- Email: [email protected]
Stability and transient behavior of homogeneous azeotropic distillation
~
Computers and Chemical Engineering Supplem ent (1999) S191-5194 (,') 1999 Elsevier Science Ltd. All rights reserved PH: S0098·1354199/00171-4
Pergamon
Stability and Transient Behavior of Homogeneous Azeotropic Distillation Cornelius Dorn, Moonyong Leet , Manfred Morari" Automatic Control Laboratory, Swiss Federal Institute of Technology ETH-Zentrum, CH-8092 Zurich, Switzerland Abstract In this article, the steady state and dynamic behavior of an azeotropic distillation column separating the homogeneous ternary mixture acetone-benzene-heptane is studied. Nonlinear phenomena repor ted in earlier studies like multiple steady states (Bekiaris et aI., 1993) and limit cycles (Lee et al., 1999) are shown for this mixture. In addition, a new phenomenon is reported: the occurrence of homoclinic bifurcations resulting in cases of only one attractor in the presence of three steady states. It is discussed how these phenomena are influenced by operating parameters such as distillate and reflux flow rate, feed composition, etc.. The steady state and dynamic behavior is classified in the parameter space. Keywords: Azeotropic distillation, Multiple steady states, Instability, Oscillation, Hopf bifurcation, Homoclinic bifurcation.
INTRODUCTION Azeotropic distillation is an important separation technique which finds wide use in industry. The steady state and dynamic behavior of azeotropic distillation have been studied extensively over the past decades. This understanding is a necessary 'prerequisite for proper column design and operation. It has been shown that azeotropic distillation columns can exhibit unusual features not observed in nonazeotropic distillation (e.g. Laroche et al. (1992». In particular, multiplicity of steady states has been the focus of much recent research . The existence of multiple solutions in azeotropic distillation is now widely accepted both by theoreticians and practitioners. Throughout this work, the term "multiple steady states" (MSS) refers to output multiplicities only. A short overview of research related to the oscillatory behavior in azeotropic distillation is given in Lee et al. (1999). Recently, Lee et al. (1999) published simulation results showing sustained oscillations (limit. cycles) in the homogeneous azeotropic distillation of methanol-methyl butyrate-toluene using a CMO model. The results obtained with the CMO model were reconfirmed with a more complex model (RADFRAC in AspenDynamics) and two different VLE models. The main emphasis in this pap er is to understand how system parameters influence the multiplicity and oscillatory behavior of a homogeneous azeotropic distillation column. It will be shown that the column can have an even more complex bifurcation behavior than shown in the earlier study by Lee et al. (1999). SIMULATION SETUP Mixture and Column Model The ternary homogeneous mixture acetone-benzeneheptane investigated by Bekiaris et al. (1993) is chosen for the simulation study. The mixture belongs to the class 001 according to Matsuyama and Nishimura (1977). Acetone and heptane form the only minimum boiling binary azeotrope in this mixture at 93.56% (molar) acetone. The constant molar over-
flow (CMO) model is chosen (for details see e.g. Lee et al. (1999». Data for the simulation are given in Table 1. Vapor pressures and liquid activity coefficients are calculated with the Antoine equation and Wilson model, respectively. The thermodynamic parameters are taken from Giittinger and Morari (1996). For the dynamic simulations the CMO model is integrated with DDASAC (Stewart et al., 1995), an index-1 DAE solver. For the steady state bifurcation analysis, the CMO model is incorporated into AUTO (Doedel et al., 1997). To investigate the local dynam ic behavior and stability, the nonlinear CMO ' model is linearized and the eigenvalues are calculated with LAPACK (Anderson et al., 1995) at every steady state. The average tray temperature T a ug is chosen as a scalar index for the bifurcation diagrams and dynamic simulations. Number of trays (including condenser and reb oiler) Feed tray (counting from condenser) Tray liquid hold-up [kmol] Condenser liquid hold-up [kmol] Reboiler liquid hold-up [krnol] Column pressure [atm] Feed flow rate [kmol/h] Table 1: Column data for simulation
46 41 3 3 3 1.0 100
Parameterization of the Column Model The number of parameters needed to specify an actual distillation column model makes a complete parametric study impractical. However, once a mixture is given, it is possible to demonstrate the more relevant parametric effects by using appropriate dimensionless variables. The set of nonlinear coupled ODEs which describe the chosen Ci\[Q model for a ternary mixture can be rewritten in a compact vector form: dx/dt' = f(x ,D',P), where t' = t/(M/ F) is the dimensionless ti me. Further, I I 2 2 N N)T' h . x = ( XL,xll'XL'XH"" 'XL 'XH IS t e composttion vector of the liquid phases, D' = D / F is the dimensionless distillate rate used as the bifurcation
tSchool of Chern. Eng. and Tech., Yeungnam University, Dae-Dong 214·1, Korea. • Author to whom correspondence should be addressed. Phone: +41 1 632-2271. Fax: +41 1 632-1211. Email: [email protected]
S 192
Computers and Chemical Engin eering Supplement (/999) SJ9J-SJ94
variable, and P := (L', xf, xlII)T is the vector of op- closest complex eigenvalue was significantly smaller. erating parameters with L' := L/F as the dimension- Thus, it is sufficient to study the dominant eigenless reflux rate, xf as the entrainer fraction in the value for a classification of the dynamic behavior. feed, and XlII := xl/xf; as the ratio of light and Regions on the steady-state branches where the domheavy component fraction in the feed. inant eigenvalue is complex (i.e. either damped or sustained osciIlation) are bounded by stars in th e biINFLUENCE OF OPERATING PARAME- furcation plots. TERS ON BIFURCATION BEHAVIOR Effect of Light/Heavy Fraction. The bifurcations In order to understand how changes in the operating plotted in Figure 1 show that the influence of XlII is parameters L', xf, and XlII influence the multiplic- mainly to shift the MSS region towards higher vality and oscillatory behavior, a parametric study is ues of D'. The characteristic features (e.g. region of carried out. osciIlation or occurrence of Hopf bifurcation points) arc nearly unaffected. MSS Region With the 00/00 analysis (Bekiaris et al., 1993) it can Effect of Entrainer Fraction. As illustrated in be predicted that for the feed mixture studied here Figure 2, the dominant eigenvalue for steady states there will be multiple st eady states for all feed com- on the low branch was always negative .real (stable positions as long as the column is sufficiently long node). For steady states on the middle branch, it and the reflux is sufficiently high . Further, the loca- was always positive real (unstable). tion of the left and right turning points D L1 and Db For large values of xf, the steady states on the high is decreased, a can be predicted for the 00/00 case (infinite number branch are all stable nodes . As region appears and becomes more apparent on the of trays, infinit e reflux flow rate) to be at: high branch where the steady states have a complex dominant eigenvalue. Time responses starting close xlu D L1 := . (l-xn (1) to those steady states wiII show underdamped oscil1 + x[JI is further decreased, two Hopf bifurlations. As F { .~. x !==f xa~ xlII:5 XL~ cation points appear in the region of oscillation. The D L2 := (2) LH FL F > az steady states on the high branch in the region be(1 - xIl XLII _ XLII tween the two Hopf bifurcation points are -unstable In the abo ve equation, x'rll denotes the ratio of light and surrounded by stable limit cycles. to heavy component in the binary acetone-heptane Next, two homoc1inic bifurcation points appear on azeotrope (:::::: 14.5). the high branch between the two Hopf bifurcation Effect of Light/Heavy Fraction. In the 00/00 points. Therefor the interval of D' where stable limit case, both turning points are moved towards larger cycles exist is disconnected by a region where the values of D' and the size of the MSS region increases limit cycles disappear, as illustrated in Figure 4 (This F • . d til F az AS XLJI F • • interval is not illustrated in Figures 1 and 2 for claras XLII IS increase un 1 X LII := XLII' IS inity) . For values of D' in this interval this results in creased further, the right turning point D L2 remains while Db is still increasing. Hence the MSS region one stable and two unstable steady states, the former becomes narrower. Figure 1 shows bifurcation dia- being the only attractor. In Figure 4, the vertical disgrams varying XlII for several values of xf. The tance between the dots depicts the amplitude of the behavior of the MSS regions is in good qualitative limit cycles. The coalescence of limit cycles with the middle branch essentially causes the limit cycles to agreement with the predictions. Effect of Entrainer Fraction. In the 00/00 case, disappear. Figure 5 iIIustrates the time response of both turning points are moved towards larger values Ta v 9 to consecutive step changes. of D' and the size of the MSS region increases as xf is As shown in Figure 2, the region limited by the Hopf decreased. The results of a bifurcation analysis vary- bifurcation points expands further until for some ing xf for several values of L' are shown in Figure value of xf the left Hopf bifurcation point and the left 2. As it can be seen ; the MSS region moves toward homoc1inic bifurcation point together reach th e left decreases for given L'. However, turning point and disappear. Then, a region where higher D' as the MSS region initially expands and then narrows the dominant eigenvalue is positive real will appear decreases whereas the 00/00 analysis predicts on the high branch between the left turning point and as a monotonic expansion. This discrepancy is due to the homoclinic point (Figure 2c) thus disconnecting the finite reflux flow rate and finite number of trays the region of oscillation. The resulting left region of osciIIation shrinks towards the left turning point and in the simulations. Effect of Reflux Rate. For the example studied disappears while the right oscillation region becomes here multiplicities occur for all possible feed compo- gradually narrower (Figure 2d). sitions if L' is sufficiently large and the column has Effect of Reflux Rate. As illustrated in Figure enough trays. As L' decreases, the MSS region be- 2, the main characteristic features of the bifurcation comes smaller and finally disappears due to insuffi- also changes significantly with increasing £'. When xf is smaller than some critical value, the qualitative cient separation. effect of increasing L' is the same as that of decreasDynamic Behavior In this section, the steady states wiII be classified ac- ing xf: first a region of osciIlation appears on the cording to the eigenvalues of the local linearizations. high branch and gets larger, then a region of unstaThe eigenvalue(s) with the largest real part will be ble steady states surrounded by stable limit cycles appears within the region of oscillation, and so on. referred to as dominant. In all cases studied where the dominant eigenvalue was real, the real part of the Effect of Other Parameters. The size of both the
xf
xf
xf
xf
Computers and Chemical Engineering Supplement (1999) S191-S194
5193
1\'ISS region and the region of oscillation is increased sharp large peak. In this part of the column there as the number of trays in the column increases. The is steep front in the temperature profile bounded by liquid hold-up directly affects the period of oscilla- well developed pinch zones. tion. CONCLUSIONS In this paper, it was shown that homogeneous azeoCLASSES OF DYNAMIC BEHAVIOR In the preceding section, it was shown that the quali- tropic distillation can exhibit complex steady state tative bifurcation behavior is mainly influenced by £' and dynamic behavior. Based on the CMO model, it and xf whereas the main effect of xfl1 is to simply was studied how process operation affects the charshift both the MSS region and the region of oscil- acteristic features of multiplicity and stability. It was lations without changing their main characteristics. observed that both xf and £' have a significant effect This allows to focus on L' and xf to characterize the on the qualitative bifurcation behavior. In comparitypes of dynamic behavior. In this section, it will be son, the effect of XfH is to shift the MSS and oscillastudied how the (£', xf) parameter space is divided tion regions. The (£', xf) parameter space is divided into regions corresponding to qualitatively different into four regions according to the main features of steady state and dynamic behavior. For the example bifurcation behavior. A new phenomenon in homostudied in this paper, the regions are illustrated in geneous distillation was discovered: the occurrence of homoclinic bifurcation points. Thus, for certain Figure 3. Region I. For values of (£', xf) in region I, there is sets of operating parameters there exist three steady a unique steady state for every D'. All steady states states with only one attractor. are stable. NOTATION Region II. In subregion II-a, MSS exist but no oscillations occur throughout the high branch. All the F Molar feed flow rate dominant eigenvalues are real. In subregion II-b, D'=Q Dimensionless distillate flow rate damped oscillation appears for some range of D' on £ -F Dimensionless reflux flow rate the high branch. Molar liq. frac. of component c on tray n x~ Region III. This region also consists of two subre- To \19 Average temperature of all trays gions. Subregion III-a is a narrow band at the border t ' -- t.E.. Dimensionless time M with region II (not illustrated separately). For values M Molar liquid holdup on a tray of L' and xf in this region, there are two Hopf bifur- N Number of trays cation points on the high branch. All steady states x F Molar feed composition between these two Hopf bifurcation points are unsta- XfH Ratio of the feed light to heavy comp, frac. ble and surrounded by stable limit cycles. x~% Molar liq. frac. of comp. c of azeotrope Subregion III-b consists of the remainder of region III x'Lit Ratio of the az. light to heavy cornp. frac. and exhibits an interesting phenomenon newly observed in distillation: the occurrence of homo clinic Acknowledgement bifurcation points. The amplitude of the limit cycles This work was supported by the KOSEF through the around the high branch grows until the limit cycles Automation Research Center at POSTECH and by touch the middle branch. The interval of D' between the KOSEF - Swiss National Science Foundation exthe Hopf bifurcation points is therefore disconnected change program. by an interval bounded by the two homo clinic bifurcation points where the limit cycles disappear, as illustrated in Figure 4. This results in one stable and REFERENCES two unstable steady states, the former being the only Anderson, E.; Bai, Z.; Bischof, C.; Demmel, J.; Donattractor. garra, J. Lapack Users' Guide : Release 2.0, InAs £' is further increased (or xf further decreased) in ternational Society for Industrial and Applied region III, the left Hopf bifurcation point and the left Mathematics. homo clinic bifurcation point approach the left turning point. Bekiaris, N.; Meski, G. A.j Radu, C. 11.1.; Morari, r-.L Region IV. For values of (£', xf) in region IV, there Ind. Eng. Chern. Res . 1993, 32(9), 2023. is only one Hopf bifurcation point on the high branch. The steady states to the right of the Hopf bifurca- Doedel, E.J.; Champneys, A.R.; Fairgrieve, T.F.; Kuznetsov, Y.A.; Standstede, B. and X. Wang tion point are stable. The interval of the high branch (1997). AUTo97: Continuation and Bifurcabetween the left turning point and the Hopf bifurcation Software for Ordinary Differential Equation point consists of two parts separated by a homotions. Concordia University. Montreal, Canada. clinic bifurcation point: The unstable steady states Software documentation. in the right part are surrounded by stable limit cycles whereas there are no limit cycles around the unstable Giittinger, T. E.; Morari, M. Ind. Eng. Chern. Res. steady states in the left part. 1996, 35(3), 2816. The simulations revealed that oscillations are most Laroche, L.; Bekiaris, N.; Andersen, H. W.; Morari, likely to occur on the high branch under the condiM. AIChE Jouma11992, 38(9), 1309. tion of small xf and large £'. The occurrence of oscillation seems to be closely related to the distribu- Lee, 11.1.; Dorn, C.; Meski, G. A.j Morari, M. Limit tion of entrainer inventory inside the column. Small Cycles in Homogeneous Azeotropic Distillation, xf and large £' result in an entrainer profile with a Accepted for Ind. Eng. Chern. Res. 1999.
'- £
5194
Computers and Chemical Engineering Supplement (/999) SJ9J-S194
Matsuyama, H.; Nishimura, H. J. Chern. Eng. Jpn. 1977, 10(3), 181. JIC
l'=48
*
L'=192
Stewart, W. E.; Caracotsios, M.; Sorensen, J. P. DDASAC Software Package Documentation, University of Wisconsin, Madison, WI 53706, USA, 1995.
X~=O.l
unslableSS 0",-
19-'
>
J
,Ill.
II
..
/? I." '
o.
Ornensoonle»cbLlateflow rate
x:., •• ) 01
) 19 ".
Figure 2: Influence of L' and xf on the multiplicity and oscillatory behaviors: Xfll=9; the oscillations occur in the region bounded by the two *s; 0 denotes Hopf bifurcation points.
,.
O~
OirrIOf1$iQn~'c1I$~IaIeIO*'rate
i r? f
II-b
,/
v
.~
01
"j
II-a
to-t 19 ".
II
()mensionJess dtstlate!low rele
Figure 1: Influence of XfH on the multiplicity and oscillatory behaviors: L'=12; the oscillations occur in the region bounded by the two *s; 0 denotes Hopf bifurcation points.
10·'
Molarentrainerfraction
Figure 3: Classification of dynamic behavior in parameter space L' - xf: Xfll=9.
:~.~~ ~:~:'1"'r"'~ 1=_.wo.LC :.~SSS I 1( ., .
L
.••••
•
,
I
............... ~~ .•..~
.
...•.....
~
, L, - -
.....
5lat>IeSS
IIIslatleSS
33Of-------'--'-----' 09
091
092
0.93
094
095
0.96
0.91
Dimensionless distillate flow rate 0
0.01
x:
OCX"
'''';; OJ)OI; 1 ( .
Figure 4: Steady state bifurcation diagram (Ta v 9 vs D') for region III: L'=48; xf=0.007; xfll=9.
..
,~!;-.-----:;---~;;----~
"
[)me~d:$lbl'
flow raJa
.•fg~ ~
~i
~:i-r
:O'~) !
)'(
$
J
•
•
•
Dimensionless time
~=__ _<-U
"1!0.-----:;---~;---~ .1 1M Dmel"\SlOt\le" d3l.J
!
.'0'
Figure 5: Response of average temperature in the column to consecutive step changes in D': £1=48, xf=0.007, xfll=9, (i' = 33.3=i = Ih),
Computers and Chemical Engineering Supplem ent (1999) S191-5194 (,') 1999 Elsevier Science Ltd. All rights reserved PH: S0098·1354199/00171-4
Pergamon
Stability and Transient Behavior of Homogeneous Azeotropic Distillation Cornelius Dorn, Moonyong Leet , Manfred Morari" Automatic Control Laboratory, Swiss Federal Institute of Technology ETH-Zentrum, CH-8092 Zurich, Switzerland Abstract In this article, the steady state and dynamic behavior of an azeotropic distillation column separating the homogeneous ternary mixture acetone-benzene-heptane is studied. Nonlinear phenomena repor ted in earlier studies like multiple steady states (Bekiaris et aI., 1993) and limit cycles (Lee et al., 1999) are shown for this mixture. In addition, a new phenomenon is reported: the occurrence of homoclinic bifurcations resulting in cases of only one attractor in the presence of three steady states. It is discussed how these phenomena are influenced by operating parameters such as distillate and reflux flow rate, feed composition, etc.. The steady state and dynamic behavior is classified in the parameter space. Keywords: Azeotropic distillation, Multiple steady states, Instability, Oscillation, Hopf bifurcation, Homoclinic bifurcation.
INTRODUCTION Azeotropic distillation is an important separation technique which finds wide use in industry. The steady state and dynamic behavior of azeotropic distillation have been studied extensively over the past decades. This understanding is a necessary 'prerequisite for proper column design and operation. It has been shown that azeotropic distillation columns can exhibit unusual features not observed in nonazeotropic distillation (e.g. Laroche et al. (1992». In particular, multiplicity of steady states has been the focus of much recent research . The existence of multiple solutions in azeotropic distillation is now widely accepted both by theoreticians and practitioners. Throughout this work, the term "multiple steady states" (MSS) refers to output multiplicities only. A short overview of research related to the oscillatory behavior in azeotropic distillation is given in Lee et al. (1999). Recently, Lee et al. (1999) published simulation results showing sustained oscillations (limit. cycles) in the homogeneous azeotropic distillation of methanol-methyl butyrate-toluene using a CMO model. The results obtained with the CMO model were reconfirmed with a more complex model (RADFRAC in AspenDynamics) and two different VLE models. The main emphasis in this pap er is to understand how system parameters influence the multiplicity and oscillatory behavior of a homogeneous azeotropic distillation column. It will be shown that the column can have an even more complex bifurcation behavior than shown in the earlier study by Lee et al. (1999). SIMULATION SETUP Mixture and Column Model The ternary homogeneous mixture acetone-benzeneheptane investigated by Bekiaris et al. (1993) is chosen for the simulation study. The mixture belongs to the class 001 according to Matsuyama and Nishimura (1977). Acetone and heptane form the only minimum boiling binary azeotrope in this mixture at 93.56% (molar) acetone. The constant molar over-
flow (CMO) model is chosen (for details see e.g. Lee et al. (1999». Data for the simulation are given in Table 1. Vapor pressures and liquid activity coefficients are calculated with the Antoine equation and Wilson model, respectively. The thermodynamic parameters are taken from Giittinger and Morari (1996). For the dynamic simulations the CMO model is integrated with DDASAC (Stewart et al., 1995), an index-1 DAE solver. For the steady state bifurcation analysis, the CMO model is incorporated into AUTO (Doedel et al., 1997). To investigate the local dynam ic behavior and stability, the nonlinear CMO ' model is linearized and the eigenvalues are calculated with LAPACK (Anderson et al., 1995) at every steady state. The average tray temperature T a ug is chosen as a scalar index for the bifurcation diagrams and dynamic simulations. Number of trays (including condenser and reb oiler) Feed tray (counting from condenser) Tray liquid hold-up [kmol] Condenser liquid hold-up [kmol] Reboiler liquid hold-up [krnol] Column pressure [atm] Feed flow rate [kmol/h] Table 1: Column data for simulation
46 41 3 3 3 1.0 100
Parameterization of the Column Model The number of parameters needed to specify an actual distillation column model makes a complete parametric study impractical. However, once a mixture is given, it is possible to demonstrate the more relevant parametric effects by using appropriate dimensionless variables. The set of nonlinear coupled ODEs which describe the chosen Ci\[Q model for a ternary mixture can be rewritten in a compact vector form: dx/dt' = f(x ,D',P), where t' = t/(M/ F) is the dimensionless ti me. Further, I I 2 2 N N)T' h . x = ( XL,xll'XL'XH"" 'XL 'XH IS t e composttion vector of the liquid phases, D' = D / F is the dimensionless distillate rate used as the bifurcation
tSchool of Chern. Eng. and Tech., Yeungnam University, Dae-Dong 214·1, Korea. • Author to whom correspondence should be addressed. Phone: +41 1 632-2271. Fax: +41 1 632-1211. Email: [email protected]
S 192
Computers and Chemical Engin eering Supplement (/999) SJ9J-SJ94
variable, and P := (L', xf, xlII)T is the vector of op- closest complex eigenvalue was significantly smaller. erating parameters with L' := L/F as the dimension- Thus, it is sufficient to study the dominant eigenless reflux rate, xf as the entrainer fraction in the value for a classification of the dynamic behavior. feed, and XlII := xl/xf; as the ratio of light and Regions on the steady-state branches where the domheavy component fraction in the feed. inant eigenvalue is complex (i.e. either damped or sustained osciIlation) are bounded by stars in th e biINFLUENCE OF OPERATING PARAME- furcation plots. TERS ON BIFURCATION BEHAVIOR Effect of Light/Heavy Fraction. The bifurcations In order to understand how changes in the operating plotted in Figure 1 show that the influence of XlII is parameters L', xf, and XlII influence the multiplic- mainly to shift the MSS region towards higher vality and oscillatory behavior, a parametric study is ues of D'. The characteristic features (e.g. region of carried out. osciIlation or occurrence of Hopf bifurcation points) arc nearly unaffected. MSS Region With the 00/00 analysis (Bekiaris et al., 1993) it can Effect of Entrainer Fraction. As illustrated in be predicted that for the feed mixture studied here Figure 2, the dominant eigenvalue for steady states there will be multiple st eady states for all feed com- on the low branch was always negative .real (stable positions as long as the column is sufficiently long node). For steady states on the middle branch, it and the reflux is sufficiently high . Further, the loca- was always positive real (unstable). tion of the left and right turning points D L1 and Db For large values of xf, the steady states on the high is decreased, a can be predicted for the 00/00 case (infinite number branch are all stable nodes . As region appears and becomes more apparent on the of trays, infinit e reflux flow rate) to be at: high branch where the steady states have a complex dominant eigenvalue. Time responses starting close xlu D L1 := . (l-xn (1) to those steady states wiII show underdamped oscil1 + x[JI is further decreased, two Hopf bifurlations. As F { .~. x !==f xa~ xlII:5 XL~ cation points appear in the region of oscillation. The D L2 := (2) LH FL F > az steady states on the high branch in the region be(1 - xIl XLII _ XLII tween the two Hopf bifurcation points are -unstable In the abo ve equation, x'rll denotes the ratio of light and surrounded by stable limit cycles. to heavy component in the binary acetone-heptane Next, two homoc1inic bifurcation points appear on azeotrope (:::::: 14.5). the high branch between the two Hopf bifurcation Effect of Light/Heavy Fraction. In the 00/00 points. Therefor the interval of D' where stable limit case, both turning points are moved towards larger cycles exist is disconnected by a region where the values of D' and the size of the MSS region increases limit cycles disappear, as illustrated in Figure 4 (This F • . d til F az AS XLJI F • • interval is not illustrated in Figures 1 and 2 for claras XLII IS increase un 1 X LII := XLII' IS inity) . For values of D' in this interval this results in creased further, the right turning point D L2 remains while Db is still increasing. Hence the MSS region one stable and two unstable steady states, the former becomes narrower. Figure 1 shows bifurcation dia- being the only attractor. In Figure 4, the vertical disgrams varying XlII for several values of xf. The tance between the dots depicts the amplitude of the behavior of the MSS regions is in good qualitative limit cycles. The coalescence of limit cycles with the middle branch essentially causes the limit cycles to agreement with the predictions. Effect of Entrainer Fraction. In the 00/00 case, disappear. Figure 5 iIIustrates the time response of both turning points are moved towards larger values Ta v 9 to consecutive step changes. of D' and the size of the MSS region increases as xf is As shown in Figure 2, the region limited by the Hopf decreased. The results of a bifurcation analysis vary- bifurcation points expands further until for some ing xf for several values of L' are shown in Figure value of xf the left Hopf bifurcation point and the left 2. As it can be seen ; the MSS region moves toward homoc1inic bifurcation point together reach th e left decreases for given L'. However, turning point and disappear. Then, a region where higher D' as the MSS region initially expands and then narrows the dominant eigenvalue is positive real will appear decreases whereas the 00/00 analysis predicts on the high branch between the left turning point and as a monotonic expansion. This discrepancy is due to the homoclinic point (Figure 2c) thus disconnecting the finite reflux flow rate and finite number of trays the region of oscillation. The resulting left region of osciIIation shrinks towards the left turning point and in the simulations. Effect of Reflux Rate. For the example studied disappears while the right oscillation region becomes here multiplicities occur for all possible feed compo- gradually narrower (Figure 2d). sitions if L' is sufficiently large and the column has Effect of Reflux Rate. As illustrated in Figure enough trays. As L' decreases, the MSS region be- 2, the main characteristic features of the bifurcation comes smaller and finally disappears due to insuffi- also changes significantly with increasing £'. When xf is smaller than some critical value, the qualitative cient separation. effect of increasing L' is the same as that of decreasDynamic Behavior In this section, the steady states wiII be classified ac- ing xf: first a region of osciIlation appears on the cording to the eigenvalues of the local linearizations. high branch and gets larger, then a region of unstaThe eigenvalue(s) with the largest real part will be ble steady states surrounded by stable limit cycles appears within the region of oscillation, and so on. referred to as dominant. In all cases studied where the dominant eigenvalue was real, the real part of the Effect of Other Parameters. The size of both the
xf
xf
xf
xf
Computers and Chemical Engineering Supplement (1999) S191-S194
5193
1\'ISS region and the region of oscillation is increased sharp large peak. In this part of the column there as the number of trays in the column increases. The is steep front in the temperature profile bounded by liquid hold-up directly affects the period of oscilla- well developed pinch zones. tion. CONCLUSIONS In this paper, it was shown that homogeneous azeoCLASSES OF DYNAMIC BEHAVIOR In the preceding section, it was shown that the quali- tropic distillation can exhibit complex steady state tative bifurcation behavior is mainly influenced by £' and dynamic behavior. Based on the CMO model, it and xf whereas the main effect of xfl1 is to simply was studied how process operation affects the charshift both the MSS region and the region of oscil- acteristic features of multiplicity and stability. It was lations without changing their main characteristics. observed that both xf and £' have a significant effect This allows to focus on L' and xf to characterize the on the qualitative bifurcation behavior. In comparitypes of dynamic behavior. In this section, it will be son, the effect of XfH is to shift the MSS and oscillastudied how the (£', xf) parameter space is divided tion regions. The (£', xf) parameter space is divided into regions corresponding to qualitatively different into four regions according to the main features of steady state and dynamic behavior. For the example bifurcation behavior. A new phenomenon in homostudied in this paper, the regions are illustrated in geneous distillation was discovered: the occurrence of homoclinic bifurcation points. Thus, for certain Figure 3. Region I. For values of (£', xf) in region I, there is sets of operating parameters there exist three steady a unique steady state for every D'. All steady states states with only one attractor. are stable. NOTATION Region II. In subregion II-a, MSS exist but no oscillations occur throughout the high branch. All the F Molar feed flow rate dominant eigenvalues are real. In subregion II-b, D'=Q Dimensionless distillate flow rate damped oscillation appears for some range of D' on £ -F Dimensionless reflux flow rate the high branch. Molar liq. frac. of component c on tray n x~ Region III. This region also consists of two subre- To \19 Average temperature of all trays gions. Subregion III-a is a narrow band at the border t ' -- t.E.. Dimensionless time M with region II (not illustrated separately). For values M Molar liquid holdup on a tray of L' and xf in this region, there are two Hopf bifur- N Number of trays cation points on the high branch. All steady states x F Molar feed composition between these two Hopf bifurcation points are unsta- XfH Ratio of the feed light to heavy comp, frac. ble and surrounded by stable limit cycles. x~% Molar liq. frac. of comp. c of azeotrope Subregion III-b consists of the remainder of region III x'Lit Ratio of the az. light to heavy cornp. frac. and exhibits an interesting phenomenon newly observed in distillation: the occurrence of homo clinic Acknowledgement bifurcation points. The amplitude of the limit cycles This work was supported by the KOSEF through the around the high branch grows until the limit cycles Automation Research Center at POSTECH and by touch the middle branch. The interval of D' between the KOSEF - Swiss National Science Foundation exthe Hopf bifurcation points is therefore disconnected change program. by an interval bounded by the two homo clinic bifurcation points where the limit cycles disappear, as illustrated in Figure 4. This results in one stable and REFERENCES two unstable steady states, the former being the only Anderson, E.; Bai, Z.; Bischof, C.; Demmel, J.; Donattractor. garra, J. Lapack Users' Guide : Release 2.0, InAs £' is further increased (or xf further decreased) in ternational Society for Industrial and Applied region III, the left Hopf bifurcation point and the left Mathematics. homo clinic bifurcation point approach the left turning point. Bekiaris, N.; Meski, G. A.j Radu, C. 11.1.; Morari, r-.L Region IV. For values of (£', xf) in region IV, there Ind. Eng. Chern. Res . 1993, 32(9), 2023. is only one Hopf bifurcation point on the high branch. The steady states to the right of the Hopf bifurca- Doedel, E.J.; Champneys, A.R.; Fairgrieve, T.F.; Kuznetsov, Y.A.; Standstede, B. and X. Wang tion point are stable. The interval of the high branch (1997). AUTo97: Continuation and Bifurcabetween the left turning point and the Hopf bifurcation Software for Ordinary Differential Equation point consists of two parts separated by a homotions. Concordia University. Montreal, Canada. clinic bifurcation point: The unstable steady states Software documentation. in the right part are surrounded by stable limit cycles whereas there are no limit cycles around the unstable Giittinger, T. E.; Morari, M. Ind. Eng. Chern. Res. steady states in the left part. 1996, 35(3), 2816. The simulations revealed that oscillations are most Laroche, L.; Bekiaris, N.; Andersen, H. W.; Morari, likely to occur on the high branch under the condiM. AIChE Jouma11992, 38(9), 1309. tion of small xf and large £'. The occurrence of oscillation seems to be closely related to the distribu- Lee, 11.1.; Dorn, C.; Meski, G. A.j Morari, M. Limit tion of entrainer inventory inside the column. Small Cycles in Homogeneous Azeotropic Distillation, xf and large £' result in an entrainer profile with a Accepted for Ind. Eng. Chern. Res. 1999.
'- £
5194
Computers and Chemical Engineering Supplement (/999) SJ9J-S194
Matsuyama, H.; Nishimura, H. J. Chern. Eng. Jpn. 1977, 10(3), 181. JIC
l'=48
*
L'=192
Stewart, W. E.; Caracotsios, M.; Sorensen, J. P. DDASAC Software Package Documentation, University of Wisconsin, Madison, WI 53706, USA, 1995.
X~=O.l
unslableSS 0",-
19-'
>
J
,Ill.
II
..
/? I." '
o.
Ornensoonle»cbLlateflow rate
x:., •• ) 01
) 19 ".
Figure 2: Influence of L' and xf on the multiplicity and oscillatory behaviors: Xfll=9; the oscillations occur in the region bounded by the two *s; 0 denotes Hopf bifurcation points.
,.
O~
OirrIOf1$iQn~'c1I$~IaIeIO*'rate
i r? f
II-b
,/
v
.~
01
"j
II-a
to-t 19 ".
II
()mensionJess dtstlate!low rele
Figure 1: Influence of XfH on the multiplicity and oscillatory behaviors: L'=12; the oscillations occur in the region bounded by the two *s; 0 denotes Hopf bifurcation points.
10·'
Molarentrainerfraction
Figure 3: Classification of dynamic behavior in parameter space L' - xf: Xfll=9.
:~.~~ ~:~:'1"'r"'~ 1=_.wo.LC :.~SSS I 1( ., .
L
.••••
•
,
I
............... ~~ .•..~
.
...•.....
~
, L, - -
.....
5lat>IeSS
IIIslatleSS
33Of-------'--'-----' 09
091
092
0.93
094
095
0.96
0.91
Dimensionless distillate flow rate 0
0.01
x:
OCX"
'''';; OJ)OI; 1 ( .
Figure 4: Steady state bifurcation diagram (Ta v 9 vs D') for region III: L'=48; xf=0.007; xfll=9.
..
,~!;-.-----:;---~;;----~
"
[)me~d:$lbl'
flow raJa
.•fg~ ~
~i
~:i-r
:O'~) !
)'(
$
J
•
•
•
Dimensionless time
~=__ _<-U
"1!0.-----:;---~;---~ .1 1M Dmel"\SlOt\le" d3l.J
!
.'0'
Figure 5: Response of average temperature in the column to consecutive step changes in D': £1=48, xf=0.007, xfll=9, (i' = 33.3=i = Ih),