多相平衡的状态方程一步算法研究
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摘要
多相平衡算法通常可以分为Gibbs自有能最小化法和解方程法，其中求解状态方程的算法一般包括相稳定性判断和平衡计算两个步骤，计算过程比较复杂。本文采用一种多相平衡的状态方程一步算法，即一种通过修正各相总摩尔分数的解方程法——??因子法。该法只需要求解一次最小化问题，就可以同时给出多相平衡时存在的相数、各相的量及组成。
Han和Rangaiah（1997）利用罚函数的思路构造目标函数并使用连续的线性规划法求解上述最小化问题。该算法包括大小两个循环，小循环利用线性规划法求解变量的迭代方向时，引入了人工变量，这样不可避免地涉及线性规划法中用到的基和基向量的确定、基的转换和运算，算法相对比较复杂。
本文采用Marquardt法求解该最小化问题，构造如下形式的目标函数：
f(w)= ()+∑hj 2(w)
其中，为权重因子，τV、τI和τII分别是汽液液三相平衡中汽相、液相I和液相II的特征参数，相平衡中的等式约束条件记作h(w) = 0，并将每一条件记作hj(w)。
通过合理的选取阻尼因子λ，Marquardt法既可以保证目标函数不断地下降，又可以确保一定的收敛速度，而且变量迭代方向的确定也非常简便。
本文对正十六烷-水-氢气、甲苯-水-氢气、二氧化碳-二甲醚和二甲醚-水等四个体系应用不同的立方型状态方程和不同的混合规则进行了多相平衡的计算：前两个体系逸度系数采用Peng-Robinson方程计算，后两个体系采用Soave-Redlich-Kwong方程计算；前三个体系采用van der Waals单流体混合规则，最后一个体系采用GE形式混合规则。对计算结果分别绘制了多组分相图，结果表明改进的??因子法对多相平衡计算是一种比较成功和可靠的算法。
Procedures for the multiphase equilibrium calculation are generally based on Gibbs free energy minimization or equation-solving methods. Equation-solving methods often involve sequential procedures for a priori the phase identification and then the phase equilibrium calculation. An one-step equation-solving algorithm (?-method), based on modifying the mole fraction summation of each phase, was adopted for the calculation of multiphase equilibria using equations of state. It requires only one-step solution of minimization problem and provides phase number present at phase equilibrium, their quantities and compositions simultaneously. And phase identification in advance is not required.
Han and Rangaiah (1997) reported an algorithm based on the penalty function method and the minimization problem was solved by a version of successive linear programming methods. Their algorithm included an outer iteration and an inner one. The artificial variables were introduced in the inner iteration for solving search direction of variables. The radius vector including artificial variables introduced should be initialized, converted and operated. The method was relatively complicated.
In this study, the Marquardt method is used for the minimization of the object function:
f(w)= ()+∑hj 2(w)
Where is a weighting parameter, τV、τI and τII are the phase characteristic variables for vapor, liquid I and liquid II respectively. Equality constraints are devoted by h(w) = 0 while any one of them is written as hj(w).
By properly selecting the damping factor-λ, the object function can be minimized with a satisfied convergence rate. And the determination for the search direction of variables becomes very convenient.
The method was verified with several typical multiphase systems, including n-C16H34-H2O-H2, C7H8-H2O-H2, CO2-DME (dimethyl ether) and DME-H2O using
different equations of state and mixing rules: the fugacity coefficients for the former two systems were calculated using the Peng-Robinson equation of state and those for the latter two were calculated using the Soave-Redlich-Kwong equation of state, and the van der Waals mixing rule was used for the former three systems and the GE model mixing rule was used for the last one. The multi-components phase diagrams were drawn and the results show that the modified ?-method is successful and reliable for multiphase equilibrium calculation.
Graduate student: Xu Huilin (Chemical Engineering) Directed by: Associate Professor Chen Jian
Han和Rangaiah（1997）利用罚函数的思路构造目标函数并使用连续的线性规划法求解上述最小化问题。该算法包括大小两个循环，小循环利用线性规划法求解变量的迭代方向时，引入了人工变量，这样不可避免地涉及线性规划法中用到的基和基向量的确定、基的转换和运算，算法相对比较复杂。
本文采用Marquardt法求解该最小化问题，构造如下形式的目标函数：
f(w)= ()+∑hj 2(w)
其中，为权重因子，τV、τI和τII分别是汽液液三相平衡中汽相、液相I和液相II的特征参数，相平衡中的等式约束条件记作h(w) = 0，并将每一条件记作hj(w)。
通过合理的选取阻尼因子λ，Marquardt法既可以保证目标函数不断地下降，又可以确保一定的收敛速度，而且变量迭代方向的确定也非常简便。
本文对正十六烷-水-氢气、甲苯-水-氢气、二氧化碳-二甲醚和二甲醚-水等四个体系应用不同的立方型状态方程和不同的混合规则进行了多相平衡的计算：前两个体系逸度系数采用Peng-Robinson方程计算，后两个体系采用Soave-Redlich-Kwong方程计算；前三个体系采用van der Waals单流体混合规则，最后一个体系采用GE形式混合规则。对计算结果分别绘制了多组分相图，结果表明改进的??因子法对多相平衡计算是一种比较成功和可靠的算法。
Procedures for the multiphase equilibrium calculation are generally based on Gibbs free energy minimization or equation-solving methods. Equation-solving methods often involve sequential procedures for a priori the phase identification and then the phase equilibrium calculation. An one-step equation-solving algorithm (?-method), based on modifying the mole fraction summation of each phase, was adopted for the calculation of multiphase equilibria using equations of state. It requires only one-step solution of minimization problem and provides phase number present at phase equilibrium, their quantities and compositions simultaneously. And phase identification in advance is not required.
Han and Rangaiah (1997) reported an algorithm based on the penalty function method and the minimization problem was solved by a version of successive linear programming methods. Their algorithm included an outer iteration and an inner one. The artificial variables were introduced in the inner iteration for solving search direction of variables. The radius vector including artificial variables introduced should be initialized, converted and operated. The method was relatively complicated.
In this study, the Marquardt method is used for the minimization of the object function:
f(w)= ()+∑hj 2(w)
Where is a weighting parameter, τV、τI and τII are the phase characteristic variables for vapor, liquid I and liquid II respectively. Equality constraints are devoted by h(w) = 0 while any one of them is written as hj(w).
By properly selecting the damping factor-λ, the object function can be minimized with a satisfied convergence rate. And the determination for the search direction of variables becomes very convenient.
The method was verified with several typical multiphase systems, including n-C16H34-H2O-H2, C7H8-H2O-H2, CO2-DME (dimethyl ether) and DME-H2O using
different equations of state and mixing rules: the fugacity coefficients for the former two systems were calculated using the Peng-Robinson equation of state and those for the latter two were calculated using the Soave-Redlich-Kwong equation of state, and the van der Waals mixing rule was used for the former three systems and the GE model mixing rule was used for the last one. The multi-components phase diagrams were drawn and the results show that the modified ?-method is successful and reliable for multiphase equilibrium calculation.
Graduate student: Xu Huilin (Chemical Engineering) Directed by: Associate Professor Chen Jian
引文
Ammar M. N. and Renon H., The isothermal flash problem: new methods for phase split calculations, AICHE J., 1987, 33: 926~939
Baker L. E., Pierce A. C. and Luks K. D., Gibbs energy analysis of phase equilibria, Soc. Petro. Eng. J., 1982, 22: 731~742
Boston J. F. and Britt H. L., A radically different formulation and solution of the single-stage flash problem, Comput. Chem. Eng., 1978, 2: 109~122
Bullard L. G. and Biegler L. T., Iterated linear programming approaches for nonsmooth simulation: continuous and mixed integer approaches, Comput. Chem. Eng., 1992, 16: 949~961
Bunz A. P., Dohrn R. and Prausnitz J. M., Three-phase flash calculations for multicomponent systems, Comput. Chem. Eng., 1991, 15: 47~51
Castier M., Rasmussen P. and Fredenslund A., Calculation of simultaneous chemical and phase equilibria in nonideal system, Chem. Engng. Sci., 1989, 44: 237~248
Castillo J. and Grossmann I. E., Computation of phase and chemical equilibria, Comput. Chem. Eng., 1981, 5: 99~108
Chen J. and Li Z., Study on the GE-type Mixing Rule for Cubic Equations of State, Tsinghua Science and Technology, 1996, 1(4): 410~415
Elbaccouch M. M. and Elliott J. R., High-pressure vapor-liquid equilibrium for dimethyl+ethanol and dimethyl ether+ethanol+water, J. Chem. Eng. Data, 2000, 45: 1080~1087
Elhassan, A. E., Tsvetkov, S. G. and Craven, R. J. B., et al., Rigorous mathematical proof of the area method for phase stability, Ind. Eng. Chem. Res., 1998, 37: 1482~1489
Fournier R. L. and Boston J. F., A quasi-Newton algorithm for solving multiphase equilibrium flash problems, Chem. Engng. Commun., 1981, 8: 305~326
Gautam R. and Seider W. D.，Computation of phase and chemical equilibrium-I. Local and constrained minima in Gibbs free energy. AICHE J.，1979, 25: 991~1015
Han G. and Rangaiah G. P.，A method for multiphase equilibrium calculations, Computers. Chem. Engng.，1998, 22: 897~911
Heidemann R. A., Computation of high pressure phase equilibria, Fluid Phase Equilibria, 1983, 14: 55~78
Hodges D., Pritchard D. W. and Anwar M. M., et al., Calculating binary and ternary multiphase equilibria: extensions of the integral area method, Fluid Phase Equilibria, 1997, 130: 101~116
Hodges D.， Pritchard D. W. and Anwar M. M., Calculating binary and ternary multiphase equilibria: the tangent plane intersection method, Fluid Phase Equilibria, 1998, 152: 187~208
Holldorff H. and Knapp H., Binary vapor-liquid-liquid equilibrium of dimethyl ether-water and mutual solubilities of methyl chloride and water, Fluid Phase Equilibria, 1988, 44: 195~209
Hua J. Z., Brennecke J. F. and Stadtherr M. A., Enhanced interval analysis for phase stability: Cubic equation of state models, Ind. Eng. Chem. Res., 1998, 37: 1519~1527
Huron M. J. and Vidal J., New mixing rules in simple equations of state for representing VLE of strongly non-ideal mixtures, Fluid Phase Equilibria, 1979, 3: 255~271
Jonasson A., Persson O. and Fredenslund A., High pressure solubility of carbon dioxide and carbon monoxide in dimethyl ether, J. Chem. Eng. Data, 1995, 40: 296~300
Kinoshita M. and Takamatsu T., A powerful solution algorithm for single-stage flash problem, Comput. Chem. Eng., 1986, 10: 352~360
Lu B. C. Y., Yu P. and Sugie A. H., Prediction of vapor-liquid-liquid equilibriaby means ofa modified regula-falsi method, Chem. Eng. Sci., 1974, 29: 321~326
Lucia A. and Taylor R., Complex iterative solutions to process model equations, Comput. Chem. Eng., 1992, 16: S387~S394
Mehra R. K., Heidemann R. A. and Aziz K., An accelerated successive substitution algorithm,Can. J. Chem. Eng., 1983, 61: 590~596
Michelsen M. L.，Calculation of multiphase equilibrium, Computers. Chem. Engng.，1994, 18: 545~550
Michelsen M. L.，The isothermal flash problem. Part I. Stability, Fluid Phase Equilibria， 1982a, 9: 1~19
Michelsen M. L.，The isothermal flash problem. Part II. Phase split calculation, Fluid Phase Equilibria，1982b, 9: 21~40
Michelsen M. L., A method for incorporating excess Gibbs energy models in equations of state, Fluid Phase Equilibria, 1990, 60: 47~58
Nelson P. A., Rapid phase determination in multiplephase flash calculations, Comput. Chem. Eng., 1987, 11: 581~591
Nghiem L. X., Aziz K. and Li Y. K., A robust iterative method for flash calculations using the Soave-Redlich-Kwong or the Peng-Robinson equation of state, Soc. Pet. Eng. J., 1983, 23: 521~530
Nghiem L. X. and Li Y. K., Computation of multiphase equilibrium phenomena with an equation of state, Fluid Phase Equilibria, 1984, 17: 77~95
Ohanomah M. O. and Thompson D. W., Computation of multicomponent phase equilibria. Part I-III, Comput. Chem. Eng., 1984, 8: 147~170
Peters C. J. and Gauter K., Occurrence of holes in ternary fluid multiphase systems of near-critical carbon dioxide and certain solutes, Chem. Rev., 1999, 99: 419~431
Phoenix A. V. and Heidemann R. A., A non-ideal multiphase chemical equilibrium algorithm, Fluid Phase Equilibria, 1998, 150: 255~265
Pozo M. E. and Streett W. B., Fluid phase equilibria for the syetem dimethyl ether/water from 50 to 220OC and pressure to 50.9 MPa, J. Chem. Eng. Data, 1984, 29: 324~329
Prausnitz J. M., Lichtenthaler R. N. and Gomes de Azevedo E., Molecular thermodynamics of fluid-phase equilibria, Prentice-Hall: Englewood Cliffs, NJ, 1986
Prausnitz J. M., 用计算机计算多元汽-液和液-液平衡(陈川美等译). 北京：化学工业出版社，1987
Soares M. E., Medina, A. G. and McDermott, C., et al., Three phase flash calculations using free energy minimization, Chem. Engng Sci., 1982, 37: 521~528
Song W. X. and Larson M. A., Phase equilibrium calculation by using large scale optimization technique, Chem. Engng. Sci., 1991, 46: 2512~2523
Stateva R. P. and Tsvetkov S. G., A diverse approach for the solution of the isothermal multiphase flash problem. Application to vapor-liquid-liquid systems, Can. J. Chem. Engng., 1994, 72: 721~734
Sun A. C. and Seider W. D., Homotopy-continuation for stability analysis in the global minimization of the Gibbs free energy, Fluid Phase Equilibria, 1995, 103: 212~249
Tsang C. and Street W., Vapor-liquid equilibrium in the system carbon dioxide/dimethyl ether, J. Chem. Eng. Data, 1981, 26: 155~159
van Pelt A.，Peters C. J. and de Swaan Arons J., et al., Global phase behavior based on the simplified-perturbed hard-chain equation of state, J. Chem. Phys., 1995, 102: 3361~3375
Whitson C. H. and Michelson M. L., The negative flash, Fluid Phase Equilibria, 1989, 53: 51~71
Wu. J. S. and Bishnoi P. R., An algorithm for three-phase equilibrium calculations, Comput. Chem. Eng., 1986, 10: 269~276
陈维扭. 超临界流体萃取的原理和应用. 北京：化学工业出版社，1998
胡英. 近代化工热力学-应用研究的新进展. 上海：上海科技文献出版社，1994
胡英. 流体的分子热力学. 北京：高等教育出版社，1983
童景山，高光华，刘裕品. 化工热力学. 北京：清华大学出版社，1995
阎炜，郭天民，汪上晓. GE型状态方程混合规则计算烃水相平衡的结果对比，石油大学学报（自然科学版），2000，24：31~35
朱文涛. 物理化学（上）. 北京：清华大学出版社，1995
朱玉山. 基于全局稳定性分析的相平衡计算：[博士学位论文]. 北京：中国科学院化工冶金研究所，1999
朱自强, 等. 流体相平衡原理及其应用. 杭州：浙江大学出版社，1990
Baker L. E., Pierce A. C. and Luks K. D., Gibbs energy analysis of phase equilibria, Soc. Petro. Eng. J., 1982, 22: 731~742
Boston J. F. and Britt H. L., A radically different formulation and solution of the single-stage flash problem, Comput. Chem. Eng., 1978, 2: 109~122
Bullard L. G. and Biegler L. T., Iterated linear programming approaches for nonsmooth simulation: continuous and mixed integer approaches, Comput. Chem. Eng., 1992, 16: 949~961
Bunz A. P., Dohrn R. and Prausnitz J. M., Three-phase flash calculations for multicomponent systems, Comput. Chem. Eng., 1991, 15: 47~51
Castier M., Rasmussen P. and Fredenslund A., Calculation of simultaneous chemical and phase equilibria in nonideal system, Chem. Engng. Sci., 1989, 44: 237~248
Castillo J. and Grossmann I. E., Computation of phase and chemical equilibria, Comput. Chem. Eng., 1981, 5: 99~108
Chen J. and Li Z., Study on the GE-type Mixing Rule for Cubic Equations of State, Tsinghua Science and Technology, 1996, 1(4): 410~415
Elbaccouch M. M. and Elliott J. R., High-pressure vapor-liquid equilibrium for dimethyl+ethanol and dimethyl ether+ethanol+water, J. Chem. Eng. Data, 2000, 45: 1080~1087
Elhassan, A. E., Tsvetkov, S. G. and Craven, R. J. B., et al., Rigorous mathematical proof of the area method for phase stability, Ind. Eng. Chem. Res., 1998, 37: 1482~1489
Fournier R. L. and Boston J. F., A quasi-Newton algorithm for solving multiphase equilibrium flash problems, Chem. Engng. Commun., 1981, 8: 305~326
Gautam R. and Seider W. D.，Computation of phase and chemical equilibrium-I. Local and constrained minima in Gibbs free energy. AICHE J.，1979, 25: 991~1015
Han G. and Rangaiah G. P.，A method for multiphase equilibrium calculations, Computers. Chem. Engng.，1998, 22: 897~911
Heidemann R. A., Computation of high pressure phase equilibria, Fluid Phase Equilibria, 1983, 14: 55~78
Hodges D., Pritchard D. W. and Anwar M. M., et al., Calculating binary and ternary multiphase equilibria: extensions of the integral area method, Fluid Phase Equilibria, 1997, 130: 101~116
Hodges D.， Pritchard D. W. and Anwar M. M., Calculating binary and ternary multiphase equilibria: the tangent plane intersection method, Fluid Phase Equilibria, 1998, 152: 187~208
Holldorff H. and Knapp H., Binary vapor-liquid-liquid equilibrium of dimethyl ether-water and mutual solubilities of methyl chloride and water, Fluid Phase Equilibria, 1988, 44: 195~209
Hua J. Z., Brennecke J. F. and Stadtherr M. A., Enhanced interval analysis for phase stability: Cubic equation of state models, Ind. Eng. Chem. Res., 1998, 37: 1519~1527
Huron M. J. and Vidal J., New mixing rules in simple equations of state for representing VLE of strongly non-ideal mixtures, Fluid Phase Equilibria, 1979, 3: 255~271
Jonasson A., Persson O. and Fredenslund A., High pressure solubility of carbon dioxide and carbon monoxide in dimethyl ether, J. Chem. Eng. Data, 1995, 40: 296~300
Kinoshita M. and Takamatsu T., A powerful solution algorithm for single-stage flash problem, Comput. Chem. Eng., 1986, 10: 352~360
Lu B. C. Y., Yu P. and Sugie A. H., Prediction of vapor-liquid-liquid equilibriaby means ofa modified regula-falsi method, Chem. Eng. Sci., 1974, 29: 321~326
Lucia A. and Taylor R., Complex iterative solutions to process model equations, Comput. Chem. Eng., 1992, 16: S387~S394
Mehra R. K., Heidemann R. A. and Aziz K., An accelerated successive substitution algorithm,Can. J. Chem. Eng., 1983, 61: 590~596
Michelsen M. L.，Calculation of multiphase equilibrium, Computers. Chem. Engng.，1994, 18: 545~550
Michelsen M. L.，The isothermal flash problem. Part I. Stability, Fluid Phase Equilibria， 1982a, 9: 1~19
Michelsen M. L.，The isothermal flash problem. Part II. Phase split calculation, Fluid Phase Equilibria，1982b, 9: 21~40
Michelsen M. L., A method for incorporating excess Gibbs energy models in equations of state, Fluid Phase Equilibria, 1990, 60: 47~58
Nelson P. A., Rapid phase determination in multiplephase flash calculations, Comput. Chem. Eng., 1987, 11: 581~591
Nghiem L. X., Aziz K. and Li Y. K., A robust iterative method for flash calculations using the Soave-Redlich-Kwong or the Peng-Robinson equation of state, Soc. Pet. Eng. J., 1983, 23: 521~530
Nghiem L. X. and Li Y. K., Computation of multiphase equilibrium phenomena with an equation of state, Fluid Phase Equilibria, 1984, 17: 77~95
Ohanomah M. O. and Thompson D. W., Computation of multicomponent phase equilibria. Part I-III, Comput. Chem. Eng., 1984, 8: 147~170
Peters C. J. and Gauter K., Occurrence of holes in ternary fluid multiphase systems of near-critical carbon dioxide and certain solutes, Chem. Rev., 1999, 99: 419~431
Phoenix A. V. and Heidemann R. A., A non-ideal multiphase chemical equilibrium algorithm, Fluid Phase Equilibria, 1998, 150: 255~265
Pozo M. E. and Streett W. B., Fluid phase equilibria for the syetem dimethyl ether/water from 50 to 220OC and pressure to 50.9 MPa, J. Chem. Eng. Data, 1984, 29: 324~329
Prausnitz J. M., Lichtenthaler R. N. and Gomes de Azevedo E., Molecular thermodynamics of fluid-phase equilibria, Prentice-Hall: Englewood Cliffs, NJ, 1986
Prausnitz J. M., 用计算机计算多元汽-液和液-液平衡(陈川美等译). 北京：化学工业出版社，1987
Soares M. E., Medina, A. G. and McDermott, C., et al., Three phase flash calculations using free energy minimization, Chem. Engng Sci., 1982, 37: 521~528
Song W. X. and Larson M. A., Phase equilibrium calculation by using large scale optimization technique, Chem. Engng. Sci., 1991, 46: 2512~2523
Stateva R. P. and Tsvetkov S. G., A diverse approach for the solution of the isothermal multiphase flash problem. Application to vapor-liquid-liquid systems, Can. J. Chem. Engng., 1994, 72: 721~734
Sun A. C. and Seider W. D., Homotopy-continuation for stability analysis in the global minimization of the Gibbs free energy, Fluid Phase Equilibria, 1995, 103: 212~249
Tsang C. and Street W., Vapor-liquid equilibrium in the system carbon dioxide/dimethyl ether, J. Chem. Eng. Data, 1981, 26: 155~159
van Pelt A.，Peters C. J. and de Swaan Arons J., et al., Global phase behavior based on the simplified-perturbed hard-chain equation of state, J. Chem. Phys., 1995, 102: 3361~3375
Whitson C. H. and Michelson M. L., The negative flash, Fluid Phase Equilibria, 1989, 53: 51~71
Wu. J. S. and Bishnoi P. R., An algorithm for three-phase equilibrium calculations, Comput. Chem. Eng., 1986, 10: 269~276
陈维扭. 超临界流体萃取的原理和应用. 北京：化学工业出版社，1998
胡英. 近代化工热力学-应用研究的新进展. 上海：上海科技文献出版社，1994
胡英. 流体的分子热力学. 北京：高等教育出版社，1983
童景山，高光华，刘裕品. 化工热力学. 北京：清华大学出版社，1995
阎炜，郭天民，汪上晓. GE型状态方程混合规则计算烃水相平衡的结果对比，石油大学学报（自然科学版），2000，24：31~35
朱文涛. 物理化学（上）. 北京：清华大学出版社，1995
朱玉山. 基于全局稳定性分析的相平衡计算：[博士学位论文]. 北京：中国科学院化工冶金研究所，1999
朱自强, 等. 流体相平衡原理及其应用. 杭州：浙江大学出版社，1990