混合物临界性质的数学模型研究与计算
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摘要
混合物临界性质的计算与预测不但有重要的理论意义而且有重要的应用价值。传统的混合物临界性质计算与预测模型或是求解过程非常复杂，或是基于经验关联方法，难以适应工程计算的需要。为了克服传统计算与预测模型的缺点，本文提出了以下三个改进的混合物临界性质的计算与预测模型。
首先，为了能够比较准确地描述临界状态流体的热力学性质，本文在原PR状态方程的基础上，以描述硬球体系最为精确的Carnahan-Starling硬球状态方程替代PR状态方程的斥力项，提出了改进的CS-PR状态方程。本文应用改进的CS-PR方程结合Gibbs严格热力学判据的严格解析方法，对二元混合物的临界性质进行了计算。计算结果表明，改进的CS-PR状态方程在混合物的临界性质计算方面比原来的PR状态方程的计算精度有较大的提高，而且保留了PR状态方程形式简单的优点，这说明CS-PR方程在混合物的临界性质计算方面是对原PR方程的一次较为成功的改进。
为进一步提高计算的精度并降低计算的复杂度，本文分别应用以启发式学习算法、BFGS算法和Levenberg-Marquardt算法改进学习算法的BP网络对二元混合物的临界性质进行计算与预测。以物系一组有代表性的几个组成点的临界性质作为训练样本对改进的BP网络进行训练，然后应用训练好的网络来预测该物系其它组成点的临界性质。van Konynenberg-Scott相图分类中各种类型物系临界性质计算结果表明，该方法的计算精度比传统方法有非常大的提高，而且实现比较简便。
上述两种方法需要物系临界性质的一组实验数据才能预测其它组成点的临界性质，这对于缺乏实验数据的物系是不适用的。因此，本文从决定物质热力学性质的分子间相互作用机理出发，提出了基于分子间相互作用参数的二元混合物临界性质预测的BP网络模型。把对二元混合物临界性质影响最大的各纯组分的临界性质、摩尔分数、分子量、偏心因子和极化率等参数作为输入参数，把混合物的临界性质作为期望输出，以若干组物系输入参数和对应的临界性质作为训练样本对网络进行训练，并用训练好的网络对其它相近类型物系的临界性质进行了预测，得到了比较令人满意的结果。
The study on the critical properties of mixtures has the significance of theory and practicability. The usual methods for calculation and prediction of the critical properties of mixtures are complicated or empirical, so these methods don't satisfy the demands of engineering calculation very well. In order to overcome the disadvantages of classical methods, in this paper, three improved calculation and prediction models of critical properties of mixtures are prompted.
First, the repulsive force item of the original PR equation of state is modified with Carnahan-Starling hard sphere equation, and the modified CS-PR equation of state is introduced. The modified CS-PR equation combined with the rigorous critical state criterion enunciated by Gibbs is applied to calculate the critical properties of binary mixtures, and the same is done with PR equation as well. The calculation results indicate CS-PR equation can calculate the critical properties of such systems better than the original PR equation, so CS-PR equation is a more desirable modification of PR equation in the field of critical properties calculation.
For the sake of more calculation accuracy and less complexity, the improved BP network with heuristic learning rules, BFGS algorithm and Levenberg-Marquardt algorithm individually is applied to predict the critical properties of binary mixtures. A group of critical properties of certain mixture as train examples are used to train the BP network, then the trained network is applied to predict the critical properties of other component points of this mixture. The calculation and prediction results of six types of binary mixtures according to van Konynenberg-Scott phase diagram classification indicate this method is more convenient, versatile and has higher calculation accuracy than other classical methods.
Two methods above are only adapt to the mixtures whose several groups of critical properties are known. In order to release from the limitation, the prediction model of critical properties based on the parameters charactering the moleculars interaction which determines the thermodynamic properties of the mixture is proposed. The critical property of certain binary mixture is looked as a function of critical properties, mole fraction, molecular weight, acentric factor and susceptibility of each pure component of the mixture, and BP network is applied to build the corresponding relation of the input parameters and the critical properties. The BP network is trained with a group of input parameters and expected output as train examples, and the critical properties of other mixtures are predicted with the trained BP network, and more desirable prediction results are gained.
首先，为了能够比较准确地描述临界状态流体的热力学性质，本文在原PR状态方程的基础上，以描述硬球体系最为精确的Carnahan-Starling硬球状态方程替代PR状态方程的斥力项，提出了改进的CS-PR状态方程。本文应用改进的CS-PR方程结合Gibbs严格热力学判据的严格解析方法，对二元混合物的临界性质进行了计算。计算结果表明，改进的CS-PR状态方程在混合物的临界性质计算方面比原来的PR状态方程的计算精度有较大的提高，而且保留了PR状态方程形式简单的优点，这说明CS-PR方程在混合物的临界性质计算方面是对原PR方程的一次较为成功的改进。
为进一步提高计算的精度并降低计算的复杂度，本文分别应用以启发式学习算法、BFGS算法和Levenberg-Marquardt算法改进学习算法的BP网络对二元混合物的临界性质进行计算与预测。以物系一组有代表性的几个组成点的临界性质作为训练样本对改进的BP网络进行训练，然后应用训练好的网络来预测该物系其它组成点的临界性质。van Konynenberg-Scott相图分类中各种类型物系临界性质计算结果表明，该方法的计算精度比传统方法有非常大的提高，而且实现比较简便。
上述两种方法需要物系临界性质的一组实验数据才能预测其它组成点的临界性质，这对于缺乏实验数据的物系是不适用的。因此，本文从决定物质热力学性质的分子间相互作用机理出发，提出了基于分子间相互作用参数的二元混合物临界性质预测的BP网络模型。把对二元混合物临界性质影响最大的各纯组分的临界性质、摩尔分数、分子量、偏心因子和极化率等参数作为输入参数，把混合物的临界性质作为期望输出，以若干组物系输入参数和对应的临界性质作为训练样本对网络进行训练，并用训练好的网络对其它相近类型物系的临界性质进行了预测，得到了比较令人满意的结果。
The study on the critical properties of mixtures has the significance of theory and practicability. The usual methods for calculation and prediction of the critical properties of mixtures are complicated or empirical, so these methods don't satisfy the demands of engineering calculation very well. In order to overcome the disadvantages of classical methods, in this paper, three improved calculation and prediction models of critical properties of mixtures are prompted.
First, the repulsive force item of the original PR equation of state is modified with Carnahan-Starling hard sphere equation, and the modified CS-PR equation of state is introduced. The modified CS-PR equation combined with the rigorous critical state criterion enunciated by Gibbs is applied to calculate the critical properties of binary mixtures, and the same is done with PR equation as well. The calculation results indicate CS-PR equation can calculate the critical properties of such systems better than the original PR equation, so CS-PR equation is a more desirable modification of PR equation in the field of critical properties calculation.
For the sake of more calculation accuracy and less complexity, the improved BP network with heuristic learning rules, BFGS algorithm and Levenberg-Marquardt algorithm individually is applied to predict the critical properties of binary mixtures. A group of critical properties of certain mixture as train examples are used to train the BP network, then the trained network is applied to predict the critical properties of other component points of this mixture. The calculation and prediction results of six types of binary mixtures according to van Konynenberg-Scott phase diagram classification indicate this method is more convenient, versatile and has higher calculation accuracy than other classical methods.
Two methods above are only adapt to the mixtures whose several groups of critical properties are known. In order to release from the limitation, the prediction model of critical properties based on the parameters charactering the moleculars interaction which determines the thermodynamic properties of the mixture is proposed. The critical property of certain binary mixture is looked as a function of critical properties, mole fraction, molecular weight, acentric factor and susceptibility of each pure component of the mixture, and BP network is applied to build the corresponding relation of the input parameters and the critical properties. The BP network is trained with a group of input parameters and expected output as train examples, and the critical properties of other mixtures are predicted with the trained BP network, and more desirable prediction results are gained.
引文
[1] J.W.Gibbs. On the equilibrium of heterogeneous substance. Collected Works. New Haven: Yale Univ. Press, 1928.55-62
[2] F.Kurata,D.L.Katz. Critical properties of volatile hydrocarbon mixtures[J]. Trans.Am.Inst. Chem.Engrs, 1942,38:995-1001
[3] E.I.Organick. Prediction of critical temperature and critical pressure of complex hydrocarbon systems[C]. Chem. Eng. Progr. Symposium ser, 1953,49(6):81-87
[4] P.C. Davis et al. The phase and volumetric behaviour of natural gases at low temperature and high pressures[J]. Petrol. Trans. A. I. M. E., 1954,201:245-252
[5] E.A.Gonzola,C.Marcelo,I.S.Stanley. Predictions of critical behavior using the Wong-Sandler mixing rule[J]. Journal of Supercritical Fluids, 1998,13:49-54
[6] L.J. Florusse, T. Fornari, S.B. Bottini, C.J. Peters. Phase behavior of the binary system near-critical dimethylether and tripalmitin: measurements and thermodynamic modeling[J]. Journal of Supercritical Fluids, 2002,22:1-13
[7] P.H. van Konynenberg, R.L. Scott. Critical lines and phase equilibria in binary van der Waals mixtures. Phil.Trans. Roy. Soc. London, Series A, 1980,298:495-506
[8] Carnahan N F, Starling K E. Equation of state for nonattraeting rigid spheres[J]. The Journal of Chemical Physics, 1969, 51(2): 635-636
[9] M. Benedict, G.B. Webb, L.C. Rubin. An empirical equation for thermodynamic properties of light hydrocarbons and their mixtures[J]. J Chem Phys, 1940,8:334-340
[10] K.E.Starling. Fluid thermodynamic properties for light petroleum systems[M]. Huston: Gulf Pubulic Co. 1973:22-29
[11] J.J.Martin,Y.C.Hou. Development of an equation of state for gases[J]. AIChE J, 1955,(1): 142-148
[12] 侯虞钧，张彬，唐宏青．马丁-侯状态方程向液相发展[J]．化工学报，1981，32(1)：1-9
[13] O. Redlich, J. N. S. Kwong, On the thermodynamics of solutions. V. An equation of state fugacities of gaseous solutions[J]. Chemical Reviews, 1949,44:233-244
[14] G. Soave. Equilibrium constants from a modified Redlich-Kwong equation of state[J].Chem Eng Sci, 1972,27:1197-1203
[15] Peng D Y, Robinson D B. A new two-constant equation of state[J]. Ind Eng Chem Fundam, 1976,15:59-64
[16] N.C.Patel,A.S.Teja. A new cubic equation of state for fluid and fluid mixtures[J]. Chemical Engineering Science, 1982,37(3): 463-473
[17] 王武谦，陈钟秀，周浩．一个修正的多参数状态方程[J]．化学工程，1996，24(1)：53-57
[18] 刘洪来，荣宗明，胡英．共聚硬球链流体及其混合物的状态方程[J]．华东理工大学学报，1995，21(5)：619-625
[19] Y.C. Chiew,S.K.H.Ting,K.K.Leong. A perturbed Lennard-Jones chain equation of state for polymer liquids[J]. Fluid Phase Equilibria, 2000,168:19-29.
[20] 李平．关于混合物高压相平衡和临界现象的研究——一个新的摄动硬球三参数状态方程[D]．大连：大连理工大学化学工程系，1990
[21] 胡仰栋，仲崇立，韩方煜．基于新α表达式的实用PT方程[J]．化学工程，1996，24(2)：54-60
[22] 云志，史美仁，时钧．四次型CCOR-PT状态方程[J]．南京化工学院学报，1995，17(2)：25-31
[23] 马涛．改进的PR状态方程及其应用研究[J]．辽宁师范大学学报(自然科学版)，2004，27：3-4
[24] Carnahan N F, Starling K E. Intermolecular repulsions and the equation of state for fluids[J]. AICHE Journal, 1972,11:1184-1189
[25] Peng D Y, Robinson D B. A rigorous method for predicting the critical properties of multicomponent systems from an equation of state[J]. AICHE Journal, 1977,(5):137-144
[26] Soave G. Equilibrium constants from a modified Redlich-Kwong equation of state[J]. Chemcal Engineering Science, 1972,27:1197-1203
[27] RAO S S. Optimization theory and application[M]. New York:Halsted Press, 1984.292-300
[28] Hicks C P, Yong C L.The gas-liquid critical properties of binary mixtures[J].Chemical Reviews, 1975,75(2):139-163
[29] 张立明．人工神经网络的模型及其应用[M]．上海：复旦大学出版社，1993：34-47
[30] D.E.Rumelhart, J.L.McCelland. Parallel distributed processing[M].Vol.1-2, MIT Press, 1986
[31] P.J.Werbos. BP and neuralcontrol: a review and prospectus. Proc.IEEE Trans.IT. 1989, 11: 209-216
[32] 向国全，董道珍．BP网络模型中的激励函数和改进的网络训练法[J]．计算机研究与发展，1997，34(2)：113-117
[33] 袁亚湘，孙文瑜．最优化理论与方法[M]．北京：科学出版社，1997：219-238
[34] 袁亚湘，孙文瑜．最优化理论与方法[M]．北京：科学出版社，1997：382-391
[35] 江学军．前馈神经网络算法研究及其应用[D]．大连：大连理工大学管理科学与工程系，1999
[36] 周公度．结构化学基础[M]．北京：北京大学出版社，1989：246-248
[37] 夏少武．简明结构化学教程[M]．北京：化学工业出版社，1995：197-198
[38] 赵广绪，R．A．格林肯．流体热力学平衡理论的导论[M]．北京：化学工业出版社，1984：119-120
[2] F.Kurata,D.L.Katz. Critical properties of volatile hydrocarbon mixtures[J]. Trans.Am.Inst. Chem.Engrs, 1942,38:995-1001
[3] E.I.Organick. Prediction of critical temperature and critical pressure of complex hydrocarbon systems[C]. Chem. Eng. Progr. Symposium ser, 1953,49(6):81-87
[4] P.C. Davis et al. The phase and volumetric behaviour of natural gases at low temperature and high pressures[J]. Petrol. Trans. A. I. M. E., 1954,201:245-252
[5] E.A.Gonzola,C.Marcelo,I.S.Stanley. Predictions of critical behavior using the Wong-Sandler mixing rule[J]. Journal of Supercritical Fluids, 1998,13:49-54
[6] L.J. Florusse, T. Fornari, S.B. Bottini, C.J. Peters. Phase behavior of the binary system near-critical dimethylether and tripalmitin: measurements and thermodynamic modeling[J]. Journal of Supercritical Fluids, 2002,22:1-13
[7] P.H. van Konynenberg, R.L. Scott. Critical lines and phase equilibria in binary van der Waals mixtures. Phil.Trans. Roy. Soc. London, Series A, 1980,298:495-506
[8] Carnahan N F, Starling K E. Equation of state for nonattraeting rigid spheres[J]. The Journal of Chemical Physics, 1969, 51(2): 635-636
[9] M. Benedict, G.B. Webb, L.C. Rubin. An empirical equation for thermodynamic properties of light hydrocarbons and their mixtures[J]. J Chem Phys, 1940,8:334-340
[10] K.E.Starling. Fluid thermodynamic properties for light petroleum systems[M]. Huston: Gulf Pubulic Co. 1973:22-29
[11] J.J.Martin,Y.C.Hou. Development of an equation of state for gases[J]. AIChE J, 1955,(1): 142-148
[12] 侯虞钧，张彬，唐宏青．马丁-侯状态方程向液相发展[J]．化工学报，1981，32(1)：1-9
[13] O. Redlich, J. N. S. Kwong, On the thermodynamics of solutions. V. An equation of state fugacities of gaseous solutions[J]. Chemical Reviews, 1949,44:233-244
[14] G. Soave. Equilibrium constants from a modified Redlich-Kwong equation of state[J].Chem Eng Sci, 1972,27:1197-1203
[15] Peng D Y, Robinson D B. A new two-constant equation of state[J]. Ind Eng Chem Fundam, 1976,15:59-64
[16] N.C.Patel,A.S.Teja. A new cubic equation of state for fluid and fluid mixtures[J]. Chemical Engineering Science, 1982,37(3): 463-473
[17] 王武谦，陈钟秀，周浩．一个修正的多参数状态方程[J]．化学工程，1996，24(1)：53-57
[18] 刘洪来，荣宗明，胡英．共聚硬球链流体及其混合物的状态方程[J]．华东理工大学学报，1995，21(5)：619-625
[19] Y.C. Chiew,S.K.H.Ting,K.K.Leong. A perturbed Lennard-Jones chain equation of state for polymer liquids[J]. Fluid Phase Equilibria, 2000,168:19-29.
[20] 李平．关于混合物高压相平衡和临界现象的研究——一个新的摄动硬球三参数状态方程[D]．大连：大连理工大学化学工程系，1990
[21] 胡仰栋，仲崇立，韩方煜．基于新α表达式的实用PT方程[J]．化学工程，1996，24(2)：54-60
[22] 云志，史美仁，时钧．四次型CCOR-PT状态方程[J]．南京化工学院学报，1995，17(2)：25-31
[23] 马涛．改进的PR状态方程及其应用研究[J]．辽宁师范大学学报(自然科学版)，2004，27：3-4
[24] Carnahan N F, Starling K E. Intermolecular repulsions and the equation of state for fluids[J]. AICHE Journal, 1972,11:1184-1189
[25] Peng D Y, Robinson D B. A rigorous method for predicting the critical properties of multicomponent systems from an equation of state[J]. AICHE Journal, 1977,(5):137-144
[26] Soave G. Equilibrium constants from a modified Redlich-Kwong equation of state[J]. Chemcal Engineering Science, 1972,27:1197-1203
[27] RAO S S. Optimization theory and application[M]. New York:Halsted Press, 1984.292-300
[28] Hicks C P, Yong C L.The gas-liquid critical properties of binary mixtures[J].Chemical Reviews, 1975,75(2):139-163
[29] 张立明．人工神经网络的模型及其应用[M]．上海：复旦大学出版社，1993：34-47
[30] D.E.Rumelhart, J.L.McCelland. Parallel distributed processing[M].Vol.1-2, MIT Press, 1986
[31] P.J.Werbos. BP and neuralcontrol: a review and prospectus. Proc.IEEE Trans.IT. 1989, 11: 209-216
[32] 向国全，董道珍．BP网络模型中的激励函数和改进的网络训练法[J]．计算机研究与发展，1997，34(2)：113-117
[33] 袁亚湘，孙文瑜．最优化理论与方法[M]．北京：科学出版社，1997：219-238
[34] 袁亚湘，孙文瑜．最优化理论与方法[M]．北京：科学出版社，1997：382-391
[35] 江学军．前馈神经网络算法研究及其应用[D]．大连：大连理工大学管理科学与工程系，1999
[36] 周公度．结构化学基础[M]．北京：北京大学出版社，1989：246-248
[37] 夏少武．简明结构化学教程[M]．北京：化学工业出版社，1995：197-198
[38] 赵广绪，R．A．格林肯．流体热力学平衡理论的导论[M]．北京：化学工业出版社，1984：119-120