Copyright Cl IFAC Dynamics and Control of Process Systems, Jejudo Island, Korea, 200 I
QUALITATIVE DYNAMIC ANALYSIS OF HOMOGENEOUS AZEOTROPIC DISTILLATION COLUMNS Cornelius Dorn *,1, Eleonora Bonanomi * , Manfred Morari .,2
• Automatic Control Laboratory, Swiss Federal Institute of Technology
Abstract: In this paper, the stability and dynamic behavior of homogeneous azeotropic distillation columns is studied with two novel tools that are based on a qualitative approach. Both methods analyze the dynamic behavior of column profiles after perturbations. The advantage of the proposed approaches is that they provide insight without detailed column simulations. In addition, they allow more general conclusions as they are not restricted to specific sets of operating conditions. The key concept of both analysis is to study the interaction of changes in the total column holdup and the shape of the column profile. This allows one to determine if the profiles are locally stable or unstable. While the first method is purely qualitative and can easily be generalized for predictions, the second method shows how it can be augmented with a numerical approach for analyses of specific cases. Copyright © 20011FAC Keywords: Distillation, Dynamic behavior, Stability, Qualitative analysis
enthalpy is neglected, all components are considered to have equal molar vaporization enthalpies, and the holdups on each stage are assumed constant. Summation of the equations of a CMO column model yields the global mass balance around the column:
Among the classical unit operations, there are many examples of processes that exhibit multiple steady state (MSS). The CSTR is the best known example (Uppal et al., 1974). There, heat produced by a chemical reaction is removed by a coolant. While the removal of heat depends linearly on the reactor temperature, heat generation depends non-linearly on the reactor temperature. At the operating point the heat produced equals the heat removed. Representing each heat flux as a function of the reactor temperature, the intersections of the two curves corresponds to steady states. If MSS are present for certain operating conditions, the so called "slope condition" can easily be applied as a necessary condition for stability. In this paper, an analogous approach for distillation columns is proposed. In the following, constant molar overflow (CMO) is assumed: mixing
where Mk is the holdup on stage k. At steady state, this global mass balance reduces to FxF = DxD + BxB , which is identical to the lever rule. The key idea of the two approaches presented in this paper is to study column profiles that do not correspond to steady states. By determining how the total column hold up composition (TCHC) changes it is possible to derive qualitative (Section 2) and semi-quantitative (Section 3) insight into the dynamic behavior of the column. As the feed composition is assumed constant, the main driver for changes of the TCHC is the difference between the actual and the steady-state product compositions.
Current affiliation: McKinsey & Company, Switzerland. to whom correspondence should be addressed: ETH Zentrum, ETL 112, CH-8092 Zurich, Switzerland. morariGaut . ee.ethz . ch. 1
the assumption of infinite reflux. Hence, the total column holdup composition will continuously change in a slow time scale as governed by Equation 1 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 total column holdup composition continuously until a steady state is reached. During the transient, the column profile readjusts after every infinitesimal change of the total column hold up composition 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 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 this section.
2. QUALITATIVE PHASE PLANE ANALYSIS: STABILITY 2.1 Column Profiles: Infinite Reflux
Directionality and time scale separation in distillation are known for a long time (Rosenbrock, 1962; Levy et al., 1969). Studying distillation column models of reduced detail, it was found that the dynamics of a column are separated into a slow and a fast part (Skogestad and Morari, 1988; Andersen et al., 1989). The slow part is caused by changes in the external flow rates D and B, while the fast part dynamics correspond to an internal redistribution of the component holdups following changes in the internal flow rates. Kumar and Daoutidis (1999) recently confirmed these results by performing a singular perturbation analysis of a distillation column model assuming CMO and constant relative volatilities. They found the fast time scale to be in the order of the residence times in 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 masstransfer" (Rosenbrock, 1962; Levy et al., 1969) . Increasing the internal flow rates (reflux and boilup) reduces the influence of the external streams on a column profile. This can be analyzed by dividing the CMO-equations by L and letting L approaching infinity. This way, many terms drop from the equations, and after some rearrangement and substitutions,
2.2 Stability Analysis
k=l, ... ,n-1. (2)
The key idea is to study the interaction of changes of the column inventory and changes of the shape of the column profile. At steady state, column profiles must satisfy a global mass balance around the column. This imposes a restriction on the column profile as its ends represent the product compositions x D and x B that appear in the steady-state global mass balance FxF = DxD + B x 13 . The perturbations of the product compositions can be characterized by their qualitative impact on the total column hold up composition. The neighborhoods of the steady-state product compositions can each be divided in six regions as illustrated in Figure 1. For example, if a perturbation caused one of the product compositions to be in its neighboring region of type 1 (synonymously denoted as "the product being in region 1"), the content of the heavy boiling component H in that product stream would be larger than at steady state while the content of L and I would be smaller. As the feed composition and all flow rates are assumed to remain constant, the result would be a depletion of H in the total column holdup and an accumulation of L and I. For a systematic approach, classes of different types of profiles are constructed. These classes can subsequently be analyzed one at a time. With the 00/00 analysis (Bekiaris et al., 1993), three
can be derived as the equation describing the column profile. Equation 2 corresponds to the definition of distillation lines. Thus, at infinite reflux the profile of a CMO tray column coincides with a section of a distillation line at any time as steady state was not assumed for the derivation of Equation 2. The previous observation can be interpreted in an interesting way: At very high internal flows in the column, holdup between stages can be exchanged very fast as the residence times (time constants) Afk of the trays are very small. At infinite reflux, the exchange of holdup between stages along the column occurs infinitely fast . Thus, if a column operated at infinite reflux is initially filled with holdups Mkxk, k = 1 ... n that do not correspond to a steady state profile, the initial total column holdup I:;=l Mkxk will be redistributed along the column infinitely fast such that the profile satisfies Equation 2 thus coinciding with a part of a distillation line. As 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. Note that the global mass balance (Equation 1) around the column also holds under 108
restricted to coincide with a section of a residue curve, it is generically not possible that the change of profile corresponding to this change in the total column holdup composition is simply a shift of the whole profile in the same direction, especially as the end of the profile is assumed to remain at pure H. An intuitive change of shape of the profile is illustrated in Figure 2b. The depletion of H in
(b) B H
Fig. 1. a) Steady-state column profile and regions qualitatively categorizing the impact of eventual non-steady-state product compositions caused by perturbations of the profile. b) Profile of type IIs in a DOl-class mixture.
types of feasible profiles for the limiting case of infinite reflux and an infinite number of trays were defined. These classes can be further subdivided according to the pinch points that the corresponding column profiles contain. Due to the limited length of this paper, only one type of profile can be studied here. This type is illustrated in Figure 1b for a 001 class mixture and is characterized by (1) H as a pinch point at the bottom, (2) monotonicity of the composition profile in L, and (3) nonmonotonicity of the composition profile in I and H. This type is called lIs (corresponding to the notation used and explained in our other publications). If a disturbance caused the distillate composition to be in region 1, the components L and I are enriched in the total column holdup, and H is depleted. As a consequence, there must be trays in the column on which L and I are enriched and H is depleted in the liquid hold up. What is the impact of this change in the holdup composition on the shape of the column profile? Figure 2a illustrates the sector of directions in the composition space in which the total column holdup composition can move undergoing this change. This sector can be constructed component-wise: The direction of the change in inventory of each component (corresponding to accumulation or depletion) determines a sector of 180 degrees each. Overlapping the sectors determined by each of the components yields a sector of 45 degrees shown in Figure 2a. As the column profile is always
Fig. 2. 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 TCHC. c) Sectors showing the direction of possible movements of the TCHC for profiles of type lIs. This coincides with the possible directions of the distillate composition. the column inventory will mostly affect the trays close to the bottom. Their composition will move away from pure H and towards I. The fraction of I in the composition on trays in the middle of the column will increase. The accumulation of L in the total column holdup composition will mostly cause trays in the top and middle section to move towards pure L. Having established the sector of directions in which the column profile and hence the distillate composition will move if x D is in region 1, the same ideas can be applied to study the behavior of column profiles of type lIs for the other five regions 2 to 6. The corresponding qualitative vector field is illustrated in Figure 2c. Thus, if the distillate composition of the perturbed profile is in one of the regions 1 to 6, this will cause a change approximately in the opposite direction. It cannot be excluded that this change in the TCHC causes the shape of the profile to change in a way such that the distillate composition enters one of the neighboring regions. How109
ever, before entering those regions, the distillate composition will have come closer to its steady state value. Hence, it can be concluded that the distillate composition is continuously approaching its steady state value, ultimately fulfilling the global mass balance and therefore finally reaching this steady state again. Qualitatively, steady states corresponding to column profiles of type lIs are therefore stable as they will be approached again after perturbation. The same methodology can also be applied to study other types of profiles. The analysis reveals that for mixtures of the class 001 , stable and unstable types of profiles exist. Profiles on the middle branch can be shown to be unstable. Further, one of the types of profiles corresponding to steady states on the high branch is unstable. Note that for the DOl-class mixtures studied here, multiple steady states exist for all feed compositions (Bekiaris et al., 1993).
Heptane mole flaction in the bottom
. .. 0.07
Acetone (mote flac.)
o.o.t Heptane (mole !rac.)
3. SEMI-QUALITATIVE STABILITY ANALYSIS In the previous section, six regions around a steady state were identified and for each a sector of directions of movement was evaluated qualitatively. In this section, a similar approach is introduced that contains more numerical features. The total column holdup of each component is governed by the global mass balance
d (~ dt L.,.-M k Xik)
-B - BXi -
...,. 0 .• 35 •• •••••••••••
-_ 2 E:
0 ... . . .
••••• ., ••• •, ••""
. . . ... . . . . . .\P.
where if is an approximate bottoms composition calculated with a McCabe procedure (Appendix A). Substituting the steady-state mass balance Fx[ - DxP = Bxf in Eq.(3) one obtains:
O· ~~05~S:-------:0-:: .06:-------:0.-::06::-S---~O.07 Heptane (mote frac .)
d ("'M k
k) _ Xi -
d ( aV9)
M tot dt
Fig. 3. (a) : Bottom molar fraction (if and xf) as a function of the distillate composition. (b) : Projection of the nullclines in the distillate composition space.
= B( XiAB
At steady state, 1tx~V9 = 0 holds for each component , and therefore if = xf . The central idea for the approach presented in this section is to calculate the nullclines 1tx~Vg = 0 for each component. Their intersections correspond to steady states and they divide the space around the steady-state distillate composition in six regions, in analogy with the qualitative analysis presented in Section 2. For each region a sector of directions of possible movements of the distillate composition is calculated as presented in the following. These sectors will correspond to the qualitative analysis introduced in the previous section.
benzene, heptane) is of the class 001 according to Matsuyama and Nishimura (1977) . The column has 21 stages. The feed of 100 kmol/h consists of 92% acetone, 7.93% benzene and 0.07% nheptane and enters on stage 15. A high reflux (reflux/feed=500) is considered, but the method is valid for any sufficiently high reflux. The distillate flow is 0.6 kmol/h. In Fig. 3(a) the surfaces xf and if for n-heptane are plotted as a function of the distillate composition xD . The intersection of the two surfaces is the nullcline for n-heptane. In the same way the nullclines for the other two components can be calculated. The three nullclines for acetone, benzene, and heptane are projected in the distillate composition space in Fig. 3(b), they intersect at the steady
3.1 Stability analysis: one steady state
In the following, the approach is illustrated with a case study. The mixture considered (acetone, lID
Fig. 4. (a),(b),(c): Contour lines of constant holdup, the semi-circle represent the sector in which the direction of movement of the distillate must be. (d) : Combination of condition (a),(b),(c) results in the sector represented. The distillate must have a direction of movement within this sector
Heptane (mole frac.]
Fig. 5. (a): Predicted direction of movement of the distillate after perturbation.
state. Notice that the regions are very similar to those represented in Fig. 2. As Fig. 3(b) shows, each nullcline divides the distillate composition space in two regions. The region where if > if corresponds to an accumulation of component i in the total column hold up when x D is in that region, and the other corresponds to depletion. This analysis is repeated for each component in each region obtaining a complete picture of the change in holdup (c.f. Fig. la). As the holdup is changing the shape of the column profile will adjust according to this change. The holdup is proportional to the average composition corresponding to the column profile obtained with the McCabe procedure: L:k MkX~ = Mtotxavg . In Fig. 4, contour lines of constant hold up composition of one component as a function of the distillate composition are shown for each component. Using these contour lines and the information about the change in the column holdup, the sectors of movement are calculated as explained in the following. Consider, for example, a perturbation that brings the distillate composition in region 5. Then there is an accumulation of heptane and acetone and a depletion of benzene. The accumulation of heptane implies movement of the distillate composition towards higher hold up for heptane; based on the contour lines, all feasible directions for increasing the hold up lie in the semi-circle "H" plotted in Fig. 4. The accumulation of acetone implies the movement of the distillate in a direction that must lie in the semi-circle "L". Finally, the depletion of benzene bring the column in the direction of lower benzene holdup, the direction is in the semicircle "I" . These three conditions together result in trajectories for the distillate restricted to fall within the cone highlighted in Fig. 4(d) .
o.a~.04!::---::O.::O,,;----,O~ .. .;---O~."';-;-,----':-O... :::---::O.="";--~ O.O·, Heplane (mole ~ac .J
Fig. 6. Nullclines and steady states for a column with MSS. The same analysis is repeated for all the six regions. In Fig. 5(a) the six sectors and the steady state distillate composition are illustrated. The arrows show the direction of the distillate in each sector: the column would move back to the steady state which is therefore stable. Analogously the analysis can be applied to other case studies where MSS or limit cycles exist. One case with MSS is shown in Fig. 6. The column has 46 stages, the feed of 100 kmoljh consists of 89, 37% acetone, 0.7% benzene and 9.93% nheptane and enters on stage 41. The reflux is 4800 kmol/h and the distillate flow rate is 7.5 kmol/h. The nullclines intersect in three points corresponding to three steady states. The stability of each equilibrium point can be studied. III
b) Stage 2
a) Total condenser
Two approaches for studying the dynamic behavior and stability of distillation columns were presented. The qualitative method allows to determine the stability of steady state column profiles assuming infinite reflux. Schematic vector fields can be constructed that allow to predict bifurcations and phenomena like limit cycles as will be shown in future publications. In addition it was shown how the qualitative method can be augmented with quantitative aspects that allow more specific and accurate predictions for specific examples.
5. REFERENCES Andersen, H. W., M. Kiimmel and S. B. J(3rgensen (1989). Dynamics and identification of a binary distillation column. Chem. Eng. Sci. 44(11),2571-2581. Bekiaris, N., G.A. Meski, C.M. Radu and M. Morari (1993) . Multiple Steady States in Homogeneous Azeotropic Distillation. Ind. Eng. Chem. Res. 32(9), 2023-2038. Kumar, A. and P. Daoutidis (1999). Nonlinear Model Reduction and Control of High-Purity Distillation Columns. In: Proceedings of the American Control Conference, San Diego, California. pp. 2057-2061. Levy, R. E., A. S. Foss and E. A. Greens (1969). Response Modes of a Binary Distillation Column. Ind. Eng. Chem. Fundam. 8(4), 765776. Matsuyama, H. and H. Nishimura (1977). Topological and Thermodynamic Classification of Ternary Vapor-Liquid Equilibria. J. Chem. Eng. Jpn. 10(3), 181-187. Rosenbrock, H. H. (1962). The Control of Distillation Columns. Trans. Inst. Chem. Engrs. 40(1), 35-53. Skogestad, S. and M. Morari (1988) . Understanding the Dynamic Behavior of Distillation Columns. Ind. Eng. Chem. Res. 27(10), 1848-1862. Stichlmair, J. and J.R. Fair (1998) . Distillation: Principles and Practice. WHey. New York etc. , NY, USA. Uppal, A., W.H. Ray and A.B. Paore (1974) . On the Dynamic Behavior of Continuous Stirred Tank Reactors. Ch em. Eng. Sci. 29, 967-985.
Fn I~ Lrectrl
Fig. A.!. Flows entering and leaving different sections of a distillation column: condenser, second stage, reboiler. until the global mass-balance is satisfied. If the procedure is applied only once to a given xD, an approximate bottoms composition fiB is obtained that can be interpreted based on the following assumption. As discussed in Section 2, the time scale separation in the transient of column profiles after perturbations results in the individual stages being approximatively in equilibrium until the profile has adjusted to fulfill the global mass balance and reached a steady state. For the approach presented here, it is assumed that during the transient all stages are in equilibrium except the reboiler in which all the residual is lumped. In particular, fiB is calculated using the following procedure. First, the vapar and liquid flows in each section of the column are determined using mass balances and assuming CMO . Then the compositions on each stage are calculated. For a total condenser (Fig. A.l(a)), the compositions y2, xl and the distillate x D are the same. Figure A.l (b) shows the flows for the first tray (stage 2). Here, the liquid composition x 2 and the temperature on the tray are evaluated through a dew point calculation for the known vapor composition y2. The composition y3 of the vapor rising from the tray below is then determined by the mass balances around the top of the column: y3 = t
ect x 2 + DxD) _1_(F V rect t t
The procedure is repeated for each stage of the rectifying section. For the feed tray (stage t), and all stages in the stripping section the mass balance changes to: HI _ _
1_(L. tri P Xit
+ D XiD
F XiF) (A .2 )
Concerning the partial reboiler (Fig. A.l(c)) , yn and x n - l are known from the previous calculations. Therefore the bottom composition fiB is computed with a dew point calculation for the known composition yn . Notice that the mass balance around the reboiler is not fulfilled as before mentioned.
Appendix A. "MCCABE-THIELE" PROCEDURE Consider a column with n stages and feed (F, x F ) . The steady state column profile can be determined with the classical McCabe-Thiele method (Stichlmair and Fair, 1998) for any given distillate and reflux flow rate. Starting with an initial guess for the distillate composition xD , an approximation of the composition profile is calculated. The procedure is iteratively repeated with updated x D 112