:

(j ) 1, 2)

(6)

and to two distinct characteristics, Γj, in the hodograph plane (c1, c2), whose slope is given by

( ) dc1 dc2

p1p2K2(H1 - ω3-j) K1(ω3-j - H2)

) Γj

(j ) 1, 2)

(7)

Along the Γj characteristic in the hodograph plane, the characteristic parameter ω3-j is constant. Therefore, Γ1 (Γ2) characteristics in the hodograph plane correspond to constant values of ω2 (ω1) and map therefore onto horizontal (vertical) straight lines in the characteristic plane of coordinates (ω1, ω2). This feature of the characteristic parameters makes their use especially advantageous. The range of values that ωi can take is given for the different isotherms by 0 < ω1 e H1 e ω2 e H2

(8)

H1 e ω1 e H2 e ω2 < +∞

(9)

0 < ω1 e H1 < H2 e ω2 < +∞

(10)

case L: case A: case M1:

(3)

νωj dτ )1+ dx d

σj )

Cj:

Γj:

where ci and ni are fluid and solid phase concentrations of solute i, x ) z/L is the dimensionless column length, τ ) tV/L is the dimensionless time, with V being the interstitial velocity, and the phase ratio is defined as ν ) (1 - ε)/ε, with ε being the void fraction of the bed. Within the frame of the equilibrium theory, the solute concentrations in the solid and fluid phases are assumed to be at equilibrium. In the case of the generalized Langmuir isotherm, the adsorbed phase concentration is given by ni ) fi(c1, c2) ) NiKici /d (i ) 1, 2)

353

2

case M2-subregion 1:

H 1 e ω1 < ω2 e H2

(11)

As discussed in the first part, in the case of the mixed M2 Langmuir isotherm, the hyperbolic region in the hodograph plane consists of four disjointed subregions.1 As in the first part, we will consider only the first subregion, as defined by eq 11, because this is the one that contains the origin and is of interest for the study of the chromatographic cycle in section 4. The shape and location of the regions in the characteristic plane defined by the four equations above is shown in Figure 1. As for an assigned composition, the values ω1 and ω2 are uniquely defined through eq 4; at the same time, a given pair of (ω1, ω2) defines uniquely a fluid phase composition and its corresponding adsorbed phase composition through the following relationships:1,2 K1c1 )

-p1H2(ω1 - H1)(ω2 - H1) ω1ω2(H2 - H1)

(12)

p2H1(ω1 - H2)(ω2 - H2) ω1ω2(H2 - H1)

(13)

K2c2 )

d) K1n1 )

H1H2 ω1ω2

(14)

-p1(ω1 - H1)(ω2 - H1) H2 - H1

(15)

p2(ω1 - H2)(ω2 - H2) H2 - H1

(16)

K2n2 )

2.3. Solution of Riemann Problems. Riemann problems are piecewise constant initial value problems, where along the initial

354


the two states that are adjacent to the transition itself in the physical plane. The image of the transition in the physical plane depends on the relative position of the states on its left- and right-hand sides in the characteristic plane with coordinates (ω1, ω2). If moving from the left to the right of the transition, i.e., from a generic state L associated with (ω1L,ω2L) to another state R associated with (ωR1 ,ωR2 ), ωj decreases and ω3-j remains constant, then the transition is continuous. It is called a Γj simple wave, and it consists of a line of straight Cj characteristics with slope σj given by σ1 )

σ2 )

Figure 1. Characteristic parameter plane, with the regions corresponding to the different generalized Langmuir isotherms according to the inequalities in eqs 8-11. Note that the point of coordinates (H1, H2) corresponds to the origin of the hodograph plane where c1 ) c2 ) 0. In each region, a point corresponding to a typical feed state in a chromatographic cycle (see section 4) is shown.

line (which consists of the positive parts of the horizontal (τ ) 0) and vertical (x ) 0) axes in the physical plane) each pair of adjacent constant states is separated by a discontinuity. Solving a Riemann problem corresponds to determining how the two constant states are connected in the physical plane (for τ g 0 and 0 e x e 1) through what the evolution of the discontinuity in the physical plane is. Let us consider a Riemann problem where the state on the left of the discontinuity, A, is characterized by the pair of characteristic parameters (ω1A, ω2A) and the state on the right, B, by the pair (ω1B, ω2B). Then, the classical solution consists of three constant states, i.e., regions of the physical plane where the composition is constant, namely, state A on the left, then an intermediate state I characterized by the pair (ω1A, ω2B), and finally, state B on the right, which are separated by two transitions that represent composition fronts traveling along the column.1,2 The transitions between A and I and between I and B map onto characteristics Γ2 and Γ1, respectively, in the hodograph plane and in the characteristic plane: Γ2

Γ1

state A (ωA1 , ωA2 ) f state I (ωA1 , ωB2 ) f state B (ωB1 , ωB2 ) (17) In the special case where the states A and B share one of the two ω values, the intermediate state and one of the transitions are missing. The mixed M2 Langmuir isotherm is special also because under certain circumstances it exhibits nonclassical solutions of Riemann problems, i.e., a δ shock and a continuous nonsimple wave, which have been proved and analyzed theoretically,13 as well as demonstrated experimentally.14 The Riemann problems leading to nonclassical solutions are not considered in this work, but we will analyze the cases where a nonclassical composition front emerges as result of the wave interaction (see section 3.3). More specifically, in the solution of a Riemann problem, each of the two transitions separating the three constant states maps onto the segment of the characteristic in the hodograph plane connecting

dτ ) 1 + Fω21ω2 dx (with ωR1 e ω1 e ωL1 and ω2 ) ωR2 ) ωL2 )

(18)

dτ ) 1 + Fω1ω22 dx (with ωR2 e ω2 e ωL2 and ω1 ) ωR1 ) ωL1 )

(19)

where F ) ν/(H1H2). If on the contrary when moving from the left to the right of the transition ωj increases (whereas ω3-j is constant), then the transition is a Σj shock (discontinuous) and consists of a straight Sj shock with slope σ˜ j given by: σ˜ 1 )

σ˜ 2 )

dτ ) 1 + FωL1 ωR1 ω2 dx (with ωR1 > ωL1 and ω2 ) ωR2 ) ωL2 )

(20)

dτ ) 1 + Fω1ωL2 ωR2 dx (with ωR2 > ωL2 and ω1 ) ωR1 ) ωL1 )

(21)

From the equations above, it is possible to derive the following general expressions for the maximum and minimum slopes of the characteristics in a transition of the jth family, which naturally coincide in the case of a shock: σmin ) 1 + Fω3-jωRj min(ωRj , ωLj ) j

(22)

σmax ) 1 + Fω3-jωLj max(ωRj , ωLj ) j

(23)

The properties above have been demonstrated for all four generalized Langmuir isotherms (as long as ω1R < ω2L and ω1L < ω2R, in the case of the mixed M2 Langmuir isotherm13). This demonstrates that the analysis of Riemann problems can be carried out in the characteristic plane (ω1, ω2) applying the results and using the rules above irrespective of the isotherm type. In particular, the solution of Riemann problems for saturation (a clean bed subject to a constant inlet composition) and desorption (a saturated bed being subjected to a pure solvent to regenerate it) were discussed earlier.15 In several problems of practical interest, such as analytical and preparative chromatography, the injected amounts are small, and the injection times are short enough to result in the interaction of different composition fronts. In this work, we address this important situation. 3. Wave Interaction Interactions between simple waves and shocks occur when the Riemann problem consists of at least three constant states separated by two discontinuities along the initial line consisting of the axes x ) 0 and τ ) 0. Since the geometry of the transitions in the physical plane, hence their phenomenology, is controlled by the values of the characteristic parameters associated with the various


constant states, then it is natural and convenient to study wave interactions in the characteristic plane (ω1, ω2), and then to map the results thus obtained onto the physical plane. The most important advantage of such an approach is that the behavior of the interactions depends only on the relative positions of the points representing the initial constant states in the characteristic plane, irrespective of the type of generalized Langmuir isotherm considered. In studying wave interactions, we can build on the established theory of wave interactions for Langmuir isotherms.2,8 In their seminal book,9 Helfferich and Klein have mentioned the possibility of using their approach to study systems with competitive-cooperative behavior (see section 5.III.3 of their book); however, they have not provided any of the detailed equations that will be derived in this section. For the sake of clarity and without a loss of generality, in the following, we call the three constant states A, P, and B, and we assign state A for x ) 0 and τ g τ0 > 0, state P for x ) 0 and 0 e τ e τ0, and state B for τ ) 0 and 0 e x e 1. Each of the two pairs of constant states adjacent to a discontinuity generates a solution of the type discussed in the last section, i.e., consisting of up to three constant states separated by up to two transitions. Therefore, the solution consists of up to five constant states separated by up to four transitions: Γ2

Γ1

state A (ωA1 , ωA2 ) f state L (ωA1 , ωP2 ) f Γ2

Γ1

state P(ωP1 , ωP2 ) f state R (ωP1 , ωB2 ) f state B (ωB1 , ωB2 )

(24)

Note that the intermediate state in the portion of the solution connecting A to P (P to B) is called state L (state R). Note also that state L (state R) might coincide with either state A (P) or state P (B) if states A and P (states P and B) share the same value of either ω2 or ω1, respectively. Note also that the transitions can be either simple waves or shocks, both of the family indicated in the previous equation. Thus, state P, i.e., the state bracketed by A on the left and by B on the right along the initial line, at least in the vicinity of the origin will be bracketed in the physical plane by two transitions. The transition on the left-hand side (left transition) is of family i and connects P to state L (possibly coinciding with A), with which it shares the ω3-i value. The right transition is of family k and connects P to state R (possibly the same as B), with which it shares the same ω3-k value. Under the proper circumstances, these two transitions may collide, and a wave interaction may develop. It is necessary to distinguish between those cases where the left and right transitions are of a different family, i.e., i * k ) 3 - i (see section 3.2), and those where they belong to the same family, i.e., i ) k (see section 3.1). Therefore, only a subset of the overall solution path in eq 24 matters when studying wave interactions, and such a subset can be represented as follows when i * k: Γleft ≡ Γi

Γright ≡ Γk

state L (ωLi , ωkP). 98 state P (ωPi , ωkP). 98 state R (ωPi , ωkR)

(25) On the other hand, the subset of interest when i ) k can be represented as Γleft ≡ Γi

state L (ωLi , ω3-i). 98 Γright ≡ Γi

state P (ωPi , ω3-i). 98 state R (ωRi , ω3-i)

(26)

355

It is worth noting that in all cases, four independent ω values only (not six) are required to define a situation where a wave interaction may occur. Moreover, in all cases, the condition for the occurrence of the interaction in the physical domain of interest is that the two transitions collide before the end of the column. This requires that the characteristic with minimum slope of the left transition reaches the column end, i.e., where x ) 1, before the characteristic with maximum slope of the right transition. Using eqs 22 and 23, such a condition can be written as follows: min max τ0 + σleft e σright

(27)

which implies τ0 e F{ω3-kωPk max(ωRk , ωPk ) - ω3-iωPi min(ωPi , ωLi )}

(28) Therefore, provided the right-hand side of the last equation is positive, there is a maximum value of τ0, below which wave interaction occurs. If the right-hand side is negative, then there can be no interaction. In the case of interaction, the coordinates of point G in the physical plane where the interaction starts are obviously given by the following equations: xG )

τG )

τ0 max σright

min - σleft

max τ0σright max min σright - σleft

(29)

(30)

All figures and numerical examples discussed below use the following set of parameters (ci and ni are in units of g/L): K1 ) K2 ) 0.1 L/g, N1 ) 10 g/L, and N2 ) 20 g/L. Moreover, ν ) 1.5 also in all examples that follow τ0 ) 1, and we let the axial coordinate x take values larger than 1, in order to allow for the interaction to take place. Another equivalent approach would have been to choose a value of τ0 small enough to fulfill eq 28. 3.1. Interaction between Waves of the Same Family. Let us consider the case corresponding to eq 26 where i ) k, i.e., the left and right transitions belong to the same family i, with i equal to either 1 or 2. No more precise specification is needed, since the analysis is the same in the two cases. The points in the characteristic plane corresponding to the states L, P, and R lie on the same characteristic (with a constant value of ω3-i) and can be arranged along the ωi axis in terms of their relative position in six different manners, as shown in the second column of Table 1 (the horizontal and vertical alignments corresponding to i ) 1 and i ) 2, respectively). In the same table, columns 3 and 4 indicate the type of the left and right transitions, i.e., either simple wave (Γi) or shock (Σi), whereas columns 5 and 6 report the slopes of the corresponding characteristics in the physical plane. Here, as in Table 2, only the variable part of the relationship giving the slope is reported; with reference to eqs 11-14, this corresponds to the expression of the slope minus 1, divided by F. Finally, columns 7 and 8 give the type of transition connecting states L and R, i.e., that prevailing after the wave interaction if it occurs, and the corresponding slopes; column 9 indicates the number of the figure that illustrates the relevant situation. In the first case, i.e., SF0 where both the left and the right transitions are simple waves, it is rather clear that the righthand side of eq 28 is zero; hence, there can be no interaction.

356


Table 1. Interaction of Waves Belonging to Same Family: Types, Nature of Interaction, and Slopes of Characteristics on the Physical Plane

Table 2. Interaction of Waves Belonging to Different Families: Types, Nature of Interaction, and Slopes of Characteristics on the Physical Plane

In all of the other cases, it is easy to see that the right-hand side of eq 28 is positive and that the left and right transitions collide provided τ0 fulfills eq 28. Two Shocks. SF1 is the simplest case, because it involves two shocks that intersect in (xG, τG), as illustrated in Figure 2a. How the interaction takes place can be easily analyzed following the classical literature on wave interaction.2 The origin of the (x, τ) plane can be shifted to point G, and a new Riemann problem centered in G having states L and R assigned as left and right states can be considered. As shown in Figure 2b, ωLi < ωRi (i ) 2 in this example); therefore, such a Riemann problem has a solution consisting of the two initial states L and R separated by a shock transition, with the slope given in column 8 of Table 1 (see Figure 2a). In other words, the two interacting shocks of the same family i meet in G and are instantly superposed to give a new shock of the same family separating states L and R. For the sake of illustration, Figure 2c shows snapshots of the composition profiles along the columns at three different dimensionless times. Note that the figure provides plots of the ωi values as a function of x to highlight the fact that one of the two values remains constant through all composition fronts in these cases, where the two interacting waves belong to the same family. Simple Wave and Shock. Cases SF2 and SF3 (a and b) involve the interaction between a simple wave and a shock, with the shock on the left and on the right, respectively, and all

three states involved, i.e., L, P, and R, belonging to the same characteristic. After the shock hits the wave in (xG, τG) (see Figures 3a-6a), the shock and the simple wave interact and cancel each other, because any transition connecting states L, P, and R must map on the same characteristic, to which they belong.2,8 The shock comes from the left-hand side of the simple wave in case SF2 (from the right-hand side in case SF3), hits it at point G, and starts cutting through it, thus yielding a transition that connects L to P′ through a discontinuity and P′ to R through a simple wave in Figures 3a and 4a (L to P′ through a simple wave and then P′ to R through a discontinuity in Figures 5a and 6a). Note that P′ is a state along the characteristic under consideration located between states P and R in case SF2 (see Figures 3b and 4b), or between states L and P in case SF3 (see Figures 5b and 6b); state P′ is defined by the characteristic parameters (ωi, ω3-i), with ωi fulfilling the condition ωRi e ωi e ωPi (ωPi e ωi e ωLi ). Since the slope of the shock in the physical plane depends on the states across it and since during the interaction the left state in case SF2 (the right state in case SF3) remains unchanged but state P′ changes, then the shock’s speed changes, and the shock path in the physical plane bends. Such a path can be calculated by enforcing the following conditions, namely, that at every point the tangent to the shock path be equal to the slope of the shock connecting L and P′ in case SF2 (P′ and R


357

Figure 2. Interaction of waves from different families, type SF1. (a) Physical plane. (b) Characteristic plane. (c) Profile of ω1 (continuous line) and ω2 (dotted line) along x for different values of τ.

in case SF3), which is indicated as σ˜ i(ωi). Here, P′ is the state that propagates along the characteristic of the original simple wave, of slope σi(ωi), which intersects the shock path at that point: dτ ) σ˜ i(ωi) dx

(31)

τ ) τ0 + σi(ωi)x

(32)

(33)

(34)

Combining these two equations, one obtains the following ordinary differential equation: {σ˜ i(ωi) - σi(ωi)}

In the latter equation, τ0 equals 0 in case SF2, whereas it is larger than 0 in case SF3. Since ωi is the only parameter varying along the shock path during the interaction, the shock path can be given a oneparameter representation in terms of ωi itself. Thus, differentiating eqs 31 and 32 with respect to ωi yields dτ dx ) σ˜ i(ωi) dωi dωi

dσi(ωi) dτ dx ) σi(ωi) +x dωi dωi dωi

dσi(ωi) dx )x dωi dωi

(35)

Integrating this equation starting from point G, i.e., subject to the initial condition x ) xG at ωi ) ωPi , yields

∫x

x G

dx ) x

∫

ωi

ωiP

dσi dωi dωi {σ˜ i(ωi) - σi(ωi)}

(36)

When x(ωi) is obtained from the latter equation, τ(ωi) can be calculated using eq 32. Equations 31 and 32 are general and can be made explicit by specializing them to cases SF2 and SF3 and to the corresponding subcases a and b, which differ by the characters

358


Figure 3. Interaction of waves from different families, type SF2a. (a) Physical plane. (b) Characteristic plane. (c) Profile of ω1 (continuous line) and ω2 (dotted line) along x for different values of τ.

either shock in subcase a or simple wave in subcase bsof the ultimate transition between states L and R after the wave interaction (see columns 7 and 8 of Table 1). Case SF2 refers to the situation where the simple wave is centered in the origin of the physical plane (τ0 ) 0 in eq 32), and the shock reaches it coming from the left, i.e., from behind (see Figures 3 and 4). In this case, σi(ωi) ) 1 + Fω3-iω2i and σ˜ i(ωi) ) 1 + Fω3-iωiωLi ; hence, the integrand on the right-hand side of eq 36 equals 2(ωLi - ωi)-1. Since ωi decreases moving from point G forward along the shock path, the shock path’s slope decreases as it bends toward the right; therefore the shock accelerates while it cancels the simple wave. Integrating eq 26 yields the following parametric equation of the shock path:

(

x(ωi) ) xG

ωPi - ωLi ωi - ωLi

)

2

(37)

(

τ(ωi) ) xG(1 + Fω3-iω2i )

ωPi - ωLi ωi - ωLi

)

2

(38)

where max(ωRi ,ωLi ) e ωi e ωPi and xG is given by eq 29. Eliminating the parameter ωi from the previous two relations yields the following explicit equation for τ in terms of x:

√τ - x ) √Fω3-i(ωLi √x - (ωLi - ωPi )√xG)

(39)

Let us consider case SF2a, where ωRi > ωLi . Then, the shock path extends between points G and I where xG e x(ωi) e xI ) xG{(ωPi - ωLi )/(ωRi - ωLi )}2; i.e., the wave interaction occupies a finite portion of the column, and accordingly it is eluted from a column in a finite period of time, as shown in Figure 3a. Figure 3c shows snapshots of the composition profiles along the columns at four different dimensionless times. In case SF2b, ωiR < ωiL. Hence, as ωi approaches ωiL, the coordinate x(ωi) becomes infinitely large, and the slope of the shock approaches that of the characteristic from the origin


359

Figure 4. Interaction of waves from different families, type SF2b. (a) Physical plane. (b) Characteristic plane. (c) Profile of ω1 (continuous line) and ω2 (dotted line) along x for different values of τ.

associated with state L, i.e., (1 + Fω3-i(ωLi )2). Under these circumstances, the wave interaction goes on indefinitely, as illustrated in Figure 4. The final solution at very large times is given by a simple wave that connects states L and R as shown in the snapshots of Figure 4c. Case SF3 is illustrated in Figures 5 and 6. Here, the simple wave is centered at (0, τ0) with τ0 > 0, whereas the shock emanates from the origin of the physical plane and travels ahead of the simple wave, but more slowly. Then, σi(ωi) ) 1 + Fω3-iω2i and σ˜ i(ωi) ) 1 + Fω3-iωiωRi ; hence, the integrand on the right-hand side of eq 36 equals 2(ωRi - ωi)-1. Now, ωi as well as the slope of the shock path increase when moving from point G forward along the shock path, which bends toward the left and decelerates. Integrating eq 36 yields the following parametric equation of the shock path:

(

x(ωi) ) xG

ωRi - ωPi ωRi - ωi

)

2

(40)

τ(ωi) ) τ0 + xG(1 + Fω3-iω2i )

(

ωRi - ωPi ωRi - ωi

)

2

(41)

with ωPi e ωi e min(ωRi ,ωLi ). The corresponding explicit equation after eliminating ωi is

√τ - τ0 - x ) √Fω3-i(ωi √x - (ωi R

R

- ωPi )√xG)

(42)

Cases SF3a and SF3b can be analyzed in a similar manner to that for cases SF2a and SF2b, leading to results that are comprehensively illustrated in Figures 5 and 6, respectively. 3.2. Interaction between Waves of Different Families. Let us consider the case where the two interacting waves belong to different families, i.e., the case corresponding to eq 25 where the left and right waves belong to families i and k, respectively, and i * k ) 3 - i. In the characteristic plane, point P is at the intersection of a characteristic i from point L and a characteristic k from point R. There are two categories of cases, namely, where i ) 2 and k ) 1, i.e., case DF0 in Table 2, and where i ) 1 and k ) 2, i.e., cases DF1 to DF4

360


Figure 5. Interaction of waves from different families, type SF3a. (a) Physical plane. (b) Characteristic plane. (c) Profile of ω1 (continuous line) and ω2 (dotted line) along x for different values of τ.

in the same table. In both categories, there are four different manners to arrange the relative positions of L, P, and R, as shown in the second column of Table 2. This leads to eight cases in all, i.e., DF0a to DF0d, where there is no interaction, and DF1 to DF4, where wave interaction indeed occurs. For all eight of these cases, Table 2 reports also the types and slopes of the left and right waves before (columns 3-6) and after (columns 7-10) the interaction, when it occurs, as well as the figure number where the relevant case is illustrated (column 11). Note that in this section we deliberately consider all generalized Langmuir isotherms but the mixed Langmuir isotherm M2 that will be dealt with in section 3.3. In all four DF0 cases, where the left wave belongs to family 2 and the right one to family 1, no interaction can take place. This can be easily seen by using the values of the characteristics reported in columns 5 and 6 of Table 2 and observing that the right-hand side of eq 28 is always negative; hence the two waves do not collide. In cases DF1 to DF4, i.e., where the left wave belongs to family 1 (i ) 1) and the right one to family 2 (k ) 2), the two waves do collide, and the coordinates of point G where the interaction starts can again be calculated using eqs 29 and 30. Note

that in these cases the characteristic parameters of the three initial states are as follows: state L, (ω1L, ω2L); state R, (ω1R, ω2R); state P, (ω1R, ω2L). It is also worth defining state Q, with characteristic coordinates (ω1L, ω2R); it is the fourth vertex of the rectangle in the characteristic plane defined by points L, P, and R. Two Shocks. Case DF1 involving two shocks is again the simplest case. As in case SF1 above, the collision of the two shocks at point G defines a new Riemann problem, whose solution connects the two states on either side of the discontinuity, i.e., states L and R (see Figure 7a). Since L and R belong to different characteristics, according to the classical theory summarized in section 2.3 the solution consists of three constant states, i.e., L, R, and an intermediate state, and two transitions. Considering the relative position of points L and R in the characteristic plane (see Figure 7b), the intermediate state is indeed state Q, and the two transitions are shocks that connect P to Q and then Q to R and have the slopes given in columns 9 and 10 of Table 2. Figure 7 shows the solution in the physical plane and in the characteristic plane, as well snapshots of the composition profiles (in terms of ωj values) along the column


361

Figure 6. Interaction of waves from different families, type SF3b. (a) Physical plane. (b) Characteristic plane. (c) Profile of ω1 (continuous line) and ω2 (dotted line) along x for different values of τ.

at different values of the dimensionless time τ. The two shocks of different families present before the interaction are transmitted through each other while interacting.2 Simple Wave and Shock. Like SF2 and SF3, cases DF2 and DF3 involve the interaction between a shock and a simple wave, where the shock is on the left-hand side and on the right-hand side of the simple wave, respectively. The relative position of points L and R in the characteristic plane (see Figures 8b and 9b) indicates that after the wave interaction, points L and R must be connected through two transitions, i.e., a simple wave and a shock in reverse order with respect to that before wave interaction, and via an intermediate state, i.e., state Q. Therefore, also in these cases, the transitions present before the interaction are transmitted through the interaction. In case DF2, the shock overtakes the simple wave (see Figure 8a), whereas the opposite occurs in case DF3 (see Figure 9a). Our goal is to determine the shock path in the physical plane, i.e., from point G where the interaction starts to point I where the interaction terminates. Let us consider once more Figures 8a and 9a and Figures 8b and 9b. Before the interaction state, L is connected to state R through the path L f P f R; after the

interaction, they are connected through the path L f Q f R. Both paths involve two transitions. During the wave interaction, states L and R can be connected through the three-transition path L f S f T f R. The first and the third transitions are simple waves of the same family as that of the incoming wave, whereas the second transition is a shock belonging to same family as that of the incoming shock. The concepts and equations that govern the evolution of the shock path, i.e., its propagation along the column, are the same as in section 2.3. However, the index in eqs 31 and 33 is that of the incoming shock, i.e., 1 in case DF2 and 2 in case DF3, whereas that in eqs 32 and 34 is the index of the incoming simple wave. If the shock belongs to the family j, then across it, parameter ωj changes from ωLj to ωRj , whereas ω3-j remains unchanged. However, ω3-j varies along the shock path between the corresponding values in L and R, and the shock path can be given a one-parameter representation in terms of it. On the basis of these considerations, eq 36 can be cast as

∫x

dx ) G x

x

∫

ω3-j

P ω3-j

dσ3-j dω3-j dω3-j {σ˜ j(ω3-j) - σ3-j(ω3-j)}

(43)

362


Figure 7. Interaction of waves from different families, type DF1. (a) Physical plane. (b) Characteristic plane. (c) Profile of ω1 (continuous line) and ω2 (dotted line) along x for different values of τ.

In case DF2, j ) 1 in the previous equation. S and T are states along the simple wave paths L f Q and P f R, respectively. Therefore, S and T are characterized by the pairs (ω1L, ω2) and (ω1R, ω2); i.e., they share the same ω2 value. The shock path G to I can be given a one-parameter representation in terms of ω2. In this case, σ2(ω2) ) 1 + Fω1Rω22 (corresponding to the characteristic associated with state T) and σ˜ 1(ω2) ) 1 + Fω1Lω1Rω2; hence the integrand on the right-hand side of eq 43 equals 2(ωL1 - ω2)-1. Integrating eq 43 yields the following parametric equation of the shock path:

(

x(ω2) ) xG

ωL2 - ωL1 ω2 - ωL1

(

τ(ω2) ) xG(1 + FωR1 ω22)

)

2

(44)

ωL2 - ωL1 ω2 -

ωL1

)

2

(45)

where ω2L ) ω2P e ω2 e ω2R, and xG is given by eq 29. Note that since ω2 decreases along the shock path, the shock accelerates during the interaction. The characteristic propagating from this point on the other side, which is associated with state S, has a slope σ2(ω2) ) 1 + Fω1Lω22. Eliminating

the parameter ω2 from the last two relations yields the following explicit equation of the shock path:

√τ - x ) √FωR1 (ωL1 √x + (ωL2 - ωL1 )√xG)

(46)

In case DF3, j ) 2 in eq 43. S and T belong to the simple wave paths L f P and Q f R, respectively, and correspond to the pairs (ω1, ω2L) and (ω1, ω2R), i.e., they have the same ω1 value. Therefore, shock path G to I can be given a one-parameter representation in terms of ω1. In case DF3, σ1(ω1) ) 1 + Fω21ωL2 (corresponding to the characteristic associated to state S) and σ˜ 2(ω1) ) 1 + Fω1ω2Lω2R; hence the integrand on the right-hand side of eq 43 equals 2(ω2R - ω1)-1. Integrating eq 43 yields the following parametric equation of the shock path:

(

x(ω1) ) xG

ωR2 - ωR1 ωR2 - ω1

(

τ(ω1) ) τ0 + xG(1 + Fω21ωL2 )

)

2

ωR2 - ωR1 ωR2 - ω1

(47)

)

2

(48)

where ω1R ) ω1P e ω1 e ω1L and xG. The corresponding explicit equation after eliminating ω1 is


363


√τ - τ0 - x ) √Fω2 (ω2 √x - (ω2 L

R

R

- ωR1 )√xG)

(49)

The shock slows down during the interaction, because ω1 increases during it. On the other side of the shock path, the characteristic associated with state T has a slope σ1(ω1) ) 1 + Fω12ω2R. Both cases DF2 and DF3 are illustrated in Figures 8 and 9. With reference to Figure 9c at τ ) 1.7, it is worth noting that during the wave interaction, one ω value changes continuously while the other exhibits a step change. Two Simple Waves. Let us consider case DF4, where two simple waves interact, namely, a Γ1 simple wave with a Γ2 simple wave traveling ahead, as shown in Figure 10a. The two simple waves are transmitted through the interaction, which consists of a mesh of C1 and C2 characteristics occupying the region G-H-I-J in the physical plane, and they switch their relative position after it. The slopes of the characteristics in the simple waves before and after the interaction are reported in Table 2. First, let us determine the slope of the G-H and G-J curved paths, which correspond to a C2 and a C1 characteristic, respectively. Point G in the physical plane has coordinates given by eqs 29 and 30 and maps onto state P associated with the characteristic parameters (ωR1 , ωL2 ). Points

H and J map onto states L and R, respectively, as shown in Figure 10c. Therefore, the C2 (C1) characteristic corresponding to path G-H (path G-J) can be given a one-parameter representation in terms of ω1 (of ω2), where its local slope is given by the expression σ2(ω1) ) 1 + Fω1(ω2L)2 (the expression σ1(ω2) ) 1 + F(ω1R)2ω2). On the basis of considerations similar to those that led to eqs 36 and 31, the following equation for paths G-H (with j ) 2) and G-J (where j ) 1) is obtained:

∫x

x G

dx ) x

∫

ω3-j

P ω3-j

dσ3-j dω3-j dω3-j {σj(ω3-j) - σ3-j(ω3-j)}

(50)

The integrands on the right-hand side of the equation for paths G-H and G-J are 2(ω2L - ω1)-1 and 2(ω1R - ω2)-1, respectively. Solving the integral and using the equation for the characteristics impinging on the path from the opposite side of the G-H-I-J interaction region (see Figure 10b) yields the following parametric and explicit equations for the path G-H:

(

x(ω1, ωL2 ) ) xG

ωL2 - ωR1 ωL2 - ω1

)

2

(ωR1 e ω1 e ωL1 )

(51)

364



τ(ω1, ωL2 )

) τ0 + xG(1 +

Fω21ωL2 )

(

ωL2 - ωR1 ωL2 - ω1

√τ - τ0 - x ) √Fω2 (ω2 √x - (ω2 L

L

L

)

2

- ωR1 )√xG)

(52)

(53)

and for the path G-J:

(

x(ωR1 , ω2) ) xG

ωR1 - ωL2 ωR1 - ω2

)

characteristics are intersected, then the C2 (C1) characteristics can be given a one-parameter representation in terms of ω1 (ω2). Therefore, eqs 19 and 18 for the slopes of the C2 and C1 characteristics in the interaction zone can be recast in the following differential form: ∂τ ∂x ) (1 + Fω1ω22) ∂ω1 ∂ω1

(57)

∂τ ∂x ) (1 + Fω21ω2) ∂ω2 ∂ω2

(58)

2

(ωR2 e ω2 e ωL2 )

(

τ(ωR1 , ω2) ) xG(1 + FωR1 ω22)

ωR1 - ωL2 ωR1

- ω2

)

(54)

2

√τ - x ) √FωR1 (ωR1 √x + (ωL2 - ωR1 )√xG)

(55)

(56)

Let us now consider the G-H-I-J interaction region, where the C1 characteristics intersect the C2 characteristics, as shown in Figure 10b. Since ω2 (ω1) remains constant along each C2 (C1) characteristic, while ω1 (ω2) changes as the C1 (C2)

Differentiating the first and the second equation with respect to ω2 and ω1, respectively, and subtracting 1 from the other yields, the partial differential equation in the unknown x is as follows:

(

)

∂2x ∂x 2 ∂x + )0 ∂ω1∂ω2 ω2 - ω1 ∂ω1 ∂ω2

(59)

This equation shall be integrated using eqs 51 and 54 as boundary conditions. It can easily be verified that the solution


365

Figure 10. Interaction of waves from different families, type DF4. (a) Physical plane. (b) Zoomed view of the physical plane. (c) Characteristic plane. (d) Profile of ω1 (continuous line) and ω2 (dotted line) along x for different values of τ.

is given by the following relationship (the reader can find the detailed derivation elsewhere2):

(

ωL2 - ωR1 x(ω1, ω2) ) xG ω2 - ω1

){ 2

1-

2(ω2 - ωL2 )(ω1 - ωR1 ) (ωL2 - ωR1 )(ω2 - ω1)

(ωR1 e ω1 e ωL1 ;

ωR2 e ω2 e ωL2 )

} (60)

Then, τ(ω1,ω2) can be calculated by integrating either eq 57 or eq 58. We integrate the first in parts using the boundary condition eq 55, thus obtaining, after some straightforward though lengthy algebra

R τ(ω1, ω2) ) τ(ω1 , ω2) +

∫

ω1

ω1R

(1 + Fωω22)

2 2 ) (1 + Fω1ω2) x(ω1, ω2) - Fω2

) (1 + Fω1ω22) x(ω1, ω2) +

( ) ∂x ∂ω1

∫

ω

ω1R

(ω1 -

dω

(ω,ω2)

x(ω, ω2) dω

ωR1 )(ω1

- ωL2 )ω22τ0

ωR1 (ω2 - ω1)2ωL2

(61)

The latter expression is equivalent to the following one, which is obtained by integrating eq 58 with the boundary conditions given by eq 52:

366


Figure 11. Interaction of waves from different families, type M2a. (a) Physical plane. (b) Characteristic plane. (c) Profile of ω1 (continuous line) and ω2 (dotted line) along x for different values of τ.

τ(ω1, ω2) ) τ0 + (1 + Fω21ω2) x(ω1, ω2) (ωL2 - ω2)(ωR1 - ω2)ω21τ0 ωR1 (ω2 - ω1)2ωL2

(62)

The coordinates of the vertices G, H, I, and J of the interaction zone are calculated by substituting the corresponding values of the characteristic parameters in eqs 60 and 61 (or 62). The coordinates of point I for instance are obtained by letting ω1 ) ω1L and ω2 ) ω2R, and it is easy to see that they are both finite. With reference to Figure 10b and c, the C2 and C1 characteristics emanating from the interaction zone are straight lines starting at any point along the curved paths H-I (parametrized by ω2) and J-I (parametrized by ω1), respectively, and having the slopes reported in columns 9 and 10 of Table 2. The evolution of the interaction is illustrated in Figure 10d, where snapshots of the composition profiles along the column in terms of ω values are plotted at six different times. It is worth noting that through the simple waves before and after the interaction only one ω value changes at a time. On the contrary, through the wave interaction zone, namely, where τG < τ < τI, e.g., at τ ) 1.6, 2.0, and 3.0 in the figure, both ω values change continuously at the same time. 3.3. Mixed Langmuir Isotherm M2: Interactions Leading to Nonclassical Waves. The mixed Langmuir isotherm M2 is the only one where the ranges of possible values of ω1 and ω2

overlap; in fact, they coincide with eq 11. Therefore, this is the only isotherm where the ω2 value associated with a given state can be smaller than the ω1 value defined by another given state. This has very important consequences, including the occurrence of nonclassical solutions of certain Riemann problems, as discussed in detail elsewhere.13 The δ shock is a traveling spike of growing strength superimposed on the discontinuity separating the initial and feed states, and it occurs when ω2A < ω1B, where the same notation used in section 2.3 is adopted. In turn, the second nonclassical solution, i.e., the continuous nonsimple wave transition, occurs when ω2B < ω1A. Considering eqs 8-11, it is clear that these two conditions cannot be fulfilled, except in the case of the mixed Langmuir isotherm M2. It is also worth noting that in this case the region of the characteristic plane that maps onto physically meaningful composition states is a right triangle with the hypotenuse given by a portion of the diagonal, as defined by eq 11, and not a rectangle as in the other three cases. Let us consider the implications of these properties on wave interaction. With reference to Table 2, particularly column 2, it can readily be observed that cases DF0a, DF0b, DF0c, DF1, DF2, and DF3 can occur also for the mixed Langmuir isotherm M2, and they lead to the same interactions discussed earlier. Cases DF0d and DF4 can also occur, but they lead to the same interactions as before if and only if ω2L > ω1R in the former case and ω2R > ω1L in the latter. Note that these two inequalities are always fulfilled in the case of the other three generalized Langmuir isotherms. However, in the case of the M2 isotherm, the opposite inequalities


367

Figure 12. Interaction of waves from different families, type M2b. (a) Physical plane. (b) Physical plane for large x. (c) Characteristic plane. (d) Profile of ω1 (continuous line) and ω2 (dotted line) along x for different values of τ.

might also be fulfilled, thus leading to two new cases, called M2a and M2b in Table 2 and in the following. Two Shocks Leading to a δ Shock. Case M2a is illustrated in Figure 11 and is characterized by a left transition that is a Σ2 shock and a right transition that is a Σ1 shock, where the inequality ω2L < ω1R is fulfilled. Contrary to the similar case DF0d where no interaction occurs, here the two shocks collide at point G, as is obvious when applying the above-mentioned condition to the slopes of the shocks reported in columns 5 and 6 of Table 2. Thus, a situation similar to the one observed in case DF1 occurs, i.e., that of an interaction between

shocks of different families. A new Riemann problem has to be considered beyond point G, but the inequality ω2L < ω1R is the very condition for the occurrence of a δ shock, having the slope given in column 9 of Table 2 (a detailed derivation of the expression of the δ shock’s rate of propagation is reported elsewhere13). It is worth noting in Figure 11c at τ ) 8.0 that across the δ shock, both ω values change discontinuously at the same time. Two Simple Waves Yielding an Unbounded Interaction Zone. Case M2b is illustrated in Figure 12. Like case DF4, it is characterized by a left transition that is a Γ1 simple wave and a

368


Table 3. Chromatographic Cycle: Sequence of Types of Wave Interactions (see Tables 1 and 2) for Chromatographic Cycles of Mixtures Subject to Different Generalized Langmuir Isotherms next interactions sequence of transitions (see eq 66)

generalized langmuir isotherm mixed M2 anti-Langmuir Langmuir mixed M1

Γ2 Σ2 Γ2 Σ2

Σ1 Σ1 Γ1 Γ1

Σ2 Γ2 Σ2 Γ2

right transition, which is a Γ2 simple wave. Contrary to case DF4, the inequality ω2R < ω1L is fulfilled. As a consequence, the relative position of the L, P, and R states in the characteristic plane is the one shown in Figure 12c, and it is clear that state Q with coordinates (ω1L, ω2R) cannot be defined. When the similarity is exploited with case DF4, it is easy to see that the derivation leading to eqs 50-55 applies unchanged, thus allowing for the calculation of paths G-H and G-J, which are shown in Figure 12b. Also, eqs 57 and 58 for the one-parameter representation of the C2 and C1 characteristics through the interaction zone still apply. However, looking at Figure 12c, one notices that the following inequalities for the feasible range of parameter values must be fulfilled: ω1R e ω1 e min(ωL1 , ω2) in the case of the C2 characteristics and max(ω1, ω2R) e ω2 e ω2L for the C1 characteristics. Therefore, these inequalities apply also to the solution of eqs 57 and 58, which is given by eq 60 for x and by eq 61 for τ. By applying these inequalities to eq 60, it is readily observed that the term (ω2 - ω1) in the denominator becomes infinitely large as the representative point in the characteristic plane approaches the diagonal (see Figure 12c). Therefore, the interaction zone is unbounded and extends to infinity. From a dynamic point of view, this implies that the interaction proceeds for very long times and at very large distances from the origin, as shown in Figure 12d. While the solution of a Riemann problem, where state L is assigned as a feed state and state R as an initial state, i.e., states A and B in the notation of section 2.3, is a continuous nonsimple wave, as discussed elsewhere,13 the wave interaction approaches such a solution only asymptotically. Note that the continuous nonsimple wave consists of three transitions that are positioned one after the other in the physical plane and have their image in the characteristic plane along the portion of the Γ2 characteristic between point L and the diagonal, along the diagonal, and along the portion of the Γ1 characteristic between the diagonal and point R. Figure 12c shows that such a path is indeed approached asymptotically by the images of the snapshot of the solution taken at increasingly large times. 4. Chromatographic Cycle The chromatographic cycle is the most common mode of operation of a chromatographic column. It is defined as the situation where a finite amount of a mixture, i.e., a binary mixture in our case, is injected into a fully regenerated column, i.e., saturated with the mobile phase only, and eluted by injecting immediately thereafter the mobile phase itself. In mathematical terms, this corresponds to a situation where in a column initially saturated with the mobile phase only, i.e., where c1 ) c2 ) 0, a finite amount of a feed mixture of composition c1 ) c1F and c2 ) c2F is injected between τ ) 0 and τ ) τ0, followed by a pure mobile phase, which is introduced to regenerate the column and to bring it back to its initial state, i.e., c1 ) c2 ) 0. This corresponds to the following piecewise constant initial conditions:

first interaction

left inter.

right inter.

Figure

DF1 DF2 DF3 DF4

SF3 SF2 SF3 SF2

SF2 SF2 SF3 SF3

13 14 15 16

Γ1 Γ1 Σ1 Σ1

at τ ) 0,

0 e x e 1:

at x ) 0,

0 < τ < τ 0:

at x ) 0,

τ g τ0:

c1 ) 0, c1 ) cF1 ,

c1 ) 0,

c2 ) 0 (state O, (H1, H2))

(63)

c2 ) cF2 (state F, (ωF1 , ωF2 )) (64)

c2 ) 0 (state O, (H1, H2))

(65)

where also the ω values associated with the feed state F and the pure mobile phase, state O, have been indicated. Therefore, the chromatographic cycle is defined similarly to the wave interaction problems studied in section 3, where both states A and B are assigned the pure mobile phase state O, whereas state P in section 3 corresponds to feed state F. Therefore, the solution of the problem before any wave interaction occurs can be represented as follows: Γ2

Γ1

Γ2

state O (H1, H2) f state P (H1, ω2F) f state F (ω1F, ω2F) f Γ1

state Q (ω1F, H2) f state O (H1, H2)

(66)

where the intermediate states have been labeled as in earlier literature,2 but differently than in the first part of this series.1 Note that such a solution is the whole solution when no wave interaction occurs, i.e., for large ratios between injected volume and column volume, or in other words between the duration of the pulse injection τ0 and residence time in the column (see the first part of this series1). In this work, we consider smaller values of such ratios, i.e., conditions that lead to wave interaction and allow for baseline separation of the two components of the feed mixture. According to eq 4, state O corresponds to the pair (H1, H2) independently of the adsorption isotherm, whereas the ω values corresponding to the feed state F depend on it. The positions of the states O and F, the latter for a specific feed composition, in the characteristic plane for the four generalized Langmuir isotherms are shown in Figure 1. While the chromatographic cycle is defined by the same initial conditions, above, for all four generalized Langmuir isotherms, this figure shows clearly that the relative position in the characteristic plane of the two states that define the chromatographic cycle, i.e., O and P, and of the two intermediate states, i.e., P and Q, is completely different in the four cases. Since the relative position of the states either separated by a discontinuity in a Riemann problem or involved in a wave interaction determines the character of either the solution or the wave interaction, then the solution of the chromatographic cycle in terms of constant states and composition transitions and its evolution along the column are different for the different generalized Langmuir isotherms. More specifically, the four transitions indicated in eq 66 are either simple waves or shocks depending on the specific adsorption isotherm, as reported in Table 3. In the following the solution of the chromatographic cycle problem for the four generalized Langmuir isotherms is derived and discussed. For each case, the solution in the physical plane along with its images in the hodograph plane and in the


characteristic (ω1, ω2) plane are shown. Further, the axial profile at specific, significant times during the development of the chromatogram is also illustrated. Finally, the solution of the equilibrium theory is compared and shown to be fully consistent with numerical simulations obtained using the same equilibrium dispersive model utilized in part 1 of this series.1 The same model parameters as in section 3 and in part 1 have been used. The duration of the injection and the feed composition are different in the different cases, and they are reported in the captions of the corresponding figures. For illustration purposes, the value of x is allowed to be larger than 1, while in practice, it will be limited to a maximum value of 1.

369

4.1. Mixed Langmuir Isotherm M2. Let us consider the chromatographic cycle for the mixed Langmuir isotherm M2, which is illustrated in Figure 13. Observing the relative positions of states O, F, P, and Q in Figure 13c, and considering the general properties of the solution of Riemann problems discussed earlier in section 2.3, it is clear that the four transitions that belong to the solution before interaction are, from left to right, a simple wave, two shocks, and a simple wave, as reported in Table 3. It is worth pointing out that the relative positions of states O and F do not allow for the formation of nonclassical interactions, i.e., of types M2a

Figure 13. Chromatographic cycle for mixed M2 Langmuir isotherm with cF1 ) 1.3, cF2 ) 1.5, and τ0 ) 0.4. (a) Image of the solution on the physical plane; (b) image of the solution on the hodograph plane; (c) image of the solution on the ω2-ω1 plane; (d) development of the chromatogram, i.e., axial profile at different times as indicated (the dashed and solid lines correspond to components 1 and 2, respectively); (e) comparison of equilibrium theory calculations (solid lines) and numerical simulations (dashed lines) for elution profiles at two values of x.

370


and M2b to occur. With reference to the analysis in section 3, the two shocks, i.e., the Σ1 on the left and the Σ2 on the right connecting states P and Q through state F, yield the case DF1, which leads always to an interaction with transmission of the two shocks (see Table 2). The point where the two shocks collide can be calculated using eqs 29 and 30, and the two shocks emerging from this point connect states P on the left to Q on the right through state O, as is obvious when looking at the characteristic plane in Figure 13c. This has the important consequence that after this point, which is called point D, the more retained component with index 2 and the less retained component with index 1 are baseline separated. This is nicely illustrated by the concentration profiles shown in Figures 13d and (e) for τ g 1.07 and for x ) 1, respectively. There are two transitions belonging to family 2 on the left of state O emerging from the interaction, and two transitions of family 1 on the right of it. The former and the latter may yield an interaction of type SF3 and of type SF2, respectively. It is worth noting that in both cases the states on the left- and on the righthand side of the interaction coincide; they are in fact both state O, i.e., a pure mobile phase. As shown in Table 1, there are interactions in both cases, the ultimate slope of the characteristics after interaction being (1 + FH1H22) and (1 + FH21H2), respectively. Since the slope on the left is larger than the slope on the right, the region of state O opens up between the two zones occupied by the interaction. The points where the interactions of types SF3 and SF2 start, namely, points G and E, can be calculated with straightforward modifications of the equations provided in section 3, and so can the equations of the shock paths that cut through the simple waves after the interactions start (see Figure 13a). All of these equations are reported in detail in Table 4. Let us consider in particular the shock path of family 1 beyond point E in Figure 13a, in order to clarify an important property of the corresponding wave interaction. Its explicit equation is obtained from eq 39 with ω3-i ) H2, ω1R ) ω1L ) H1, and ω1P ) ω1F. The parametric equation for x in terms of ω1 is eq 37, where the parameter’s lower bound is H1 and x grows indefinitely as ω1 approaches H1. A similar analysis and the same conclusion can be drawn for the shock path of family 2 beyond point G. The fact that the two wave interactions SF3 and SF2 occurring during the chromatographic cycle go on indefinitely makes perfect sense from a physical point of view. This means in fact that there are two peaks in the chromatogram, i.e., one for each component, at any time in accordance with the principle of mass conservation that requires that whatever is injected does not disappear. 4.2. Anti-Langmuir and Langmuir Isotherms. The chromatographic cycle for mixtures subject to the anti-Langmuir and to the Langmuir isotherm is illustrated in Figures 14 and 15, respectively. On the basis of the relative position of states F and O in the characteristic plane, it is readily seen that the four transitions before the interaction are those reported in Table 3. As a consequence, the first interaction between the waves originated in the origin of the physical plane and, at point A (0,τ0), involves a simple wave and a shock, and it is a transmissive interaction of type DF2 for the anti-Langmuir isotherm and of type DF3 for the Langmuir isotherm. Following the theory presented in section 3, it is straightforward to determine the coordinates of point B, where the interaction starts; the equations of the shock path through the interaction zone; and the coordinates of point D, where the interaction is over and baseline separation is attained (see

Table 4. Mixed Langmuir Type M2 Isotherm: Coordinates and Equations of Key Points and Trajectories of the Two Shocks on the Physical Plane (see Figure 13) point D

xD )

τ0 FωF1 ωF2 (H2 - H1)

; τD )

point E

xE )

xG )

FωF1 ωF2 (H2 - H1)

(ωF2 - H1)τ0 FωF1 ωF2 (H1

- H2)(H1 - ωF1 )

τE )

point G

(1 + FωF1 ωF2 H2)τ0

(ωF2 - H1)(1 + F(ωF1 )2H2)τ0 FωF1 ωF2 (H1 - H2)(H1 - ωF1 )

(H2 - ωF1 )τ0 FωF1 ωF2 (H2 - H1)(H2 - ωF2 )

(

τG ) 1 +

;

;

(H2 - ωF1 )(1 + FH1(ωF2 )2) FωF1 ωF2 (H2 - H1)(H2 - ωF2 )

)

Σ1 shock AD

τ ) (1 + FH1ωF1 ωF2 )x + τ0

Σ1 shock DE

τ ) (1 + FH1H2ωF1 )(x - xD) + τD

Σ1 beyond E

√τ - x ) H1√FH2x - (H1 - ωF1 )√FH2xD

Σ2 shock OD

τ ) (1 + FH2ωF1 ωF2 )x

Σ2 shock DG

τ ) (1 + FH1H2ωF2 )(x - xD) + τD

Σ2 beyond G

√τ - x - τ0 ) H2√FH1x - (H2 - ω2)√FH1xD

Γ1

τ(ω1) ) (1 + FH2ω21)x

Γ2

τ(ω2) ) (1 + FH1ω22)x + τ0

τ0

F

Figures 14a and 15a). All of these equations are reported in Tables 5 and 6. State O emerging beyond point D of the incipient baseline separation is connected on the left to state P and on the right to state Q, through either a simple wave or a shock. Let us consider the shocks that separate O from Q to the right in Figure 14a (of family 1) and O from P to the left in Figure 15a (of family 2). The two shocks interact with a centered simple wave of the same family through an interaction of type SF2 and SF3, respectively. The two situations are identical to those encountered in the case of the mixed Langmuir isotherm M2; hence the governing equations reported in Table 5 for the Σ1 path beyond point E and in Table 6 for the Σ2 path beyond point G are the same as those in Table 4. The situation is more complex for the simple waves that separate O from P on the left in Figure 14a (of family 2) and O from Q on the right in Figure 15a (of family 1). Both interact with a shock of the same family, but none of them is centered, because they both originate from the wave interaction, and their characteristics start from different points along the shock path B-D. Therefore, the calculation of the Σ2 path beyond point G in Figure 14a and of the Σ1 path beyond point E in Figure 15a require a modification of the equations derived in section 3. Let us consider in detail the former interaction for the antiLangmuir isotherm, by first noting that the coordinates of points


371

Figure 14. Chromatographic cycle for the anti-Langmuir isotherm with cF1 ) 2.5, cF2 ) 3.5, and τ0 ) 0.2. (a) Image of the solution on the physical plane; (b) image of the solution on the hodograph plane; (c) image of the solution on the ω2-ω1 plane; (d) development of the chromatogram, i.e., axial profile at different times as indicated (the dashed and solid lines correspond to components 1 and 2, respectively); (e) comparison of equilibrium theory calculations (solid lines) and numerical simulations (dashed lines) for elution profiles at two values of x.

B and G are reported in Table 5. The coordinates of the points along the DF2 interaction path B-D are the following special cases of eqs 44 and 45 for the DF2 interaction: xBD(ω2) ) xB

(

ωF2 - H1 ω2 - H1

)

2

)

τ0(ωF2 - H1) FωF1 ωF2 (ω2 - H1)2

τBD(ω2) ) (1 + FωF1 ω22)xBD(ω2)

(67)

equation can be written in terms of the parameter ω2 as τ ) σ2(ω2)x + τBD(ω2) - σ2(ω2)xBD(ω2) ) σ2x + Fω22(ωF1 - H1)xBD ) σ2x + Fq(ω2)

(69)

(68)

Since the characteristics of the simple wave involved in the interaction start from the shock path B-D, their

where the relevant expressions for the slopes of the characteristics have been used and the last equality serves as a definition of the new auxiliary function q(ω2). In the study

372


Figure 15. Chromatographic cycle for competitive Langmuir isotherm with cF1 ) 3.5, cF2 ) 2.5, and τ0 ) 0.2. (a) Image of the solution on the physical plane; (b) image of the solution on the hodograph plane; (c) image of the solution on the ω2-ω1 plane; (d) development of the chromatogram, i.e., axial profile at different times as indicated (the dashed and solid lines correspond to components 1 and 2, respectively); (e) comparison of equilibrium theory calculations (solid lines) and numerical simulations (dashed lines) for elution profiles at two values of x.

of the wave interaction of type SF2, the last equation replaces eq 32; accordingly, eq 34 can be recast as dσ2 dτ dx dq ) σ2 +x +F dω2 dω2 dω2 dω2

(70)

The equation of the shock path through the interaction beyond point G is obtained by eliminating the derivative of τ between the last equation and eq 33, thus obtaining an ordinary

differential equation in x that can be integrated starting from point G. One obtains first (σ2 - σ˜ 2)

dσ2 dx dq +x ) -F dω2 dω2 dω2

(71)

Noting that σ2 ) 1 + FH1ω22 and σ˜ 2 ) 1 + FH1H2ω2, the equation above can be made exact by multiplying throughout by (ω2 - H2)/ (FH1ω2), thus obtaining


373

Table 5. Anti-Langmuir Isotherm: Coordinates and Equations of Key Points and Trajectories of the Two Shocks on the Physical Plane (See Figure 14) τ0

xB )

point D

xD )

point G

xG )

point E

xE )

Σ1 shock AB

τ ) (1 + FωF1 ωF2 H1)x + τ0

Σ1 shock BD

√τ - x ) H1√FωF1 x + (ωF2 - H1)√FωF1 xB

Σ1 shock DE

τ ) (1 + FωF1 H1H2)(x - xD) + τD

Σ1 beyond E

√τ - x ) H1√FH2x - (H1 - ωF1 )√FH2xE

Σ2 shock AG

τ ) (1 + FH1H2ωF2 )x + τ0

Σ2 beyond G

x(ω2) )

FωF1 ωF2 (ωF2

- H1 )

;

τB )

(1 + FωF1 H1ωF2 )τ0

point B

(ωF2 - H1)τ0

FωF1 ωF2 (ωF2 - H1)

τD )

; FωF1 ωF2 (H1 - H2)2

(ωF2 - H1)(1 + FωF1 H22)τ0 FωF1 ωF2 (H2 - H1)2

(ωF2 - ωF1 )τ0 FωF1 ωF2 (H1

-

-

ωF2 )(H2

(ωF2 - H1)τ0 FωF1 ωF2 (H2 - H1)(ωF1 - H1)

τ0 FH1ωF1 ωF2 (ω2 - H2)2 τ(ω2) )

[

;

τG ) 1 +

;

τE )

ωF2 )

(ωF2 - ωF1 )(1 + FH1H2ωF2 ) FωF1 ωF2 (H1 - ωF2 )(H2 - ωF2 )

]

τ0

(ωF2 - H1)[1 + F(ωF1 )2H2]τ0 FωF1 ωF2 (H2 - H1)(ωF1 - H1)

{

ωF1 (ωF2 - H2) -

(H1 - ωF1 )(H1 - ωF2 )ω22τ0 ωF1 ωF2 (H1 - ω2)2

(ωF1 - H1){ω2(ω2 - H2)(ωF2 - H1) + H2(ω2 - H1)(ωF2 - ω2)} (ω2 - H1)2

}

;

+ (1 + FH1ω22)x(ω2)

τ(ω1) ) (1 + FH2ω21)x

Γ1

Γ2 before interaction with Σ1 τ(ω2) ) (1 + FωF1 ω22)x Γ2 after interaction with Σ1

τ(ω2) ) (1 + FH1ω22)x + F(ωF1 - H1)ω22xB

(

H1 - ωF2 H1 - ω2

)

2

d((ω2 - H2)2x) H2 - ω2 dq ) dω2 H1ω2 dω2

(72)

This equation can be easily integrated by parts from the initial point x ) xG and ω2 ) ω2F, thus obtaining through cumbersome but straightforward algebra x(ω2) )

τ0 FH1ωF1 ωF2 (ω2 - H2)2

{

ωF1 (ωF2 - H2) -


}

(73)

Finally, τ is obtained by substituting the last equation into eq 69. In the case of the Langmuir isotherm, i.e., the classical adsorption isotherm for which these results had already been obtained earlier,2 a similar analysis can be carried out, thus leading to the following equation for the axial coordinate along the SF3 interaction shock path beyond point E in Figure 15a: x(ω1) )

τ0 FH2ωF1 ωF2 (ω1 - H1)2

{

ωF2 (H1 - ωF1 ) -

(H2 - ωF2 ){ω1(H1 - ω1)(H2 - ωF1 ) + H1(H2 - ω1)(ω1 - ωF1 )} (H2 - ω1)2

}

(74)

The time coordinate of the interaction shock path beyond point E is calculated by substituting the last equation into the equation for the characteristics in the Γ1 simple wave emerging from the DF3 interaction path B-D (see Figure 15a):

374


Table 6. Langmuir Isotherm: Coordinates and Equations of Key Points and Trajectories of the Two Shocks on the Physical Plane (See Figure 15) τ0

xB )

point D

xD )

point G

xG )

point E

xE )

Σ2 shock OB

τ ) (1 + FωF1 ωF2 H2)x

Σ2 shock BD

√τ - x - τ0 ) H2√Fω2x - (H2 - ω1)√Fω2xB

Σ2 shock DG

τ ) (1 + FH1H2ωF2 )(x - xD) + τD

Σ1 shock OE

τ ) (1 + FH1ωF1 H2)x

Σ2 beyond G

√τ - x ) H2√FH1x - (H2 - ωF2 )√FH1xG

Σ1 beyond E

x(ω1) )

FωF1 ωF2 (H2

-

ωF1 )

;

(H2 - ωF1 )τ0 FωF1 ωF2 (H2

- H 1)

τB )

(1 + FωF1 ωF2 H2)τ0

point B

2

;

τD )

FωF1 ωF2 (H2 - ωF1 )

{

(H2 - ωF1 )τ0 FωF1 ωF2 (H2

- H1)(H2 -

ωF2 )

(ωF2 - ωF1 )τ0 FωF1 ωF2 (H2 - ωF1 )(H1 - ωF1 )

1+

;

τG )

;

τE )

F

- H 1)

FωF1 ωF2 (H2 - H1)2

{

1+

2

{

F 2 Γ1 before interaction with Σ2 τ(ω1) ) (1 + Fω2 ω1)x + τ0

Γ1 after interaction with Σ2

τ(ω1) ) (1 + FH2ω21)x + F(ωF2 - H2)ω21xB

Γ2

τ(ω2) ) (1 + FH1ω22)x + τ0

(75)

4.3. Mixed Langmuir Isotherm M1. The chromatographic cycle for the mixed Langmuir isotherm M1 is finally illustrated in Figure 16, while Table 7 provides the coordinates of all of the important points defined by the wave interaction in the physical plane and the equations of all of the relevant lines. The relative position of states F and O in the characteristic plane of Figure 16c shows that two simple waves of different families are involved in the first interaction, which is of type DF4 (see Table 3). With reference to Figure 16a, the wave interaction occupies the region BFDH where for each pair (ω1, ω2) the coordinates of the corresponding point in the physical plane are given by eqs 60 and 61 (or 62), where ω1R and ω2L are substituted by ω1F and ω2F, respectively. The two simple waves are transmitted through the interaction zone. Both emerging simple waves collide with the two shocks of the same family originating at point A and at the origin, thus giving rise to two new

(H2 - ωF1 )[1 + FH1(ωF2 )2] FωF1 ωF2 (H2 - H1)(H2 - ωF2 )

}

τ0

F

τ(ω1) ) τ0 + (1 + Fω21H2)x(ω1) +

ωF1 ωF2 (H2 - ω1)2

τ0

FωF1 ωF2 (H2 - ωF1 )(H1 - ωF1 )

ωF2 (H1 - ωF1 ) -

τ(ω1) ) τ0 + (1 + Fω21H2)x(ω1) + τ0ω21(ωF2 - H2)(H2 - ωF1 )

}

(ωF2 - ωF1 )(1 + FH1H2ωF1 )τ0

F

τ0 FH2ωF1 ωF2 (ω1

(H2 - ωF1 )[1 + F(H1)2ωF2 ]

(


}

;

τ0ω21(ωF2 - H2)(H2 - ωF1 ) ωF1 ωF2 (H2 - ω1)2

H2 - ωF1 H2 - ω1

)

2

+ τ0

interactions of types SF2 and SF3, which are located above and on the right-hand side of the region BFDH and start at points G and E, respectively. Note that also in this case, point D in the physical plane of Figure 16a is where baseline separation is attained. The SF2 wave interaction involves characteristics that originate from the F-D border of the region BFDH, where ω1 ) H1 and ω2 is the independent parameter. Noticing that along the straight characteristic F-G ω2 ) ωF2, it is easy to show that the coordinates of point G are the same as for the same point in the case of the anti-Langmuir isotherm (compare Tables 5 and 7). In fact, all characteristics in the Γ2 simple wave emerging from line F-D are defined by the same eq 69 as in the anti-Langmuir case. Since also the shock Σ2 originating at point A has the same equation, it turns out that the equations of the shock path beyond point G are exactly the same as in the anti-Langmuir case, i.e., eqs 69 and 73. This is a rather remarkable result, which applies also to the SF3 wave interaction between the simple wave emerging from the H-D border of the region BFDH and Σ1 stemming from the origin of the physical plane. The coordinates of point E where the interaction starts as well as the equations of the characteristics constituting


375

Figure 16. Chromatographic cycle for mixed M1 Langmuir isotherm with cF1 ) 2.5, cF2 ) 3.5, and τ0 ) 0.2. (a) Image of the solution on the physical plane; (b) image of the solution on the hodograph plane; (c) image of the solution on the ω2-ω1 plane; (d) development of the chromatogram, i.e., axial profile at different times as indicated (the dashed and solid lines correspond to components 1 and 2 respectively); (e) comparison of equilibrium theory calculations (solid lines) and numerical simulations (dashed lines) for elution profiles at two values of x.

the simple wave are the same as in the case of the Langmuir isotherm, where the same situation is encountered. As a consequence, the equation of the shock path beyond point E is also the same and is given by eqs 74 and 75. 5. Concluding Remarks The generalized binary Langmuir isotherm combined with the analysis of the solutions of the equilibrium theory model using the method of characteristics provides a powerful tool to study chromatographic systems of increasingly complex be-

havior with the simple tools that have been familiar in the case of the binary Langmuir isotherm for several decades. The property that characteristics in the hodograph plane are straight lines not only for the Langmuir isotherms but also for the other three isotherms of the generalized Langmuir family was demonstrated in the first part of this series, where also the Riemann problem and the saturation and regeneration problems were thoroughly solved.1 In this paper, we have studied the chromatographic cycle for the four isotherms, using the preliminary study of wave interaction as a starting

376


Table 7. Mixed Langmuir Type M1 Isotherm: Coordinates and Equations of Key Points and Trajectories of the Two Shocks on the Physical Plane (See Figure 16) point B

xB )

point H

xH )

point F

xF )

point D

xD )

τ0 FωF1 ωF2 (ωF2

-

ωF1 )

τB )

;

(ωF2 - ωF1 )τ0

FωF1 ωF2 (ωF2

- H 1)

2

FωF1 ωF2 (ωF2 - ωF1 ) (ωF2 - ωF1 )(1 + FωF1 H22)τ0

τH )

; FωF1 ωF2 (H2 - ωF1 )2 (ωF2 - ωF1 )τ0

[1 + FωF1 (ωF2 )2]τ0

τF )

;

FωF1 ωF2 (H2 - ωF1 )2 (ωF2 (1 + FH21ωF2 ) + ωF1 (F(ωF2 )3 - 2FH1(ωF2 )2 - 1))τ0 FωF1 ωF2 (H1 - ωF2 )2

(H1(2H2 - ωF1 - ωF2 ) + 2ωF1 ωF2 - H2(ωF1 + ωF2 ))τ0 FωF1 ωF2 (H1 - H2)3

τD )

[(H31 + (H2 - 2(ωF1 + ωF2 ))H21 + 3ωF1 ωF2 H1 - H2ωF1 ωF2 )FH22 + 2(H1H2 + ωF1 ωF2 ) - (H1 + H2)(ωF1 + ωF2 )]τ0 FωF1 ωF2 (H1 - H2)3 (ωF2 - ωF1 )τ0

point G

xG )

point E

xE )

Σ1 shock OE

τ ) (1 + FωF1 H1H2)x + τ0

Σ2 shock AG

τ ) (1 + FH1H2ωF2 )x

Σ1 beyond E

x(ω1) )

FωF1 ωF2 (H1 - ωF2 )(H2 - ωF2 ) (ωF2 - ωF1 )τ0 FωF1 ωF2 (H1

-

-

ωF1 )(H2

τ0 FH2ωF1 ωF2 (ω1

- H 1)

2

ωF1 )

;

τG )

;

τE )

(ωF2 - ωF1 )(1 + FH1H2ωF2 )τ0 FωF1 ωF2 (H1 - ωF2 )(H2 - ωF2 )

x(ω2) )

τ0 FH1ωF1 ωF2 (ω2 τ(ω2) )

mesh BFDH

- H 2)

2

FωF1 ωF2 (H1 - ωF1 )(H2 - ωF1 )

{

ωF2 (H1 - ωF1 ) -

{

ωF1 (ωF2 - H2) -

(H1 - ωF1 )(H1 - ωF2 )ω22τ0 ωF1 ωF2 (H1 - ω2)2

xM(ω1, ω2) ) xB

(

ωF2 - ωF1 ω2 - ω1

)[ 2

1-2


}

;

τ0ω21(ωF2 - H2)(H2 - ωF1 ) ωF1 ωF2 (H2 - ω1)2


}

;

+ (1 + FH1ω22) x(ω2)

(ω2 - ωF2 )(ω1 - ωF1 ) (ωF2 - ωF1 )(ω2 - ω1)

]

τM(ω1, ω2) ) τ0 + (1 + Fω21ω2)x(ω1, ω2) -

Γ1 before interaction with Γ2

τ(ω1) ) (1 + FωF2 ω21)x + τ0

Γ1 after interaction with Γ2

τ(ω1) ) (1 + FH2ω21)(x - xM(ω1, H2)) + τM(ω1, H2)

Γ2 before interaction with Γ1

τ(ω2) ) (1 + FωF1 ω22)x

Γ2 after interaction with Γ1

τ(ω2) ) (1 + FH1ω22)(x - xM(H1, ω2)) + τM(H1, ω2)

point. The solution of the chromatographic cycle was available for the Langmuir isotherm,2 and we have shown the differences and similarities in the case of the three other isotherms. In all cases, wave interactions and chromatographic cycles admit solutions that can be formulated as explicit equations describing characteristics and interaction

+ τ0

(ωF2 - ωF1 )(1 + FH1ωF1 H2)τ0

τ(ω1) ) τ0 + (1 + Fω21H2)x(ω1) +

Σ2 beyond G

;

(ωF2 - ω2)(ωF1 - ω2)ω21τ0 ωF1 (ω2 - ω1)2ωF2

lines or zones in the physical plane, in terms of adsorption isotherm parameters and compositions. We believe that the comprehensive set of results and relationships provided in this paper are going to be useful for the practitioner. Generalized Langmuir isotherms different from the classical Langmuir isotherm are starting to be used


in our laboratory and in others to provide a rather simple description of systems where at least one species exhibits an anti-Langmuir behavior. While explicit criteria for the design of simulated moving bed operations were already developed and reported,15 the results presented in this paper allow for a rather straightforward prediction of the behavior of single chromatographic column elution of an injected pulse, i.e., the most universal operation in preparative chromatography. Such a prediction is extremely accurate, as demonstrated by the comparisons between equilibrium theory results and detailed simulations presented in Figures 13e-16e. The deeper understanding of the behavior of competitive/ co-operative systems that has been achieved by studying the general Langmuir isotherms has allowed for the prediction of, theoretically, the existence of new types of composition fronts, particularly the δ shock,13 and for the discovery of their experimental occurrence in the laboratory.14 These striking findings provide motivation to study such systems both theoretically and experimentally for shear scientific interest or for potential applicability. We believe that the theory and the results reported in this paper will be of tremendous help in this endeavor, as the corresponding equations for the binary Langmuir isotherm were in the past and still are. Appendix Notation ci ) fluid phase concentration of component i Cj ) jth characteristics in the physical plane d ) denominator in the generalized Langmuir isotherm, defined by eq 3 Hi ) Henry’s constant of component i Ki ) adsorption equilibrium constant of component i L ) length of the adsorption column ni ) adsorbed phase concentration of component i Ni ) adsorbed phase saturation concentration of component i pi ) parameter in the generalized Langmuir isotherm, pi ) (1 t ) time V ) interstitial velocity x ) dimensionless axial coordinate, x ) z/L z ) axial coordinate Greek Letters Γj ) jth characteristics in the hodograph plane

377

ε ) overall void fraction ν ) external phase ratio, ν ) (1 - ε)/ε F ) parameter defined as ν/(H1H2) σj ) slope of a Cj simple wave in the physical plane σ˜ j ) slope of a Sj shock in the physical plane Σj ) image of a Sj shock in the hodograph plane τ ) dimensionless time, τ ) tV/L ω ) characteristic parameter defined by eq 4 Subscripts and Superscripts F ) feed state i ) component index j ) eigenvalue and eigenvector index L ) state on the LHS of a simple wave or a shock transition R ) state on the RHS of a simple wave or a shock transition

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ReceiVed for reView July 23, 2010 ReVised manuscript receiVed October 17, 2010 Accepted October 20, 2010 IE1015798

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