同伦法在共沸物预测中的应用
摘要
在共沸精馏模拟中共沸点的准确预测起着举足轻重的作用，传统的预测方法，有的采用经验关联式，但是其适用范围有限；而采用基于多元汽－液相平衡数据的共沸点预测方法，目前一般采用局部收敛性的Newton法求解。同伦方法具有大范围收敛性，对初值要求不高的优点，但收敛速度慢；Newton法具有收敛速度快的优点，但对初值要求苛刻。本文将两种算法优势互补，提出了预测共沸点的新算法同伦-Newton联合算法。
同伦－Newton联合算法首先采用同伦方法把满足共沸条件的方程转化为微分连续同伦函数，并由此变成一个常微分方程的初值问题。对于转化以后的常微分初值问题，分别采用了中点求积公式和Euler预估－Newton校正法对其进行求解。以此解作为牛顿法的初值，然后采用Newton方法得到方程的精确解。
在本文中，分别采用Newton法、同伦法、同伦-Newton联合算法预测了二元物系和三元物系等十余种强非理想性物系的共沸点，结果表明，相对牛顿法，同伦-Newton联合算法的收敛范围有了明显的扩大，收敛迭代过程的稳定性也有了很大的改善；而与同伦法相比，同伦-Newton联合算法的收敛速度和精度明显地提高。
It is important to predict azeotropic temperature and composition in distillation. The traditional predictive methods include experiential methods and multicomponent vapor-liquid equilibrium methods, etc. But the applicable scope is finite, or the convergence scope is finite. Because the convergence of homotopy methods is larger than Newton methods, so it can make up for this disadvantage. In this thesis, authors combine the advantage of homotopy methods with Newton methods to form new homotopy-Newton united methods.
In the first, the equations are being transformed to a differential homotopy-continuation function, and being a initial value problem of ordinary differential equation. Prediction-corrector methods are adopted in this thesis to solve ordinary differential equation. At last we adopt Newton methods to get the precision answer.
In the thesis, new homotopy-Newton united methods、Newton methods and homotopy methods are adopted to predict azeotropic temperature and composition. Compare with Newton methods, the convergence scope of the new homotopy-Newton united methods is being enlarge and robust. Compare with homotopy methods, The new homotopy-Newton united methods, precision and rapidity are make a progress.
同伦－Newton联合算法首先采用同伦方法把满足共沸条件的方程转化为微分连续同伦函数，并由此变成一个常微分方程的初值问题。对于转化以后的常微分初值问题，分别采用了中点求积公式和Euler预估－Newton校正法对其进行求解。以此解作为牛顿法的初值，然后采用Newton方法得到方程的精确解。
在本文中，分别采用Newton法、同伦法、同伦-Newton联合算法预测了二元物系和三元物系等十余种强非理想性物系的共沸点，结果表明，相对牛顿法，同伦-Newton联合算法的收敛范围有了明显的扩大，收敛迭代过程的稳定性也有了很大的改善；而与同伦法相比，同伦-Newton联合算法的收敛速度和精度明显地提高。
It is important to predict azeotropic temperature and composition in distillation. The traditional predictive methods include experiential methods and multicomponent vapor-liquid equilibrium methods, etc. But the applicable scope is finite, or the convergence scope is finite. Because the convergence of homotopy methods is larger than Newton methods, so it can make up for this disadvantage. In this thesis, authors combine the advantage of homotopy methods with Newton methods to form new homotopy-Newton united methods.
In the first, the equations are being transformed to a differential homotopy-continuation function, and being a initial value problem of ordinary differential equation. Prediction-corrector methods are adopted in this thesis to solve ordinary differential equation. At last we adopt Newton methods to get the precision answer.
In the thesis, new homotopy-Newton united methods、Newton methods and homotopy methods are adopted to predict azeotropic temperature and composition. Compare with Newton methods, the convergence scope of the new homotopy-Newton united methods is being enlarge and robust. Compare with homotopy methods, The new homotopy-Newton united methods, precision and rapidity are make a progress.
引文
[1] 时均，汪家鼎，余国琮等，化学工程手册[上]，化学工业出版社，1996
[2] 熊洪允，曾绍标，毛云英，应用数学基础, 天津：津大学出版社，1994
[3] Wayburn T.L, Seader J.D., Homotopy continuation methods for computer-aided process design, Computer&Chemical engineering，1987，11(1):7-25
[4] 刘庆林，韩方煜，丁惠华，同伦函数法用于非理想物系分离的计算，化学工程，1996,24(3):65-70
[5] 李庆扬，关治，白峰杉，数值计算原理，北京：清华大学出版社，2000
[6] Kovach J.W., Seader W.D., Heterogeneous azeotropic distillation homotopy continuation methods, Computer&Chemical engineering, 1987,11(6):593-605
[7] 周爱月，刘成，陈洪钫，用同伦延拓算法模拟复杂精馏过程(1)模型与算法，化学工程，1995,46(4): 431-438
[8] 周爱月，刘成，陈洪钫，用同伦延拓算法模拟复杂精馏过程(2)模型与算法，化学工程，1995,46(4):439-445
[9] Fidkowski Z.T., Malone M.F., Doherty M.F., Computing azeotropes in mult
icomponent mixture, Computer&Chemical engineering, 1993,17(12): 1141-1155
[10] Eckert E., Kubicek M., Computing heterogeneous azeotropes in multicomponent mixtures, Computer&Chemical engineering, 1997,21(3): 347-350
[11] John E.T.,Paul I.B., Computation of heteroazeotropes, Computer&Chemical engineering, 1998,22(suppl): s61-s68
[12] John E.T., Paul I.B., Computation of heteroazeotropes PartⅠtheory，Chemical engineering science, 2000,35(18): 3817-3834
[13] John E.T., Paul I.B., Computation of heteroazeotropes PartⅡ: efficient calculation of changes in phase equilibrium structure, Chemical engineeringscience, 2000,35(18): 3835-3853
[14] Nikolaos B,Thomas E.G., Manfred M., Multiple steady states in distillation effect of VL(L)E inaccuracies, AIChE Journal, 2000, 46(5): 955-979
[15] Taylor K., Krishna R.,Modelling reactive distillation. Chemical engineering sciences, 2000,55(22): 5183-5229
[16] Reneaume J.M.,Meyer M., Letourneaut J.J., et.al. A global MINLP approach for phase equilibrium calculations, Computers&Chemical engineering，
1996,20(suppl): s303-s308
[17] Amy C.S., Warren D.S., Homotopy continuation methods for stability analysis in the global minimization of the Gibbs free energy, Fluid phase equilibria, 1995,103(2): 213-249
[18] Jalali F., Seader J.D., Homotopy continuation method in multi-phase multi-reaction equilibrium system Computer&Chemical engineering, 1999,23(19): 1319-1331
[19] Jurgen B., Wolfgang M., Quick and reliable phase stability test in VLLE flash calculation by homotopy continuation, Computer&Chemical engineering, 2000, 24(11): 2447-2456
[20] Sven H,Kai F.,Jurgen G., Measurement and calculation of critical points
for binary and ternary mixture, AICHE Journal,2002,48(10):2350-2356
[21] Heidemann R.A.,khalil A.M.,The calculation of critical points, AIChE Journal, 1980,26(5): 769-779
[22] Wang M.C,David S.H.W,Chen H.,et al.Homotopy continuation methods for calculation critical loci of binary mixture, Chemical engineering science,
1999,54(17): 3873-3883.
[23] Andrzej S.,Reactive separation for process intensification: an industrial perspective, Chemical engineering and processing, 2003, 42(3): 137-144
[24] Sebastian E.,Gregor F.,Control of a reactive separation process, Chemical engineering and processing 2003,42(3):201-210
[25] 刘劲松，白鹏，朱思强，等反应精馏过程的研究进展，化学工业与工程，
2002，19(1)：101-106
[26] Chang Y.A., Seader J.D., Simulation of continuous reactive distillation by a homotopy-continuation methods, Computer&Chemical engineering, 1988,
12(12): 1243-1255
[27] Jin-Ho L., Dudukovic M.P.,A comparison of the equilibrium and nonequilibrium models for a multicomponent reactive distillation column, Computer&Chemical engineering,1999,23(1):159-172
[28] Mathew J.D., Michale F.D., Thermodynamic behavior of reactive azeotropes, AICHE Journal, 1997, 43(9): 2227-2238.
[29] Shenbo Y.,Aiyue Z., Qiu T., Simulation of multistage catalytic stripping with anonequilibrium stage model, Computer&Chemical engineering,1997,21(4):409-415
[30] Erik.A.W., Sigurd,S., Operation of integrated three-product (Petlyuk) distillation, Industrial&engineering chemistry research,1995,34(6):2094-2103
[31] Atle C.C., Sigurd S., Kristian L., Complex distillation arrangements: extending the Petlyuk ideas, Computer&Chemical engineering, 1997,21(suppl): s237-s242
[32] Young H.K., Rigorous design of fully thermally coupled distillation column,Journal of chemical engineering of Japan, 2001,34(2):236-243
[33] Piyush B.S., Antonis C.K., New synthesis framework for the optimization of complex distillation system, AIChE Jorunal,2002,48(3):527-550
[34] Ben G.R., Andezej K., Synthesis of functionally distinct thermally coupled coch, 2003,42(6): 1204-1214
[35] 杨友麟，热偶精馏塔操作特性的模拟研究，化工学报,1990,41(4):491-497
[36] Rafael C.C., Seader J.D., Thomas L.W., Multiple steady-state solutions for interlinked separation systems, Industrial&engineering chemistry fundamentals, 1986,25(4): 566-576
[37] Wen-Jing L., Seader J.D., Wayburn T.L., Computing multiple solutions to systems of interlinked separation column AIChE Journal, 1987,33(6): 886-897
[38] Kyung T.H., Kang J.L., Simulation and analysis of complex distillation column using continuation methods, Computer&Chemical engineering,1993,17(suppl): s479-s484
[39] 周爱月，王瑜，崔玉林，用同伦延拓算法模拟热偶精馏过程,化工学报,1999,50(1): 1-7
[40] 陈洪钫，刘家祺,化工分离过程，化学工业出版社，1995
[41] 秦奎德，复杂精馏过程的模拟:[硕士学位论文],天津：天津大学化工学院，
1995
[42] 刘成，同伦方法在精馏模拟计算中的应用：[硕士学位论文]，天津：天津大学化工学院，1993
[43] 朱水根，龚时霖，计算方法引论及例题选讲，天津科学技术出版社，1990
[44] John E.Tolsma, Paul I.Barton, Computation of heteroazeotropes Part I: Theory, Chemical Engineering Science 55(2000): 3817-3834
[45] John E.Tolsma, Paul I.Barton, Computation of heteroazeotropes PartⅡ:Efficient calculation of changes in phase equilibrium structure, Chemical Engineering Science 55(2000):3835-3853
[46] 黄承遇，陈钟秀，顾飞燕，化工热力学，化学工业出版社，1990
[47] 郭天民等，多元汽－液相平衡和精馏，化学工业出版社，1983
[2] 熊洪允，曾绍标，毛云英，应用数学基础, 天津：津大学出版社，1994
[3] Wayburn T.L, Seader J.D., Homotopy continuation methods for computer-aided process design, Computer&Chemical engineering，1987，11(1):7-25
[4] 刘庆林，韩方煜，丁惠华，同伦函数法用于非理想物系分离的计算，化学工程，1996,24(3):65-70
[5] 李庆扬，关治，白峰杉，数值计算原理，北京：清华大学出版社，2000
[6] Kovach J.W., Seader W.D., Heterogeneous azeotropic distillation homotopy continuation methods, Computer&Chemical engineering, 1987,11(6):593-605
[7] 周爱月，刘成，陈洪钫，用同伦延拓算法模拟复杂精馏过程(1)模型与算法，化学工程，1995,46(4): 431-438
[8] 周爱月，刘成，陈洪钫，用同伦延拓算法模拟复杂精馏过程(2)模型与算法，化学工程，1995,46(4):439-445
[9] Fidkowski Z.T., Malone M.F., Doherty M.F., Computing azeotropes in mult
icomponent mixture, Computer&Chemical engineering, 1993,17(12): 1141-1155
[10] Eckert E., Kubicek M., Computing heterogeneous azeotropes in multicomponent mixtures, Computer&Chemical engineering, 1997,21(3): 347-350
[11] John E.T.,Paul I.B., Computation of heteroazeotropes, Computer&Chemical engineering, 1998,22(suppl): s61-s68
[12] John E.T., Paul I.B., Computation of heteroazeotropes PartⅠtheory，Chemical engineering science, 2000,35(18): 3817-3834
[13] John E.T., Paul I.B., Computation of heteroazeotropes PartⅡ: efficient calculation of changes in phase equilibrium structure, Chemical engineeringscience, 2000,35(18): 3835-3853
[14] Nikolaos B,Thomas E.G., Manfred M., Multiple steady states in distillation effect of VL(L)E inaccuracies, AIChE Journal, 2000, 46(5): 955-979
[15] Taylor K., Krishna R.,Modelling reactive distillation. Chemical engineering sciences, 2000,55(22): 5183-5229
[16] Reneaume J.M.,Meyer M., Letourneaut J.J., et.al. A global MINLP approach for phase equilibrium calculations, Computers&Chemical engineering，
1996,20(suppl): s303-s308
[17] Amy C.S., Warren D.S., Homotopy continuation methods for stability analysis in the global minimization of the Gibbs free energy, Fluid phase equilibria, 1995,103(2): 213-249
[18] Jalali F., Seader J.D., Homotopy continuation method in multi-phase multi-reaction equilibrium system Computer&Chemical engineering, 1999,23(19): 1319-1331
[19] Jurgen B., Wolfgang M., Quick and reliable phase stability test in VLLE flash calculation by homotopy continuation, Computer&Chemical engineering, 2000, 24(11): 2447-2456
[20] Sven H,Kai F.,Jurgen G., Measurement and calculation of critical points
for binary and ternary mixture, AICHE Journal,2002,48(10):2350-2356
[21] Heidemann R.A.,khalil A.M.,The calculation of critical points, AIChE Journal, 1980,26(5): 769-779
[22] Wang M.C,David S.H.W,Chen H.,et al.Homotopy continuation methods for calculation critical loci of binary mixture, Chemical engineering science,
1999,54(17): 3873-3883.
[23] Andrzej S.,Reactive separation for process intensification: an industrial perspective, Chemical engineering and processing, 2003, 42(3): 137-144
[24] Sebastian E.,Gregor F.,Control of a reactive separation process, Chemical engineering and processing 2003,42(3):201-210
[25] 刘劲松，白鹏，朱思强，等反应精馏过程的研究进展，化学工业与工程，
2002，19(1)：101-106
[26] Chang Y.A., Seader J.D., Simulation of continuous reactive distillation by a homotopy-continuation methods, Computer&Chemical engineering, 1988,
12(12): 1243-1255
[27] Jin-Ho L., Dudukovic M.P.,A comparison of the equilibrium and nonequilibrium models for a multicomponent reactive distillation column, Computer&Chemical engineering,1999,23(1):159-172
[28] Mathew J.D., Michale F.D., Thermodynamic behavior of reactive azeotropes, AICHE Journal, 1997, 43(9): 2227-2238.
[29] Shenbo Y.,Aiyue Z., Qiu T., Simulation of multistage catalytic stripping with anonequilibrium stage model, Computer&Chemical engineering,1997,21(4):409-415
[30] Erik.A.W., Sigurd,S., Operation of integrated three-product (Petlyuk) distillation, Industrial&engineering chemistry research,1995,34(6):2094-2103
[31] Atle C.C., Sigurd S., Kristian L., Complex distillation arrangements: extending the Petlyuk ideas, Computer&Chemical engineering, 1997,21(suppl): s237-s242
[32] Young H.K., Rigorous design of fully thermally coupled distillation column,Journal of chemical engineering of Japan, 2001,34(2):236-243
[33] Piyush B.S., Antonis C.K., New synthesis framework for the optimization of complex distillation system, AIChE Jorunal,2002,48(3):527-550
[34] Ben G.R., Andezej K., Synthesis of functionally distinct thermally coupled coch, 2003,42(6): 1204-1214
[35] 杨友麟，热偶精馏塔操作特性的模拟研究，化工学报,1990,41(4):491-497
[36] Rafael C.C., Seader J.D., Thomas L.W., Multiple steady-state solutions for interlinked separation systems, Industrial&engineering chemistry fundamentals, 1986,25(4): 566-576
[37] Wen-Jing L., Seader J.D., Wayburn T.L., Computing multiple solutions to systems of interlinked separation column AIChE Journal, 1987,33(6): 886-897
[38] Kyung T.H., Kang J.L., Simulation and analysis of complex distillation column using continuation methods, Computer&Chemical engineering,1993,17(suppl): s479-s484
[39] 周爱月，王瑜，崔玉林，用同伦延拓算法模拟热偶精馏过程,化工学报,1999,50(1): 1-7
[40] 陈洪钫，刘家祺,化工分离过程，化学工业出版社，1995
[41] 秦奎德，复杂精馏过程的模拟:[硕士学位论文],天津：天津大学化工学院，
1995
[42] 刘成，同伦方法在精馏模拟计算中的应用：[硕士学位论文]，天津：天津大学化工学院，1993
[43] 朱水根，龚时霖，计算方法引论及例题选讲，天津科学技术出版社，1990
[44] John E.Tolsma, Paul I.Barton, Computation of heteroazeotropes Part I: Theory, Chemical Engineering Science 55(2000): 3817-3834
[45] John E.Tolsma, Paul I.Barton, Computation of heteroazeotropes PartⅡ:Efficient calculation of changes in phase equilibrium structure, Chemical Engineering Science 55(2000):3835-3853
[46] 黄承遇，陈钟秀，顾飞燕，化工热力学，化学工业出版社，1990
[47] 郭天民等，多元汽－液相平衡和精馏，化学工业出版社，1983