用CPA方程结合重整化群理论计算缔合流体的热力学性质 许心皓

1、中国工程热物理学会 工程热力学与能源利用学术会议论文 编号：091213用CPA方程结合重整化群理论计算缔合流体的热力学性质* 基金工程：国家重点根底研究开展方案资助 (2021CB219805)第一 许心皓(1985)，男，江苏南京人，博士研究生，主要从事热力学与流体热物性研究许心皓 段远源(清华大学 热科学与动力工程教育部重点实验室，北京 100084)(Tel:E-mail: )摘要：醇，水，氨等缔合流体在工业上有着广泛的应用，而近年来，在临界点附近的物性越来越受到关注。本文结合重整化群理论和适用于缔合流体的CPA状态方程计算了甲醇，水和氨的热力学性质。与原C

2、PA方程相比，这种结合的方法显著改善了临界参数和近临界区的气液相平衡数据的描述情况，在单相区的预测精度也较高。关键词：重整化群理论；CPA状态方程；缔合流体 中图分类号：TK121；TK123Investigation of the thermodynamic properties of associating fluids using CPA EOS + renormalization group theoryXU Xinhao, DUAN Yuanyuan(Key Laboratory for Thermal Science and Power Engineering of Ministr

3、y of Education, Tsinghua University, Beijing 100084, China)Abstract: Associating fluids, such as alcohols, water and ammonia, have wide application in industry. The thermophysical properties near the critical point have received increasing attention in the past years. The thermodynamic properties of

4、 methanol, water and ammonia was investigated by combining the renormalization group theory and the CPA equation of state extended for associating fluids. The results showed significant improvement in the representation of the critical property and the phase equilibrium in the region near critical p

5、oint compared with the original CPA EOS. Good results were also obtained in the single-phase region.Key words: renormalization group theory; CPA EOS; associating fluids0 前言状态方程是研究流体物性和相平衡性质的重要工具。其中，SRKSoave-Redlich-Kwong方程以其形式简单，通用性强，计算精度较高而得到了广泛的应用。但它在计算缔合流体时精度较差。CPACubic-Plus-Association状态方程 ADDIN

6、 EN.CITE Kontogeorgis1996797917Kontogeorgis, G. M.Voutsas, E. C.Yakoumis, I. V.Tassios, D. P.Natl tech univ athens,dept chem engn,sect 2,gr-15773 athens,greece.An equation of state for associating fluidsIndustrial & Engineering Chemistry ResearchInd. Eng. Chem. Res.Industrial & Engineering Chemistry

7、 ResearchInd. Eng. Chem. Res.Industrial & Engineering Chemistry ResearchInd. Eng. Chem. Res.4310-43183511PHASE-EQUILIBRIAHYDROGEN-FLUORIDEMIXTURESSYSTEMSPREDICTIONMOLECULESMETHANOLMODELWATER1996Nov0888-5885ISI:A1996VR70500057Article:/A1996VR70500057 English1是应用微扰理论，将SRK状态方程与Wertheim缔合项相结合建立的新型状态方程。它

8、不仅保持了SRK方程形式的简洁，同时引入了对氢键缔合作用的考虑，目前已成功的应用于包含缔合流体和非缔合流体的体系 ADDIN EN.CITE Kontogeorgis2006808017Kontogeorgis, G. M.Michelsen, M. L.Folas, G. K.Derawi, S.von Solms, N.Stenby, E. H.Tech Univ Denmark, Ctr Phase Equilibria & Separat Proc IVC SEP, Dept Chem Engn, DK-2800 Lyngby, Denmark.Kontogeorgis, GM, Te

9、ch Univ Denmark, Ctr Phase Equilibria & Separat Proc IVC SEP, Dept Chem Engn, DK-2800 Lyngby, Denmark.gkkt.dtu.dkTen years with the CPA (Cubic-Plus-Association) equation of state. Part 1. Pure compounds and self-associating systemsIndustrial & Engineering Chemistry ResearchInd. Eng. Chem. Res.Indust

10、rial & Engineering Chemistry ResearchInd. Eng. Chem. Res.Industrial & Engineering Chemistry ResearchInd. Eng. Chem. Res.4855-48684514VAPOR-LIQUID-EQUILIBRIUMTEMPERATURE MUTUAL SOLUBILITIESPHASE-EQUILIBRIASAFT EQUATIONGENERALIZED EQUATIONPHYSICAL-PROPERTIESPROPANE-METHANOLALKANE MIXTURESBINARY-MIXTUR

11、ESCARBON-DIOXIDE2006Jul0888-5885ISI:000238590400001Review:/000238590400001 10.1021/ie051305vEnglish2。然而，与其它基于平均场理论的经典状态方程一样，CPA方程没有考虑密度的涨落，因而无法正确描述流体在临界点附近的性质。重整化群renormalization group，RG理论 ADDIN EN.CITE Wilson1971424217Wilson, K. G.Renormalization Group and Critical PhenomenaPhysical Review BPhysic

12、al Review BPhys. Rev. B3174-3205491971ISI:A1971K734100043Article:/A1971K734100043 English3考虑了临界点附近强烈的密度涨落，是处理流体的临界性质的有效途径之一。最初的RG理论只适用于非常接近临界点的区域。White等 ADDIN EN.CITE Salvino1992373717Salvino, L. W.White, J. A.American univ,dept phys,washington,dc 20016.Calculation of Density Fluctuation Contributio

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18、luidsJournal of Chemical PhysicsJ. Chem. Phys.Journal of Chemical PhysicsJ. Chem. Phys.J Chem PhysJournal of Chemical PhysicsJ. Chem. Phys.J Chem Phys1922-19281035HIERARCHICAL REFERENCE THEORY1995Aug0021-9606ISI:A1995RL76700023Article:/A1995RL76700023 EnglishWhite1998696917White, J. A.Zhang, S.Amer

19、Soc Mech Engineers, Heat Transfer Div Standing Comm ThermophysProperties, Natl Inst StandTechnol, Chem SciTechnol Lab, PhysChem Properties, DivRenormalization group theory for fluids to greater density distances from the critical pointInternational Journal of ThermophysicsInternational Journal of Th

20、ermophysicsInt. J. Thermophys.1019-1027194critical pointdensity fluctuationsnonuniversal thermal behaviorradial distribution functionrenormalization groupvolumetricproperties1998Jun 22-27Boulder, ColoradoSpringer NetherlandsISI:000077286300002:/000077286300002 English4-8开展了RG理论，建立了Helmholtz自由能的迭代计算方

21、法，使RG理论可以用于预测实际流体在临界点附近和远离临界点的全范围的热力学性质。这种方法得到了广泛的关注，目前已被应用到了立方型状态方程 ADDIN EN.CITE Cai2004707017Cai, J.Prausnitz, J. M.Univ Calif Berkeley, Dept Chem Engn, Berkeley, CA 94720 USA. Lawrence Berkeley Lab, Div Chem Sci, Berkeley, CA 94720 USA.Prausnitz, JM, Univ Calif Berkeley, Dept Chem Engn, Berkeley

22、, CA 94720 USA.Thermodynamics for fluid mixtures near to and far from the vapor-liquid critical pointFluid Phase EquilibriaFluid Phase Equilib.Fluid Phase EquilibriaFluid Phase Equilib.Fluid Phase EquilibFluid Phase EquilibriaFluid Phase Equilib.Fluid Phase Equilib205-2172192renormalization group th

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26、ION-GROUPTHEORYLENNARD-JONES FLUIDCRITICAL REGIONCRITICAL-POINTCRITICALEXPONENTSCROSSOVERTEMPERATURESSIMULATIONS2006Mar0378-3812ISI:000236637200024Article:/000236637200024 10.1016/j.fluid.2005.11.003EnglishLlovell2021727217Llovell, F.Vega, L. F.Seiltgens, D.Mejia, A.Segura, H.Llovell, F.; Vega, L. F

27、. CSIC, Inst Ciencia Mat Barcelona, ICMAB, E-08193 Barcelona, Spain. Vega, L. F. MATGAS Res Ctr, Barcelona 08193, Spain. Seiltgens, D.; Mejia, A.; Segura, H. Univ Concepcion, Dept Ingn Quim, Concepcion, Chile.Vega, LF, CSIC, Inst Ciencia Mat Barcelona, ICMAB, Campus UAB, E-08193 Barcelona, Spain.lve

28、gaicmab.esAn accurate direct technique for parametrizing cubic equations of state - Part III. Application of a crossover treatmentFluid Phase EquilibriaFluid Phase Equilib.Fluid Phase EquilibriaFluid Phase Equilib.Fluid Phase EquilibFluid Phase EquilibriaFluid Phase Equilib.Fluid Phase Equilib201-21

29、02641-2cubic equation of statesoft-SAFFcrossovercritical regionn-alkanes1-alkanolscarbon dioxidewaterRENORMALIZATION-GROUP THEORYASSOCIATING FLUID THEORYINCLUDINGCRITICAL REGIONCRITICAL-POINTOF-STATETHERMODYNAMIC PROPERTIESBINARY-MIXTURESPURE FLUIDSSAFTPREDICTION2021Mar0378-3812ISI:000253670200021Ar

30、ticle:/000253670200021 10.1016/j.fluid.2007.11.006English9-11和SAFT状态方程 ADDIN EN.CITE Llovell2004737317Llovell, F.Pamies, J. C.Vega, L. F.CSIC, Inst Ciencia Mat Barcelona, E-08193 Barcelona, Spain.Vega, LF, CSIC, Inst Ciencia Mat Barcelona, Campus UAB, E-08193 Barcelona, Spain.lvegaicmab.esThermodyna

31、mic properties of Lennard-Jones chain molecules: Renormalization-group corrections to a modified statistical associating fluid theoryJournal of Chemical PhysicsJ. Chem. Phys.Journal of Chemical PhysicsJ. Chem. Phys.J Chem PhysJournal of Chemical PhysicsJ. Chem. Phys.J Chem Phys10715-1072412121EQUATI

32、ON-OF-STATEDIRECTIONAL ATTRACTIVE FORCESVAPOR-LIQUID-EQUILIBRIAMULTIPLE BONDING SITESSOFT-SAFT EQUATIONHEAVY N-ALKANESCRITICAL REGIONPERTURBATION-THEORYPHASE-EQUILIBRIACRITICAL-POINT2004Dec0021-9606ISI:000225136300050Article:/000225136300050 10.1063/1.1809112EnglishMi2004777717Mi, J. G.Zhong, C. L.L

33、i, Y. G.Chen, J.Beijing Univ Chem Technol, Key Lab Nanomat, Minist Educ, Dept Chem Engn, Beijing 100029, Peoples R China. Tsing Hua Univ, State Key Lab Chem Engn, Beijing 100084, Peoples R China.Zhong, CL, Beijing Univ Chem Technol, Key Lab Nanomat, Minist Educ, Dept Chem Engn, POB 100, Beijing 1000

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35、RE-WELL FLUIDSCROSSOVEREQUATIONTHERMAL-PROPERTIESCRITICAL-BEHAVIORMIXTURESTHERMODYNAMICSSIMULATIONSSAFT2004Oct0301-0104ISI:000224321000004Article:/000224321000004 10.1016/j.chemphys.2004.06.031EnglishMi2005767617Mi, J. G.Zhong, C. L.Li, Y. G.Beijing Univ Chem Technol, Dept Chem Engn, Minist Educ, Ke

36、y Lab Nanomat, Beijing 100029, Peoples R China.Zhong, CL, Beijing Univ Chem Technol, Dept Chem Engn, Minist Educ, Key Lab Nanomat, Beijing 100029, Peoples R C Renormalization group theory for fluids including critical region. II. Binary mixturesChemical PhysicsChem. Phys.Chemical PhysicsChem. Phys.C

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38、0004Article:/000228206700004 10.1016/j.chemphys.2004.11.018EnglishLlovell2006747417Llovell, F.Vega, L. F.CSIC, Inst Ciencia Mat Barcelona, E-08193 Barcelona, Spain.Vega, LF, CSIC, Inst Ciencia Mat Barcelona, Campus UAB, E-08193 Barcelona, Spain.lvegaicmab.esGlobal fluid phase equilibria and critical

39、 phenomena of selected mixtures using the crossover soft-SAFT equationJournal of Physical Chemistry BJ. Phys. Chem. BJournal of Physical Chemistry BJ. Phys. Chem. BJournal of Physical Chemistry BJ. Phys. Chem. B1350-13621103DIRECTIONAL ATTRACTIVE FORCESVAPOR-LIQUID-EQUILIBRIARENORMALIZATION-GROUP CO

40、RRECTIONSLENNARD-JONES CHAINSHEAVYN-ALKANESOF-STATECRITICAL REGIONTHERMODYNAMIC PROPERTIESCRITICAL-POINTBINARY-MIXTURES2006Jan1520-6106ISI:000235046300040Article:/000235046300040 10.1021/jp0551465EnglishBymaster2021787817Bymaster, A.Emborsky, C.Dominik, A.Chapman, W. G.Bymaster, Adam; Emborsky, Chri

41、s; Dominik, Aleksandra; Chapman, Walter G. Rice Univ, Dept Chem & Biomol Engn, Houston, TX 77005 USA.Chapman, WG, Rice Univ, Dept Chem & Biomol Engn, 6100 S Main St, Houston, TX 77005 USA.Renormalization-group corrections to a perturbed-chain statistical associating fluid theory for pure fluids near

42、 to and far from the critical regionIndustrial & Engineering Chemistry ResearchInd. Eng. Chem. Res.Industrial & Engineering Chemistry ResearchInd. Eng. Chem. Res.Industrial & Engineering Chemistry ResearchInd. Eng. Chem. Res.6264-62744716EQUATION-OF-STATEDIRECTIONAL ATTRACTIVE FORCESLENNARD-JONES CH

43、AINSVAPOR-LIQUID-EQUILIBRIAMULTIPLE BONDING SITESPHASE-EQUILIBRIASAFTEQUATIONTHERMODYNAMIC PROPERTIESN-ALKANESTHERMAL-PROPERTIES2021Aug0888-5885ISI:000258400300054Article:/000258400300054 10.1021/ie8001167English12-16上。本文将White开展的RG 理论与CPA状态方程相结合, 计算了甲醇，氨和水这三种具有代表性的缔合流体在临界点附近和远离临界点的热力学性质。1 CPA状态方程CP

44、A方程形式如下：(1)式中：p为压力，R为通用气体常数，T为温度，v为摩尔体积，为密度；a、b是SRK方程的参数，XA为分子在A位上没有参加缔合作用的比例。SRK方程的能量参数a的表达式如下：(2)式中：a0、c1是系数；Tr为比照温度T / Tc。缔合作用项XA可通过下式得到：(3)A位与B位间的缔合强度AB可通过下式得到(4)式中：AB和AB分别为缔合能和相互作用体积，g为径向分布函数。Kontogeorgis等 ADDIN EN.CITE Kontogeorgis1999818117Kontogeorgis, Georgios M.V. Yakoumis, IakovosMeijer,

45、HenkHendriks, EricMoorwood, TonyMulticomponent phase equilibrium calculations for water-methanol-alkane mixturesFluid Phase EquilibriaFluid Phase EquilibriaFluid Phase Equilib.Fluid Phase Equilib201-209158-160Equation of stateLiquid-liquid equilibriaAssociationMulticomponentAlcoholWater1999 :/ scien

46、cedirect /science/article/B6TG2-3Y9G7WW-3V/2/c9ff1bbdf1de340eab409275a115880d 17提出了g的近似表达式，形式如下：(5a)(5b)式中，为硬球流体的比照密度。2 CPA结合RG的方法根据White对RG理论的开展，Helmholtz自由能密度的迭代计算过程如下：(6)(7)(8)(9) (10) (11)(12)(13)式中：下标l 和s 分别表示长波和短波作用。L 是截断长度，是密度涨落初始最短波长的函数。在本文计算中，这两个参数均为可调参数。针对CPA方程，本文具体采用了如下一些处理。-2表示平均场Helmhol

47、tz自由能密度的引力局部，它取决于具体的分子势能模型。本文采用Cai ADDIN EN.CITE Cai2004707017Cai, J.Prausnitz, J. M.Univ Calif Berkeley, Dept Chem Engn, Berkeley, CA 94720 USA. Lawrence Berkeley Lab, Div Chem Sci, Berkeley, CA 94720 USA.Prausnitz, JM, Univ Calif Berkeley, Dept Chem Engn, Berkeley, CA 94720 USA.Thermodynamics for

48、fluid mixtures near to and far from the vapor-liquid critical pointFluid Phase EquilibriaFluid Phase Equilib.Fluid Phase EquilibriaFluid Phase Equilib.Fluid Phase EquilibFluid Phase EquilibriaFluid Phase Equilib.Fluid Phase Equilib205-2172192renormalization group theorycontinuous thermodynamicscriti

49、calstateequation-of-statepolydisperse mixtureEQUATION-OF-STATERENORMALIZATION-GROUP THEORYLENNARD-JONES FLUIDCRITICAL REGIONPHASE-EQUILIBRIACRITICAL EXPONENTSCROSSOVERBEHAVIORTEMPERATURESSIMULATIONS2004May0378-3812ISI:000221516300012Article:/000221516300012 10.1016/j.fluid.2004.01.033EnglishCai20067

50、17117Cai, J.Qiu, D. L.Zhang, L. N.Hu, Y.E China Univ Sci & Technol, Dept Chem, Shanghai 200237, Peoples R China.Cai, J, E China Univ Sci & Technol, Dept Chem, Shanghai 200237, Peoples R China.tsaijunonline.sh Vapor-liquid critical properties of multi-component fluid mixtureFluid Phase EquilibriaFlui

51、d Phase Equilib.Fluid Phase EquilibriaFluid Phase Equilib.Fluid Phase EquilibFluid Phase EquilibriaFluid Phase Equilib.Fluid Phase Equilib229-2352411-2renormalization groupcriticalmulti-componentfluid mixtureEQUATION-OF-STATEGLOBAL THERMODYNAMIC BEHAVIORRENORMALIZATION-GROUPTHEORYLENNARD-JONES FLUID

52、CRITICAL REGIONCRITICAL-POINTCRITICALEXPONENTSCROSSOVERTEMPERATURESSIMULATIONS2006Mar0378-3812ISI:000236637200024Article:/000236637200024 10.1016/j.fluid.2005.11.003English9, 10使用的简化表达方法，将其与CPA方程中的参数a相关联：(14)类似的，本文将max与CPA方程中的参数b相关联，为了防止计算值的发散，本文取：(15)迭代初始项，推导可得：(16)式中：M为分子上的缔合位数。式10的积分采用梯形法计算，密度区域分

53、为1000个区间。每一个迭代步得到的自由能密度用三次样条将其转化为光滑曲线，并在下一步中使用。当迭代步数n5时， ADDIN EN.CITE Llovell2004737317Llovell, F.Pamies, J. C.Vega, L. F.CSIC, Inst Ciencia Mat Barcelona, E-08193 Barcelona, Spain.Vega, LF, CSIC, Inst Ciencia Mat Barcelona, Campus UAB, E-08193 Barcelona, Spain.lvegaicmab.esThermodynamic properties

54、 of Lennard-Jones chain molecules: Renormalization-group corrections to a modified statistical associating fluid theoryJournal of Chemical PhysicsJ. Chem. Phys.Journal of Chemical PhysicsJ. Chem. Phys.J Chem PhysJournal of Chemical PhysicsJ. Chem. Phys.J Chem Phys10715-1072412121EQUATION-OF-STATEDIR

55、ECTIONAL ATTRACTIVE FORCESVAPOR-LIQUID-EQUILIBRIAMULTIPLE BONDING SITESSOFT-SAFT EQUATIONHEAVY N-ALKANESCRITICAL REGIONPERTURBATION-THEORYPHASE-EQUILIBRIACRITICAL-POINT2004Dec0021-9606ISI:000225136300050Article:/000225136300050 10.1063/1.1809112EnglishLlovell2006747417Llovell, F.Vega, L. F.CSIC, Ins

56、t Ciencia Mat Barcelona, E-08193 Barcelona, Spain.Vega, LF, CSIC, Inst Ciencia Mat Barcelona, Campus UAB, E-08193 Barcelona, Spain.lvegaicmab.esGlobal fluid phase equilibria and critical phenomena of selected mixtures using the crossover soft-SAFT equationJournal of Physical Chemistry BJ. Phys. Chem

57、. BJournal of Physical Chemistry BJ. Phys. Chem. BJournal of Physical Chemistry BJ. Phys. Chem. B1350-13621103DIRECTIONAL ATTRACTIVE FORCESVAPOR-LIQUID-EQUILIBRIARENORMALIZATION-GROUP CORRECTIONSLENNARD-JONES CHAINSHEAVYN-ALKANESOF-STATECRITICAL REGIONTHERMODYNAMIC PROPERTIESCRITICAL-POINTBINARY-MIX

58、TURES2006Jan1520-6106ISI:000235046300040Article:/000235046300040 10.1021/jp0551465English12, 15。本文中取n=5时，迭代结束。3 结果与讨论结合了重整化群理论的CPA状态方程共有7个可调参数：b，a0，c1，AB，AB，L，。本文将NIST提供的饱和蒸气压和饱和液相密度数据 ADDIN EN.CITE 2021878745NIST Chemistry Webbook2021 :/chemistry :/chemistry18作为基准值，回归得到了甲醇，水和氨的相关参数，其中甲醇采用2B缔合方式，水和氨

59、采用3B缔合方式。回归结果如表1所示。表1 CPA+RG方程参数Table 1 Parameters of CPA+RG (m3 mol-1)a0(Pa m6 mol-2)c1AB(Pa m3 mol-1)AB103 (m)甲醇3.15020.434750.4683523647水1.47230.225980.837881791875.00.74.43氨1.97690.18041.29865579247图1-3比拟了SRK，CPA，CPA结合RG理论计算的气液相平衡结果与NIST提供的结果。可以看到SRK方程在液相区有较大偏差，原始的CPA方程虽然成功再现了远离临界区的数据，但在近临界区计算的压力和温度都偏高。而采用本文建立CPA结合RG理论的方法，能够同时在接近临界点和远离临界点的区域再现NIST的数据，精度较高。(a)(b)图1 甲醇的相平衡 (a) T-图 (b) p-图Fig. 1 Vapor-liquid coexistence lines for methanol (a) T- diagram (b) p-diagram图2 水的相平衡 (a) T-图 (b) p-图Fig. 2 Vapor-liquid coexistence lines for water (a) T- diagram (b) p-diagram图3 氨的相平衡 (a) T-图 (b)