The geometry of separation boundaries: Four-component mixtures
The determination of distillation boundaries for four‐component mixtures that exhibit azeotropy is studied. It is shown that these distillation boundaries correspond to local maxima in surface areas under conditions of Levi‐Civita parallelism. An extension of the optimization methodology given in Lu...
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| Published in | AIChE journal Vol. 53; no. 7; pp. 1770 - 1778 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Hoboken
Wiley Subscription Services, Inc., A Wiley Company
01.07.2007
Wiley Subscription Services American Institute of Chemical Engineers |
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| Abstract | The determination of distillation boundaries for four‐component mixtures that exhibit azeotropy is studied. It is shown that these distillation boundaries correspond to local maxima in surface areas under conditions of Levi‐Civita parallelism. An extension of the optimization methodology given in Lucia and Taylor is used to determine local maxima in surface areas by repeatedly computing maximum line integrals or distances in one spherical coordinate over a set of initial conditions that span the range of the second spherical coordinate. An optimization formulation that defines the determination of local maxima in surface areas subject to residue curve and stable node constraints is given. Numerical measurement of surface areas is accomplished using a triangulation procedure. Several numerical examples of varying complexity involving four‐component mixtures that exhibit azeotropes are presented to show the efficacy of the proposed optimization methodology. Geometric illustrations are used throughout to highlight key features of our methodology for determining distillation boundaries in four‐component mixtures. © 2007 American Institute of Chemical Engineers AIChE J, 2007 |
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| AbstractList | The determination of distillation boundaries for four-component mixtures that exhibit azeotropy is studied. It is shown that these distillation boundaries correspond to local maxima in surface areas under conditions of Levi-Civita parallelism. An extension of the optimization methodology given in Lucia and Taylor is used to determine local maxima in surface areas by repeatedly computing maximum line integrals or distances in one spherical coordinate over a set of initial conditions that span the range of the second spherical coordinate. An optimization formulation that defines the determination of local maxima in surface areas subject to residue curve and stable node constraints is given. Numerical measurement of surface areas is accomplished using a triangulation procedure. Several numerical examples of varying complexity involving four-component mixtures that exhibit azeotropes are presented to show the efficacy of the proposed optimization methodology. Geometric illustrations are used throughout to highlight key features of our methodology for determining distillation boundaries in four-component mixtures. The determination of distillation boundaries for four‐component mixtures that exhibit azeotropy is studied. It is shown that these distillation boundaries correspond to local maxima in surface areas under conditions of Levi‐Civita parallelism. An extension of the optimization methodology given in Lucia and Taylor is used to determine local maxima in surface areas by repeatedly computing maximum line integrals or distances in one spherical coordinate over a set of initial conditions that span the range of the second spherical coordinate. An optimization formulation that defines the determination of local maxima in surface areas subject to residue curve and stable node constraints is given. Numerical measurement of surface areas is accomplished using a triangulation procedure. Several numerical examples of varying complexity involving four‐component mixtures that exhibit azeotropes are presented to show the efficacy of the proposed optimization methodology. Geometric illustrations are used throughout to highlight key features of our methodology for determining distillation boundaries in four‐component mixtures. © 2007 American Institute of Chemical Engineers AIChE J, 2007 The determination of distillation boundaries for four-component mixtures that exhibit azeotropy is studied. It is shown that these distillation boundaries correspond to local maxima in surface areas under conditions of Levi-Civita parallelism. An extension of the optimization methodology given in Lucia and Taylor is used to determine local maxima in surface areas by repeatedly computing maximum line integrals or distances in one spherical coordinate over a set of initial conditions that span the range of the second spherical coordinate. An optimization formulation that defines the determination of local maxima in surface areas subject to residue curve and stable node constraints is given. Numerical measurement of surface areas is accomplished using a triangulation procedure. Several numerical examples of varying complexity involving four-component mixtures that exhibit azeotropes are presented to show the efficacy of the proposed optimization methodology. Geometric illustrations are used throughout to highlight key features of our methodology for determining distillation boundaries in four-component mixtures. [PUBLICATION ABSTRACT] |
| Author | Lucia, Angelo Bellows, Meghan |
| Author_xml | – sequence: 1 givenname: Meghan surname: Bellows fullname: Bellows, Meghan organization: Dept. of Chemical Engineering, University of Rhode Island, Kingston, RI 02881 – sequence: 2 givenname: Angelo surname: Lucia fullname: Lucia, Angelo email: lucia@egr.uri.edu organization: Dept. of Chemical Engineering, University of Rhode Island, Kingston, RI 02881 |
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| Cites_doi | 10.1016/S0009-2509(03)00018-6 10.1016/0009-2509(79)85164-7 10.1002/aic.690440616 10.1002/aic.10663 10.1002/aic.10668 10.1021/ie00052a021 10.1016/0009-2509(88)87014-3 10.1016/0009-2509(90)87059-2 10.1016/0009-2509(78)80086-4 10.1016/0009-2509(88)87015-5 10.1016/0009-2509(90)87058-Z 10.1021/ie030303c 10.1021/ie9604379 10.1016/0009-2509(88)85144-3 10.1016/0009-2509(84)85057-5 |
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| Keywords | Azeotropic mixture Distillation four-component mixtures surface areas distillation boundaries Levi-Civita parallelism Surface area Optimization |
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| References | Krolikowski LJ. Determination of distillation regions for non-ideal ternary mixtures. AIChE J. 2005; 52: 532-544. Foucher ER, Doherty MF, Malone MF. Automatic screening of entrainers for homogeneous azeotropic distillation. Ind Eng Chem Res. 1991; 29: 760-772. Schreinemakers FAH. Dampfdrucke ternarer gemische. III. Theoretischer teil. Z Phys Chem. 1901; 36: 710-740. Doherty MF, Perkins JD. On the dynamics of distillation processes. I. The simple distillation of multicomponent, nonreacting, homogeneous mixtures. Chem Eng Sci. 1978; 33: 281-301. Ostwald W. Dampfdrucke ternarer gemische. Abahndlungen der Mathematisch-Physischen der Konig Sachsischen. Gesellschaft Wissenschaften. 1900; 25: 413. Schreinemakers FAH. Dampfdrucke ternarer gemische. I. Theoretischer teil. Z Phys Chem. 1901; 36: 257-289. Peterson EJ, Partin LR. Temperature sequences for categorizing all ternary distillation boundary maps. Ind Eng Chem Res. 1997; 36: 1799-1811. Kiva VN, Hilmen EK, Skogestad S. Azeotropic phase equilibrium diagrams: a survey. Chem Eng Sci. 2003; 58: 1903-1953. Thorpe, JA. Elementary Topics in Differential Geometry. Springer-Verlag: New York, 45-47 Doherty MF, Perkins JD. On the dynamics of distillation processes. III. The topological structure of ternary residue curve maps. Chem Eng Sci. 1979; 34: 1401-1414. Pham HN, Doherty MF. Design and synthesis of heterogeneous azeotropic distillations. II. Residue curve maps. Chem Eng Sci. 1990; 45: 1837-1844. Barbosa D. Doherty MF. The influence of chemical reactions on vapor-liquid phase diagrams. Chem Eng Sci. 1988; 45: 529-540. Barbosa D, Doherty MF. Design and minimum reflux calculations for single-feed multicomponent reactive distillation columns. Chem Eng Sci. 1988; 43: 1523-1537. Prausnitz JM, Anderson TF, Grens EA, Eckert CA, Hsieh R, O'Connell JP. Computer Calculations for Multicomponent Vapor-Liquid and Liquid-Liquid Equilibria. Engelwood Cliffs: Prentice-Hall, 1980. Lucia A, Taylor R. The geometry of separation boundaries. I. Basic theory and numerical support. AIChE J. 2006; 52: 582-594. Van Dongen DB, Doherty MF. On the dynamics of distillation processes vs. the topology of the boiling temperature surface and its relation to azeotropic distillation. Chem Eng Sci. 1984; 39: 883-892. Pöpken T, Gmehling J. Simple method for determining the location of distillation region boundaries in quaternary systems. Ind Eng Chem Res. 2004; 43: 777-783. Schreinemakers FAH. Dampfdrucke ternarer gemische. II. Theoretischer teil. Z Phys Chem. 1901; 36: 413-449. Ostwald W. Lehrbuk der Allgemeinen Chemie. Leipzig, Germany: Engelmann, 1902. Barbosa D, Doherty MF. The simple distillation of homogeneous reactive mixture. Chem Eng Sci. 1988; 43: 541-550. Pham HN, Doherty MF. Design and synthesis of heterogeneous azeotropic distillations. I. Heterogeneous phase diagrams. Chem Eng Sci. 1990; 45: 1823-1836. Rooks RE, Julka V, Doherty MF, Malone MF. Structures of distillation regions for multicomponent azeotropic mixtures. AIChE J. 1988; 44: 1382-1391. 2004; 43 1901; 36 1991; 29 2006; 52 1979; 34 1990; 45 1978; 33 1984; 39 1997; 36 1988; 43 2003; 58 1988; 44 2005; 52 1988; 45 1900; 25 1980 1902 Prausnitz JM (e_1_2_14_23_2) 1980 Ostwald W (e_1_2_14_2_2) 1900; 25 Schreinemakers FAH (e_1_2_14_4_2) 1901; 36 e_1_2_14_10_2 e_1_2_14_12_2 e_1_2_14_20_2 e_1_2_14_11_2 e_1_2_14_21_2 Schreinemakers FAH (e_1_2_14_6_2) 1901; 36 e_1_2_14_14_2 e_1_2_14_13_2 Schreinemakers FAH (e_1_2_14_5_2) 1901; 36 e_1_2_14_16_2 e_1_2_14_15_2 Ostwald W (e_1_2_14_3_2) 1902 e_1_2_14_18_2 e_1_2_14_17_2 e_1_2_14_19_2 Thorpe JA (e_1_2_14_22_2) e_1_2_14_7_2 e_1_2_14_9_2 e_1_2_14_8_2 |
| References_xml | – reference: Pham HN, Doherty MF. Design and synthesis of heterogeneous azeotropic distillations. I. Heterogeneous phase diagrams. Chem Eng Sci. 1990; 45: 1823-1836. – reference: Prausnitz JM, Anderson TF, Grens EA, Eckert CA, Hsieh R, O'Connell JP. Computer Calculations for Multicomponent Vapor-Liquid and Liquid-Liquid Equilibria. Engelwood Cliffs: Prentice-Hall, 1980. – reference: Barbosa D, Doherty MF. The simple distillation of homogeneous reactive mixture. Chem Eng Sci. 1988; 43: 541-550. – reference: Ostwald W. Dampfdrucke ternarer gemische. Abahndlungen der Mathematisch-Physischen der Konig Sachsischen. Gesellschaft Wissenschaften. 1900; 25: 413. – reference: Peterson EJ, Partin LR. Temperature sequences for categorizing all ternary distillation boundary maps. Ind Eng Chem Res. 1997; 36: 1799-1811. – reference: Kiva VN, Hilmen EK, Skogestad S. Azeotropic phase equilibrium diagrams: a survey. Chem Eng Sci. 2003; 58: 1903-1953. – reference: Foucher ER, Doherty MF, Malone MF. Automatic screening of entrainers for homogeneous azeotropic distillation. Ind Eng Chem Res. 1991; 29: 760-772. – reference: Pham HN, Doherty MF. Design and synthesis of heterogeneous azeotropic distillations. II. Residue curve maps. Chem Eng Sci. 1990; 45: 1837-1844. – reference: Doherty MF, Perkins JD. On the dynamics of distillation processes. III. The topological structure of ternary residue curve maps. Chem Eng Sci. 1979; 34: 1401-1414. – reference: Schreinemakers FAH. Dampfdrucke ternarer gemische. I. Theoretischer teil. Z Phys Chem. 1901; 36: 257-289. – reference: Lucia A, Taylor R. The geometry of separation boundaries. I. Basic theory and numerical support. AIChE J. 2006; 52: 582-594. – reference: Barbosa D. Doherty MF. The influence of chemical reactions on vapor-liquid phase diagrams. Chem Eng Sci. 1988; 45: 529-540. – reference: Doherty MF, Perkins JD. On the dynamics of distillation processes. I. The simple distillation of multicomponent, nonreacting, homogeneous mixtures. Chem Eng Sci. 1978; 33: 281-301. – reference: Van Dongen DB, Doherty MF. On the dynamics of distillation processes vs. the topology of the boiling temperature surface and its relation to azeotropic distillation. Chem Eng Sci. 1984; 39: 883-892. – reference: Pöpken T, Gmehling J. Simple method for determining the location of distillation region boundaries in quaternary systems. Ind Eng Chem Res. 2004; 43: 777-783. – reference: Thorpe, JA. Elementary Topics in Differential Geometry. Springer-Verlag: New York, 45-47 – reference: Schreinemakers FAH. Dampfdrucke ternarer gemische. II. Theoretischer teil. Z Phys Chem. 1901; 36: 413-449. – reference: Barbosa D, Doherty MF. Design and minimum reflux calculations for single-feed multicomponent reactive distillation columns. Chem Eng Sci. 1988; 43: 1523-1537. – reference: Krolikowski LJ. Determination of distillation regions for non-ideal ternary mixtures. AIChE J. 2005; 52: 532-544. – reference: Ostwald W. Lehrbuk der Allgemeinen Chemie. Leipzig, Germany: Engelmann, 1902. – reference: Schreinemakers FAH. Dampfdrucke ternarer gemische. III. Theoretischer teil. Z Phys Chem. 1901; 36: 710-740. – reference: Rooks RE, Julka V, Doherty MF, Malone MF. Structures of distillation regions for multicomponent azeotropic mixtures. AIChE J. 1988; 44: 1382-1391. – volume: 43 start-page: 1523 year: 1988 end-page: 1537 article-title: Design and minimum reflux calculations for single‐feed multicomponent reactive distillation columns publication-title: Chem Eng Sci. – volume: 39 start-page: 883 year: 1984 end-page: 892 article-title: On the dynamics of distillation processes vs. the topology of the boiling temperature surface and its relation to azeotropic distillation publication-title: Chem Eng Sci. – volume: 43 start-page: 541 year: 1988 end-page: 550 article-title: The simple distillation of homogeneous reactive mixture publication-title: Chem Eng Sci. – volume: 45 start-page: 1823 year: 1990 end-page: 1836 article-title: Design and synthesis of heterogeneous azeotropic distillations. I. Heterogeneous phase diagrams publication-title: Chem Eng Sci. – volume: 45 start-page: 1837 year: 1990 end-page: 1844 article-title: Design and synthesis of heterogeneous azeotropic distillations. II. Residue curve maps publication-title: Chem Eng Sci. – year: 1980 – volume: 34 start-page: 1401 year: 1979 end-page: 1414 article-title: On the dynamics of distillation processes. III. The topological structure of ternary residue curve maps publication-title: Chem Eng Sci. – volume: 52 start-page: 582 year: 2006 end-page: 594 article-title: The geometry of separation boundaries. I. Basic theory and numerical support publication-title: AIChE J. – volume: 58 start-page: 1903 year: 2003 end-page: 1953 article-title: Azeotropic phase equilibrium diagrams: a survey publication-title: Chem Eng Sci. – volume: 29 start-page: 760 year: 1991 end-page: 772 article-title: Automatic screening of entrainers for homogeneous azeotropic distillation publication-title: Ind Eng Chem Res. – volume: 25 start-page: 413 year: 1900 article-title: Dampfdrucke ternarer gemische. Abahndlungen der Mathematisch‐Physischen der Konig Sachsischen publication-title: Gesellschaft Wissenschaften. – volume: 33 start-page: 281 year: 1978 end-page: 301 article-title: On the dynamics of distillation processes. I. The simple distillation of multicomponent, nonreacting, homogeneous mixtures publication-title: Chem Eng Sci. – volume: 36 start-page: 710 year: 1901 end-page: 740 article-title: Dampfdrucke ternarer gemische. III. Theoretischer teil publication-title: Z Phys Chem. – volume: 44 start-page: 1382 year: 1988 end-page: 1391 article-title: Structures of distillation regions for multicomponent azeotropic mixtures publication-title: AIChE J. – start-page: 45 end-page: 47 – year: 1902 – volume: 52 start-page: 532 year: 2005 end-page: 544 article-title: Determination of distillation regions for non‐ideal ternary mixtures publication-title: AIChE J. – volume: 45 start-page: 529 year: 1988 end-page: 540 article-title: The influence of chemical reactions on vapor–liquid phase diagrams publication-title: Chem Eng Sci. – volume: 43 start-page: 777 year: 2004 end-page: 783 article-title: Simple method for determining the location of distillation region boundaries in quaternary systems publication-title: Ind Eng Chem Res. – volume: 36 start-page: 1799 year: 1997 end-page: 1811 article-title: Temperature sequences for categorizing all ternary distillation boundary maps publication-title: Ind Eng Chem Res. – volume: 36 start-page: 257 year: 1901 end-page: 289 article-title: Dampfdrucke ternarer gemische. I. Theoretischer teil publication-title: Z Phys Chem. – volume: 36 start-page: 413 year: 1901 end-page: 449 article-title: Dampfdrucke ternarer gemische. II. Theoretischer teil publication-title: Z Phys Chem. – volume: 36 start-page: 413 year: 1901 ident: e_1_2_14_5_2 article-title: Dampfdrucke ternarer gemische. II. Theoretischer teil publication-title: Z Phys Chem. – volume: 25 start-page: 413 year: 1900 ident: e_1_2_14_2_2 article-title: Dampfdrucke ternarer gemische. Abahndlungen der Mathematisch‐Physischen der Konig Sachsischen publication-title: Gesellschaft Wissenschaften. – ident: e_1_2_14_8_2 doi: 10.1016/S0009-2509(03)00018-6 – ident: e_1_2_14_10_2 doi: 10.1016/0009-2509(79)85164-7 – ident: e_1_2_14_19_2 doi: 10.1002/aic.690440616 – ident: e_1_2_14_20_2 doi: 10.1002/aic.10663 – ident: e_1_2_14_7_2 doi: 10.1002/aic.10668 – ident: e_1_2_14_17_2 doi: 10.1021/ie00052a021 – volume-title: Computer Calculations for Multicomponent Vapor–Liquid and Liquid–Liquid Equilibria year: 1980 ident: e_1_2_14_23_2 – ident: e_1_2_14_13_2 doi: 10.1016/0009-2509(88)87014-3 – ident: e_1_2_14_12_2 doi: 10.1016/0009-2509(90)87059-2 – ident: e_1_2_14_9_2 doi: 10.1016/0009-2509(78)80086-4 – volume-title: Lehrbuk der Allgemeinen Chemie year: 1902 ident: e_1_2_14_3_2 – ident: e_1_2_14_14_2 doi: 10.1016/0009-2509(88)87015-5 – start-page: 45 volume-title: Elementary Topics in Differential Geometry ident: e_1_2_14_22_2 – ident: e_1_2_14_11_2 doi: 10.1016/0009-2509(90)87058-Z – ident: e_1_2_14_21_2 doi: 10.1021/ie030303c – volume: 36 start-page: 257 year: 1901 ident: e_1_2_14_4_2 article-title: Dampfdrucke ternarer gemische. I. Theoretischer teil publication-title: Z Phys Chem. – ident: e_1_2_14_18_2 doi: 10.1021/ie9604379 – volume: 36 start-page: 710 year: 1901 ident: e_1_2_14_6_2 article-title: Dampfdrucke ternarer gemische. III. Theoretischer teil publication-title: Z Phys Chem. – ident: e_1_2_14_15_2 doi: 10.1016/0009-2509(88)85144-3 – ident: e_1_2_14_16_2 doi: 10.1016/0009-2509(84)85057-5 |
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| Snippet | The determination of distillation boundaries for four‐component mixtures that exhibit azeotropy is studied. It is shown that these distillation boundaries... The determination of distillation boundaries for four-component mixtures that exhibit azeotropy is studied. It is shown that these distillation boundaries... |
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| SubjectTerms | Applied sciences Boundaries Chemical engineering Distillation distillation boundaries Exact sciences and technology four-component mixtures Geometry Levi-Civita parallelism Methods Optimization Surface area surface areas |
| Title | The geometry of separation boundaries: Four-component mixtures |
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