Ind. Eng. Chem. Res. 2006, 45, 4429-4432
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RESEARCH NOTES A Fast Method To Calculate Residue Curve Maps Claudia Gutie´ rrez-Antonio,† Miguel Vaca,†,‡ and Arturo Jime´ nez*,† Departamento de Ingenierı´a Quı´mica, Instituto Tecnolo´ gico de Celaya, AV. Tecnolo´ gico y Garcı´a Cubas s/n, 38010, Celaya, Guanajuato, and DiVisio´ n de Ingenierı´a Quı´mica y Bioquı´mica, Tecnolo´ gico de Estudios Superiores de Ecatepec, AV. Tecnolo´ gico s/n, Valle de Ana´ huac, 55210, Ecatepec, Me´ xico
Residue curve maps are used in the preliminary stages of azeotropic distillation problems to establish feasible separation targets. Their calculations involve an iterative procedure, because of bubble point determinations. In this work, a simplified method for the generation of residue curve maps is presented. The method is based on the use of a mean relative volatility, independent of temperature, that avoids iterative calculations. Several case studies are used to show the effectiveness of the proposed method. 1. Introduction Residue curves represent the change in composition of the residue of a simple distillation process with time.1 A distinction between residue curves and distillation lines should be made. Distillation lines represent the liquid compositions whose vapor compositions at equilibrium also lie on the same line. Unlike the residue curve, the distillation line is the operating line at total reflux.1 However, frequently, the assumption that residue curves approximate the profiles in staged columns at total or finite reflux is made, because that provides a convenient tool in the preliminary steps of the design of distillation columns (azeotropic and reactive), either packed or staged.2 For instance, for azeotropic separations, a residue curve map (RCM) helps the designer to visualize feasible splits, and to discard infeasible specifications; once this is accomplished, one can determine the feasible product region, which requires the calculation of the distillation line passing through the feed point.3 It is well-known that relative volatilities are less dependent on temperature than composition for almost-ideal mixtures. Also, temperature effects on relative volatilities are not as significant for nonideal mixtures at low pressures. In this work, we take advantage of this behavior and calculate RCMs, neglecting the effect of temperature on the relative volatilities, but preserving the composition effect. Although this assumption has been widely used for ideal mixtures, it has seldom been applied for nonideal mixtures, regardless of whether they are zeotropic or azeotropic. A comparison of the resulting RCMs with those evaluated rigorously is also presented. 2. Residue Curve Calculation The equation to determine the residue curves is a modification of the Rayleigh equation:2
dxi ) xi - yi dξ
(1)
where xi and yi are the molar compositions of component i in * To whom correspondence should be addressed. Tel. (+52-461) 611-7575, Ext. 130. Fax: (+52-461) 611-7744. E-mail: arturo@ iqcelaya.itc.mx. † Departamento de Ingenierı´a Quı´mica, Instituto Tecnolo´gico de Celaya. ‡ Divisio´n de Ingenierı´a Quı´mica y Bioquı´mica, Tecnolo´gico de Estudios Superiores de Ecatepec.
the liquid and vapor phase, respectively, and ξ is a dimensionless time. If eq 1 is integrated forward and backward in time from a given initial composition, a residue curve is obtained. A small integration step is usually recommended to generate the RCM. Each residue curve requires a great number of points to be constructed, each one involving the calculation of the vaporphase composition in equilibrium with the liquid phase. Bubble point calculations are needed to obtain the vapor-phase composition. Because such calculations are iterative, the generation of RCM requires considerable numerical effort.4 In this work, we propose the use of relative volatilities for RCM calculations. Because relative volatilities show a rather weak dependence on temperature, such an effect on the calculation of residue curves can be neglected. The calculation of vapor-phase compositions is as follows. At low to moderate pressures, the modified Raoult’s law properly represents the vapor-liquid equilibrium of nonideal mixtures:4
yi )
γiPsat i xi P
(2)
where γi is the liquid-phase activity coefficient of component i, Psat i is the saturated pressure of pure component i, and P is the total pressure of the system. After substituting the total pressure and rearranging the equation, one obtains the following equation for the vapor-phase composition:
Ri,Rxi
yi )
(3)
C
∑Rk,RxK
k)1
where C is the number of components, and Ri,R is the relative volatility of component i, with respect to a reference component R, which is given by
Ri,R )
Psat i γi sat γ P R
(4)
R
Because of the small temperature effect on the relative volatilities,4 eq 3 has been widely applied to ideal mixtures (γi ) 1.0). For nonideal mixtures (zeotropic or azeotropic), eq 3
10.1021/ie051132+ CCC: $33.50 © 2006 American Chemical Society Published on Web 05/04/2006
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Ind. Eng. Chem. Res., Vol. 45, No. 12, 2006
Figure 2. Residue curve map (RCM) for the acetone-2-propanol-water mixture.
3. Results and Discussion
Figure 1. Vapor-liquid equilibrium for two binary mixtures using rigorous bubble-point estimations and the constant relative volatility method.
has not had the same degree of use, because of the effect of the liquid composition. When the temperature effect over Ri,R is neglected, a good estimation of the bubble temperature (TB) is sufficient to obtain a good description of the vapor-phase equilibrium composition. In this work, a weighted sum of the pure-component boiling temperatures (Tbi) is used: N
TB )
Tb xi ∑ i)1 i
(5)
After eq 5 is used, the vapor-phase composition is calculated with eqs 4 and 3. Because of the fact that the bubble point temperature is not determined with an iterative procedure, a significant reduction in computer time should be observed. To show the use of eq 4 on vapor-liquid equilibrium calculations, two binary systemssthe acetone-benzene and 2-propanol-water systemssare analyzed. Figure 1 shows the equilibrium curves (vapor mole fraction versus liquid mole fraction) for each mixture at 101.3 kPa. The curve labeled “rigorous” was calculated using eq 2, with rigorous bubble temperature calculations, whereas the curve labeled “short” was determined with eq 3, where relative volatilities were calculated at the bubble temperature given by eq 4. An excellent agreement between the rigorous and short curves for both mixtures can be observed. The differences in boiling temperatures of the pure components may affect the agreement between both approaches. As a reference, such differences are 24 K for the acetonebenzene system (a zeotropic mixture), and 18 K for the 2-propanol-water system (an azeotropic mixture). Under these conditions, the agreement proved to be very good.
In this section, the results obtained from the application of the proposed short method for the calculation of RCMs (hereafter referenced as “short curves”) are shown, and they are compared to those obtained with the traditional method based on iterative bubble-point calculations (labeled as “rigorous curves”). RCMs were calculated for five ternary azeotropic mixtures at atmospheric pressure (101.3 kPa). Three mixtures have only one distillation boundary: the acetone-2-propanolwater (M1), acetone-chloroform-benzene (M2), and methanol2-propanol-water (M3) systems. The other two ternary mixtures have four separatrices: the methanol-chloroform-methyl acetate (M4) system and the acetone-chloroform-methanol (M5) system. The NRTL solution model was used to describe the liquid-phase nonideality, whereas the vapor phase was considered as an ideal gas, which is an implicit assumption in eq 2. Binary interaction parameters, which are required by the NRTL model, were taken from the Dechema Collection.5 The results for ternary mixtures that have one boundary are discussed first. Figure 2 shows the RCMs for the acetone-2-propanol-water (M1) mixture. An excellent agreement between both residue curves is observed; for practical uses, both residue curves can be considered as overlaid. In Figure 3, the RCMs for the acetone-chloroform-benzene (M2) mixture are presented. In this case, the distillation boundary has a pronounced curvature. Nevertheless, an excellent agreement is again observed between the rigorous curve and the short residue curve. The RCMs for the methanol-2-propanol-water (M3) mixture are shown in Figure 4. Although the short curve does not match the rigorous curve as well as it does for mixtures M1 and M2, it may provide a proper use for preliminary applications, because a good description of the space composition is observed. One aspect worth of mention is the reduction in computer time achieved by the proposed method. Table 1 presents the computer times involved with both types of residue curves for mixtures M1-M3. For each mixture, 17 residue curves were calculated (although not all of them are included in Figures 2-4, for the sake of clarity). Table 1 shows that the computer time required for the calculation of the RCM with the proposed method is 10%-14% of the time with the rigorous method. Such a reduction is obtained without a practical loss of the quality of each RCM.
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Figure 3. RCM for the acetone-chloroform-benzene mixture.
Figure 4. RCM for the methanol-2-propanol-water mixture. Table 1. Computer Times Required for Residue Curve Map (RCM) Calculations mixture M1 M2 M3
Computer Time (CPU s) rigorous RCM short RCM 353.6 180.5 208.5
49.1 18.4 26.9
short RCM/rigorous RCM ratio 0.14 0.10 0.13
Mixtures M4 and M5 are more complex, because they involve a higher number and different types of nodes. Figure 5 shows the RCM for mixture M4. Four separatrices arise, because of the existence of three binary azeotropes and one ternary azeotrope. The binary azeotropes consist of two unstable nodes and one stable node, whereas the ternary azeotrope is a saddle node.2 There is a good agreement between the short curves and that based on iterative calculations. The residue curves for mixture M5 are shown in Figure 6. This mixture shows four azeotropic nodes: two unstable nodes, one saddle node, and one stable node. In addition, pure methanol provides a stable node, and chloroform and methyl acetate are saddle nodes. The complex nature of these mixtures is reflected in the shapes of their RCMs. Four regions are identified. Note that (i) the residue curves end in the methanol node and (ii) they just pass through the chloroform and methyl acetate nodes. Also, there is a short residue curve that seemingly is not calculated with the rigorous model; such a rigorous curve is actually
Figure 5. RCM for the acetone-chloroform-methanol mixture.
Figure 6. RCM for the methanol-chloroform-methyl acetate mixture.
located in a neighbor distillation region (an inconsistent estimation). This happened because the boundary prediction differed from each method. Nonetheless, one can still observe an overall good agreement between the short and rigorous curves. One last observation is in place. RCMs are calculated using an approximate TB value that is bounded by the pure-component boiling temperatures (eq 5). For mixtures M1-M3, the azeotrope for each mixture has a TB value within the interval of the boiling temperatures for the pure components. The proposed method provides good results in those cases. However, in cases M4 and M5, there are azeotropes with temperatures outside the range given by the bubble points of the pure components. Even under those conditions, the method provides residue curves that satisfactorily match those provided by rigorous calculations. This observation seems to indicate that the proposed method offers good robustness properties. 4. Concluding Remarks Residue curve maps (RCMs) are important tools to analyze the feasibility of a proposed split for nonideal azeotropic separations in an easy, fast, and qualitative format. In this paper, the use of a temperature-independent (but liquid-phase-composition-dependent) relative volatility has been proposed to calculate residue curves. The results indicate that a good representation
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Ind. Eng. Chem. Res., Vol. 45, No. 12, 2006
of the residue curve maps is obtained with the simplified method, even for azeotropic mixtures with more than one separatrix. The region in which some differences are sometimes observed is located in the neighborhood of the distillation boundaries. This does not seem to be a major problem, because the use of RCM is quite restricted to assessing the preliminary characteristics of azeotropic separation problems, as opposed to their use within a formal numerical method; a quick approach to obtain residue curves such as that presented in this work can be particularly helpful during those preliminary design stages for azeotropic distillation problems. However, if a better representation in the neighborhood of the distillation boundaries is important, a hybrid approach can be used: the boundary and two additional neighbor curves, for instance, can be determined with the rigorous model, whereas the rest of the space composition can be described with the proposed method. For example, if we take a typical calculation of 17 curves (as we did for some of the case studies), then three of those curves would be calculated rigorously and the other 14 would be calculated with the short method. In this form, both a reduction in computer time and a good description of the RCMs would be preserved. Acknowledgment Financial support from CONACYT, Mexico (project number SEP-2003-C02-43898) is gratefully acknowledged. Also, C.G.A. and M.V. were supported with scholarships from CONACYT. M.V. is also grateful to the Instituto Mexicano del Petro´leo for financial support.
Nomenclature C ) number of components Pisat ) saturated pressure of the pure component i P ) total pressure TB ) bubble-point temperature Tbi ) boiling temperature of component i xi ) liquid-phase mole fraction of component i yi ) vapor-phase mole fraction of component i Ri,R ) relative volatility of component i, with respect to component R γi ) liquid-phase activity coefficient of component i ξ ) dimensionless time Literature Cited (1) Seider, W.; Widagdo, S. Azeotropic Distillation. AIChE J. 1996, 42 (1), 96-130. (2) Doherty, M. F.; Malone, M. F. Conceptual Design of Distillation Systems; McGraw-Hill: New York, 2001. (3) Fair, J. R.; Stichlmair, J. G. Distillation: Principles and Practice; Wiley: New York, 1998. (4) Henley, E. J.; Seader, J. D. Equilibrium Stage Separation Operations in Chemical Engineering; Wiley: New York, 1981. (5) Gmehling, J.; Onken, U. Vapor-Liquid Equilibrium Data Collection, Dechema; Chemistry Data Series; DECHEMA: Frankfurt/Main, Germany, 1977.
ReceiVed for reView October 10, 2005 ReVised manuscript receiVed February 22, 2006 Accepted April 20, 2006 IE051132+
