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A simple exact method for calculating tangent pinch points in multicomponent nonideal mixtures by bifurcation theory
Chemicd Engineering Scimre, Printed in Great Britain.
Vol
41, No
12, pp. 31X-3160.
1986.
OOW-2509/86 S3.00f0.00 Pergamon Journals Ltd.
A SIMPLE EXACT METHOD FOR CALCULATING TANGENT PINCH POINTS IN MULTICOMPONENT NONIDEAL MIXTURES BY BIFURCATION THEORY SANFORD G. LEVY and MICHAEL F. DOHERTY Chemical Engineering Department, University of Massachusetts, Amherst, MA 01003, U.S.A. (Received 24 May
1985)
Abstract-The tangent pinch problem for multicomponent systems has been investigated. It is shown that the notion of a binary tangent pinch, which is well understood, logically extends to the multicomponent case. A bifurcation analysis provides further insight into the mathematical conditions that apply at a tangent pinch, along with a procedure which serves to locate the minimum reflux ratio under tangent pinch conditions.
the operating line and equilibrium curve [see Fig. 1, (b) and (d)]. For a ternary system, the analog would be the meeting of an equilibrium surface and an operating line. The effect of changing reflux ratio in a multicomponent case is identical to a binary example. There is some critical reflux where the column profile suddenly pinches much earlier than expected. That is, if r is decreased by some small amount, the pinch (end-point) of one section of the column profile jumps to a new position.
INTRODUCTION
When designing a single-feed distillation column separate a nonideal binary mixture, it is not unusual
to to
encounter a tangent pinch. In this situation, there is a tangency between an operating line and the equilibrium curve on a McCabe-Thiele diagram at minimum reflux. However, because of the way that nonideal multicomponent distillations have been handled in the past, design procedures for systems of three or more components did not explicitly encounter tangent pinches. The design technique (e.g. Levy et al., 1985) that we use, however, will display the existence of a tangent pinch in exactly the same fashion as in a McCabeeThiele analysis. As in the binary case, there will be only one pinch zone at minimum reflux instead of one in each section of the column. Most often, this pinch occurs in the rectifying section, although it may be found in either section of the column. We have found tangent pinches to occur quite frequently during our investigation of multicomponent nonideal and azeotropic column a more EFFECX
design and therefore, the subject is deserving of detailed discussion than is currently available. OF A TANGENT
PINCH
ON
COLUMN
PROFILES
tangent pinch is of interest because of its effect on the calculations for distillation design. In the following discussion, binary vapor-liquid equilibrium diagrams will be shown to illustrate many of the The
concepts. The use of ternary equilibrium diagrams is not possible because of the existence of two independent variables (x1, x1) and two dependent variables (yl, y2). This is presumably one reason that the idea of a tangent pinch has been solely associated with binary design problems. When using our differential method or a plate-to-plate method to determine the column design parameters for a ternary system, it is common to find a region on a liquid-phase composition diagram which requires many trays to move past (the profile is near a pinch). As the reflux ratio is slightly decreased, the end-point of the column profile, x,, may suddenly jump to this pinch [see Fig. l(a) and (c)l. In two-component design, this can be explained by the close approach and then tangent intersection between
When a column end-pinch position The discontinuous
is designed, it is expected that the moves as a continuous function of r. movement seen in Fig. 1 is charac-
teristic of a tangent pinch. In the three examples to follow, a small decrease in I produces a drastic change in the position of the end-point. In all three cases, the standard
geometrical
representation
of
minimum
reflux (Levy et al., 1985) does not appear. It is not possible to find a reflux ratio such that there will be a pinch in each section of the column. Figure 2 for a nonideal system and Figs 3 and 4 for azeotropic systems all show the effect of a tangent pinch. The van Laar model was used to represent the liquid-phase nonidcalities in the water/acetone/isopropanol system, the others were represented by the two parameter Margules model. All vapor-liquid equilibrium calculations are carried out at a constant pressure of 1 atm. In each of these examples, when r is less than the tangent pinch value, one of the profiles ends abruptly before meeting the other and when r is above it, the profiles cross by a large amount. For these cases, the minimum
reflux ratio would
be the lowest reflux where
the separation was still feasibIe (i.e. the profiles cross). Notice that this implies that we do not have an infinite number of trays in both sections of the column under minimum reflux conditions. MATHEMATICAL
ANALYSIS
Underlying this distillation theory are some simple mathematical concepts which we can exploit. A tangent pinch is a point of bifurcation. This can be seen in the following manner. When the reflux ratio is greater than the minimum, a rectifying section phase plane [Fig.
3155
3156
SANFORD
G.
LEVY
and MICHAEL
F. DOHERTY
d.
C.
r = honpent Pinch Fig. 1. Comparison
of a binary
--E
and a ternary tangent
pinch.
5(a)] shows a simple structure with only one stable node (i.e. the end-point pinch of the profile). A phase plane is produced by integrating the equations describing the column section beginning at several different initial conditions, recognizing that only one initial condition, x,, = xD, actually makes physical sense. Just
As in reactor analysis, and bifurcation theory in general, it is well known that the Jacobian matrix of the first-order derivatives will be singular at the bifurcation point, i.e. det J = 0. This provides the starting point for some analysis. The most convenient formulation is to use the differential rectifying section equations derived
below rmio, the phase plane has two nodes and one saddle point [Fig. 5(c)]. Therefore, a node-saddle must exist at rmin as demonstrated in Fig. 5(b). For binary systems, the analogs to Figs 5(a j(c) can be seen in Figs 6(at(c). Three intersections, corresponding to a nodesaddle-node structure, occur if the reflux ratio is less than the tangent pinch value. If r is greater than the tangent pinch value, only one node exists. This problem is reminiscent of the stability analysis for a nonisothermal stirred tank reactor with the reaction A + B. On the ternary system phase planes, as the reflux ratio is increased from below the tangent pinch value, the node to the left and the saddle approach each other. Exactly at rmin (the tangency), they coincide to form a nonelementary node-saddle before disappearing as r becomes greater than r,,_,,”_
by Levy et al. (1985). These equations are exact under pinch conditions (i.e. where dxi/dh = 0) and therefore, introduce no additional approximations into the tangent pinch analysis. The equations are dxi r-t1 dh = xi--yYi+;Y,,i we find
r. J=
or
dyl
I
ax,
1
dY1 r+ 1
r+l r
dx,
l--axP 2 r+l r
Y
(1)
1
r (2)
aYzr+l
aYzr+l -___ ax, r J=I--
1,2
i=
r (3)
Calculation of tangent pinch points
3157
(a) Methanol 10
(b)
(b) Acetone
Methanol
0
02’
rsoproponol
Fig. 2. Composition
profile at (a) r = 0.568, (b) r = 0.570.
y=
where
[
2 J
1 P,X’
det.I=det(I-FY)=O, This can be rearranged det
into the equivalent
form (5)
From eq. (5), we see that at the point of bifurcation, one of the eigenvalues of the matrix Y becomes equal to r/(r+ 1). This is a natural extension of the binary tangent pinch condition. Since for binary mixtures the matrix Y reduces to dy, /dxl, the eigenvalue condition for binary mixtures is simply dy,/dx, = r/r + 1 at the tangent pinch. This requires that the slope of the equilibrium curve be equal to the slope of the operating line at tangent pinch conditions, which is the standard McCabe-Thiele construction. As an aside, it is worth
Fig. 3. Composition
04
06
X,
0.8
Woter
profile at (a) r = 0.76, (b) r = 0.77
noting that for multicomponent mixtures, the eigenvalues of Y will generally be strictly positive and distinct. For multicomponent nonideal mixtures with constant latent heat, this property of Y is exact (see Doherty, 1985). Therefore, at a tangent pinch, there will be a single zero eigenvalue, which is well known to correspond to the simplest kind of bifurcation structures in the phase plane. In particular, this kind of bifurcation generates node-saddle structures exactly as observed in Fig. 5(b). Because the bifurcation is uniquely described by eq. (5), an algorithm can be constructed which easily locates the tangent pinch. This algorithm alleviates the need for a tedious trial-and-error search to determine whether a tangent pinch occurs and if it does, to locate the proper rmj, (as done in Figs 2, 3 and 4). Since the reflux ratio located is exactly at the tangent pinch, any increase in r will allow the column to be operated with a finite number of plates. Our analysis produces a system of three equations in three unknowns (r, x,, 1 and x,, z)
SANFORD
G.
LEW
and
Acetone
Chloroform
MICHAEL
F. DOHERTY
lsoproponol
(b)
(b)
Benzene
Acetone
‘Oil
08
06
X2
04 02 Iv
Acetone
Chloroform Fig. 4. Composition
which
can
profile
be readily
at (a) r = 7.50, (b) r = 7.70
solved
root finding techniques conditions. The equations are:
to
using standard find
the
tangent
det J = 0 r+l Xi-y_Vt+iyO,i
=O
nonlinear pinch
lsoproponol
(cl Acetone
(64 i =
1,2.
(6b)
When this method is used to locate the tangent pinch for the examples of Figs 2, 3 and 4, the results given in Table 1 are obtained. Comparing these results to Figs 24, we see that all of the pinched figures [Figs 2(a), 3(a), 4(a)] are below the calculated tangent pinch values, while the figures which display a feasible column [Figs 2(b), 3(b), 4(b)] are all at a reflux ratio slightly greater than the tangent pinch value. This procedure provides a noniterative technique which locates the precise reflux ratio corresponding to the tangent pinch. The method as discussed here was applied to three
lsopropanol Fig. 5. Phase plane at (a) r = 0.85, (b) r = 0.84,
Water (c) r = 0.80
3159
Calculation of tangent pinch points
(b)
(01
Fig. 6. Structures analogous to Fig. 5 for the binary case.
Table 1. Calculated System Acetaldehyde methanol water Water acetone isopropanol Acetone benzene chloroform
of a binary tangent pinch, which is well understood, logically extends to the multicomponent case. A bifur-
values of the tangent pinch rmin
xe,
1
xe.
2
0.569
0.156
0.617
0.765
0.376
0.161
7.560
0.498
0.016
component systems. It is easily extended for a C component mixture to a system of C equations and C unknowns. There will be C-l mass balance equations and one equation of the type det J = 0. Also, since no assumption was ever made about the vapor-liquid equilibrium model, these ideas and the methods may be applied with any model. CONCLUSIONS The tangent pinch problem for multicomponent systerns has been investigated. It is shown that the notion
cation analysis provides further insight into the mathematical conditions that apply at a tangent pinch, along with a procedure which serves to locate the minimum reflux ratio under tangent pinch conditions. Acknowledgements-This research was supported by the National Science Foundation (grant number CfE-8406983) and by the Petroleum Research Fund, administered by the American Chemical Society.
NOTATION h
xi
continuous plate number identity matrix Jacobian matrix of first derivatives pressure reflux ratio composition of species i in the liquid
y1 Y
composition of species i in the vapor phase matrix of derivatives of y with respect to x
I J P r
phase
3160 Subscripts D distillate product e end point i component number min minimum n node 0 initial condition
SANFORD
G. LEVY
and MICHAELF.DOHERTY REFERENCES Doherty, M. F., 1985, Properties of liquid-vapour composition surfaces for multicomponent mixtures with constant latent heat. Chem. Engng Sci. 40, 1979-1980. Levy, S. G., Van Dongen, D. B. and Doherty, M. F., 1985, Design and synthesis of homogeneous azeotropic distillations--II. Minimum reflux calculations for nonideal and azeotropic columns. Ind. Engng Chem. Fundam. 24, 463474.
Vol
41, No
12, pp. 31X-3160.
1986.
OOW-2509/86 S3.00f0.00 Pergamon Journals Ltd.
A SIMPLE EXACT METHOD FOR CALCULATING TANGENT PINCH POINTS IN MULTICOMPONENT NONIDEAL MIXTURES BY BIFURCATION THEORY SANFORD G. LEVY and MICHAEL F. DOHERTY Chemical Engineering Department, University of Massachusetts, Amherst, MA 01003, U.S.A. (Received 24 May
1985)
Abstract-The tangent pinch problem for multicomponent systems has been investigated. It is shown that the notion of a binary tangent pinch, which is well understood, logically extends to the multicomponent case. A bifurcation analysis provides further insight into the mathematical conditions that apply at a tangent pinch, along with a procedure which serves to locate the minimum reflux ratio under tangent pinch conditions.
the operating line and equilibrium curve [see Fig. 1, (b) and (d)]. For a ternary system, the analog would be the meeting of an equilibrium surface and an operating line. The effect of changing reflux ratio in a multicomponent case is identical to a binary example. There is some critical reflux where the column profile suddenly pinches much earlier than expected. That is, if r is decreased by some small amount, the pinch (end-point) of one section of the column profile jumps to a new position.
INTRODUCTION
When designing a single-feed distillation column separate a nonideal binary mixture, it is not unusual
to to
encounter a tangent pinch. In this situation, there is a tangency between an operating line and the equilibrium curve on a McCabe-Thiele diagram at minimum reflux. However, because of the way that nonideal multicomponent distillations have been handled in the past, design procedures for systems of three or more components did not explicitly encounter tangent pinches. The design technique (e.g. Levy et al., 1985) that we use, however, will display the existence of a tangent pinch in exactly the same fashion as in a McCabeeThiele analysis. As in the binary case, there will be only one pinch zone at minimum reflux instead of one in each section of the column. Most often, this pinch occurs in the rectifying section, although it may be found in either section of the column. We have found tangent pinches to occur quite frequently during our investigation of multicomponent nonideal and azeotropic column a more EFFECX
design and therefore, the subject is deserving of detailed discussion than is currently available. OF A TANGENT
PINCH
ON
COLUMN
PROFILES
tangent pinch is of interest because of its effect on the calculations for distillation design. In the following discussion, binary vapor-liquid equilibrium diagrams will be shown to illustrate many of the The
concepts. The use of ternary equilibrium diagrams is not possible because of the existence of two independent variables (x1, x1) and two dependent variables (yl, y2). This is presumably one reason that the idea of a tangent pinch has been solely associated with binary design problems. When using our differential method or a plate-to-plate method to determine the column design parameters for a ternary system, it is common to find a region on a liquid-phase composition diagram which requires many trays to move past (the profile is near a pinch). As the reflux ratio is slightly decreased, the end-point of the column profile, x,, may suddenly jump to this pinch [see Fig. l(a) and (c)l. In two-component design, this can be explained by the close approach and then tangent intersection between
When a column end-pinch position The discontinuous
is designed, it is expected that the moves as a continuous function of r. movement seen in Fig. 1 is charac-
teristic of a tangent pinch. In the three examples to follow, a small decrease in I produces a drastic change in the position of the end-point. In all three cases, the standard
geometrical
representation
of
minimum
reflux (Levy et al., 1985) does not appear. It is not possible to find a reflux ratio such that there will be a pinch in each section of the column. Figure 2 for a nonideal system and Figs 3 and 4 for azeotropic systems all show the effect of a tangent pinch. The van Laar model was used to represent the liquid-phase nonidcalities in the water/acetone/isopropanol system, the others were represented by the two parameter Margules model. All vapor-liquid equilibrium calculations are carried out at a constant pressure of 1 atm. In each of these examples, when r is less than the tangent pinch value, one of the profiles ends abruptly before meeting the other and when r is above it, the profiles cross by a large amount. For these cases, the minimum
reflux ratio would
be the lowest reflux where
the separation was still feasibIe (i.e. the profiles cross). Notice that this implies that we do not have an infinite number of trays in both sections of the column under minimum reflux conditions. MATHEMATICAL
ANALYSIS
Underlying this distillation theory are some simple mathematical concepts which we can exploit. A tangent pinch is a point of bifurcation. This can be seen in the following manner. When the reflux ratio is greater than the minimum, a rectifying section phase plane [Fig.
3155
3156
SANFORD
G.
LEVY
and MICHAEL
F. DOHERTY
d.
C.
r = honpent Pinch Fig. 1. Comparison
of a binary
--E
and a ternary tangent
pinch.
5(a)] shows a simple structure with only one stable node (i.e. the end-point pinch of the profile). A phase plane is produced by integrating the equations describing the column section beginning at several different initial conditions, recognizing that only one initial condition, x,, = xD, actually makes physical sense. Just
As in reactor analysis, and bifurcation theory in general, it is well known that the Jacobian matrix of the first-order derivatives will be singular at the bifurcation point, i.e. det J = 0. This provides the starting point for some analysis. The most convenient formulation is to use the differential rectifying section equations derived
below rmio, the phase plane has two nodes and one saddle point [Fig. 5(c)]. Therefore, a node-saddle must exist at rmin as demonstrated in Fig. 5(b). For binary systems, the analogs to Figs 5(a j(c) can be seen in Figs 6(at(c). Three intersections, corresponding to a nodesaddle-node structure, occur if the reflux ratio is less than the tangent pinch value. If r is greater than the tangent pinch value, only one node exists. This problem is reminiscent of the stability analysis for a nonisothermal stirred tank reactor with the reaction A + B. On the ternary system phase planes, as the reflux ratio is increased from below the tangent pinch value, the node to the left and the saddle approach each other. Exactly at rmin (the tangency), they coincide to form a nonelementary node-saddle before disappearing as r becomes greater than r,,_,,”_
by Levy et al. (1985). These equations are exact under pinch conditions (i.e. where dxi/dh = 0) and therefore, introduce no additional approximations into the tangent pinch analysis. The equations are dxi r-t1 dh = xi--yYi+;Y,,i we find
r. J=
or
dyl
I
ax,
1
dY1 r+ 1
r+l r
dx,
l--axP 2 r+l r
Y
(1)
1
r (2)
aYzr+l
aYzr+l -___ ax, r J=I--
1,2
i=
r (3)
Calculation of tangent pinch points
3157
(a) Methanol 10
(b)
(b) Acetone
Methanol
0
02’
rsoproponol
Fig. 2. Composition
profile at (a) r = 0.568, (b) r = 0.570.
y=
where
[
2 J
1 P,X’
det.I=det(I-FY)=O, This can be rearranged det
into the equivalent
form (5)
From eq. (5), we see that at the point of bifurcation, one of the eigenvalues of the matrix Y becomes equal to r/(r+ 1). This is a natural extension of the binary tangent pinch condition. Since for binary mixtures the matrix Y reduces to dy, /dxl, the eigenvalue condition for binary mixtures is simply dy,/dx, = r/r + 1 at the tangent pinch. This requires that the slope of the equilibrium curve be equal to the slope of the operating line at tangent pinch conditions, which is the standard McCabe-Thiele construction. As an aside, it is worth
Fig. 3. Composition
04
06
X,
0.8
Woter
profile at (a) r = 0.76, (b) r = 0.77
noting that for multicomponent mixtures, the eigenvalues of Y will generally be strictly positive and distinct. For multicomponent nonideal mixtures with constant latent heat, this property of Y is exact (see Doherty, 1985). Therefore, at a tangent pinch, there will be a single zero eigenvalue, which is well known to correspond to the simplest kind of bifurcation structures in the phase plane. In particular, this kind of bifurcation generates node-saddle structures exactly as observed in Fig. 5(b). Because the bifurcation is uniquely described by eq. (5), an algorithm can be constructed which easily locates the tangent pinch. This algorithm alleviates the need for a tedious trial-and-error search to determine whether a tangent pinch occurs and if it does, to locate the proper rmj, (as done in Figs 2, 3 and 4). Since the reflux ratio located is exactly at the tangent pinch, any increase in r will allow the column to be operated with a finite number of plates. Our analysis produces a system of three equations in three unknowns (r, x,, 1 and x,, z)
SANFORD
G.
LEW
and
Acetone
Chloroform
MICHAEL
F. DOHERTY
lsoproponol
(b)
(b)
Benzene
Acetone
‘Oil
08
06
X2
04 02 Iv
Acetone
Chloroform Fig. 4. Composition
which
can
profile
be readily
at (a) r = 7.50, (b) r = 7.70
solved
root finding techniques conditions. The equations are:
to
using standard find
the
tangent
det J = 0 r+l Xi-y_Vt+iyO,i
=O
nonlinear pinch
lsoproponol
(cl Acetone
(64 i =
1,2.
(6b)
When this method is used to locate the tangent pinch for the examples of Figs 2, 3 and 4, the results given in Table 1 are obtained. Comparing these results to Figs 24, we see that all of the pinched figures [Figs 2(a), 3(a), 4(a)] are below the calculated tangent pinch values, while the figures which display a feasible column [Figs 2(b), 3(b), 4(b)] are all at a reflux ratio slightly greater than the tangent pinch value. This procedure provides a noniterative technique which locates the precise reflux ratio corresponding to the tangent pinch. The method as discussed here was applied to three
lsopropanol Fig. 5. Phase plane at (a) r = 0.85, (b) r = 0.84,
Water (c) r = 0.80
3159
Calculation of tangent pinch points
(b)
(01
Fig. 6. Structures analogous to Fig. 5 for the binary case.
Table 1. Calculated System Acetaldehyde methanol water Water acetone isopropanol Acetone benzene chloroform
of a binary tangent pinch, which is well understood, logically extends to the multicomponent case. A bifur-
values of the tangent pinch rmin
xe,
1
xe.
2
0.569
0.156
0.617
0.765
0.376
0.161
7.560
0.498
0.016
component systems. It is easily extended for a C component mixture to a system of C equations and C unknowns. There will be C-l mass balance equations and one equation of the type det J = 0. Also, since no assumption was ever made about the vapor-liquid equilibrium model, these ideas and the methods may be applied with any model. CONCLUSIONS The tangent pinch problem for multicomponent systerns has been investigated. It is shown that the notion
cation analysis provides further insight into the mathematical conditions that apply at a tangent pinch, along with a procedure which serves to locate the minimum reflux ratio under tangent pinch conditions. Acknowledgements-This research was supported by the National Science Foundation (grant number CfE-8406983) and by the Petroleum Research Fund, administered by the American Chemical Society.
NOTATION h
xi
continuous plate number identity matrix Jacobian matrix of first derivatives pressure reflux ratio composition of species i in the liquid
y1 Y
composition of species i in the vapor phase matrix of derivatives of y with respect to x
I J P r
phase
3160 Subscripts D distillate product e end point i component number min minimum n node 0 initial condition
SANFORD
G. LEVY
and MICHAELF.DOHERTY REFERENCES Doherty, M. F., 1985, Properties of liquid-vapour composition surfaces for multicomponent mixtures with constant latent heat. Chem. Engng Sci. 40, 1979-1980. Levy, S. G., Van Dongen, D. B. and Doherty, M. F., 1985, Design and synthesis of homogeneous azeotropic distillations--II. Minimum reflux calculations for nonideal and azeotropic columns. Ind. Engng Chem. Fundam. 24, 463474.