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Prediction of Molar Volumes, Vapor Pressures and Supercritical Solubilities of Alkanes by Equations of State

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Prediction of Molar Volumes, Vapor Pressures and Supercritical Solubilities of Alkanes by Equations of State Preprint Chemical Engineering Communications Volume 173, Issue 1, pp. 23-42, 1999 DOI: 10.1080/00986449908912774 Prediction of molar volumes, vapor pressures and supercritical solubilities of alkanes by equations of state Ratnawati Hartono*, G.Ali Mansoori** Thermodynamics Research Laboratory, University of Illinois at Chicago (M/C 063) Chicago, IL 60607-7052 USA and Aryadi Suwono Thermodynamics Research Laboratory, Inter University Center for Engineering Science, Bandung Institute of Technology, Jl. Tamansari No. 126 Bandung, Indonesia, [email protected] ABSTRACT A generalized cubic equation of state which can represent all the cubic equations is introduced and thermodynamic property relations for it are presented. Five cubic equations of states with respective mixing rules are used to predict molar volumes and vapor pressures of pure alkanes (from methane till n-tritriacontane) and solubilities of solid wax components (high molecular weight alkanes) in supercritical solvents. They are the RK (Redlich-Kwong), MMM (Mohsennia- Modarress-Mansoori), RM (Riazi-Mansoori), PR (Peng-Robinson), and SRK (Soave-Redlich- Kwon) equations of state. The experimental data necessary to compare the equations of state are taken from the literature. It is demonstrated that the SRK equation of state is more accurate for predicting vapor pressures of alkanes. The RM equation of state is shown to be more accurate for predicting molar volumes of saturated and sub-cooled liquid alkanes as well as molar volumes of alkanes in their supercritical condition. For the solubility of wax components in supercritical solvents it is shown that the MMM equation of state gives the least AAD% for the 270 data points of 10 binary systems studied consisting of a high molecular weight alkane and supercritical ethane and carbon dioxide. * Permanent address: Chemical Engineering Department, Diponegoro University, Kampus Tembalang, Semarang, Indonesia, [email protected] ** Corresponding author, email: [email protected] R. Hartono, G.A. Mansoori, A. Suwono Prediction of molar volumes, vapor pressures and supercritical solubilities of alkanes by equations of state Chem. Eng. Comm. 173(1): 23-42, 1999 DOI: 10.1080/00986449908912774 INTRODUCTION Petroleum wax contains hundreds to thousands of components higher than C16 (Himran et al., 1994). These components can be classified into two major classes, i.e., macrocrystalline or paraffinic and microcrystalline or naphthenic waxes. The macrocrystalline wax is composed of mainly straight-chain paraffins and microcrystalline or amorphous wax contains isoparaffins and naphthenes. Macrocrystalline wax is more valuable than microcrystalline or crude wax. In order to characterize the petroleum wax and perform various operations on wax mixtures, such as wax fractionation, it is necessary to be able to predict thermodynamic properties of wax. In this report we examine the accuracy of three well-known equations of state RK (Redlich-Kwong), SRK (Soave-Redlich-Kwong), PR (Peng-Robinson) as they are reported in numerous textbooks (Walas, 1985) and two newly developed equations of state namely RM (due to Riazi and Mansoori, 1993) and MMM (due to Mohsennia, et al. (1995) for prediction of molar volumes, vapor pressures and supercritical solubilities of alkanes. It should pointed out that all these five equations of state are cubic. CUBIC EQUATIONS OF STATE AND THEIR GENERALIZATION Cubic equations of state are widely used in phase equilibrium calculation because of their simplicity and, in general, good performance. The simplest and one of the widely known equations of state is that of van der Waals. The van der Waals equation of state is a simple model that incorporates some corrections to the ideal gas law for attraction and repulsion between molecules. However, this equation of state is not accurate enough to predict thermodynamic properties of most fluid. Inspired by the van der Waals model, investigators have proposed several equations of state through the years. Almost every equation of state has been claimed to be superior in some respects 1 R. Hartono, G.A. Mansoori, A. Suwono Prediction of molar volumes, vapor pressures and supercritical solubilities of alkanes by equations of state Chem. Eng. Comm. 173(1): 23-42, 1999 DOI: 10.1080/00986449908912774 to the earlier ones. The RK equation that is a modification of the van der Waals equation, was a considerable improvement over other equations of relatively simple forms at the time of its introduction. In the SRK equation the temperature-dependent term of a/T0.5 of the RK equation is replaced by a function denoted by α that depends on the acentric factor of the compound and temperature. The PR equation is another cubic equation of state involving acentric factor (Walas, 1985). Recently, Riazi and Mansoori (1993) stated that parameter b of the RK equation is more effective for calculating liquid densities because it represents the volume of the molecule. They modified the parameter b of the RK equation by introducing a function, denoted by β, that depends on the refractive index of the compound. Also recently, by another approach Mohsennia et al.(1995) modified the repulsive part of the RK equation. Their modification was based on the statistical mechanical information available for the repulsive thermodynamic functions and the phenomenological knowledge of the attractive intermolecular potential tail contributions to the thermodynamic properties. It should be pointed out that the RK and MMM equations are two- constant-parameter equations of state, while the RM, PR and SRK are three-constant-parameter equations. All the above mentioned five equations of state can be written in the following generalized form: v+ γ b av RT Z = − (1) v− b Tε ()() v+η c v + λ c 2 (2+ε) where a = Ωa α R Tc / Pc and b = c = Ωb β R Tc / Pc. 2 R. Hartono, G.A. Mansoori, A. Suwono Prediction of molar volumes, vapor pressures and supercritical solubilities of alkanes by equations of state Chem. Eng. Comm. 173(1): 23-42, 1999 DOI: 10.1080/00986449908912774 Parameters Ωa, Ωb, γ, η and λ are component-independent constants, while α and β are component-dependent constants, and their numerical values for various equations of states are given in Table I. Parameters a, b and c are dependent on critical properties and (in some cases) on temperature. In extending the equations of state to mixtures parameters a, b, and c are replaced with am, bm, and cm with the following expressions (mixing rules): = = = RK, PR, SRK, RM-1: am ∑∑yi y j a ij bm c m ∑ yi b ii ij i = = + = MMM: am ∑∑yi y j a ij bm 3∑∑yi y j b ij ∑ yi b ii 4 cm ∑ yi b ii ij ij i i For the RM equation there is another alternative in extending it to mixtures by replacing Tc and Pc with Tcm and Pcm as given below (Riazi and Mansoori, 1993): = 2 RM-2: Tcm ∑∑yi y j T cij P cij ∑∑yi y j T cij P cij ij ij 2 = 2 * = * Pcm ∑∑yi y j T cij P cij ∑∑yi y j T cij P cij Rm ∑∑yi y j R ij ij ij ij These above equations of state are used to calculate the properties of the pure components, i.e. vapor pressure, molar volumes of liquid at saturated, sub-cooled and supercritical conditions as well as the solubility of wax components in supercritical solvents. 3 R. Hartono, G.A. Mansoori, A. Suwono Prediction of molar volumes, vapor pressures and supercritical solubilities of alkanes by equations of state Chem. Eng. Comm. 173(1) : 23-42, 1999 DOI: 10.1080/00986449908912774 To calculate both the vapor pressure and the saturated liquid molar volume using equation of state, we need to know the fugacity coefficients for pure compound. The fugacity coefficient for pure compound is in the following generalized form: b a v+η b lnφ =ZZ − 1 − ln −() 1 +γ ln 1 − − ()ln (2) v ()η− λ bRT 1+ε v+ λ b To calculate the solubility of solid wax components (yi) in a gas phase the following expression can be used (Walas, 1985; Schulz, et al., 1988): P sub vS () P− P sub y = i exp i i (3) i φ V i P RT V The fugacity coefficient of a compound in a mixture (φi ) derived from the generalized equation of state is in the following form: ∂()nb ∂ n a v ∂()nc lnφV =() 1 + γ m i −ln()v − b v −ln Z − m m i − m ()1+ε ()()+η + λ ∂ v bm cm RT v cm v cm ni a 1 1 ∂()n2 a 1 ∂()nc v+η c − m m − m ln m (4) ()η− λ ()1+ε ∂ ∂ + λ cm RT am n ni cm ni v cm where for RK, PR, SRK, RM-1: 4 R. Hartono, G.A. Mansoori, A. Suwono Prediction of molar volumes, vapor pressures and supercritical solubilities of alkanes by equations of state Chem. Eng. Comm. 173(1): 23-42, 1999 DOI: 10.1080/00986449908912774 1 ∂(n2 a ) ∂()nb ∂()nc m = 2∑ y a m = m = b ∂ j ij ∂ ∂ ii n ni j ni ni for MMM: 1 ∂(n2 a ) ∂()nb ∂()nc m = 2∑ y a m = 3 2∑∑∑y b− y y b + b 4 m = b ∂ j ij ∂ j ij i j ij ii ∂ ii n ni j ni jij ni and for RM-2: 1 ∂()n2 a m = Ω RT 0.5 3∑ y T2 P− T∑ y T P ∂ a cm j cij cij cm j cij cij n ni j j ∂()nb ∂()nc ∂β m = m = −β + n ∑∑y y T P + 2β ∑ y T P ∂ ∂ ∂ i j cij cij j cij cij ni ni ni ij j In order to test the relative accuracy of various equations of state the above generalized expressions are used to calculate vapor pressures and molar volumes of saturated, sub-cooled and supercritical pure liquid n-alkanes [from methane (C1) till n-tritriacontane (C33)] and carbon dioxide as well as the solubilities of wax components in supercritical solvents.
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