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:
+ γ bv RTva Z = − (1) − bv ε ()()++ λη cvcvT
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 ∑∑ ayy ijji mm ∑ bycb iii ij i
= = + = MMM: am ∑∑ ayy ijji bm 3∑∑ ijji ∑ bybyy iii 4 m ∑ byc iii 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 ∑∑ PTyy cijcijji ∑∑ PTyy cijcijji ij ij
2 = 2 * = * Pcm ∑∑ PTyy cijcijji ∑∑ PTyy cijcijji Rm ∑∑ Ryy ijji 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 +ηbv φ ZZ ()γ 1ln1ln1ln −+−−−= − ()ln (2) v ()− λη bRT 1+ε + λbv
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 S ()− PPv 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:
()∂∂ nnb a v ∂()nc V ()1ln += γφ im ln()−− ln Zvbv −− m m i − m ()1+ε ()()++ λη ∂ bv m m RTc m cvcv m ni
a 11 ∂()2 an 1 ∂()nc +ηcv − m m − m ln m (4) ()− λη ()1+ε ∂ ∂ + λ m RTc m na i cn m ni cv m
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 ∂( 2 an ) ∂()nb ∂()nc m = 2∑ ay m = m = b ∂ ijj ∂ ∂ ii n ni j ni ni
for MMM:
1 ∂( 2 an ) ∂()nb ∂()nc m = 2∑ ay m = 23 ∑∑∑− + bbyyby 4 m = b ∂ ijj ∂ ijj ijji ii ∂ ii n ni j ni jij ni
and for RM-2:
1 ∂()2 an m Ω= RT 5.0 3∑ 2 − ∑ PTyTPTy ∂ cma cijcijj cijcijjcm n ni j j
∂()nb ∂()nc ∂β m = m β +−= n ∑∑ PTyy + 2β ∑ PTy ∂ ∂ ∂ cijcijji cijcijj 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. Comparisons are made between the calculated results and the available experimental data.
5 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
RESULTS AND DISCUSSION
To calculate liquid molar volume and vapor pressure using equations of state, the data of critical temperature and pressure, acentric factor, and molar refraction are needed. The experimental critical properties of carbon dioxide and n-alkanes up to C24 are taken from the literature (Nikitin t al., 1994; Frenkel et al., 1997a&b), while those of n-alkanes higher than C24 are predicted using a correlation developed by Twu (1984). The acentric factors are calculated using Pitzer correlation.
The molar refractions of all compounds are taken from the literature (Frenkel et al., 1997a&b).
The molar volumes of saturated liquids calculated using various equations of state are compared to those using an accurate correlation proposed by Hankinson and Thomson (1979). The percentage of average absolute deviations (AAD%) of the equations are presented in Table II.
According to this table, the RM equation of state is better than the other equations compared. The
RM equation is actually one order of magnitude more accurate than the other equations. It must be also pointed out that the two-constant-parameter MMM equation is more accurate than the two- constant-parameter RK equation and the three-constant-parameter SRK equation.
The AAD% of various equations of state in predicting molar volumes of liquid in sub- cooled and supercritical conditions are presented in Table III. According to this table also the RM equation is one order of magnitude more accurate than the other equations. It must be also pointed out that the two-constant-parameter MMM equation is more accurate than the two-constant- parameter RK equation and three-constant-parameter PR and SRK equations.
The AAD% of various equations of state in predicting vapor pressures of carbon dioxide and n-alkanes are presented in Table IV. According to this table prediction by the SRK equation is one order of magnitude superior to the other three-constant equations (RM and PR) and it is two orders
6 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 of magnitude better than the two-constant-parameter equations (RK and MMM). However, it must be pointed out that the PR and SRK equations are de veloped by fitting to the vapor pressure data.
To calculate the solubility of solid wax components in supercritical solvent, the data of molar volume and sublimation pressure of the solid wax components are needed, as well as the critical properties, acentric factors, and molar refractions of all compounds involved. The molar volumes of the solid are taken from the literature, and the sublimation pressures are calculated using correlations developed by Moradinia and Teja (1986 & 1988)
The solubility of n-tritriacontane (n-C33H68) in supercritical ethane is depicted in Figure 1 along with the predictions obtained from various equations of state. According to this figure, among the two-constant-parameter equations, the MMM equation is more accurate than the RK equation. The SRK equation of state is closer to the experimental data than the other equations are.
It is shown in figure 1 that there is a cross over region; at Pr ~ 1.33 the experimental solubility at
308 K is higher than that at 318 K is, while at Pr > 2 the solubilities at 318 K are higher than those at
308 K are. The equations of state also show cross over regions. Figure 2 depicts the solubility of n- tritriacontane in supercritical carbon dioxide. It is shown that the MMM equation is better than the
RK equation, and the RM-1 equation is better than the MMM, RM-2 and the SRK equations. The
PR equation of state is closer to the experimental data than the other equations are. In both figures we assume that the interaction parameter kij = 0.
The solubilities of solid wax components in supercritical solvents are also calculated with kij best fitted to experimental data. Table V shows the interaction parameters of various equations of state for a number of systems at various temperatures along with the AAD%. According to this table, the MMM equation of state gives the least value of AAD%. For 270 data points of 10 systems, this equation gives the AAD% of 20.3. A major advantage of the MMM equation is that it
7 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 is a two-constant-parameter equation of state; it does not need the acentric factor or molar refraction data which are rare for heavy molecular weight compounds.
As it was mentioned above for all the wax components experimental critical properties and acentric factor data are not available and estimation methods are used for their calculations. The molar volume and vapor pressure of liquids calculated using equations of state are influenced by the critical properties and acentric factor used. To test the effect of these properties on such calculations, we performed error calculation for molar volumes and vapor pressures of various wax components. As an example, in Table VI we report the errors which will be caused due to inaccuracies in critical properties and acentric factor on saturated liquid molar volume and vapor pressure of n-octacosane at 583 K. According to this table an error of 0.7% in Tc, an error of 5% in
Pc and an error of 1% in ω will result in 10% maximum error in saturated liquid molar volume calculation regardless of the equation of state used. For vapor pressure calculation the percentage error by various equations of state are different and varies from 18 to 35 %.
To evaluate the effect of the input data (critical properties, acentric factor and sublimation pressure of the solid) on the supercritical solubility calculation, the solubility of n-tritriacontane in supercritical carbon dioxide at 308 K are calculated for four cases using four different sets of input data. In case 1, the critical properties of the solid are predicted using Twu (1984) method and the sublimation pressure is predicted using Moradinia-Teja (1988) method. In case 2 the critical properties are calculated using Twu method and the sublimation pressure is calculated using
Pouillot et al. (1996) method. In case 3 the critical properties are calculated using Constantinou-
Gani (1994) method and the sublimation pressure is calculated using Moradinia-Teja method. In case 4 the critical properties are calculated using Constantinou-Gani method and the sublimation pressure is calculated using Pouillot et al. method. In all cases the interaction parameter kij are best
8 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 fitted to the experimental data. The results of the above mentioned four cases for different equations of state are reported by Figures 3-8. According to Figures 3, 6 and 7 the supercritical solubility calculations by RK, PR and SRK equations are quite sensitive to the input data and they do not fit well to the experimental data. According to Figures 4, 5 and 8 the supercritical solubility calculations by MMM, RM-1 and RM-2 are less sensitive to the variations of the input data and they fit well to the experimental data.
The liquid molar volumes and vapor pressures of carbon dioxide and n-alkanes as well as the solubility of solid wax components in supercritical solvents have been calculated using various equations of state. The RM equation of state is more accurate for predicting molar volume, while the SRK equation of state is more accurate for predicting vapor pressure. Using kij = 0, the SRK
EOS is closer to the experimental solubility of wax components in supercritical ethane data than the other equations, while for carbon dioxide-wax component systems, the PR EOS is closer to the experimental data than the others. The MMM EOS gives the least value of AAD% when kij is best fitted to the experimental data.
ACKNOWLEDGEMENT
This research is supported by the Directorate General of Higher Education, Ministry of
Education and Culture, Indonesia, through URGE project, under grant No. 019/HTPP-
II/URGE/1996.
9 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
Symbols and Nomenclature
MMM Mohsennia-Moddaress-Mansoori n number of mole in the system
P pressure [bar]
PR Peng-Robinson
R universal gas constant
R* ratio of molar refraction of a compound to that of a reference (methane)
RK Redlich-Kwong
RM Riazi-Mansoori
SRK Soave-Redlich-Kwong
T temperature [K] v molar volume [l/mol] y mole fraction
Z compressibility factor
Greek Letters
φ fugacity coefficient
ω acentric factor
Subscripts c critical ij component indices m mixture
10 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 r reduced
Superscripts
L liquid phase
S solid phase sat saturated sub sublimation
V vapor phase
11 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
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12 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
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13 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
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14 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
Table I. Parameters of the generalized equation of state
Eq. of State → RK MMM RM PR SRK Parameters ↓ γ 0 1.3191 0 0 0 η 1111+√2 0 λ 0001-√2 0 ε 0.5 0.5 0.5 0 0
Ωa 0.42748 0.487480 0.42748 0.45724 0.42748 Ωb 0.08664 0.064462 0.08664 0.07780 0.08664 α 111αPR αSRK β 11βRM 11
2 0.5 2 αPR = [1+(0.37464+1.524226ω-0.26992ω )(1-Tr )]
2 0.5 2 αSRK = [1+(0.48508+1.55171ω-0.15613ω )(1-Tr )]
-1 * (βRM) = 1+{0.02[1-0.92 exp(-1,000 |Tr-1|)] - 0.035 (Tr-1)} (R -1)
15 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
Table II. The Average Deviations of Various Equations of State in Predicting Saturated Liquid Molar Volumes of Pure Compounds Compared with Those Calculated Using the Hankinson and Thomson (1979) Correlation.
AAD % Compound Tr range RK MMM RM PR SRK CO2 0.71-1.00 19.5 8.8 19.5 4.7 14.7 CH4 0.48-0.99 4.5 13.9 4.5 8.6 4.5 C2H6 0.33-0.99 10.3 11.8 6.4 6.0 9.2 C3H8 0.35-0.98 11.2 10.8 3.9 5.3 9.2 n-C4H10 0.36-0.96 13.3 9.0 3.0 3.6 10.3 n-C5H12 0.47-0.99 16.8 7.5 2.4 3.4 12.5 n-C6H14 0.39-0.99 19.9 6.9 2.2 2.2 14.8 n-C7H16 0.41-0.99 22.3 7.4 1.4 2.7 16.0 n-C8H18 0.41-0.99 24.7 6.9 1.2 4.2 17.7 n-C9H20 0.42-0.98 26.8 7.8 0.8 5.1 18.7 n-C10H22 0.43-0.98 29.9 10.5 0.7 7.0 20.8 n-C11H24 0.55-0.78 31.1 10.5 0.3 7.4 21.3 n-C12H26 0.54-0.89 35.3 14.7 0.4 10.1 24.3 n-C13H28 0.56-0.80 37.6 16.5 0.2 11.4 25.8 n-C14H30 0.54-0.85 42.8 21.4 1.2 15.0 29.9 n-C15H32 0.58-0.82 45.8 24.1 1.2 16.7 31.8 n-C16H34 0.56-0.81 49.7 26.9 1.8 19.7 35.1 n-C17H36 0.59-0.83 56.2 32.9 3.9 24.3 40.3 n-C18H38 0.55-0.84 59.5 35.5 4.0 26.9 43.1 n-C19H40 0.56-0.85 63.5 39.0 4.4 29.6 46.1 n-C20H42 0.57-0.85 67.9 42.9 4.8 32.2 49.0 n-C22H46 0.55-0.81 57.6 31.8 4.5 25.1 40.6 n-C24H50 0.56-0.82 66.2 39.2 2.9 31.1 47.2 n-C28H58 0.58-0.84 62.9 36.4 11.2 27.7 43.1
Overall 36.5 19.7 3.6 13.8 26.1
16 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
Table III. The Average Deviations of Various Equation of State in Predicting Molar Volumes of Liquids in Sub-cooled and Supercritical Conditions Compared with Experimental Data
AAD % Experimental data Compound Tr range Pr range RK MMM RM PR SRK No. of data pts. ref CO2 0.7-2.2 1.0-13.6 5.0 4.8 5.0 2.9 6.1 447 a CH4 0.5-2.6 0.0-15.2 2.0 11.1 2.0 7.4 2.4 459 b C2H6 0.3-2.3 0.0-14.3 3.8 11.2 1.7 5.6 4.1 474 c C3H8 0.2-1.9 0.0-16.5 5.9 9.6 1.8 4.2 5.9 533 d n-C4H10 0.3-1.7 0.0-18.5 8.4 8.4 1.6 3.8 7.9 638 e n-C5H12 0.4-0.7 0.0-71.3 11.2 7.6 0.8 2.5 9.4 880 f n-C6H14 0.4-0.7 0.0-332 14.7 5.8 2.0 2.5 12.6 510 f n-C7H16 0.6-1.1 1.8-183 14.5 5.6 2.3 3.3 13.5 70 f,g n-C9H20 0.5-1.0 2.2-218 18.8 3.1 2.2 5.1 17.1 66 g n-C11H24 0.5-0.9 2.6-259 25.1 3.1 2.6 10.4 23.1 70 g n-C13H28 0.4-0.9 3.0-303 32.3 7.9 3.2 16.7 30.1 70 g n-C17H36 0.4-0.8 4.1-410 48.5 19.8 7.2 31.0 45.9 60 g n-C20H42 0.4-0.8 5.0-500 59.9 28.6 9.9 41.1 57.1 50 g n-C30H62 0.4-0.8 6.8-682 62.7 28.9 6.8 43.5 59.8 50 g
Overall 22.3 11.1 3.5 12.8 21.0 4377
aAngus et al., 1976. bGoodwin, 1974. cGoodwin and Roder, 1976. dGoodwin and Haynes, 1982. eHaynes and Goodwin, 1976. fFrenkel et al., 1997a. gDoolittle, 1964.
17 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
Table IV. The Average Deviations of Various Equations of State in Predicting Vapor Pressures of Pure Compounds Compared with the Experimental Data
Compound Tr range AAD % Experimental data RK MMM RM PR SRK No. of data pts. ref CO2 0.71-1.00 19.1 4.0 19.1 0.8 0.5 47 a CH4 0.48-0.99 17.2 50.0 17.2 0.7 2.9 84 b,c C2H6 0.33-0.99 11.5 36.3 11.9 3.0 2.6 114 b,d C3H8 0.35-0.98 19.8 39.5 8.3 3.0 1.9 101 b,e n-C4H10 0.36-0.96 43.2 32.4 7.0 5.6 1.9 130 b,f n-C5H12 0.47-0.99 63.4 19.2 9.9 0.8 1.5 91 b,h n-C6H14 0.39-0.99 >100 15.8 16.5 3.1 1.9 88 b,h n-C7H16 0.41-0.99 >100 11.0 20.7 2.4 1.2 80 b,h n-C8H18 0.41-0.99 >100 18.5 28.9 2.7 1.2 87 b,h n-C9H20 0.42-0.98 >100 33.2 33.3 2.1 1.5 82 c,h n-C10H22 0.43-0.98 >100 51.4 37.4 3.1 1.2 86 b,g,h n-C11H24 0.55-0.78 >100 81.7 37.7 6.6 4.4 27 b n-C12H26 0.54-0.89 >100 89.9 31.7 2.9 0.4 40 b,g,h n-C13H28 0.56-0.80 >100 >100 28.8 3.3 0.5 27 b n-C14H30 0.54-0.85 >100 >100 30.3 5.8 2.5 42 b,g,h n-C15H32 0.58-0.82 >100 >100 23.4 4.0 0.6 27 b n-C16H34 0.56-0.81 >100 >100 29.3 5.6 0.9 45 b,g,h n-C17H36 0.59-0.83 >100 >100 23.9 6.2 1.6 43 b n-C18H38 0.55-0.84 >100 >100 30.2 9.2 3.0 44 b,g n-C19H40 0.56-0.85 >100 >100 30.4 10.4 3.6 42 b,g n-C20H42 0.57-0.85 >100 >100 33.0 8.8 2.4 42 b,g n-C22H46 0.55-0.81 >100 >100 69.1 18.6 1.2 21 b,g n-C24H50 0.56-0.82 >100 >100 79.4 24.7 1.5 22 b,g n-C28H58 0.58-0.84 >100 >100 57.9 33.9 1.7 23 b,g n-C29H60 0.54-0.84 >100 >100 82.5 42.5 2.7 12 b n-C30H62 0.54-0.84 >100 >100 75.8 44.5 2.7 12 b n-C32H66 0.55-0.85 >100 >100 61.5 48.9 3.2 12 b n-C33H68 0.55-0.85 >100 >100 56.9 51.8 3.9 12 b
Overall ~ 2,100 ~ 650 35.4 12.7 2.0 1483
aAngus et al., 1979. bFrenkel at al., 1997a. cGoodwin, 1974. dGoodwin and Roder, 1976. eGoodwin and Haynes, 1982. fHaynes and Goodwin, 1976. gMorgan and Kobayashi, 1994. hSalerno et al., 1986.
18 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
Table V. Interaction Parameter (k12) of Some Systems exp. data TP k12 AAD % no.of System [K] [bar] RK MMM RM-2 PR SRK RM-1 RK MMM RM-2 PR SRK RM-1 data pts. ref. C2H6 - n-C28H58 308.2 56-240 -0.4638 -0.2099 0.0807 -0.0553 -0.0189 -0.1283 46.2 27.0 14.4 38.5 43.5 25.3 20 a,b,c C2H6 - n-C29H60 308.2 65-240 -0.4146 -0.1618 0.1215 -0.0131 0.0260 -0.0701 53.1 29.9 23.9 48.5 54.0 28.7 12 c,d C2H6 - n-C30H62 308.2 66-200 -0.4777 -0.2066 0.1025 -0.0571 -0.0206 -0.1121 25.0 22.8 57.2 22.9 26.8 39.9 6 b,c 313.2 66-136 -0.4738 -0.2137 0.0901 -0.0517 -0.0139 -0.1233 13.9 30.8 29.7 18.9 20.1 35.0 4 b C2H6 - n-C32H66 308.2 66-240 -0.5124 -0.2259 0.1020 -0.0707 -0.0297 -0.1214 46. 43.0 56.2 41.8 45.4 55.2 17 a,b,c 313.2 66-200 -0.5011 -0.2264 0.1050 -0.0658 -0.0263 -0.1131 24.4 34.5 40.2 20.6 25.7 51.4 6 b 318.2 80-240 -0.5248 -0.2438 0.0966 -0.0762 -0.0347 -0.1142 45.2 17.7 22.8 32.0 40.0 42.7 9 b,c 319.2 80-136 -0.4872 -0.2241 0.1087 -0.0565 -0.0157 -0.1219 23.6 37.4 39.3 28.5 29.4 44.6 4 b C2H6 - n-C33H68 308.2 65-240 -0.4632 -0.1933 0.1100 -0.0286 0.0137 -0.0779 50.2 34.4 50.7 43.1 48.3 44.5 13 c,d 313.2 65-202 -0.4459 -0.1845 0.1433 -0.0240 0.0193 -0.0580 39.6 21.5 28.9 25.9 30.5 45.3 6 d 318.2 65-240 -0.4506 -0.1918 0.1433 -0.0203 0.0228 -0.0530 45.7 24.2 28.0 42.7 47.9 44.0 13 c,d CO2 - n-C28H58 307.2 123-181 -0.3458 -0.0936 0.2487 0.0110 0.0507 0.0211 51.0 7.5 34.4 45.3 51.3 33.5 10 e 308.2 80-240 -0.3161 -0.0901 0.2477 0.0296 0.0708 0.0194 52.3 18.3 35.0 49.2 52.5 34.3 16 f,g 313.2 90-275 -0.2910 -0.0835 0.2532 0.0365 0.0765 0.0286 46.7 25.0 44.5 39.1 41.7 43.4 10 f 318.2 100-250 -0.2915 -0.0867 0.2504 0.0359 0.0746 0.0232 64.7 13.4 31.0 47.8 49.7 30.4 16 f,g 318.6 119-284 -0.3067 -0.0859 0.2531 0.0347 0.0736 0.0278 53.7 8.2 33.2 45.4 48.6 32.6 7 e 323.4 125-327 -0.2973 -0.0869 0.2552 0.0385 0.0764 0.0314 62.5 5.8 28.9 53.7 56.4 29.2 22 e 325.2 121-284 -0.2946 -0.0867 0.2540 0.0321 0.0690 0.0287 64.2 7.3 22.6 45.5 49.3 23.5 18 e CO2 - n-C29H60 308.2 100-240 -0.2751 -0.0530 0.2782 0.0645 0.1075 0.0670 71.8 21.5 12.9 67.4 70.5 12.9 7 g 318.2 100-240 -0.1961 -0.0540 0.2789 0.0818 0.1451 0.0672 81.2 22.2 6.9 76.7 76.3 7.1 7 g CO2 - n-C30H62 308.2 90-250 -0.3254 -0.1141 0.2481 0.0327 0.0779 0.0084 69.6 17.1 28.8 66.4 67.2 30.0 9 f 318.2 105-250 -0.3125 -0.1197 0.2439 0.0273 0.0700 -0.0005 67.5 8.3 28.7 57.8 58.2 30.8 9 f CO2 - n-C32H66 308.2 120-240 -0.4140 -0.1500 0.2448 -0.0162 0.0316 -0.0139 59.5 8.1 24.1 54.8 58.7 27.0 6 g 318.2 140-240 -0.3913 -0.1462 0.2483 -0.0035 0.0426 -0.0092 67.3 6.7 22.3 55.8 58.7 25.3 6 g 328.2 140-240 -0.3777 -0.1345 0.2596 0.0044 0.0477 0.0104 57.5 9.2 27.1 40.2 43.2 30.0 5 g CO2 - n-C33H68 308.2 120-240 -0.3461 -0.1051 0.2825 0.0262 0.0754 0.0496 68.4 25.0 11.3 64.3 68.0 15.0 6 g 318.2 140-240 -0.3384 -0.1043 0.2832 0.0280 0.0872 0.0494 65.0 22.9 4.8 61.4 63.5 7.6 5 g 328.2 140-240 -0.3057 -0.0990 0.2878 0.0428 0.0876 0.0557 67.4 18.5 3.5 60.1 62.1 7.9 5 g
Overall 60.0 20.3 28.3 46.2 49.6 31.3 270
aKalaga and Trebble, 1997. bMoradinia and Teja, 1986. cSuleiman and Eckert, 1995. dMoradinia and Teja, 1988. eMcHugh et al., 1984. fReverchon et al., 1993. gChandler et al., 1996
20 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
Table VI. Saturated Liquid Molar Volume and Vapor Pressure n-Octacosane at 583 K Calculated Using Various Equations of State with Tc = (829.2±±±6.0) K, Pc = (7.550±±±0.358) bar, ω = (1.1772±±±0.0129)
Eq. of state vsat Psat [l/mol] [bar] RK 1.1705 ± 0.1148 0.6494 ± 0.1143 MMM 0.9907 ± 0.0958 0.3714 ± 0.0732 RM 0.6400 ± 0.0630 0.0590 ± 0.0138 PR 0.9037 ± 0.0913 0.0621 ± 0.0201 SRK 1.0138 ± 0.1026 0.0510 ± 0.018
Experimental 0.6516 0.0514
21 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
Table VII. Physical Properties of n-Tritriacontane Used in Sensitivity Test of Various Equations of State in Predicting Supercritical Solubility
sub S Tc Pc ωωω P v [K] [bar] [-] [bar] [l/mol] Case 1 866.72 (a) 6.578 (a) 1.3004 (a) 1.5226 × 10-14 (c) 0.5710 (c) Case 2 866.72 (a) 6.578 (a) 1.3004 (a) 9.2618 × 10-16 (d) 0.5710 (c) Case 3 854.02 (b) 5.949 (b) 1.3665 (b) 1.5226 × 10-14 (c) 0.5710 (c) Case 4 854.02 (b) 5.949 (b) 1.3665 (b) 9.2618 × 10-16 (d) 0.5710 (c) aTwu (1984). bConstantinou and Gani (1994). cMoradinia and Teja (1988). dPouillot et al. (1996).
22 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
0 RM-2 318 K
SRK 318 K -2 RM-2 308 K RM-1 318 K
SRK 308 K
-4 RM-1 308 K PR 318 K PR 308 K
-6
Log (y) MMM 318 K
MMM 308 K -8
RK 318 K -10
RK 308 K
-12 0123456 Pr
Figure 1 Solubility of n-tritriacontane (n-C33H68) in supercritical ethane at various temperatures. The experimental data are taken from Chandler et al. (1996): (o) 308 K, (x) 318 K. The lines are the solubility calculated using various equation of state with kij=0
23 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
0
RM-2 318 K RM-1 318 K -2 RM-1 308 K RM-2 308K
SRK 318K -4 SRK 308K
PR 318 K
-6 PR 308K Log(y) MMM 318K
-8 MMM 308K
-10 RK 318K
RK 308K
-12 012345 Pr
Figure 2 Solubility of n-tritriacontane (n-C33H68) in supercritical carbon dioxide at various temperatures. The experimental data are taken from Chandler et al. (1996): (o) 308 K, (x) 318 K. The lines are the solubility calculated using various equation of state with kij=0
24 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
-4 RK Eq. of State
-5
-6
-7
-8
Log (y) -9
-10
-11 Case 1, kij=-0.3461 Case 2, kij=-0.4196 -12 Case 3, kij=-0.3726 Case 4, kij=-0.4328
-13 01234
Pr
Figure 3 Sensitivity of calculations of solubility of n-tritriacontane (n-C33H68) in supercritical carbon dioxide at 308 K by the RK equation of state to variations of the input. The input data for various cases are reported in Table 7. The experimental solubility data is taken from Chandler et al., 1996.
25 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
-4 MMM Eq. of State
-5
-6
-7
-8
Log (y) -9
-10
-11 Case 1, kij=-0.1051 Case 2, kij=-0.1521 -12 Case 3, kij=-0.1166 Case 4, kij=-0.1700
-13 01234
Pr
Figure 4 The same as in Figure 3 but for the MMM equation of state.
26 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
-4
RM-2 Eq. of State -5
-6
-7
-8
Log (y) -9
-10
-11 Case 1, kij=0.2825 Case 2, kij=0.2526 -12 Case 3, kij=0.2925 Case 4, kij=0.2644
-13 01234
Pr
Figure 5 The same as in Figure 3 but for the RM-2 equation of state
27 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
-4 PR Eq. of State
-5
-6
-7
-8
Log (y) -9
-10
Case 1, kij=0.0262 -11 Case 2, kij=-0.0148 Case 3, kij=0.0256
-12 Case 4, kij=-0.0245
-13 01234
Pr
Figure 6 The same as in Figure 3 but for the PR equation of state
28 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
-4 SRK Eq. of State
-5
-6
-7
-8
Log (y) -9
-10 Case 1, kij=0.0754 Case 2, kij=0.0348 -11 Case 3, kij=0.0804 Case 4, kij=0.0396 -12
-13 01234 Pr
Figure 7 The same as in Figure 3 but for the SRK equation of state
29 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
-4 RM-1 Eq. of State
-5
-6
-7
-8
Log (y) -9
-10
Case 1, kij=0.0495 -11 Case 2, kij=-0.0060 Case 3, kij=0.0378 -12 Case 4, kij=-0.0160
-13 01234 Pr Figure 8 The same as in Figure 3 but for the RM-1 equation of state
30