Incorporating Tangent Pinch Points into the Conceptual Modeling of

Ind. Eng. Chem. Res. 2009, 48, 857–869

857

Incorporating Tangent Pinch Points into the Conceptual Modeling of Batch Distillations: Ternary Mixtures Karina Andrea Torres and Jose´ Espinosa* INGAR-CONICET, AVellaneda 3657, S3002 GJC Santa Fe, Argentina

This contribution explores the influence of tangent pinch points on the performance of batch distillations of highly nonideal ternary mixtures and its incorporation into a conceptual modeling framework under the assumption of a rectifier with an infinite number of stages. The maximum feasible distillate composition on the line of preferred separation and its corresponding limiting reflux ratio are first determined with the aid of bifurcation analysis of reversible distillation profiles. Then, the dependence of feasible distillate mole fractions on reflux ratios above the limiting one is calculated by solving a nonlinear equation system, which incorporates the tangency condition. Results obtained from the conceptual model for instantaneous column performance are in excellent agreement with those calculated from rigorous simulation. Two highly nonideal ternary mixtures are studied. 1. Conceptual Models in Batch Distillation and Problem Statement Calculation of the maximum feasible distillate composition on the equilibrium vector or preferred separation line for a given instantaneous still composition is the key ingredient of a conceptual model which attempts to incorporate the influence of tangent pinches on the performance of batch distillations. This problem is trivial for mixtures whose separation is not controlled by tangent pinch points. In this case and focusing our attention on ternary mixtures, a rectifier with an infinite number of equilibrium stages shows a pinch at the lower end with a composition identical to that of the liquid mixture in the still. The maximum feasible distillate composition on the preferred line is found in the intersection of the equilibrium vector with the binary axis corresponding to the two more volatile components. The corresponding value of the limiting reflux ratio R* is then calculated from the lever arm rule, i.e., the mass balance around the rectifier. For reflux ratios between 0 and the limiting one, the corresponding distillate compositions are located on the preferred separation line with the invariant pinch point behaving as a stable node. For values of the reflux ratio r above R*, feasible distillates formed by binary mixtures of the two more volatile component are expected as shown in Figure 1. This concept is an extension of the behavior of continuously operated columns at minimum reflux1 to the instantaneous performance of batch rectifiers with infinite numbers of trays.2 Duessel2 was the first to extend the ideas of Offers et al.1 from continuous to batchwise operation of a rectifier with distillate cuts of constant composition (i.e., xD in Figure 1). In his work, Duessel2 proposed to solve the eigenvalue problem of the Jacobian of the equilibrium function in the instantaneous still composition xB to obtain a linearization of the adiabatic profile in the neighborhood of the still composition and, hence, an estimation of the instantaneous minimum reflux ratio. Figure 1 shows the basic ideas of the approach developed by Duessel in graphic form. Note that, for an instantaneous still composition xB, the feed to the column is a vapor stream whose composition y/xB is in phase equilibrium with the composition of the liquid in the still and the liquid leaving the lower column end xN is * To whom correspondence should be addressed. E-mail: destila@ santafe-conicet.gov.ar.

located on the linear approximation of the adiabatic profile departing from xB. The mass balance envelope used in the conceptual model is shown in Figure 2. On the other hand, Espinosa and Salomone3 adapted the approach to handle the operation at constant reflux ratio. To this end, they identified three different situations: (i) For values of the reflux ratio r between 0 and R* ) (xD/ y/xB)/(y/xB - xB), the distillate compositions are aligned with the equilibrium vector or preferred separation line, starting from y/xB when the reflux equals 0 up to the point xD/ where the composition of the heaviest component becomes 0. All these separations are controlled by a pinch at the rectifier lower end whose composition is identical to that of the mixture in the still. (ii) For values of the reflux ratio r between R* and R** ) (xD// - y/xB)/(y/xB - xN,max), the distillate compositions are located in the binary edge corresponding to the lighter components between xD/ andxD//. xN,max is calculated from the intersection of vectors (xD// - y/xB) and xN ) xB + λν, which results from the

Figure 1. Instantaneous minimum reflux ratio for given still and distillate compositions calculated from the linearization of the adiabatic profile in the neighborhood of the still composition. Region of feasible distillate mole fractions.

10.1021/ie801169x CCC: $40.75  2009 American Chemical Society Published on Web 12/15/2008

858 Ind. Eng. Chem. Res., Vol. 48, No. 2, 2009

Figure 2. Envelope of the mass balance used in the conceptual modeling of batch distillations.

linearization of the adiabatic profile in the neighborhood of the still composition. There is no more a pinch at the end of the column, but a binary saddle pinch xPII remains with invariant composition. (iii) For r g R** the distillate composition equals xD//, the composition of the more volatile species with a pinch at the top of the column controlling the separation. Depending on the value of the reflux ratio, the conceptual model estimates the value of the instantaneous distillate composition. To approximate the complex behavior of highly nonideal mixtures, both azeotrope4,5 and distillation region6,7 calculations are incorporated into the conceptual modeling framework to take into account linear approximations of unstable distillation boundaries. Curved still paths following part of stable distillation boundaries are enforced by constraining instant feasible distillates to be located on the line of preferred separation.8 Only two equilibrium calculations are needed to determine a swap in the unstable node (light pure component or azeotrope), which in turn indicates the crossing of a stable boundary.9 Once this condition is verified, only step I of the algorithm above is performed to determine the feasible distillate composition. The conceptual model can also be extended to estimate instantaneous column performance for highly nonideal or even azeotropic mixtures. Figure 3 shows the performance of a column separating a mixture acetone/chloroform/benzene/ toluene. Linearization of the internal profile in the neighborhood of the composition of the mixture in the boiler produces a plane which approximates very well the behavior of the adiabatic profile simulated in Hysys.10 Besides the linearization of the adiabatic profile in the neighborhood of the still composition (Offers et al.1 and Duessel2 for operation at constant distillate composition; Espinosa and Salomone3 for operation at constant reflux ratio), there is another approach to calculate the instantaneous rectifier performance, which is based on the calculation of rectification bodies.11 While linearization needs the solution of an eigenvalue problem of the Jacobian of the equilibrium function in xB, the rectification body method approximates the manifold of all potential profiles by linearly connecting controlling pinch points.12 Both approaches make the assumption that the controlling pinch points are invariant, which is true for ideal mixtures but only an approximation for highly nonideal systems. In order to model different operation modes (constant distillate composition, constant reflux ratio, or a combination of both) of

Figure 3. System acetone/chloroform/benzene/toluene. Minimum reflux ratio for instantaneous still composition xB. The composition xN of the stream leaving the rectifier lower end belongs to the plane defined by the controlling pinch points xB, xPIII, and xPII. Intersection between the mass balance line xD-yx/B and the hyperplane formed by the controlling pinch points gives the exact location of xN.

a batch rectifier, the following set of algebraic and differential equations, which are written in terms of component recoveries and rectification advance, must be integrated:13 xDi dσDi ) B dη x

(1)

i0

1 - σDi (2) 1-η Instant values of either the minimum reflux ratio or the distillate composition that are required to integrate the equation system (1) and (2) are obtained from the instantaneous column performance model explained in this section. The results obtained from the whole conceptual model (differential equations and instantaneous column performance) for the different cuts can then be used as initialization of a rigorous dynamic optimization of a batch rectifier with a finite number of equilibrium stages.14 Even when conceptual models have demonstrated to be powerful tools in the design of separations by batch distillation, some issues still remain as subjects of study. In this contribution, we will focus on the incorporation of tangent pinches into the conceptual modeling of batch distillations. We will demonstrate that the occurrence of tangent pinches in batch distillation of highly nonideal mixtures affects the maximum distillate composition achievable on the preferred line. Similarly to ideal mixtures, the lower pinch controls the separation for values of the reflux ratio within an interval with bounds [0;rbif]. At r ) rbif, the lever arm rule determines the maximum feasible distillate composition xDbif for which the corresponding pinch point curve shows a saddle-node bifurcation at the instantaneous still composition xB. This contribution explores different methods to determine [rbif;xDbif] from the detection of saddle-node bifurcations in reversible profiles. As operation at reflux ratios above the bifurcation value is controlled by tangent pinches of unknown composition, a method to determine the dependence of feasible distillate mole fractions on the rectifier energy demand is also proposed. Thus, as in the case of ideal distillation, the whole xBi ) xBi0

Ind. Eng. Chem. Res., Vol. 48, No. 2, 2009 859

range of feasible distillate compositions corresponding to an instantaneous still composition is obtained. 2. Quantitative Determination of Profiles of Reversible Distillation From a qualitative standpoint, a profile of reversible distillation corresponding to a given distillate composition consists of a series of liquid compositions that fulfills two conditions: (i) Each composition represents a pinch (i.e., it is in equilibrium with its vapor). (ii) Each composition obeys the mass balance around the rectifier (i.e., the liquid, its vapor in equilibrium, and the distillate are aligned in the composition simplex). Reversible profiles or pinch point curves have been profusely used in minimum energy demand calculations of columns operated either continously12 or batchwise,11 taking into account that points belonging to a pinch point curve of a given distillate composition represent end points of adiabatic profiles calculated from the same distillate composition for reflux ratios varying from zero to infinity. In order to calculate the entire reversible profile for a given distillate composition p, let us define x, y, and p as the compositions of a liquid, vapor, and distillate, respectively. Vectors x - p and y - x must be aligned according to the mass balance. In mathematical terms x - p ) s(y - x)

(3)

where s is the ratio of the lengths of the vectors. Let us define xk as the kth component of x, s ) (xk - pk)/(yk - xk), and then, for i * k, eq 3 can be rewritten as xi - pi )

xk - pk (y - xi) yk - xk i

or alternatively as (xi - pi)(yk - xk) - (xk - pk)(yi - xi) ) 0

(4)

The index k must be chosen in such a way that both yk - xk and xk - pk are different from 0. As each liquid composition represents a pinch, mole fractions of liquid and vapors must satisfy the equilibrium equation yi )

γip0i x ) κixi (i ) 1, ..., c) p i

(5)

where γi is the activity coefficient of component i, p0i is the vapor pressure of component i, p is the system total pressure, and κi is the corresponding equilibrium constant. Inserting eq 5 into eq 4 and choosing k ) c, it follows the equation developed by Poellmann and Blass:15 fc,i ) κcxc(xi - pi) - κixi(xc - pc) + xcpi - xipc ) 0 (i ) 1, ..., c - 1) (6) Equation 6 is a nonlinear algebraic system of equations: bf c(T, b x) ) b 0 which implicitly defines the profile of reversible distillation in the composition space as a function of temperature. By way of implicit differentiation, the mentioned authors obtained the total differential of eq 6 that can be interpreted as an inhomogeneous system of linear equations for the derivative (dx/dT)Σ of the reversible profile with respect to the temperature:

( )(

∂ bf c dbf c(T, b x) ∂fc,i )b 0) + dT ∂T ∂xj

Σ

dx b dT

Σ

)

(7)

where both ∂fbc/∂T and (∂fc,i/∂xj)Σ can be analytically calculated. The superscript “Σ” in eq 7 indicates that the mole fraction summation over all components is accounted for. All partial derivatives are analytically calculated as if all mole fractions were independent, but one has to subtract the cth column of the matrix of partial derivatives from all other columns to obtain physically correct derivatives.15,16 We refer the interested reader to the paper of Poellmann and Blass15 for a more detailed analysis of this subject. Summarizing, calculation of both ∂fbc/∂T and (∂fc,i/∂xj)Σ allows the numerical integration of the derivative (dx/dT)Σ and, therefore, the calculation of the entire reversible profile for a given distillate p. Typically, the initial integration point corresponds to a singular point such as a pure component or azeotrope, which is a trivial pinch point. Figure 4 presents two common ways of showing reversible profiles for a given distillate mole fraction: light species composition versus intermediate component mole fraction and the composition of any of the components of the mixture versus the reflux ratio. In this figure, two stable zones and one saddle region are shown. Stability is related to the behavior of adiabatic profiles departing from a distillate of composition p. Evaluation of the tray-by-tray profiles starting from p shows that only pinch points on the stable regions can be reached. For values of the reflux ratio between 0 and 0.68, tray-by-tray profiles end at the upper stable section of Figure 4a. For r > 0.68, on the other hand, the adiabatic profiles have their termination points at the lower stable zone. Two solutions appear at the limiting reflux ratio, with the upper (lower) solution in Figure 4a-c (Figure 4d) corresponding to a saddle-node bifurcation of the pinch profile. As we will see, the local maximum of the pinch curve is responsible for the appearance of tangent pinch points in batch distillations of highly nonideal mixtures. 3. Behavior of Reversible Profiles along the Line of Preferred Separation For ideal systems, reversible profiles calculated for different distillate compositions on the line of preferred separation have two characteristics in common: all of them travel across the still composition and the corresponding distillate mole fraction is achievable from xB; i.e., it behaves like a stable node for the adiabatic profile. In other words, calculation of an adiabatic profile using a tray-by-tray procedure from a distillate mole fraction xD pertaining to the line determined from the equilibrium vector (y/xB - xB) and at a reflux ratio r ) (xD - y/xB)/(y/xB - xB) gives rise to a curve in the composition simplex with the termination point at x ) xB. Strictly speaking, if we take into account that in batch distillation the feed to the column is the vapor in equilibrium with xB, the still composition behaves like an unstable node. For the sake of simplicity, however, we will maintain throughout this paper the terminology adopted in continuous distillation.17 For highly nonideal mixtures, the behavior of pinch point curves can be totally different. Figure 5 depicts the behavior of reversible profiles corresponding to three distillate mole fractions belonging to the line of preferred separation: xD,1, xD,bif, and xD,2. For xD,1 (feasible) < xD,bif, xB is located in the upper stable zone as shown in Figure 5a,b, and therefore, it is a stable node (9). For the corresponding value of the reflux ratio r < rbif, there exist two additional saddle solutions (2). For xD,2 (unfeasible) > xD,bif, on the other hand, xB is a saddle node and


Figure 4. Reversible profile corresponding to p ) [0.4800961 0.35012158 0.16997894]. (a) xMeOH vs xIPA; (b) xIPA vs r; (c)xMeOH vs r; (d) xW vs r. Table 1. Methanol/2-Propanol/Water System: (a) Composition of the Still, Its Vapor in Equilibrium, Activity Coefficient and Vapor Pressure [kPa] of Each Component;a (b) Analytical Derivatives of Activity Coefficients and Vapor Pressure at xB xB

yx/B

γi

pi0

(a) 0.253 0.427 0.32

0.368 461 0.387 858 0.243 681

0.932 227 1.158 557 1.897 742

∂γi/∂xj

158.255 737 79.421 318 40.648 415 ∂γi/∂T

dpi0/dT

0.001 303 -0.000 208 -0.006 465

5.779 228 3.302 38 1.686 483

(b) -0.740 221 -1.408 074 -1.661 396 a

-1.132 999 -1.552 579 -0.713 378

-0.816 127 -0.435 512 -3.664 991

Tb is 349.421 K and p is 101.3 kPa.

the pinch point curve presents two extra solutions for r > rbif: one above xB (stable) and the other below xB (saddle). For xD (feasible) ) xD,bif (r ) rbif) the solutions of the upper stable zone and the intermediate saddle region collide with each other at the composition corresponding to the mixture in the still, giving rise to a saddle-node bifurcation of the reversible profile,

which characterizes the maximum feasible distillate composition on the line of the preferred separation for ternary mixtures.11 4. Physical Property Data for the Components and Mixtures of This Work 4.1. System Methanol/2-Propanol/Water at 101.3 kPa. The Appendixlists the molar volume and Antoine equation coefficients for each component of the mixture (Table A1 and eq A2). Equation A1 shows the Wilson parameters. The system presents an unstable distillation boundary beginning from the vertex of the composition triangle corresponding to pure methanol and ending at the azeotrope 2-propanol/water. The term “unstable” means that the boundary limits the feasible distillate compositions in continuous distillation as well as in the batchwise operation of a rectifier. Table 1a presents the instantaneous still composition to be analyzed and its vapor in equilibrium together with the vapor pressure and the activity coefficient of each component of the mixture. Table 1b shows the analytical derivatives of the activity coefficients and vapor pressure at xB. These data will be used in the next section to find the maximum distillate composition on the line of preferred separation. According to the liquid-vapor equilibrium, the line of preferred separation has the following vectorial equation:


Figure 5. (a) Pinch point curve xMeOH vs r for three distillate compositions on the preferred line. (b) Reversible profile xMeOH vs xIPA for xD < xbif D . (c) bif Reversible profile xMeOH vs xIPA for xD ) xbif D . (d) Reversible profile xMeOH vs xIPA for xD > xD .

(x, y, z) ) (0.368461 0.387858 0.243681) + λ(0.115461 -0.039142 -0.076319) where λ ∈ R. 4.2. System Acetone/Chloroform/Benzene at 101.3 kPa. As a second mixture test, the system acetone/chloroform/ benzene at 101.3 kPa is analyzed. The necessary parameters to model the liquid-vapor behavior are shown in the Appendix (Table A2 and eqs A3 and A4). The mixture presents a stable distillation boundary with its origin at the azeotrope acetone/ chloroform and its end at pure benzene. The term “stable” indicates that the boundary limits the feasible still path (bottom composition) in the batchwise (continuous) operation of a distillation rectifier (column). Data in Table 2 will be used in the next section. In this case, the line of preferred separation has the following vectorial equation: (x, y, z) ) (0.559973 0.265787 0.174240) + λ(0.109973 -0.059213 -0.050760) where λ ∈ R.

Table 2. Acetone/Chloroform/Benzene System: (a) Composition of the Still, Its Vapor in Equilibrium, Activity Coefficient and Vapor Pressure [kPa] of Each Component;a (b) Analytical Derivatives of Activity Coefficients and Vapor Pressure at xB xB

yx/B

γi

0.45 0.325 0.225

0.559 973 0.265 787 0.174 240

0.954 589 0.743 860 1.255 318

pi0

(a)

∂γi/∂xj

135.058 0 113.940 5 63.996 375 ∂γi/∂T

dpi0/dT

-0.001 250 0.001 329 0.001 942

4.345 195 3.632 322 2.219 874

(b) -0.844 896 -0.980 441 -0.967 128 a

-1.258 192 -0.374 619 -1.356 885

-0.735 440 -0.804 046 -1.684 987

Tb is 337.884 K and p is 101.3 kPa.

5. Quantitative Determination of the Maximum Feasible Distillate Composition on the Line of Preferred Separation 5.1. Region Elimination Method (REM). As mentioned in section 3, the maximum feasible distillate mole fraction on the


Figure 6. Pinch point curve for x/D ) [0.454636 0.358644 0.186722] (REM). Figure 7. Pinch point curve for x/D ) [0.875536 0.095881 0.028589] (REM).

line of preferred separation is characterized by a saddle-node bifurcation of the corresponding reversible profile at the instantaneous still composition. From Figure 5a, it is clear that the occurrence of a saddle-node bifurcation is closely related to the appearance of a maximum in the curve r vs xMeOH. This fact can be used to determine the limiting distillate composition xD/ and its corresponding reflux ratio r* ) rbif. Given a distillate p belonging to the line of preferred separation, the main idea of this method is to first calculate two points x+ B and xB of the reversible profile around xB by numerically integrating the equation system (7) for given values of the temperature increments ∆T1 and ∆T2 ) -∆T1. Then, reflux ratios are obtained from the lever arm rule for the three compositions, and finally, values of the derivatives ∆r1/∆x1k ) + 2 + (rxB - rxB)/(xB,1 - xB,1) and ∆r2/∆xk ) (rxB - rxB)/(xB,1 - xB,1)

Table 4. Maximum Feasible Distillate Composition and Limiting Reflux Ratio from the Region Elimination Algorithm and the Analytical Method for the Two Highly Nonideal Mixtures under Study system

x/D

r* ) rbif

[0.454636 0.358644 0.186722]

0.746

[0.454392 0.358727 0.186880] [0.875536 0.095881 0.028589]

0.744 2.89

[0.875721 0.095777 0.028501]

2.87

method

methanol/2-propanol/ REM water analytical acetone/chloroform/ REM benzene analytical

on both sides of the still composition are estimated. If the signs of the derivatives differ from each other, a saddle-node bifurcation at instantaneous still composition is found with the corresponding distillate mole fraction as the maximum feasible

Table 3. Determination of x/D from the Region Elimination Method (REM) for the System Acetone/Chloroform/Benzene ∆r1/∆xk1 xDA 0.559 973 0.937 469 0.748 721 0.843 095 0.890 282 0.866 688 0.878 485 0.872 587 0.875 536

∆T ∆T ∆T ∆T ∆T ∆T ∆T ∆T ∆T ∆T ∆T ∆T ∆T ∆T ∆T ∆T ∆T ∆T ∆T ∆T ∆T ∆T ∆T ∆T ∆T ∆T ∆T

) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )

∆T

pinch compositions pointing to p

0.0020 0 -0.0020 0.0020 0 -0.0020 0.0020 0 -0.0020 0.0020 0 -0.0020 0.0020 0 -0.0020 0.0020 0 -0.0020 0.0020 0 -0.0020 0.0020 0 -0.0020 0.0020 0 -0.0020

[0.449835 [0.450000 [0.450165 [0.449833 [0.450000 [0.450166 [0.449850 [0.450000 [0.450149 [0.449840 [0.450000 [0.450160 [0.449830 [0.450000 [0.450169 [0.449836 [0.450000 [0.450164 [0.449833 [0.450000 [0.450166 [0.449835 [0.450000 [0.450165 [0.449834 [0.450000 [0.450166

0.324880 0.325000 0.325119 0.324872 0.325000 0.325128 0.324989 0.325000 0.325011 0.324917 0.325000 0.325083 0.324853 0.325000 0.325146 0.324888 0.325000 0.325111 0.324872 0.325000 0.325128 0.324880 0.325000 0.325119 0.324876 0.325000 0.325124

0.225285] 0.225000] 0.224715] 0.225294] 0.225000] 0.224706] 0.225161] 0.225000] 0.224839] 0.225243] 0.225000] 0.224757] 0.225316] 0.225000] 0.224684] 0.225275] 0.225000] 0.224724] 0.225294] 0.225000] 0.224706] 0.225285] 0.225000] 0.224715] 0.225289] 0.225000] 0.224710]

∆r2/∆xk2

relative error

-0.003 539 -0.293 091 0.562 587 0.271 429 -7.078 795 -7.157 728 -2.418 508 -2.697 269 1.793 215

0.111 937 0.053 002

1.499 558 0.109 831 35 -0.196 769 10 0.562 587

0.027 223 0.013 428

0.271 429 -0.003 539 -0.293 091 0.275 647 -0.014 735

0.006 759 0.003 368


one. Otherwise, another distillate composition must be picked up until convergence. Beginning with two candidate distillate compositions, a region elimination method (REM) is used to determine xD/ : Step 1. Select p1 ) y/xB and p2 ) zD as the lower and upper bounds of the search interval, respectively. zD represents the intersection of the equilibrium vector with either the binary axis corresponding to the more volatile components (zeotropic systems) or the pinch distillation boundary, PDB (azeotropic mixtures with unstable distillation boundaries).17 + Step 2. Calculate xB and xB from equation system (7) for given ∆T1 and ∆T2 ) -∆T1. + Step 3. Solve the liquid-vapor equilibrium for xB and xB , to + / + obtain first the corresponding reflux ratios rxB ) (xD,1 - yxB,1 )/ + / / ) and r(yx/B,1+ - xB,1 xB ) (xD,1 - yxB,1-)/(yxB,1- - xB,1), and then 1 the derivatives ∆r1/∆xk ) (rxB - rxB)/(xB,1 - xB,1) and ∆r2/∆x2k + ) (rxB - r+ xB)/(xB,1 - xB,1) around xB. If both derivatives have negative values for p1 and positive values for p2, there is a tangent pinch controlling the separation; go to step 4. Otherwise, assign the value of zD to the maximum feasible distillate composition xD/ ; stop. Step 4. Obtain p3 from the region elimination method (REM) 2 and calculate ∆r1/∆x1k ) (rxB - rxB)/(xB,1 - xB,1) and ∆r2/∆xk ) + + (rxB - rxB )/(xB,1 - xB,1) corresponding to the new distillate composition. If the signs of the derivatives differ from each other, a saddle-node bifurcation at instantaneous still composition is found with xD/ ) p3. Otherwise, select a new search interval and repeat step 4 until convergence. In step 2, the algorithm stops when both derivatives corresponding to p2 have negative values. In this case, the reversible profile resembles either a pinch point curve without a tangent pinch point or a reversible profile showing a noncontrolling tangent pinch like the one shown in Figure 5b. In other words, xB behaves like a stable node. 5.1.1. System Methanol/2-Propanol/Water at 101.3 kPa. Taking into account the data in section 4.1, the region elimination algorithm needs five iterations for ∆T1 ) 0.002 K to obtain the maximum distillate composition xD/ ) [0.454636 0.358644 0.186722] together with its limiting reflux ratio r/ ) rbif ) 0.74. Figure 6 shows the projection of the reversible profile on the space light species composition versus reflux ratio. Note, however, that the algorithm does not need to calculate the complete curve to determine both xD/ and r/. 5.1.2. System Acetone/Chloroform/Benzene at 101.3 kPa. Table 3 shows in detail the steps of the algorithm for the instantaneous still composition and data given in Table 2 for the mixture acetone/chloroform/benzene. As in the previous example, the value of the differential temperature ∆T is set to 0.002. The bifurcation values for the distillate composition and reflux ratio are xD/ ) [0.875536 0.095881 0.028589] and r* ) rbif ) 2.89, respectively. The reversible profile xA vs r is shown in Figure 7. Numerical results for different values of ∆T show a tradeoff between the number of steps to achieve convergence and the relative error in the composition of the light species defined as |(xDit - xDit-1)/xDit|. The selected value for ∆T allows for accurate results, while the number of iterations is always below 10. 5.2. Analytical Method. The occurrence of a local maximum in both projections of the x vs r diagram and its relation with the appearance of tangent pinch points was thoroughly analyzed by Fidkowski et al.18 in the case of continuous distillation. The authors extended the “zero volume method”, developed by Julka

19

and Doherty, in order to determine the minimum reflux ratio for the design of a continuous column controlled by a tangent pinch point. The minimum reflux ratio situation for known values of the compositions of feed, distillate, and bottom streams corresponds to a “turning point” in the volume vs reflux ratio diagram defined as V ) det[e1, e2, ..., ec-1] where e1 ) xˆ1,s - xˆ2,r ;

e2 ) xˆc,r - xˆ2,r ;

e3 ) xˆ3,r - xˆ2,r ;

...;

ec-1 ) c-1,r 2,r

xˆ - xˆ In the equations above, the superscripts “s” and “r” refer to the stripping zone and the rectifying section, respectively. The different controlling pinch points corresponding to a given value of the reflux ratio r are used to define the vectors ei. According to the authors, the minimum reflux condition is characterized by a turning point or local maximum in each one of the projections of the r vs x diagram corresponding to the column section where the tangent pinch takes place. Moreover, they derived sufficient conditions, which prove that if a pinch point curve turns in r-x space, the turning point is visible in all its projections at the same value of r ) r* ) rbif. Finally, Fidkowski et al.18 also derived conditions which guarantee that a turning point on a curve of pinch points implies the same behavior in the volume vs reflux ratio diagram. We refer the interested reader to the Appendix of the mentioned paper for details of derivations. As a consequence of the analysis referred above, the following equation system represents the tangent pinch situation in a given column section (e.g., for the rectifying section): r 1 x + p )0 (8) r+1 k r+1 k r I )0 det Y (9) r+1 where Y ) [∂yi/∂xj]xΣˆ i*j is the Jacobian matrix of the equilibrium function and xk, y(xk), and pk are the kth components of the bifurcation point, its vapor in equilibrium, and the distillate, respectively. While from eq 8 it follows that b x, b y(x), and b p must be aligned, eq 9 represents the sufficient condition for the occurrence of a turning point in the diagram of pinch composition vs reflux ratio. Getting back to the case of batch distillation, the equation system (8) and (9) holds for a rectifying section showing a tangent pinch point. However, while in continuous distillation the controlling pinch composition does not correspond with either the mole fraction of the feed or the composition of any of the column products (i.e., it is unknown), in batch distillation of ternary mixtures operating at a reflux ratio r e rbif, the bifurcation point coincides with the instantaneous still composition xB. Therefore, a simple method to obtain the maximum feasible composition in the line of preferred separation consists of the following steps: (i) Calculate the elements of the Jacobian matrix of the equilibrium function from their analytical expressions.15,16 (ii) Solve the eigenvalue problem given by eq 9 at xB to obtain the bifurcation value of the reflux ratio r ) r/ ) rbif. (iii) Calculate the maximum feasible distillate composition p ) xD/ ) xDbif through the lever arm rule, eq 8. It is interesting to note that two values of the reflux ratio are obtained from solving eq 9. For systems where the reversible -y(xk) +

(

)


profile resembles either a pinch point curve without a tangent pinch point or a reversible profile showing a noncontrolling tangent pinch, the reflux ratio values will take either negative values or positive ones, giving rise in the last case to distillate compositions outside the composition simplex. 5.2.1. System Methanol/2-Propanol/Water at 101.3 kPa. Taking into account the data given in Table 1 for the mixture methanol/2-propanol/water, eq 9 transforms into

(

1.253009 -

r -0.134703 r+1

-0.593788

0.523481 -

)

)0 r r+1 From eq 9 two values of the reflux ratio, which make 0 the value of the determinant, are obtained. Only the positive one r ) r* ) rbif ) 0.744 240 is selected to calculate x/D ) [0.454391 0.358768 0.186915] from eq 8. 5.2.2. System Acetone/Chloroform/Benzene at 101.3 kPa. For this mixture, eq 9 takes the form det

det

(

r 0.945477 -0.272860 r+1 -0.184856

0.989179 -

r r+1

)

)0

By solving this equation, a bifurcation value for the reflux ratio of 2.871 150 is obtained. The corresponding maximum distillate mole fraction on the line of preferred separation is then calculated from the mass balance around the rectifier by inserting the value of the reflux ratio rbif in eq 8. A distillate mole fraction xD/ ) [0.875721 0.095777 0.028501] with a small amount of benzene is obtained even though the mixture does not present any distillation boundary for the distillate mole fractions. 6. Comparison of Results Table 4 summarizes the results obtained from both methods. An excellent agreement is found. While the analytical method may be used to determine limiting values of the reflux ratio and distillate composition due to both its simplicity and accuracy, the importance of the region elimination method, which calculates part of reversible profiles for different distillate mole fractions, is based on its characterization of the problem as a saddle-node bifurcation of the pinch point curve departing from the limiting distillate composition. 7. Quantitative Determination of the Curve of Feasible Distillate Compositions for Reflux Ratios Greater Than rbif Up to this point the discussion has covered the behavior of the distillate mole fraction on the line of preferred separation in the presence of controlling tangent pinch points. At instantaneous operation of the rectifier for values of the reflux ratio lower than or equal to the bifurcation value rbif, the controlling pinch has the composition of the liquid in the still, the distillate mole fractions are located on the line of preferred separation, and the composition of the liquid at the rectifier lower end is identical to that of the still (xN ) xB) (see Figure 2). For the case studies, the appearance of a tangent pinch at rbifgives rise to limiting distillate compositions with considerable amounts of the heavy components benzene (system acetone/chloroform/ benzene) or water (mixture methanol/2-propanol/water). In other words, the limiting distillate mole fraction does not correspond to the intersection of the equilibrium vector with either the binary axis corresponding to the light components acetone and chloroform or the unstable distillation boundary joining metha-

nol with the azeotrope 2-propanol/water. Thus, the behavior shown in Figure 1 for ideal systems is not valid anymore. For reflux ratios above rbif, the instantaneous operation is controlled by unknown values of tangent pinches xp, lower end liquid compositions xN * xB, and distillate mole fractions xD. Therefore, the trajectory of each of these variables versus the reflux ratio must be determined. To do this and given a value of the reflux ratio r g rbif, the composition of the controlling tangent pinch point xp must be first determined and then the maximum feasible distillate composition xD on the line determined by the equilibrium vector y/xp - xp. Although the composition of the liquid at the rectifier lower end xN is unknown, it is possible to estimate the locus on which it lies by linearization of the adiabatic profile in the neighborhood of xB, i.e., by solving the eigenvalue problem of the Jacobian of the equilibrium function in xB, (∂yi/∂xj)Σ|xB, to obtain λ∈R

xN ) xB + λν,

ν is the eigenvector of (∂yi/∂xj)Σ|xB, which corresponds to the lower eigenvalue.3 In addition, the feasible values of xN (see eq 13) are bounded by [xB,xN,max], where xN,max corresponds to the intersection of the eigenvector with the line defined from xD// - y/xB. Typically, xD// represents the unstable node (pure component or azeotrope) corresponding to the distillation region of the still composition (Figure 1). Taking into account the basic ideas presented above, the equation system (10)-(14) must be solved to determine a point in the trajectory of feasible distillate compositions. Equations 10 and 11 correspond to the mass balance around the rectifier and an envelope from column top until the tangent pinch, respectively. Equation 12 assumes liquid-vapor equilibrium at the tangent pinch composition xp, while eq 13 defines the composition at the rectifier lower end xN from known values of the instantaneous still composition xB and the direction of the eigenvector ν. Finally, the tangency condition given by eq 14 completes the nonlinear conceptual model of a column controlled by a tangent pinch of composition xp * xB: -yx/B +

xD r x + )0 r+1 N r+1

(10)

-yx/p +

xD r x + )0 r+1 p r+1

(11)

yx/p )

γi(xp, T) p0i (T) xp p

xN ) xB + λν

(

det Y -

r I r+1

)

xp

(12) (13)

)0

(14)

The equation system (10)-(14) has 13 unknown variables, including y/xp, xN, xD, xp, and T. The same number of equations is obtained if the closure equations for mole fractions are added to the nine equations above. Therefore, both system consistency and solution unicity are guaranteed. 7.1. Internal Profiles from Hysys Simulations. In order to make a comparison between the results from the solution of the conceptual model and actual profiles of a column having 200 equilibrium stages (as an approximation to a column with an infinite number of trays), rigorous simulations with Hysys are performed for different values of the reflux ratio r g rbif. According to Figure 2, the feed to the rectifier is the vapor y/xB in equilibrium with the still composition xB. To run the simulations, an arbitrary value of the vapor flow rate is set (i.e.,

Ind. Eng. Chem. Res., Vol. 48, No. 2, 2009 865 Table 5. Distillate Compositions Calculated from Hysys by Given Total Number of Trays (Nstages ) 200), Reflux Ratio r, and Feed Composition and Flow Rate (yx/B, V ) 100 kmol/h) profile

distillate mole fraction

r

1 2 3

[0.90438231 0.0774 0.0182] [0.93242633 0.0569 0.0107] [0.97609351 0.0203 0.00356]

3.2 3.5 4

Table 6. Estimated Tangent Pinch Points from Hysys for the System Acetone/Chloroform/Benzene profile tolerance

tangent pinch stage

1

0.0007

131

2

0.0007

100

3

0.0009

72

[ ][ ] [ ][ ] [ ][ ] 0.486069 0.352089 0.161842 0.519272 0.376739 0.10399 0.553725 0.402295 0.0440

tangency condition

yx/p

xp

T

T

T

0.586528 0.286489 0.126983 0.611627 0.305609 0.0828 0.638434 0.325996 0.0356

T

0.001 343

T

0.000 366

T

0.000 022

V ) 100 kmol/h). With these data and for a given value of the reflux ratio, the simulation provides composition and flow rates of the distillate stream (xD, D), lower column end stream (xN, L), and the corresponding values for each equilibrium stage. From the analysis of the calculated internal profile it is possible to establish an approximate value of the tangent pinch, in case of occurrence, with the following procedure: (i) Obtain the Jacobian of the equilibrium function for each liquid composition on a tray and evaluate the tangency condition (eq 14) for the selected reflux ratio. (ii) Estimate the distillate composition from the lever arm rule for each liquid composition, its vapor in equilibrium, and given reflux ratio. (iii) Set a tolerance limit for the difference between the distillate mole fraction calculated from Hysys and the distillate composition estimated from the lever arm rule in step ii. Select the liquid compositions whose estimated distillate compositions fall within the accepted error tolerance. (iv) From the set of liquid compositions obtained in step iii, select the one with the minimum absolute value of eq 14.

Figure 8. (a) xA vs xC for adiabatic profile 1; (b) xA vs xC for adiabatic profile 2; (c) xA vs xC for adiabatic profile 3. (d) Tangent pinch points and adiabatic profiles for three different values of the reflux ratio.


Figure 9. (a) Pinch point curve for xD ) [0.9102 0.0750 0.0148] (∆r > 0). (b) Pinch point curve for xD ) [0.8949 0.0636 0.0415] (∆r < 0).

Figure 10. Error function ∆r vs xAN: (a) r ) 3.2; (b) r ) 3.5; (c) r ) 4.

Results obtained from simulation in Hysys for the system acetone/chloroform/benzene are shown in Tables 5 and 6, and in Figure 8. 7.2. Improved Memory Method To Solve Equation System (10)-(14). The numerical method implemented to obtain the controlling tangent pinch is based on the improved memory method developed by Shacham20 for the solution of a nonlinear equation or error function.

The method is used to solve nonlinear equations of the form f(x) ) 0 by approximating the inverse function of f(x), namely x ) Ψ(f), through inverse interpolation with continued fractions and evaluating the inverse function for f ) 0. The x found value is the root of the nonlinear equation; i.e., x* ) Ψ(0). The algorithm requires the evaluation of a series of points (x0,f0), (x1,f1),..., (xn,fn), and it demands the smallest number of function evaluations in comparison with other methods as a


Figure 11. Pinch point curve for xD ) [0.9044 0.0774 0.0182] at r ) 3.2. Table 7. Tangent Pinch Point and Distillate Compositions Obtained from Hysys and the Improved Memory Method r ) 3.2

[ [ [ [

0.4860 xp ) 0.3520 0.1618 0.9044 xD ) 0.0774 0.0182 0.4981 xp ) 0.3598 0.1420 0.9089 xD ) 0.0741 0.0169

r ) 3.5

] ] ] ]

[ [ [ [

Hysys

T

T

T

T

r)4

] ] ] ]

[ [ [ [

0.5192 T 0.5537 xp ) 0.3767 xp ) 0.4022 0.1039 0.0044 0.9324 T 0.9761 xD ) 0.0569 xD ) 0.0203 0.0107 0.0036 Improved Memory Method 0.5248 T 0.5555 xp ) 0.3784 xp ) 0.3991 0.0969 0.0454 0.9375 T 0.9819 xD ) 0.0530 xD ) 0.0155 0.0094 0.0025

] ] ] ]

T

T

< 0 and the operating reflux ratio r. The selected point corresponds to a saddle-node bifurcation of the reversible profile (see Figure 9). Thus, eqs 11 and 12 are satisfied while the error function ∆r replaces the tangency condition, eq 14. However, the bifurcation reflux differs in general from the operation reflux r as shown in Figure 9. Step 4. If ∆r(xN,1) · ∆r(xN,2) > 0, initialize the improved memory method by taking [xN,1,xN,2] as the initial search interval until determining the composition at the lower end column xN for which the error function ∆r = 0. Otherwise, go to step 1, select a new value for xN, and repeat steps 2 and 3 until an appropriate search interval is found. From the analysis of the algorithm above it is clear that the method searches for a distillate composition whose corresponding pinch point at the saddle-node bifurcation [xp, y/xp, rbif, Tp] has a reflux ratio rbif such that rbif ) r. In other words, it searches for the root of the error function ∆r ) ∆r(xN). Figure 10 shows the curves ∆r vs xN for all the feasible range of xN [xB,xN,max]. Note, however, that the improved memory method needs relatively little evaluation of the error function to converge. Figure 10 also shows the compositions at the rectifier lower end that make 0 the error function ∆r. Figure 11 presents the reversible profile corresponding to the distillate obtained from solution of the conceptual model at an operating reflux ratio r ) 3.2 with the aid of the improved memory method. As shown, the reflux at bifurcation coincides with the operating reflux ratio. A comparison between results obtained from the conceptual model and simulations in Hysys is shown in Table 7. From the analysis of the results, the effectiveness of the proposed algorithm is highlighted.

T

8. Conclusions and Future Work T

consequence of using the information from previous iterations to generate greater order estimations of the inverse function (lineal, quadratic, etc.). Two initial points (x0,f0) and (x1,f1) must be calculated to start the algorithm in such a way that f(x1) · f(x2) < 0. More details can be found elsewhere.20 The application of this method to solve the equation system (10)-(14) is appropriate because the number of pinch point curve evaluations that are necessary to find the tangent pinch composition will tend to its minimum value. However, a variable of the set [y/xp, xN, xD, xp, Tp] must be selected as the independent one to transform the solution of the equation system (10)-(14) into an iterative solution of an error function. We select the composition of the light species at the lower column end as the independent variable. The preprocessing algorithm needed to generate the two initial points in the improved memory algorithm is shown below. There, both the error function selected and the proposed way to solve the conceptual model are emphasized: Step 1. Given a value for the reflux ratio r, select values for xN,1 and xN,2 belonging to the eigenvector within the interval [xB,xN,max]. Therefore, eq 13 is obeyed. Step 2. Solve eq 10 for xD,1 and xD,2 from the estimated values xN,1 and xN,2. Step 3. For each distillate composition obtained in step 2 calculate, by integrating eq 7, the pinch point curve, and ∆r, defined as the difference between the reflux ratio corresponding to the first ith point of the pinch curve for which r(i+1) - r(i)

Figure 12a shows the curve of feasible distillate mole fractions corresponding to an instantaneous still composition xB for the mixture acetone/chloroform/benzene. The products belong to the line of preferred separation at reflux ratios r e rbif. In this case, xB behaves like a stable node for the adiabatic profiles departing from the corresponding distillate compositions. For r ) rbif, the reversible profile corresponding to the limiting composition xD/ presents a saddle-node bifurcation in xB giving rise to the appearance of a tangential pinch. The separation for reflux ratios above the bifurcation value r > rbif is controlled by tangent pinches located along a curved trajectory. Calculation of the saddle-node bifurcation for either of the proposed methods (REM, analytical method) determines not only the type of pinch point that controls the separation but also the conceptual model to be used, i.e., the lever arm rule or the conceptual model formed by eqs 10-14. The equation system (10)-(14) replaces the model based on the assumption of the invariance of a controlling saddle pinch, which is valid for ideal systems and nonideal mixtures without tangent pinch points (section 1, step ii). Figure 12 emphasizes the variation of the composition of the tangential pinch points for reflux ratios r g rbif. A very interesting aspect to be considered emerges from the analysis of Figure 13 and Table 8. There, the design of a column continuously operated with a feed composition that coincides with the instantaneous still composition xB (system acetone/ chloroform/benzene) is presented. The selected distillate mole fraction pertains to the preferred separation line, and it has trace amounts of the heavy species benzene. At first sight, it seems that the feasible design shown in Figure 13 and Table 8 contradicts the results for batch rectification. Note, however, that the adiabatic profile begining at the distillate composition


Figure 12. (a) Trajectory of feasible distillate composition corresponding to a still composition xB for reflux ratios between 0 and r**. (b) Three different tangential pinches for reflux ratios above the bifurcation one. Mixture acetone/chloroform/benzene.

ceptual model and consider a generalization of the developed ideas to mixtures with more than three components. Acknowledgment The authors gratefully acknowledge the financial support of this work to CONICET, UNL, and ANPCyT. Appendix: Liquid-Vapor Equilibrium Data A.1. System Methanol/2-Propanol/Water. Table A1 lists the Wilson molar volumes for the methanol/ 2-propanol/water system. Table A1. Wilson Molar Volumes for the Methanol/2-Propanol/ Water System

Figure 13. Feasible design for a continuously operated column. The feed to the column has the same composition as the instantaneous still mole fraction xB. System acetone/chloroform/benzene. Table 8. Mass Balance and Reflux Ratio for a Feasible Design of a Continuously Operated Column for the System Acetone/Chloroform/Benzenea xB xs xD r

[0.45 0.325 0.225] [0.28505 0.41382 0.30114] [0.93747 0.06253 0] 6

a

The feed to the column has the same composition as the instantaneous still mole fraction xB.

is far from the eigenvector that, in turn, represents the path of the adiabatic profile in batch distillation. In other words, instantaneous batch rectification is not comparable to continuous distillation. While in continuous distillation it is possible to find a design with a distillate mole fraction with only traces of the heavy component benzene by increasing the reflux ratio until intersection of both the rectifying and stripping sections, in batch rectification the adiabatic profiles follow another path (Figure 8), giving rise to a curved trajectory of distillate compositions. The results found encourage future research work in order to incorporate nonlinear distillation boundaries into the con-

component

Wilson molar vol [cm3/gmol]

methanol 2-propanol water

40.729 76.916 18.069

The Wilson binary coefficents are given by

(

0 -242.76 -105.9234 556.390 W(i, j) ) 859.450 0 684.0054 1294.715 0

)

and the extended Antoine vapor pressure coefficients are

(

(A1)

)

16.57227019 -3626.55 -34.29 0 0 2 0 A(i, j) ) 16.67767019 -3640.20 -53.54 0 0 2 0 16.28837019 -3816.44 -46.13 0 0 2 0 (A2) A.2. System Acetone/Chloroform/Benzene. Table A2 lists the Wilson molar volumes for the acetone/ chloroform/benzene system. Table A2. Wilson Molar Volumes for the Acetone/Chloroform/ Benzene System component

Wilson molar vol [cm3/gmol]

acetone chloroform benzene

74.05 80.67 89.41


The Wilson binary coefficients are given by

(

0 116.1171 547.0188 -208.50 W(i, j) ) -506.8513 0 -216.6089 148.44 0

)

(A3)

and the extended Antoine vapor pressure coefficients are A(i, j) ) 71.3031 -5952 0 -8.53128 0.00000782393 2 0 73.7058 -6055.60 0 -8.91890 0.00000774407 2 0 169.650 -10314.8 0 -23.5895 0.0000209442 2 0 (A4)

(

)

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(9) Salomone, E.; Espinosa, J. Conceptual Dynamic Models for the Design of Batch Distillations. AIChE Symposium Series; Malone, M., Trainham, J. A., Carnahan, B., Eds.; American Institute of Chemical Engineers: New York, 2000; Vol. 96, pp 342-345. (10) Hysys User Manual; Hyprotech Ltd.: Calgary, Canada, 1999. (11) Espinosa, J.; Brueggemann, S.; Marquardt, W. Application of the Rectification Body Method to Batch Rectification. In ESCAPE-15; Puigjaner, L., Espun˜a, A., Eds.; Elsevier: New York, 2005; pp 757-762. (12) Bausa, J.; Watzdorf, R. V.; Marquardt, W. Shortcut methods for nonideal multicomponent distillation: 1. Simple columns. AIChE J. 1998, 44 (10), 2181–2198. (13) Salomone, H. E.; Chiotti, O. J.; Iribarren, O. A. Short-Cut Design Procedure for Batch Distillations. Ind. Eng. Chem. Res. 1997, 36 (1), 130– 136. (14) Brueggemann, S.; Oldenburg, J.; Marquardt, W. Combining Conceptual and Detailed Methods for Batch Distillation Process Design. In Proceedings of FOCAPD; Floudas, C. A., Agrawal, R., Eds.; Elsevier: New York, 2004; pp 247-250. (15) Poellmann, P.; Blass, E. Best Products of Homogeneous Azeotropic Distillations. Gas Sep. Purif. 1994, 8 (4), 194–228. (16) Taylor, R.; Kooijman, H. A. Composition Derivatives of Activity Coefficient Models. Chem. Eng. Commun. 1991, 102, 87–106. (17) Brueggemann, S. Rapid Screening of Conceptual Design Alternatives for Distillation Processes. Ph.D. Thesis, RWTH Aachen, Germany, 2005. (18) Fidkowski, Z. T.; Malone, M. F.; Doherty, M. F. Nonideal Multicomponent Distillation: Use of Bifurcation Theory for Design. AIChE J. 1991, 37 (12), 1761–1779. (19) Julka, V.; Doherty, M. F. Geometric Behavior and Minimum Flows for Nonideal Multicomponent Distillation. Chem. Eng. Sci. 1990, 45, 1801– 1822. (20) Shacham, M. An Improved Memory Method for the Solution of a Nonlinear Equation. Chem. Eng. Sci. 1989, 44 (7), 1495–1501.

ReceiVed for reView July 30, 2008 ReVised manuscript receiVed October 14, 2008 Accepted October 27, 2008 IE801169X

Incorporating Tangent Pinch Points into the Conceptual Modeling of

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