Qualitative Analysis for Homogeneous Azeotropic Distillation. 1. Local

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Qualitative Analysis for Homogeneous Azeotropic Distillation. 1. Local Stability† Cornelius Dorn‡ and Manfred Morari* Automatic Control Laboratory, Swiss Federal Institute of Technology, ETHZ, CH-8092 Zu¨ rich, Switzerland

A novel approach is presented for the qualitative analysis of the dynamic behavior of homogeneous azeotropic distillation columns of finite length. The methodology can be used to study the dynamic behavior of steady-state column profiles after perturbations. The key concept is to study the interaction of changes in the total column holdup composition and the shape of the column profile. This allows one to qualitatively determine if the profiles are locally stable or unstable. It is shown that the stability of the steady states is dominantly governed by the external mass balance around the column. This is highlighted by studying the case of infinite internal flows where the internal dynamics of the column become infinitely fast while the external dynamics remain at a slow time scale. The analysis of the local stability of column profiles presented in this work will be extended to construct bifurcation diagrams in the accompanying paper (Dorn, C.; Morari, M. Ind. Eng. Chem. Res. 2002, 41, XXXX). 1. Introduction The steady-state and dynamic behavior of azeotropic distillation has been studied extensively over the past decades because their understanding is a necessary prerequisite for proper column design and operation. Laroche et al.2 have shown that azeotropic distillation columns can exhibit unusual features not observed in zeotropic distillation. In particular, multiplicity of steady states has been a subject of much recent research interest. An overview of steady-state multiplicity in distillation is given by Bekiaris et al.3 In this paper, the term “multiple steady states” (MSS) refers to output multiplicities only, i.e., that a system with as many inputs specified as there are degrees of freedom exhibits different solutions at steady state.4 Only multiplicities of type II5 will be considered. This type of multiplicity is caused by the vapor-liquid equilibrium (VLE) as well as the complex recycle structure between the stages. In connection with MSS, the instability of steady states in distillation was studied by Jacobsen and Skogestad,6 who rigorously showed that in the case of MSS of type I (caused by the nonlinear transformation between mass and molar flow rates) the equilibria on the middle branch are open-loop unstable. For MSS of type II in homogeneous azeotropic distillation, Dorn et al.7 demonstrated the instability of the equilibria on the middle branch by dynamic simulations and experiments. In addition to MSS, another well-known phenomenon which nonlinear systems can exhibit is sustained oscil† I (M.M.) met Jim Douglas first when I was a graduate student at the University of Minnesota. He visited there to present a seminar on process control in an interdisciplinary series organized by George Stephanopoulos. We had numerous interactions and discussions ever since including overlapping Sabbaticals at ICI in 1982 during a period of revolutionary developments in process synthesis. I always greatly appreciated Jim’s insights and suggestions. It is my pleasure and honor to dedicate this series of articles to Jim Douglas. * To whom correspondence should be addressed. Phone: +41 1 632-2271. Fax: +41 1 632-1211. E-mail: morari@ aut.ee.ethz.ch. ‡ Current affiliation: McKinsey & Company, Switzerland.

lations (limit cycles). An overview of the oscillatory behavior in distillation is given by Lee et al.8 To the authors’ knowledge, the first dynamic simulation results showing underdamped oscillation in heterogeneous distillation were published by Widagdo et al.9 Recently, Lee et al.8 published the first simulation results showing sustained oscillations (limit cycles) in the homogeneous azeotropic distillation of methanol-methyl butyratetoluene using a CMO (constant molar overflow) model. The limit cycles surround unstable steady states on the high branch between two Hopf bifurcation points. In a subsequent publication,10 the mixture acetone-benzeneheptane was studied. It was found that the column can have a much more complex bifurcation behavior than observed before. These different new types of complex dynamic behavior of distillation columns were reported, but no physical explanation or prediction was given. In this and the accompanying1 paper, new tools and methods are presented for a qualitative stability and bifurcation analysis for homogeneous azeotropic distillation columns. In section 2, the methodology underlying the following sections is introduced. It will be shown how the complexity of the analysis of the dynamic behavior of a distillation column can be reduced with simplifying assumptions. Building on this basis, concepts are presented in section 3 for a qualitative analysis of the local stability of steady-state column profiles. In the same way as the ∞/∞ analysis,4,11 the qualitative stability analysis is based on graphical arguments only. The key idea is to study the interaction of changes in the column inventory and changes of the shape of the column profile. This allows one to identify global mass balances as the primary source of instability and oscillations. The theory is presented for mixtures of the class 001 according to the classification by Matsuyama and Nishimura.12 This class can be regarded as the least complex case of a ternary homogeneous mixture forming azeotropes. An example for this class is the mixture acetone (L)benzene (I)-heptane (H). The only azeotrope is minimum boiling between acetone and heptane (93.56 mol % acetone). The residue curve map at p ) 1 bar is

10.1021/ie010725r CCC: $22.00 © 2002 American Chemical Society Published on Web 07/12/2002

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separated into a slow part and a fast part.16,17 The slow part is caused by changes in the external flow rates D and B, while the fast part results from changes in the internal flow rates L and V. Using a mixing-tank approach, Skogestad and Morari18 derived an expression for the dominant time constant τc of the slow dynamics after “large” perturbations that change the external mass balances and the component holdups:

τc ) “change in holdup of one component [kmol]” “imbalance in the supply of this component [kmol/h]” (1)

Figure 1. Residue curve diagram of the mixture acetone (L)benzene (I)-heptane (H) at p ) 1 bar.

illustrated in Figure 1. An extension of the methodology for a qualitative analysis of mixtures whose residue curve maps have more complex topologies will be published separately.13 To make the qualitative analysis in this paper more manageable, two main simplifications are made: (1) The impact of heat effects, volumetric holdups, etc., is ignored (CMO model). (2) Because the column will be studied in an input-outputoriented approach, most of the details of its complex inner structure are ignored. 2. Unsteady-State Column Profiles 2.1. Overview. Directionality and time-scale separation in distillation are well-known. Rosenbrock14 published two key findings about column behavior. He noted a basic difference in the steady-state effects of flow rates in high-purity distillation columns. While the product compositions are sensitive to changes in the external flows, they are comparatively insensitive to changes in the internal flows. In this context, “changes in the internal flows” refer to changes in the reflux and boilup flow rates such that the distillate and bottoms flow rates are not altered. Further, Rosenbrock distinguished two phases in the response of a distillation column to a step change in the inputs or disturbances. The first phase is dominantly governed by so-called “secondary effects” such as hydraulic delays, level control, or pressure drops. The second phase is “determined chiefly by mass transfer”. For a specific example column, Levy et al.15 performed a modal analysis of column models of different complexity. In all cases, they found the dominant (slowest) mode to correspond mainly to composition effects. This “composition mode” is nearly unaffected by flow dynamics, and assuming a constant molar overflow, the composition mode is completely decoupled from the flow dynamics. This led to the conjecture that the dominant part of the column dynamics can be captured by modeling only the composition dynamics. In the study of distillation column models of reduced detail (not including the effects that Rosenbrock called “secondary”14), it was found that the dynamics of a column in the phase governed by “mass transfer” are

Equation (1) does not apply to changes in the internal flows as both the numerator and denominator approach zero. The reason is the aforementioned fact that the steady-state effect of changes in the internal flows causes little change in the product composition and the component inventories because it mainly sharpens the profile. This indicates that the fast dynamics correspond to an internal redistribution of the component holdups following changes in the internal flow rates. Kumar and Daoutidis19 recently confirmed these results by performing a singular perturbation analysis of a distillation column model assuming constant molar overflow and constant relative volatilities. They found the fast time scale to be on the order of the residence times of the individual stages in the column. In the following, a more radical approach that does not attempt to derive any quantitative expressions will be pursued which, in the limiting case, includes all other work on time-scale separation and directionality of the phase “governed by mass transfer”. By starting with the equations defining a CMO model at finite reflux, it will be shown how the infinite reflux approach (extensively studied in the past for steady states) carries over to predict also the dynamic behavior of a column. 2.2. Finite Reflux. The CMO model of a column with n trays separating a single liquid feed consisting of nc components (without loss of generality) entering on tray kF will be revisited. Using the notation listed at the end of this paper, the equations are

Top tray including the total condenser: M1 d 1 V 2 D x ) yi - x1i - y1i L dt i L L

(2)

Rectifying section (k ) 2, ..., kF - 1): Mk d k V V x ) xk-1 + yk+1 - xki - yki i L dt i L i L

(3)

Feed tray: F

F Mk d k F V F L + F kF V kF F F xi ) xki -1 + yki +1 xi - yi + xi L dt L L L L (4)

Stripping section (k ) kF + 1, ..., n - 1): Mk d k L + F k-1 V k+1 L + F k V k x ) xi + yi xi - yi L dt i L L L L

(5)

Bottom tray including reboiler: Mn d n L + F n-1 V n B n x ) xi - yi - xi L dt i L L L (For all equations, i ) 1, ..., nc - 1.)

(6)

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It will later be required to assume an equal distribution of the total column holdup (TCH) along the column, i.e., M1 ) M2 ) ... ) Mn. When the TCH is defined as n Mk and the TCH of component i as xavg ∑k)1 i TCH, where the vector of the TCH composition (TCHC) is defined as n

xavg )

∑ xkMk k)1 n

Mk ∑ k)1 the global mass balance around the column (by summation of eqs 2-6) can be written as n

d dt

xavg

Mk ) FxF - DxD - BxB ∑ k)1

(7)

where xD ) y1 and xB ) xn. Note that the TCH of component i is proportional to the TCHC of component i because the holdup on all trays was assumed to be constant. 2.3. Infinite Reflux. The dynamic behavior of the distillation column at finite reflux is defined through eqs 2-6. Increasing the internal flow rates (reflux and boilup) reduces the influence of the external streams on the column profile. For large L, the terms for the feed flow (F/L)xF and for the products, (D/L)y1 and (B/ L)xn, drop out. Also, simplifications such as limLf∞ (V/ L) ) 1 and limLf∞ [(L+F)/L] ) 1 hold. Further, the time constants Mk/L f 0 as L f ∞. Hence, the equations describing the column profile reduce to

Top tray including total condenser: 0 ) y2i - x1i

(8)

Rectifying section (k ) 2, ..., kF - 1): 0 ) xk-1 + yk+1 - xki - yki i i

(9)

Feed tray: 0 ) xki

F-1

+ yki

F+1

F

- xki - yki

F

(10)

Stripping section (k ) kF + 1, ..., n - 1): 0 ) xk-1 + yk+1 - xki - yki i i

(11)

Bottom tray including reboiler: - yni 0 ) xn-1 i

(12)

Starting with eq 8, iteratively inserting eqs 8-12 (and some rearranging) yields

yk+1 ) xk

k ) 1, ..., n - 1

(13)

Equation 13 is the equation defining distillation lines.20 Thus, at infinite reflux the profile of the tray column defined above will coincide with a section of a distillation line at any time because steady state was not assumed for the derivation of eq 13. This can be interpreted in an interesting way: At high internal flows in the column, holdup between stages can be exchanged fast because the residence times (time constants) Mk/L of the

trays are small. At infinite reflux, the exchange of holdup between stages along the column occurs infinitely fast (time constants “zero”). Thus, if a column operated at infinite reflux is initially filled with holdups Mkxˆ k(0), k ) 1, ..., n, that do not correspond to a steadyn Mkxˆ k(0) will be redisstate profile, the initial TCH ∑k)1 tributed along the column infinitely fast such that the profile satisfies eq 13, thus coinciding with a part of a distillation line. (For some applications such as startup procedures for distillation columns, it would be interesting to examine whether more than one column profile can correspond to the same TCHC.) Because the coincidence of the column profile with a section of a distillation line holds at any time, it also holds at steady state, where this coincidence is well-known. By assuming infinite reflux, the terms for the external streams and for the derivatives were eliminated from eqs 2-6. However, by summation of all of eqs 2-6 before letting L f ∞, the global mass balance (eq 7) around the column is found which is unaffected by the assumption of infinite reflux. Hence, the TCHC will continuously change at a slow time scale as governed by eq 7 until the steady-state global mass balance is satisfied. For a given initial holdup and operating conditions, the column will generically not satisfy the steady-state global mass balance in the beginning. Thus, after the first infinitely fast transient, the column will undergo a second phase of transient with dynamics of finite speed. During this transient, the residual of the finite flows entering and leaving the column changes the TCHC continuously until a steady state is reached. During the transient, the column profile readjusts after every infinitesimal change of the TCHC such that it always coincides with a part of a distillation line. It seems justified to expect that the scenario described above carries over to the finite case of sufficiently high internal flows in the column as compared to the external flows. Then, there will be a time-scale separation in the transient with fast dynamics during which the column profile approaches a distillation line and with slow dynamics during which the whole profile adjusts such that finally a global mass balance around the column is satisfied and a steady state is reached. This agrees with the earlier findings on time-scale separation in distillation mentioned at the beginning of section 2.16,17 In this section, it is assumed that the column finally approaches a steady state during the slow part of the transient. This, however, implied that such a steady steady exists and is attracting. While the existence of at least one steady state for a homogeneous azeotropic distillation column is not questioned here, the attractor to which the column is converging is not necessarily a stable steady state but, depending on the situation, can also be a stable limit cycle. The concepts presented above can also be carried over to the case of a packed distillation column whose profile will coincide, under certain assumptions, with a section of a residue curve if the reflux is infinite. For convenience, however, but without loss of generality, residue curves will be used in the following to approximate the profiles of tray columns. This approximation is valid under additional assumptions21 and studied in numerous simulations, experiments, and applications.22,23 In the following, the time-scale separation is illustrated in a dynamic simulation. A distillation column separating the mixture acetone (L)-benzene (I)-heptane (H) is initially loaded with holdups of composition


xkL ) 0.8 -

0.65 (k - 1) 45

k ) 1, ..., 46

xkI ) 0.05 +

0.65 (k - 1) 45

k ) 1, ..., 46

xkH ) 0.15 k ) 1, ..., 46 This initial profile forms a straight line in the composition space and does not correspond to a steady state. Performing the dynamic simulation with the CMO model (compare Appendix A) for the operating conditions D/F ) 0.9, L/F ) 100, and (xFL, xFI , xFH) ) (0.75, 0.1, 0.15) yields the results illustrated in Figures 2-4. The time response of the TCHC is shown in Figure 2. A steady state is reached after about 50 h. Because the first phase of the transient corresponds to an internal redistribution of the holdup, it does not appear in the plot of the TCHC. Figure 2 can only illustrate the second phase of the transient. The first part of the transient lasts about 0.1 h. Corresponding profiles are shown in Figure 3. At t ) 0 h, the profile has the initial shape of a straight line. At t ) 0.08 h, the shape of the profile is very similar to one of the residue curves shown in the same subfigure. For t g 0.08 h, the profile will approximately coincide with a section of a residue curve. In the second part of the transient as illustrated in Figure 4, the TCHC changes until a steady-state profile is reached. In all subfigures, a straight line is drawn connecting the distillate and bottoms composition to illustrate that at the beginning the feed composition is not on that line. At t ) 5 h in Figure 4, the feed composition is on a straight line connecting the distillate and the bottoms composition. However, this is only a necessary but not a sufficient condition for the steadystate global mass balance to be fulfilled. For sufficiency, also

Figure 3. Column profiles at selected times during the first part of a transient. At t ) 0.08 h, residue curves are also shown (the x axis is the mole fraction of H, the y axis is the mole fraction of L, and the distillate and bottoms compositions are marked with O and ]).

B F D |x - x |2 ) D B |x - xF|

2

must hold. 2.4. Methodology. In the following sections, the qualitative assessment of the local dynamic behavior of homogeneous azeotropic distillation columns will be described. In this work, two approaches will be used in

Figure 4. Column profiles at selected times during the second part of a transient (the x axis is the mole fraction of H, the y axis is the mole fraction of L, the distillate, bottoms, and feed compositions are marked with O, ], and +, respectively.)

Figure 2. TCHC during a transient.

parallel: infinite reflux in columns of infinite and finite lengths. The first approach, the ∞/∞ analysis,4 is helpful for predicting feasible steady-state column profiles by exploiting two facts: (1) For infinite reflux, column profiles must coincide with sections of residue curves. (2) For infinite column length, the column profile must contain at least one pinch point in which the infinite separation power is consumed. Candidates for pinch points are singular points in the residue curve map, i.e., pure components and azeotropes. These observations significantly reduce the set of all feasible profiles such that it becomes possible to determine all of them graphically. Variation of a parameter, usually the distillate flow rate, within its interval of feasible values results in the continuation of all feasible product compositions (both distillate and bottoms) that are called the product paths. Assuming infinite column length, however, has serious drawbacks: the TCH is infinitely large. Hence, it

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Figure 5. Profiles in columns with an increasing and infinite number of trays, operated at infinite reflux.

is not meaningful to study the dynamics of changes of that holdup because the external streams have only finite flow rates. In addition, it is nearly impossible to focus on specific sections of the column or locations in the column profile. For infinite reflux in a column of finite length (compare section 2.3), the dynamic behavior of the column can be analyzed because the TCH as well as the flow rates of the external streams are finite entities. However, it is not obvious how to determine the feasible profiles because they are not restricted to contain a pinch point. One possibility is to study the case of infinite reflux in a column of finite but sufficiently large length. Then, the column profile coincides with a section of a residue curve, and it approximately contains a pinch point. The composition along the column is on none of the trays equal to any of the singular points of the residue curve map, but it comes close to at least one of them. In Figure 5, column profiles are sketched. A feed F is separated into a distillate D and a bottoms B where D ) B is assumed. At infinite reflux, the compositions of both products must be on the same residue curve. The length of the column (number of trays) determines with which residue curve the column profile coincides. The profile for the ∞/∞ case contains pure I as a pinch point. Assuming infinite reflux flow rate, some profiles of columns of finite length are illustrated in Figure 5. Because the relative volatilities approach 1 in the neighborhood of pure I, the number of trays necessary in the column increases for profiles closer to I. However, if the distillate and bottoms composition are not too close to pure I, they come very close to the boundaries for a sufficiently high but finite number of trays. This is a consequence of the structure of the residue curve map. In conclusion, the product compositions of columns of finite length are restricted in the same way as the product compositions in the ∞/∞ case even for moderate column lengths. In contrast, the number of trays required for the column profile to approach a pinch point increases rapidly. Differences between the product paths of the columns of finite and infinite lengths must be expected where the product paths come near or go through singular points. These differences will be highlighted where appropriate in the later sections and addressed in more detail in the following paper.1 In this and the accompanying paper, columns of finite length will be studied assuming infinite reflux unless stated otherwise. Hence, pinch points in the column profile will refer to pinch points in a column of finite length as illustrated in Figure 5. This is illustrated with another example that will repeatedly be referred to in

Figure 6. Pinch point at the end of a column profile (finite case): the bottoms composition (here tray 1) is (nearly) unaffected by a change in the profile.

the following sections. Consider a column profile ending at pure H as illustrated in Figure 6. In the ∞/∞ case, the bottoms composition will be exactly pure H. There will be an infinitely long section with the composition of pure H at the end of the column profile. In the finite case, a long (but finite) section of the column profile is nearly at the same (pinch) composition. If for some reason the shape of the column profile changes in a way that more and more trays move away from pure H, the bottoms composition will remain relatively unchanged as long as the change of the profile is not too big (see Figure 6b). 3. Qualitative Stability Analysis In this section, the dynamic behavior of a distillation column (of finite length operated at infinite reflux) after a perturbation will be analyzed. The key idea is to study how changes of the column inventory and changes of the shape of the column profile interact. After the basic principles of this approach are presented, the feasible column profiles will be categorized into different types. The classification of profiles extends the three types originally proposed by Bekiaris et al.4 Subsequently, each type of profile can be analyzed. For brevity, only two types of profiles will be analyzed in this paper. At steady state, column profiles must satisfy a global mass balance around the column. This imposes a restriction on the column profile because its ends represent the product compositions xD and xB, which !

appear in the steady-state global mass balance FxF ) DxD + BxB. As discussed in section 2, any perturbation will cause the column profile to adjust its shape immediately such that it coincides again with a section of a residue curve. Generically, both product compositions will not be at their steady-state values after that first transient. Furthermore, any perturbation of the product compositions will generically cause a violation of the mass balance around the column, i.e., D∆xD + B∆xB * 0. Hence, perturbations of the column profile and of the product compositions are not independent. Therefore, any generic perturbation of the column profile will result in a perturbation of the TCHC as described by eq 7. Note that the TCH does not change because constant molar holdup on all trays was assumed. For the following, it is useful to categorize the perturbations of the product compositions by characterizing their qualitative impact on the TCHC. With the natural selection of the three pure components, the neighborhoods of the steady-state product compositions can each be divided into six regions as


Figure 8. Product paths in the ∞/∞ case for a 001 class mixture. Px corresponds to the product P (D or B) of a profile of type x (I, II, ...). Figure 7. Steady-state column profile and regions qualitatively categorizing the impact of eventual non-steady-state product compositions caused by perturbations of the profile. Table 1. Impact of Perturbations in the Product Compositions (Figure 7) on the TCHC total holdup composition region

L

I

H

1 2 3 4 5 6

accumulation depletion depletion depletion accumulation accumulation

accumulation accumulation accumulation depletion depletion depletion

depletion depletion accumulation accumulation accumulation depletion

illustrated in Figure 7. (Note that the profile in Figure 7 does not comply with the assumptions discussed in section 2.4 because it does not contain a pinch point. It was chosen solely for its illustrative value.) For example, if a perturbation caused one of the product compositions to be in its neighboring region of type 1 (in the following, we will synonymously denote this case as the product being in region 1), the content of the heavy boiling component H in that product stream would be higher than that at steady state while the contents of L and I would be smaller. Because the feed composition and all flow rates are assumed to remain constant, the result would be a depletion of H in the TCH and an accumulation of L and I. The six types of neighboring regions have the characteristics listed in Table 1. 3.1. Classification of Profiles. Classes of different types of profiles are defined that can subsequently be analyzed one at a time. Bekiaris et al.4 defined three different classes of ∞/∞ column profiles. These classes will be further subdivided according to the pinch points that the corresponding column profiles contain. Later, further subdivisions will be introduced that include geometric characteristics. In this paper, profiles will be denoted to start at the top of the column (distillate) and to end at the bottom. In Figure 8, the product paths for a given feed of a 001 class mixture are segmented depending on the type of profile they correspond to. The properties of the different types of profiles are summarized in Table 2. For example, profiles of type I start at the L-H azeotrope

Table 2. Properties of the Types of Profiles Feasible for 001 Class Mixturesa type

contained pinch point

distillate composition

bottoms composition

I IIIa IIIb IIIc II

Az L L+I I H

Az (Az, L) (Az, L) (L, I) ternary

ternary (L, I) (I, H) (I, H) H

a Notation for the product composition: (I, H) corresponds to compositions on the boundary between I and H.

(the unstable node of the residue curve map) and end inside the composition space, i.e., at a composition that is not on a singular point or on a simple distillation boundary. This categorization of ∞/∞ column profiles can also be applied to profiles of columns of finite length (compare section 2.4). For simplicity, the qualitative study of the dynamic behavior of a distillation column after a perturbation will start with the restriction that only one of the product compositions is perturbed. This is justified for all steady-state profiles of types I and II because they contain a pinch point at one of their ends. For a discussion of profiles of type III, this restriction is eased.23 For brevity, the qualitative stability analysis will only be demonstrated for profiles of type II in this paper. Before proceeding, the profiles of type II must be further subdivided because they can have completely different dynamic behavior. A suitable approach is to study the monotonicity of the individual component fractions in the composition profile along the column. Four cases can be distinguished as shown in Table 3. A detailed analysis of the stability of the four cases of profiles reveals that types IIa, IIb, and IIc are in fact very similar: they will turn out to be stable. In contrast, an analysis of profiles of type IId will show them to be unstable. In the following, two types will be distinguished: stable profiles of type IIs, as shown in Figure 9a, and unstable profiles of type IIu (Figure 9b). For profiles of type IIs, type IIb was chosen as a prototype. The main difference between profiles of types IIs and

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Table 3. Subcases of Profiles of Type II Depending on the Monotonicity Properties of the Composition Profile (m ) monotonic, n ) nonmonotonic) component

type

L

I

H

detailed

stability

m m m n

m n n n

m m n n

IIa IIb IIc IId

IIs IIs IIs IIu

IIu is that the latter are nonmonotonic in the light boiling component. 3.2. Type IIs. Assuming that the distillation column is at a steady state corresponding to a profile of type IIs, we can proceed to study the six possible perturbed profiles corresponding to the distillate composition xD being in one of the six regions shown in Figure 7 and characterized in Table 1. If the distillate composition is in region 1, the components L and I are enriched in the TCH and H is depleted. Figure 10a illustrates the sector of directions in the composition space in which the TCHC can move undergoing this change. This sector can be constructed componentwise: The direction of the change in inventory of each component (corresponding to accumulation or depletion) determines a sector of 180° each. Overlapping the sectors determined by each of the components yields the respective sector. As shown in Figure 10a, the sector corresponding to region 1 spans 45°. Note that the angle of the sector depends on the way in which the composition space is drawn. Because the column profile is always restricted to coincide with a section of a residue curve, it is not possible that the change of profile corresponding to this change in the TCHC is simply a shift of the whole profile in the same direction, especially because the end of the profile is assumed to remain at pure H. An intuitive change of shape of the profile is illustrated in Figure 10b. The depletion of H in the column inventory will mostly affect the trays close to the bottom. Their composition will move away from pure H and toward I. The fraction of I in the composition on trays in the middle of the column will increase. The accumulation of L in the TCHC will mostly cause trays in the top and middle sections to move toward pure L. Situations in which the shape of the profile might not change in this intuitive way will be discussed at the end of this section. Having established the sector of directions in which the column profile and hence the distillate composition will move if xD is in region 1, the same ideas can be applied to study the behavior of column profiles of type IIs for the other five regions 2-6. It is straightforward that the TCHC will change in a direction within the sector “opposite” to the sector in which the distillate composition currently is. Also, for any of the six sectors of possible directions for the change of the TCHC, it is intuitive that the distillate composition will move in a direction within the same sector. The corresponding schematic vector field is illustrated in Figure 11. Thus, if the distillate composition of the perturbed profile is in one of the regions 1-6, this will cause a change approximately in the opposite direction. Hence, it can be concluded that xD is continuously reapproaching its steady-state value. Qualitatively, steady states corresponding to column profiles of type IIs are therefore stable because they will be approached again after perturbations. While approaching its steady-state value,

Figure 9. Profiles of type II in a 001 class mixture: (a) type IIs (IIb); (b) type IIu (IId).

the distillate composition might enter neighboring regions. Therefore, the equilibrium can be either a node or a focus. The qualitative behavior of profiles of type IIs is confirmed with dynamic simulations for an example profile obtained with the CMO model (ABH; compare Appendix A) for the operating conditions D/F ) 0.7778, L/F ) 500, and (xFL, xFI , xFH) ) (0.5444, 0.1556, 0.3000). Dynamic simulations are performed for a number of different initial conditions, each time until steady state is reached. The idea is to initialize the column with profiles that approximately coincide with residue curves but are not at steady state. These profiles begin in the neighborhood of the steady-state distillate composition and end at pure H. The initial profiles correspond to steady states themselves, however, to different feed compositions and distillate flows. In Figure 12, the steady-state profile and the distillate compositions of the initial profiles are shown. The transients of all 16 cases confirm that every time the same steady state is reached. The transients of the


Figure 10. (a) Change of TCHC if the distillate composition is in region 1. (b) Change of the shape of the profile caused by this change of the column holdup composition.

Figure 11. Sectors showing the direction of possible movements of the TCHC for profiles of type II when the distillate composition is in one of the regions 1-6. For profiles of type IIs, this coincides with the possible directions of the distillate composition.

Figure 12. Confirmation of the qualitative predictions: dynamic simulations corresponding to the steady state are initialized with different neighboring profiles. Of those profiles, only the distillate composition is shown. The bottoms composition is always H.

distillate compositions (Figure 13) show that there is very good agreement between the simulations and the qualitative predictions. The same agreement was also found for steady-state profiles of the subtypes IIa and IIc of type IIs (compare Table 3). Also, the predictions carried over to cases of lower reflux-to-feed ratios. However, the predicted sections and the behavior found in the simulations do not match as closely as the reflux is reduced. As mentioned before, it is possible to construct situations where the shape of the profile does not change in this intuitive way. Figure 14 illustrates a situation where the TCH is not equally distributed among the trays. Tray 3 is assumed to have a holdup much bigger than those of the other trays. When profiles 1 and 2 are compared, the fraction of L on tray 4, here assumed to be the distillate composition, decreases although the total inventory of L increases. To avoid this scenario,

an equal distribution of the TCH along the column is assumed throughout this work. Nonintuitive movement of the shape of the profile in response to a change of the TCHC could also be caused by less “regular” residue curve maps as sketched in Figure 15. Such cases will be excluded in this work. Although it seems impossible to reject the existence of mixtures with such residue curve maps, common VLE models such as Wilson or UNIQUAC do not exhibit this “irregular” behavior in the authors’ experience. Indeed, they may not be able to exhibit such behavior at all because their internal structures restrict the set of behaviors they can model. 3.3. Type IIu. In this section, profiles are studied that are nonmonotonic (compare Figure 9b) in all three components along the column profile and end at pure H.

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Figure 13. Transients of the distillate composition for the initial profiles illustrated in Figure 12.

Figure 14. Comparison of the column profiles before (1) and after (2) an accumulation of L and I in the TCH and a depletion in H as in Figure 10. In this case, the holdup on tray 3 is assumed to be much bigger than the other tray holdups.

Figure 15. Residue curve map with “irregular” behavior near A and B.

Assuming that the distillate composition is in region 1 (Figure 16a), the TCHC will change, independent of the actual profile shape, in the same direction as illustrated in Figure 11: L and I accumulate in the TCH, and H will be depleted. However, the impact of these changes on the shape of the profile is different compared to the profiles of type IIs. The accumulation of I in the TCH causes the profile to shift onto a section of an “outer” residue curve. By “outer”, a residue curve is meant that is closer to the boundary AzLH-L-I-H. The “innermost” residue curve is therefore the boundary between the L-H azeotrope and pure H. Thus, the peak compositions of L and I along the residue curve onto which the column profile has shifted increase. Hence, also the peak compositions of L and I along the column profile increase. In other words, the profile “enlarges” toward the pure I corner in the composition space. However, because the number of trays in the column is finite and constant, the profile cannot get “longer”. Therefore, the ends of the profile will be pulled in the direction of the profile (like the ends

of a rubber band under tension). Because a pinch point was assumed to be at the end of the profile at pure H, the bottoms composition will remain nearly unchanged. (Remember that this is a pinch point in a column of finite length; compare section 2.4.) The distillate composition will move in the direction of the profile and, because it is not at a pinch point, hence change significantly compared to the bottoms composition. This is illustrated in Figure 16a: xD will increase in L and decrease in I and H. The decrease of I in xD will be comparably small to the changes in L and H because it results mostly from the column profile moving to an outer residue curve, closer to the boundary, while the changes in L and H result from xD moving along that residue curve due to the limited number of trays. Because the column profile is continuously changing its shape as long as the distillate composition xD is away from the steady-state value, xD will ultimately enter region 2. Similar arguments hold for the case when the distillate composition is in region 2 (Figure 16b): because of the accumulation of I in the TCH, the profile will shift onto an outer residue curve, closer to the boundary AzLH-L-I-H. Because also L will deplete in region 2, the effect on xD will be even stronger. Coming from region 2, the distillate composition will enter region 3. In region 3, L is depleted and both I and H are enriched in the TCHC. The result is a movement of the distillate composition in the direction as illustrated in Figure 16c. However, two effects can cause a change of the direction. (1) As the profile is shifted more and more to an outer residue curve closer to the AzLH-L-I-H boundary, the pinch sections for L and I in the column profile become more prominent. (2) Resulting from the change of the TCHC in region 3 (Table 1), the composition on more and more trays increases in both I and H. The result of both effects is that the finite number of trays in the column will ultimately limit the separation power of the column in a way such that the distillate composition moves along the direction of the residue curves. Hence, the direction in which xD moves in region 3 changes as xD moves away from the boundary between regions 2 and 3 and approaches pure L. Therefore, the distillate composition will finally enter region 4. If the distillate composition is in region 4, 5, or 6, the TCH in I decreases. The same arguments that were used to predict the movement of the distillate composition in regions 1-3 can be “inverted” for a prediction in regions 4-6. When the results are combined for each region, a schematic vector field can be constructed. From the arguments given above, it can be concluded that the distillate composition is continuously moving away from its steady-state value after any initial perturbation. Qualitatively, steady states corresponding to column profiles of type IIu are therefore unstable. Note that this is a statement about the local behavior of the distillation column near the equilibrium. At this point of the discussion, no information is available whether the unstable equilibrium has the characteristics of a node or a focus and whether it is surrounded by a stable limit cycle or not. Again, the qualitative behavior of profiles of type IIu is confirmed with dynamic simulations using the CMO model (ABH; compare Appendix A) to study an example profile obtained for the operating conditions D/F ) 0.9564, L/F ) 100, and (xFL, xFI , xFH) ) (0.8990, 0.0010, 0.1000). Dynamic simulations are performed for a


Figure 16. Change of the shape of the profile of type IIu caused by the distillate composition being in region 1, 2, or 3.

Figure 17. (Left) Profile of type IIu whose instability is to be confirmed with dynamic simulations. Shown are the distillate compositions of different neighboring profiles. The bottoms composition is always H. (Right) Zoom around the initial distillate compositions.

number of different initial conditions, each time until a steady state is reached. In Figure 17, the steady-state profile and the distillate compositions of the initial profiles are shown.

The transients of all 16 cases confirm that in all cases the distillate composition moves away from the steadystate value. The transients of the distillate compositions (Figure 18) and the initial phase of the transients of the

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Figure 18. Transients of the distillate composition for the initial profiles illustrated in Figure 17. (Left) Close-up of the transients near the initial distillate compositions. The arrows indicate the initial direction of movement. (Right) All trajectories approach the same final distillate composition.

Figure 19. Initial phase of the transients of the TCHC for the initial profiles illustrated in Figure 17. All subfigures are close-ups of the composition space as shown in Figure 18. The arrows indicate the initial direction of movement.

TCHCs (Figure 19) show that there is very good agreement between the simulations and the qualitative predictions. Note that the initial directions of movement are predicted. Further note that the initial direction of movement of the TCHC depends only on the distillate composition but not on the shape of the profile. As the profile drifts away from the steady-state profile of type IIu, the assumptions are not met and therefore the

directions of movement of the TCHC in the composition space change. The right part of Figure 18 shows that in all cases the distillate composition converges toward the same final composition close to the L-H azeotrope; i.e., the column converges toward the stable profile of type I. 3.4. Other Types. Because of constraints on the length of this paper, the detailed stability analysis

Ind. Eng. Chem. Res., Vol. 41, No. 16, 2002 3941 Table 4. Stability of the Types of Feasible Profiles for 001 Class Mixtures type

stability

type

stability

I IIIa IIIb

stable stable unstable

IIIc IIs (IIa, IIb, IIc) IIu (IId)

stable stable unstable

cannot be presented here for all types of profiles. Following the same steps of analysis as shown for the subtype IIb of profiles of type IIs (section 3.2), it is straightforward to confirm that profiles of the subtypes IIa and IIc are stable. For profiles of type I, subtypes could be introduced according to monotonicity properties of the composition profiles, but because they are all stable, this is not necessary. The analysis of profiles of type I follows basically the same lines as those for profiles of type IIs. Studying the stability of profiles of type III is more involved, although basically the same ideas are used. While it is comparably easy to show that the subtypes IIIa and IIIc are stable, it is much more complex to derive the instability of profiles of type IIIb.23 The qualitative predictions of the dynamic behavior of profiles of types I-III can be confirmed with dynamic simulations for all of their subclasses in the same way as presented in sections 3.2 and 3.3. The stability properties of all subtypes of profiles are listed in Table 4. Note that, for the 001 class mixtures studied here, MSS exist for all feed compositions.4 The unstable profiles of types IIIb and IIu correspond to steady states on the middle branch and the high branch, respectively. 4. Conclusion A novel approach for the qualitative analysis of the dynamic behavior of homogeneous azeotropic distillation columns was presented. It was shown that, at infinite column reflux, dynamically evolving column profiles coincide with a section of a distillation line at any time. When steady-state column profiles are perturbed, studying global mass balances and how they interact with the shape of the column profile allowed one to determine the direction in which the product compositions will move in the composition space. When schematic vector fields are constructed, the local stability of the steadystate column profile can be determined. After a thorough classification of different types of feasible column profiles, the new approach was demonstrated for two profile types, one of which was found to be unstable. To validate the schematic vector fields, a method for their quantitative construction was developed.24 The insight gained from the qualitative stability analysis will be compiled to construct qualitative bifurcation diagrams in an accompanying paper.1 It will be shown how the methods presented in sections 2 and 3 can be used to predict phenomena like Hopf bifurcations, limit cycles, and homoclinic bifurcations. An extension of the methodology for a qualitative analysis of mixtures whose residue curve maps have more complex topologies will be published separately.13 As with the ∞/∞ analysis, it is important to point out that the insight is gained on physical grounds without numerical simulations of the distillation column. Instead, all results can be derived by interpretation of the residue curve map of the corresponding mixture. Note that the residue curve map illustrates some physical properties of a mixture and therefore does not depend on the thermodynamical model used because it could be measured directly.

Table 5. Column Design Used for the Simulations no. of trays (including the condenser and reboiler) feed tray (counting from the condenser) tray liquid holdup [kmol] condenser liquid holdup [kmol] reboiler liquid holdup [kmol] column pressure [atm] feed flow rate [kmol/h]

46 41 3 3 3 1.0 100

Acknowledgment The authors thank Jan Ulrich for his support in the preparation of the final version of this paper. Further, the authors thank Thomas E. Gu¨ttinger, who programmed the distillation column model used for the dynamic simulations. Appendix A. Simulation Details All simulations were performed using the CMO model as described in eqs 2-6. For the dynamic simulations, the CMO model is integrated with DDASAC,25 an index-1 DAE solver. The simulations are based on the column design given in Table 5. Vapor pressures and liquid activity coefficients are calculated with the Antoine equation and the Wilson model, respectively. The thermodynamic parameters for the mixture acetonebenzene-heptane are taken from Gu¨ttinger and Morari.26 Notation n ) number of trays nc ) number of components L ) light boiling component I ) intermediate boiling component H ) heavy boiling component Mk ) holdup on tray k kF ) feed tray F ) feed flow rate xF ) feed composition L, V ) reflux and boilup flow rates D, xD ) distillate flow rate and composition B, xB ) bottoms flow rate and composition xk ) liquid composition on tray k yk ) vapor composition in equilibrium with xk xki ) fraction of component i in xk Abbreviations CMO ) constant molar overflow TCH ) total column holdup TCH in i ) total column holdup in component i TCHC ) total column holdup composition ABH ) acetone-benzene-heptane

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Received for review September 4, 2001 Revised manuscript received April 23, 2002 Accepted May 15, 2002 IE010725R

Qualitative Analysis for Homogeneous Azeotropic Distillation. 1. Local

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