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Research into the Polynomial Alpha Function for the Cubic Equation of State wenying zhao, XiaoYan Sun, Li Xia, and Shuguang Xiang Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b02549 • Publication Date (Web): 17 Aug 2018 Downloaded from http://pubs.acs.org on August 18, 2018
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Industrial & Engineering Chemistry Research
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Research into the Polynomial Alpha Function for the Cubic
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Equation of State
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Wenying Zhao1, Xiaoyan Sun1, Li Xia1, and Shuguang Xiang1
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People’s Republic of China.
Faculty of Chemical Engineering, Qingdao University of Science & Technology, Qingdao,
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Corresponding author: Shuguang Xiang E-mail:
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Abstract: Alpha functions usually aim at predicting the thermodynamic properties of specific
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compounds in a limited range of reduced temperatures. The differences between polynomial
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alpha functions are mainly the functional forms and the parameter correlations. In this work, the
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functional forms, the parameter correlations, and the scope of 64 polynomial alpha functions are
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reviewed. The inherent relationships between the polynomial alpha functions, as well as
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developing trends are also discussed. These functions were modified by changing the coefficients,
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exponents, or variables terms. The parameter formulas were re-correlated with the acentric
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factors or the other natural parameters. In recent years, research into polynomial alpha functions
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has mainly focused on reservoir fluids, and the alpha functions have been modified by adding
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characteristic constants of pseudo-components to the equations or the parameter correlations.
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The Soave-type function has significant research value and influence the development of
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polynomial alpha functions in the coming years.
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Keywords: Polynomial alpha functions; Functional forms; Parameter correlations; Research
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progress
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1. Introduction
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Equations of state (EoS) play an essential role in chemical simulation and design. An appropriate
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EoS can accurately calculate vapor–liquid equilibria (VLE) and the thermodynamic properties of
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pure compounds and mixtures in a wide temperature and pressure region. Since van der Waals
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(1873)1 first proposed a simple cubic equation of state for real gases, thermodynamic models
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have developed rapidly and represent a milestone in the development of thermodynamics.2-5 The
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cubic EoS contains an attractive term and a repulsive term. The attractive term is a significant
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factor that influenced the prediction of the saturated vapor pressure.6,7 In 1949, Redlich and
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Kwong2 first modified the van der Waals (VDW) EoS by introducing a reduced temperature
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function to the attractive term. Wilson (1964)8 proposed an alpha function with physical
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properties and reduced temperature as variables, but the prediction of the vapor pressures of pure
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fluids was poor, and this concept was ignored for a long time9,10 until Soave proposed a
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generalized alpha function in 1972.3 Soave’s 1972 alpha function was a qualitative leap in the
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modification of the EoS and gave direction for the modification and development of the EoS.
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Since then, the alpha function has been vigorously developed.
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Formally, alpha functions can be divided into polynomial and exponential functions. In this work,
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we review and analyze the functional forms, parameter correlations, and scope of the polynomial
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alpha functions developed over the past 40 years. The research methods and developing trend for
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this kind of function are also reported. A review of exponential alpha functions will be made in a
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separate article.
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2. A general overview of the alpha function
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2.1. The classical cubic equations of state
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Although the cubic EoS proposed by Van der Waals in 18731 cannot be used for realistic
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simulation and calculation, the VDW EoS laid the foundation for the development of the cubic
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EoS. In 1949, Redlich and Kwong2 modified the attractive term and employed Tr0.5 as a function 3
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of the attractive term coefficient, which improved the predictive accuracy for the vapor pressure
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for gases and light hydrocarbons. However, there were still significant deviations in the
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prediction of the vapor pressure of different compounds with the Redlich–Kwong (RK) EoS.
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Soave (1972)3 found a linear relationship between α0.5 and (1–Tr0.5) for hydrocarbons, whose m
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slope was correlated with the acentric factor of hydrocarbons; thus, he proposed a new alpha
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function for the RK EoS and obtained the famous Soave–Redlich–Kwong (SRK) EoS.
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P = − ( )
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()
(1)
() = ( , )
(2)
( ) = 1 + (1 − . )
(3)
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= 0.480 + 1.574 − 0.176
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The SRK EoS can accurately predict the vapor pressures of non-polar and weakly polar
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compounds. However, for liquid densities, the predicted values are often larger than the
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experimental results. The volume translation concept proposed by Péneloux helps improving the
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volumetric behavior of the SRK EoS but such a correction is not sufficient to have satisfactory
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volumetric properties at near-critical conditions.11-13 With increasing acentric factor, the deviation
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in the predicted liquid densities increases. To improve the accuracy of liquid density prediction,
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Peng and Robinson (1976) modified the SRK EoS by adding a b(v-b) term into the denominator
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of the attractive term.4 Thus, the Peng–Robinson (PR) EoS was obtained. Although the
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compression factor of the PR EoS (0.307) is smaller than that of the RK EoS (0.333) and is
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closer to the real compression factor of hydrocarbons (0.27–0.29), the predicted liquid densities
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still deviate greatly from the experiments14,15. The fundamental reason lies in the constant
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compression factor of the EoS. Patel and Teja (1982)5 proposed a new equation of state and
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correlated the compression factor with the acentric factor. Therefore, the compression factor
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became a variable in the Patel–Teja (PT) EoS. Most modified alpha functions are based on these
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three classical cubic equations of state.
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2.2 The defects of Soave’s 1972 alpha function
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Soave’s 1972 alpha function has a simple functional form with only one parameter. This function
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showed improved predictive accuracy for the vapor pressures of non-polar and weakly polar
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compounds. However, it is restricted by the research objects and the functional form, and two
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defects are worth mentioning: First, the m parameter of Soave’s 1972 alpha function was just the
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functional value, calculated at Tr = 0.7. Thus, this EoS can be used to predict the vapor pressure
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of non-polar and weakly polar compounds in the reduced temperature range of 0.7–1.0.16 This
(4)
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alpha function deviates significantly from experimental results for the prediction of vapor
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pressures below the normal boiling point17, and the prediction of the vapor pressures of polar
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compounds and heavy hydrocarbons shows significant deviation from experimental values.4,17,18
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Secondly, the quadratic form of Soave’s 1972 function has an inherent defect. According to van
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der Waal’s theory, the functional value of the attractive term should monotonically decrease with
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increasing temperature,19,20 and tend to zero at high reduced temperatures,21-23 except for the
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quantum fluids (e.g., H2 or He)19,24. However, the value of Soave’s 1972 alpha function for most
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of the compounds decreases at first and then rises with increasing temperature,19,24,25 which is
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not appropriate for the attractive term with varying temperature.26,27 Because of this defect, the
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prediction of the Boyle temperature has no physical meaning. In addition, there are several
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critical points,28 and the high and low isotherms intersect.17 Twu (1988),29 and Coquelet et al.
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(2004)30 successively proposed constraints for the alpha function and its derivatives to ensure the
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correct trend in the predicted thermodynamic properties. Le Guennec et al. (2016)19,31 concluded
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and presented the list of constraints for an alpha function: (1 ) ≥ 0 and (1 ) continuous 0 :; :; . : ≤ 0 and : continuous .
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For all :
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The set of conditions reported in Eq.(5) were the “consistency test for an alpha function”. In
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reality, the polynomial alpha functions cannot satisfy the requirements of the consistency test
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because of these inherent defects.19,32-36 Although the Soave’s 1972 alpha function has the defects
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mentioned above, it is still widely used in the simulation and design of chemical and petroleum
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processes because the turning temperature is often higher than the maximum temperature in real
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applications.28,32,37 For example, the turning temperature of RK Soave's alpha function for
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n-alkanes (C1–C20), cycloalkanes (C5–C8), and aromatics (from benzene to naphthalene) were
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around 1800, 1300, and 2100 K, respectively, and for the polar compounds (methanol, ethanol,
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acetone, and water), the extrema were about 1500–2600 K.32,38
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There have been several articles evaluating the predictive capability of polynomial functions for
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the thermodynamic properties of pure fluids or the VLE, Liquid–liquid equilibria (LLE), and
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Vapor–liquid–liquid equilibria (VLLE) of mixtures, such as those by Kleiman et al. (2002),39
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Luo et al. (2008),40 Hong et al. (2012),9 and Young et al. (2016).36 In addition, Mathias (1994)10
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made an in-depth analysis of the advantages and disadvantages of the Stryjek–Vera (1986a,
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1986b) alpha functions in cubic EoS. Wang et al. (2004)41 reviewed the explored alpha functions
