Ind. Eng. Chem. Res. 2001, 40, 1975-1984
1975
Use of the Refractive Index in the Estimation of Thermophysical Properties of Hydrocarbons and Petroleum Mixtures Mohammad R. Riazi* and Yousef A. Roomi Chemical Engineering Department, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait
In this paper we show application of the refractive index in the estimation of various thermodynamic and physical properties of pure hydrocarbons and undefined petroleum mixtures. In particular, we demonstrate how one can use the refractive index to estimate equation of state parameters, critical constants, the composition of petroleum mixtures, and temperaturedependent properties such as heat capacity, PVT, and transport properties of hydrocarbon fluids. The refractive index is an easily measurable property; however, when it is not available, methods of estimation of this property through other available parameters are also presented. Introduction Accurate knowledge of volumetric, physical, and thermodynamic (known as thermophysical) properties of hydrocarbons, petroleum fractions, crude oils, and petroleum gases is important in optimum design and operation of equipments related to petroleum production, processing, transportation, and related industries. Because experimental measurement of these properties for various mixtures under different conditions is difficult, time-consuming, and expensive, methods of estimation of these properties have become increasingly important. Estimation methods for most of the thermophysical properties of petroleum fluids are given in the API Technical Data Book.1 Reid et al.2 also provide estimation methods for many properties of hydrocarbons as well as non-hydrocarbon compounds and their mixtures. However, in many cases parameters needed to estimate a property may not be available or easily measurable. For example, Reid et al.2 have shown that, in order to estimate the viscosity of liquid hydrocarbons by the relation proposed by van Velzen et al.,3 seven different propertiessthe critical volume, critical temperature, critical pressure, freezing point, molar volume at the freezing point, molecular weight, and acentric factorsare needed as input parameters. All of these parameters are not available for petroleum mixtures, and most of these parameters should be estimated with great uncertainty. Volumetric properties and most of the thermodynamic properties are calculated through equations of state (EOSs) or generalized correlations.1 These correlations require critical constants and acentric factors as their input parameters. However, as explained by Riazi and Sahhaf,4,5 most of the estimation methods for critical constants of hydrocarbons heavier than C20 are not reliable. Therefore, estimation of thermophysical properties through critical constants for such compounds or their mixtures may not be accurate. Riazi et al.6 have shown that small errors in the estimation of critical constants significantly increase errors in the estimation of the desired thermodynamic properties. * To whom all correspondence should be addressed. Tel: (+965) 4817662. Fax: (+965) 4811772. E-mail: riazi@ kuc01.kuniv.edu.kw. Homepage: http://kuc01.kuniv.edu.kw/ ∼riazi.
The refractive index, n, is defined as the ratio of velocity of light in the vacuum to the velocity of light in the medium and, therefore, for a fluid it is greater than unity. The refractive index is a thermodynamic property and is a state function, which for a pure fluid depends on temperature and pressure. For gases, the refractive index is very close to unity, but for liquids, it is greater than 1. The refractive index or refractivity (n) can be easily measured by the sodium D line of a simple refractometer at a temperature of interest. Values of n at 20 and 25 °C are given by the API Technical Data Book1 for many different hydrocarbons. The main objective of this work is to use the refractive index, which is easily measurable, to estimate the thermophysical properties of petroleum fluids. Properties such as critical constants, EOS parameters, heat capacity, and transport properties are related to the refractive index. In addition, application of the refractive index in the estimation of the composition of undefined petroleum mixtures as well as methods of the estimation of the refractive index itself is also demonstrated. Theoretical Developments Intermolecular forces existing between various molecules of a fluid determine the nature of the properties of the fluid. For nonpolar compounds such as hydrocarbons, the main intermolecular force is the London dispersion force, which is characterized by the factor polarizability, R, defined as7,8
R)
( )( )(
)
3 M n2 - 1 4πNA d n2 + 2
(1)
where NA is Avogadro’s number, M is the molecular weight, d is the density, and n is the refractive index. Molar refraction Rm is defined as
Rm ) VI
(2)
where V is the molar volume and I is the refractive index parameter:
V ) M/d I)
n2 - 1 n2 + 2
10.1021/ie000419y CCC: $20.00 © 2001 American Chemical Society Published on Web 03/14/2001
(3) (4)
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Ind. Eng. Chem. Res., Vol. 40, No. 8, 2001
Table 1. Parameters and Constants in Equation 5 for Various Properties: θ ) a0 exp(b0θ1 + c0I20 + d0θ1I20)θ1e0I20f0 a θ
θ1
a0
b0
c0
d0
e0
f0
no. of data points
AAD %
Tc Tc Tc Pc Pc Pc Vc Vc Vc M M Tb Tb S S S ∆H
Tb M ν Tb M ν Tb M ν Tb ν M ν Tb M ν Tb
4.4876 × 105 1.3474 × 106 2.4522 × 103 8.4027 × 1023 2.025 × 1016 393.306 6.712 × 10-6 6.3429 × 10-8 2.01 × 10-3 8.9205 × 10-6 4 × 10-9 75.775 5.063 × 10-2 2.4381 × 107 1.1284 × 106 3.8083 × 107 39.741
-1.3171 × 10-3 2.001 × 10-4 -0.0291 -1.2067 × 10-2 -0.01415 0 -2.72 × 10-3 -2.0208 × 10-3 -0.16318 15.5833 × 10-6 -8.9854 × 10-2 0 -6.5236 × 10-2 -4.194 × 10-4 -1.588 × 10-3 -6.1406 × 10-2 0
-16.9097 -13.049 -1.2664 -74.5612 -48.5809 0 0.91548 14.1853 36.09011 4.2376 38.106 0 14.9371 -23.5535 -20.594 -26.3935 0
4.5236 × 10-3 0.0 0.0 0.0342 0.0451 0 7.92 × 10-3 4.5318 × 10-3 0.4608 0 0 0 6.029 × 10-2 3.9874 × 10-3 7.344 × 10-3 0.2533 0
0.6154 0.2383 0.1884 -1.0303 -0.8097 -0.4974 0.5775 0.2556 0.1417 2.0935 0.6675 0.4748 0.3228 -0.3418 -7.71 × 10-2 -0.02353 1.13529
4.3469 4.0642 0.7492 18.4330 12.9148 2.052 -2.1548 -4.60413 -10.65067 -1.9985 -10.6 0.4283 -3.8798 6.9195 6.3028 8.04224 0.02414
143 143 45 143 143 45 111 134 45 147 45 147 45 147 147 45 113
0.6 0.8 1.0 2.6 2.3 4.2 1.8 2.1 1.7 2.3 3.5 1.1 1.6 0.5 0.5 0.5 1.6
a
Data source: API Technical Data Book.1
As is clear from eqs 1 and 2, the polarizability, R, is directly related to the molar refraction, Rm. Although both the density (d) and the refractive index (n) are functions of temperature, the molar refraction (Rm) is nearly independent of the temperature for a substance. V is the apparent molar volume of the fluid, but Rm is the actual molar volume of molecules. Therefore, according to eq 2, parameter I represents the fraction of fluid occupied by the molecules. As expected, the value of the refractive index for a gas is less than that for a liquid and parameter I for gases is less than its value for liquids. Because n for gases is close to unity, parameter I for gases is close to zero, while for liquids, I is greater than zero but less than unity. Values of n and d at a reference temperature of 20 °C for many pure hydrocarbons are given in the API Technical Data Book1 and by Reid et al.2 As discussed by Hirschfelder et al.,7 two-parameter potential energy relations such as the Lennard-Jones potential describe the intermolecular forces for nonpolar compounds such as hydrocarbons. From statistical mechanics, one can show that for such compounds there must be a universal EOS in terms of the two parameters in the potential energy function. One of these parameters represents the molecular size, and the other one represents the energy parameter. In fact, on this basis many thermophysical properties of hydrocarbons have been related to the boiling point (Tb) and the specific gravity (S), as shown by Riazi and Daubert.9-11 However, because Tb can represnt the energy parameter and I can represent the size parameter for a nonpolar compound, we may develop similar relations in terms of Tb and I for various thermodynamic properties as shown in the following sections. These relations have been developed based on the properties of pure hydrocarbons; however, they are also applicable to narrowcut petroleum fractions and products using average bulk properties for such mixtures. Estimation of the Basic Parameters. The basic parameters are those parameters which are independent of temperature or defined at a standard temperature of 20 °C and are used to estimate other thermodynamic or physical properties. These parameters are mainly critical constants [namely, critical temperature (Tc), critical pressure (Pc), and critical volume (Vc)], molecular weight (M), density at 20 °C (d20), and boiling point (Tb). Critical constants are important in using generalized correlations or EOSs to estimate the ther-
modynamic and physical properties of fluids. The molecular weight is needed to convert molar properties into mass-based specific properties. The density and boiling point are needed to estimate other properties.9,10 The refractive index parameter (I) is needed to estimate the refractive index, n, when it is not available. With an approach similar to that used in previous developments,9,10 the following relation can be used to estimate critical constants, molecular weight, density, and boiling point in terms of parameter I20, which is considered as a size parameter 0 θ ) a0 exp(b0θ1 + c0I20 + d0θ1I20)θe10If20
(5)
where θ is a property such as Tc (K), Pc (bar), Vc (m3/ kg), Tb (K), S, or ∆Hv. S is the specific gravity at 15.5 °C (60 °F), and ∆Hv is the heat of vaporization at the normal boiling point in kJ/kmol. θ1 is a characterizing parameter that is available such as the normal boiling point, Tb. In cases when Tb is not available, other parameters such as the molecular weight (M) or kinematic viscosity (ν, cSt) at 37.8 °C (100 °F) may also be used to estimate various basic properties. Constants a0, b0, ..., f0 for various properties are given in Table 1 for the carbon number range of C5-C20 which is nearly equivalent to molecular weights of 70-300. Data from the API Technical Data Book1 have been used in obtaining the constants. Using eq 5 and constants given in Table 1, critical constants, molecular weight, boiling point, or specific gravity may be estimated from the refractive index at 20 °C and another available parameter. Average absolute deviations (AADs) for various parameters for pure hydrocarbons are also given in Table 1. AADs are generally within 1-2% of their reported values. Constants given in Table 1 for the estimation of Tb from M and I were obtained as independent of the constants for the estimation of M from Tb and I. However, both sets of constants give similar results when used in eq 5. Equation 5 with Table 1 can be applied to narrowboiling-range undefined petroleum fractions using the average properties of the mixture. For petroleum fractions, Tb in eq 5 is the mean-average boiling point; however, for relatively narrow-boiling-range petroleum mixtures, it is very close to the 50% ASTM-D86 temperature. For most samples evaluated in this work, the boiling temperature at 50% volume vaporized was used for the Tb of the mixture. Results for the application of
Ind. Eng. Chem. Res., Vol. 40, No. 8, 2001 1977 Table 2. Evaluation of the Proposed Modified RK-EOS for the Liquid Density of Heavy Hydrocarbonsa AAD% compound n-heptane (n-C7) n-nonane (n-C9) n-undecane (n-C11) n-tridecane (n-C13) n-heptadecane (n-C17) n-eicosane (n-C20) n-triacontane (n-C30) n-tetracontane (n-C40) total a
Tc, K
P c, bar
ω
Rm (20 °C), cm3/g‚mol
r
540.3 594.6 638.8 676.0 733.0 767.0 842.0 887.0
27.4 22.9 19.7 17.2 13.0 11.1 6.7 4.4
0.349 0.445 0.535 0.619 0.770 0.907 1.210 1.500
34.554 43.840 53.135 62.427 80.948 95.418 141.298 187.690
4.945 6.274 7.605 8.935 11.585 13.656 20.223 26.862
no. of data points
proposed EOS
RK
SRK
PR
35 35 35 30 30 20 20 20 225
0.6 0.6 1.7 2.8 1.2 2.8 0.6 4.1 1.6
12.1 15.5 18.0 20.3 27.3 29.5 41.4 50.9 24.3
10.5 13.4 15.5 17.7 24.8 26.7 39.4 49.4 22.1
1.4 3.4 5.4 7.9 16.0 18.2 32.5 44.4 13.3
Temperature range: 303-573 K. Pressure range for all compounds: 50-500 bar. Data source: Doolittle.21
eq 5 to some narrow-boiling-range undefined petroleum fractions are given in the Application to Undefined Petroleum Fractions section. Methods of estimation of properties of wide-boiling-range fractions have been discussed by Riazi and Daubert12 and for crude oils are given by Riazi.13,14 Refractive Index Based EOS. EOSs are powerful tools in the estimation of PVT and thermodynamic properties of fluids. One of the main difficulties with the use of existing cubic EOSs for property and phase behavior calculations is their low accuracy for calculation of the density of liquids. For example, the RedlichKwong EOS15 (RK-EOS), a simple and widely used EOS, is relatively accurate for the gas phase, but it is not accurate enough for liquid systems or the saturated region.16 The RK-EOS is given in the following form:
P)
RT a V - b T1/2V(V + b) 2
2.5
(6)
1/β ) 1 + {0.02[1 - 0.92 exp(-1000|Tr - 1|)] 0.035(Tr - 1)}(r - 1) (10) in which Tr is the reduced temperature and r is defined as
r ) Rm/Rm,ref
(7)
bRK ) 0.08664RTc/Pc
(8)
in which V is the molar volume. A modified RK-EOS proposed by Soave, named SRK, and another cubic EOS by Peng and Robinson (PR-EOS) are typically used for hydrocarbons.17,18 The SRK and PR equations, however, break down for C10 and heavier hydrocarbons when used for liquid density calculations. Parameter b in an EOS represents the actual volume of molecules, and it plays an important role in the calculation of liquid densities.19 Both SRK and PR equations use a third parameter, namely, acentric factor, to improve phase behavior calculations. However, their modifications do not significantly improve the liquid density calculation especially for heavy hydrocarbons. Peneloux et al.20 suggested a fourth parameter in terms of the Rackett parameter to improve the accuracy of the SRK equation for liquid volumes. However, the fourth parameter is not available for heavy compounds. Riazi et al.19 have shown that parameter b in RKEOS can be related to molar refraction. In fact, calculations show that RK-EOS is more accurate than both PR and SRK equations for simple fluids such as methane, and for this reason, we choose methane as the reference compound for modification of RK-EOS. Parameter b in eq 8 may be modified in the following form:19
(9)
where parameter β for the reference compound of
(11)
Rm is independent of the temperature and can be estimated from the liquid density at 20 °C (d20) and the refractive index parameter at 20 °C (I20):
Rm ) MI20/d20
a ) 0.42748R Tc /Pc
b ) βbRK
methane is unity. For other fluids, β is proportional to the molar refraction (Rm) defined in eq 2. On the basis of the PVT data for hydrocarbons from C2 and C8 for pressures up to 700 bar, the following relation was developed to estimate parameter β:
(12)
Rm,ref is the value of Rm for the reference fluid of methane which is 6.987. Equation 10 was developed based on PVT data for compounds from C2 to C8, but because of the linear relationship between 1/β and r, it can be easily applied to heavier hydrocarbons (with higher r values).19 Values of Tc, Pc, and r for some hydrocarbons from C7 to C40 are given in Table 2. Values of Tc, Pc, d20, and I20 for heavy compounds were calculated from equations provided for heavy hydrocarbons.4 A summary of results for the estimation of liquid densities from proposed equations (eqs 6-12) is also given in Table 2. Experimental data on liquid densities were taken from Doolittle.21 These data were not used in the development of eq 10. As shown in Table 2, the proposed modified RK-EOS (eqs 6-12) is significantly more accurate than SRK or PR equations for calculation of liquid densities of heavy hydrocarbons. In addition in this equation, the refractive index is used instead of the acentric factor. When the proposed modified RKEOS is used to calculate densities of both liquid and gases for hydrocarbons from C1 to C40 and pressures from 1 to 700 bar, it gives an average deviation of 1.3% for 1750 data points. For the same database, SRK and PR equations give deviations of 7.4% and 4.6%, respectively. It was noticed that the numerical value of 0.02 in eq 10 is not the best value for all compounds. For heavier compounds, values lower than 0.02 increase the accuracy of the proposed method. For example, for n-C17 and n-C20 compounds, if a value of 0.018 is used instead of 0.02, errors in density calculations for these compounds reduce from 2% to 0.5%. This indicates that if the numerical value of 0.02 in eq 10 is replaced with a
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Ind. Eng. Chem. Res., Vol. 40, No. 8, 2001
Table 3. Prediction of the Critical Compressibility Factor Using Different Cubic EOSsa no.
compound 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
methane ethane propane isobutane n-butane n-pentane n-hexane cyclohexane benzene toluene n-heptane n-octane n-nonane n-decane n-undecane n-dodecane n-tridecane n-tetradecane n-pentadecane n-hexadecane n-heptadecane
parameter r 1.000 1.620 2.259 2.955 2.929 3.616 4.281 3.966 3.748 4.45 4.945 5.608 6.274 6.941 7.605 8.273 8.942 9.597 10.263 10.933 11.593 overall AAD %
exp Zc
calcd Zc, proposed method
0.288 0.284 0.280 0.282 0.274 0.269 0.264 0.273 0.271 0.264 0.263 0.259 0.255 0.249 0.243 0.238 0.236 0.234 0.228 0.225 0.217
0.333 0.3 0.282 0.280 0.278 0.271 0.266 0.269 0.270 0.265 0.262 0.258 0.254 0.250 0.247 0.245 0.242 0.240 0.238 0.235 0.233
% abs dev proposed method PR 15.6 5.6 0.7 0.7 1.5 0.7 0.7 1.5 0.4 0.4 0.4 0.4 0.4 0.4 1.6 2.9 2.6 2.5 4.3 4.2 7.4 2.6
6.6 8.1 9.6 8.9 12.0 14.1 16.3 12.5 13.3 16.3 16.7 17.0 18.8 23.3 26.3 29.0 30.1 31.2 34.6 36.4 41.5 20.1
SRK 15.6 17.3 18.9 18.1 21.5 23.8 26.1 22.0 22.9 26.1 26.6 28.6 30.6 33.7 37.0 39.9 41.1 42.3 46.1 48.0 53.5 30.5
a Data for Z are taken from API Technical Data Book.1 Calculated values of Z from SRK and PR EOSs for all compounds are 0.333 c c and 0.307, respectively.
r-dependent term, it may further improve the accuracy of the proposed equation especially for heavy compounds. The proposed method also improves the estimation of the critical density or the critical compressibility factor (Zc). For example, for 20 hydrocarbons from C2 to C17, PR and SRK equations estimate critical compressibilities with AADs of 18% and 28%, respectively. The proposed modified RK-EOS estimates the critical compressibility of these compounds with a AAD of 2.3%. Details of these evaluations with values of parameter r are given in Table 3. It should be noted that when eq 10 is applied at the critical point (Tr ) 1), parameter β becomes only a function of parameter r. This means that Zc is only a function of r and, therefore, eq 6 generates different values for Zc of different compounds, while PR and SRK equations produce constant values of 0.307 and 0.333 for Zc of all compounds. However, for methane the proposed method (eqs 6-12) gives the same Zc as RK- or SRK-EOSs, because it was chosen as the reference compound with r ) 1. Equations 6-12 estimate the gas density, liquid density, and critical density with good accuracy especially for heavy hydrocarbons, but these equations are not suitable for non-hydrocarbons and they cannot be used for fugacity calculations. Therefore, the proposed modified RK equation is unsuitable for phase equilibrium calculations. However, because in fugacity calculations the densities of both liquid and vapor phases are required, the proposed equation indirectly improves the phase behavior and vapor-liquid equilibria (VLE) calculations of hydrocarbon fluids. Estimation of the Transport Properties and Heat Capacity. Viscosity (µ), thermal conductivity (k), and diffusion coefficient (D) known as transport properties are important in the calculations related to fluid flow and heat and mass transfer. The kinetic theory of gases leads to some theoretically based relations for the estimation of these properties.7,22 However, for liquid systems, predictive methods are usually based on some empirical relations and they are less accurate.2
Hildebrand23 developed a relation for the viscosity of liquids and suggested that the fluidity (1/µ) of liquids varies linearly with the molar volume:
(
)
V - V0 1 ) B0 µ V0
(13)
where B0 is a constant, V is the molar volume, and V0 is the molar volume at zero fluidity. The Hildebrand relation is, in fact, consistent with the structure theory of liquids.24 V0 in eq 13 approximately represents the volume of molecules (the molar volume at zero fluidity); therefore, it is expected that the ratio of V0/V representing the fractional volume occupied by molecules would be similar in nature to parameter I, and they should be proportional to each other. Equation 13 in terms of parameter I becomes
1 1 ) C0 - 1 µ I
(
)
(14)
This equation represents the relationship between the viscosity and the refractive index at the same temperature. The linear relationship between 1/µ and 1/I is confirmed in Figure 1 based on experimental data for n-pentane and cyclopentane compounds. Modified versions of the Hildebrand equation were also proposed for thermal conductivity and diffusivity by other investigators.25,26 The inter-relationship between viscosity, thermal conductivity, and diffusivity was developed in previous publications.27,28 A generalization can be made for all three transport properties in the form29,37
θtr ) A
(1I - 1) + B
(15)
where θtr is a transport property such as 1/µ, 1/k, or D at the same temperature as I. According to the above relation, as parameter I increases, the viscosity and thermal conductivity also increase while the diffusion coefficient decreases. Constants A and B for various
Ind. Eng. Chem. Res., Vol. 40, No. 8, 2001 1979
Figure 1. Relationship between the viscosity and the refractive index.
hydrocarbons are given by Riazi et al.29 Equation 15 gives an AAD of less than 1% for some 60 compounds for viscosity, thermal conductivity, and diffusivity of liquid hydrocarbons as well as some non-hydrocarbons. It is also shown that eq 15 can be further improved by adding another parameter to consider the effects of polarity, and it can be applied to non-hydrocarbon systems as well.29 Riazi30 has shown that the heat capacity (Cp) of liquid hydrocarbons is related to the molar volume at the same temperature. In fact, it is appropriate to say that the heat capacity can be related to the free space between the molecules similar to thermal conductivity. As the free space increases, the heat capacity decreases. This can be formulated through parameter I in a fashion similar to that for thermal conductivity. On this basis, the following equation was developed to estimate the heat capacity of hydrocarbons in terms of the refractive index for both liquid and solid phases.
(1 -I I) + B
Cp ) A1
(16)
1
Constants A1 and B1 have been determined for pure hydrocarbons from various groups, and they are independent of temperature. Furthermore, these constants have been related to the molecular weight for each homologous hydrocarbon group in a linear form, and eq 16 in terms of the molecular weight becomes
Cp ) (a1M + b1)
(1 -I I) + c M + d 1
1
(17)
Constants a1, b1, c1, and d1 for n-alkanes, 1-alkenes, n-alkylcyclopentanes, n-alkylcyclohexanes, and n-alkylbenzenes are given in Table 4. Data on the heat capacity of solids for n-alkanes were also used to evaluate eq 17. In using eq 17 for solids, because of the lack of experimental data, values of refractive indices of solid alkanes at temperatures below the freezing point were obtained through linear extrapolation of values of refractive indices of liquids at temperatures above the freezing point. This approach does not give true values of refractive indices of solids, but it works for use of eq 17 for solids in a fashion similar to that for the liquids. Data for the heat capacities were obtained from the
Figure 2. Relationship between the heat capacity and the refractive index.
DIPPR database,31 and the results of evaluations are also given in Table 4. The average error is about 2% for a large number of data points. A comparison between eq 17 and the experimental data is also shown in Figure 2 for n-eicosane. Estimation of the Composition of Petroleum Fractions. Petroleum fractions and crude oils are mixtures of many hydrocarbons of different types. Hydrocarbon groups generally found in most petroleum fractions are from paraffinic, naphthenic, and aromatic groups. The quality and characteristics of petroleum products mainly depend on the types of compounds in the mixture. Knowledge of the composition of these mixtures as well as the amount of sulfur in the mixture is extremely important in the evaluation of the quality of a petroleum product. In addition, such information may be used for a more accurate estimation of mixture physical properties.4,5 Determination of the exact composition of a petroleum mixture requires analytical tools such as GC-MS, elemental analyzers, etc., and such measurements on every petroleum fraction are practically impossible. Riazi and Daubert32,33 have developed several predictive methods for the composition of petroleum products. It is quite important for process engineers to be able to estimate the composition of a petroleum fraction from its bulk properties easily measurable in a laboratory. Parameters useful for the characterization of types of hydrocarbons must have the ability to clearly identify molecular types. In addition, they should be easily measurable, or one has to be able to estimate them accurately. For example, the refractivity intercept RI and parameter m have been shown to be capable of such characterization. These parameters are both defined in terms of the sodium D line refractive index at 20 °C, n20.
RI ) n20 - d20/2
(18)
m ) M(n20 - 1.475)
(19)
where M is the molecular weight and d20 is the liquid density at 20 °C and 1 atm. The beauty of these parameters is that they give a specific value for each hydrocarbon group. Variations in the values of RI and
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Ind. Eng. Chem. Res., Vol. 40, No. 8, 2001
Table 4. Constants for the Estimation of the Heat Capacity from the Refractive Index (Eq 17): Cp/R ) (a1M + b1)(I/1 - I) + c1M + d1a group
state
carbon range
temp range, °C
a1
b1
c1
d1
no. of data points
AAD %
MAD %
n-alkanes 1-alkenes n-alkylcyclopentane n-alkylcyclohexane n-alkylbenzene n-alkanes
liquid liquid liquid liquid liquid solid
C5-C20 C5-C20 C5-C20 C6-C20 C6-C20 C5-C20
-15 to +344 -60 to +330 -75 to +340 -100 to +290 -250 to +354 -180 to +3
-0.9861 -1.533 -1.815 -2.725 -1.149 -1.288
-43.692 40.357 56.671 165.644 4.357 -66.33
0.6509 0.836 0.941 1.270 0.692 0.704
5.457 -21.683 -28.884 -68.186 -3.065 14.678
225 210 225 225 225 195
0.89 1.5 1.05 1.93 1.06 2.3
1.36 5.93 2.7 2.3 4.71 5.84
a
AAD %: average absolute deviation percent. MAD %: maximum absolute deviation percent. Data are taken from DIPPR.31
Table 5. Variation of Parameters RI and m for Different Hydrocarbon Groups hydrocarbon type
variation of RI
variation of m
paraffins naphthenes aromatics
1.048-1.05 1.03-1.046 1.07-1.105
-9 to -8 -5 to -4 2 to 44
XS ) 177.448 - 170.946RI + 0.2258m + 4.054S (26)
m for paraffins, naphthenes, and aromatics are given in Table 5. As is clear from these values, RI and m very clearly characterize hydrocarbon types. For example, a hydrocarbon whose RI value is 1.08 has to be aromatic; it cannot be paraffinic or naphthenic. Similarly, paraffins have a negative m value. Parameter m can even characterize various aromatic groups. For example, benzenes (monoaromatics) have m values of 2-3, while polycyclic aromatics have m values of about 40-45. So, if m for a hydrocarbon is about 40, it is for sure a polyaromatic hydrocarbon. Using the average values of RI and m for each hydrocarbon group, the following relations have been developed for the composition of petroleum fractions with molecular weights of less than 250:
XP ) 325.74 - 348.148S + 1.1666m
(20)
XN ) -195.71 + 263.853S - 3.992m
(21)
XA ) 100 - (XP + XN)
(22)
In these relations S is the specific gravity at 15.5 °C and parameter m is defined in eq 19. XP, XN, and XA are volume percents of paraffins, naphthenes, and aromatics, respectively. Equations 20-22 give average errors of about 5% for about 100 fractions. If the calculated value for any of XP, XN, or XA is negative, then it should be set equal to zero and other values should be adjusted accordingly. Relations to estimate the composition of heavier petroleum fractions in terms of the refractive index are given in previous publications.33 Because parameter m also characterizes types of aromatics, it can be used to determine various types of aromatics especially for fractions with high aromatic contents. For fractions with molecular weights of less than 250, the following relations can be used to estimate the monoaromatics (XMA) and polyaromatics (XPA) in terms of RI and m.
XMA ) -6282.45 + 5990.816RI - 2.4833m
(23)
XPA ) 1188.175 - 1122.13RI + 2.3745m (24) XA ) XMA + XPA
molecular weights of less than 250, the sulfur weight percent (XS) is given by the following relation:
(25)
Average deviations for these relations are about 5-6%.33 Most recently, Riazi et al.34 have developed relations for the estimation of the sulfur contents of petroleum fractions and crude oils. For fractions with
The accuracy of this relation in the estimation of the sulfur content of petroleum fractions is about 0.1% for 76 fractions. Methods of estimation of the sulfur contents of heavy fractions and crude oils have been discussed in detail in our previous publication.34 If the calculated X value from any of the above relations (eqs 20-26) becomes negative, then the value of X should be set equal to zero. In these relations the main input parameters are the refractive index at 20 °C (n20) and the liquid density at 20 °C (d20) in g/cm3. In cases when d20 is not available, it may be estimated from the following relations:
for M e 300
d20 ) 0.9837Tb0.002S1.005
(27)
for M > 300
d20 ) 2.8309M0.04I201.1354
(28)
In these relations Tb is in degrees Kelvin and M is the molecular weight. The accuracy of these relations is about 0.3% for pure hydrocarbons. Estimation of the refractive index, n20, is discussed in the following section. Estimation of the Refractive Index. In this paper we have demonstrated that the thermodynamic and physical properties of petroleum fractions and hydrocarbons can be estimated through the relations in which the refractive index is one of the input parameters. The refractive index can be experimentally determined with good accuracy; however, in cases when it is not available, it can be estimated through other parameters such as the boiling point (Tb), specific gravity (S), molecular weight (M), or kinematic viscosity at 37.8 °C (ν). For pure hydrocarbons and narrow-boiling-range petroleum fractions, the following relations for prediction of parameter I20 are recommended for light and heavy fractions according to the availability of input parameters:
for M e 300
I20 ) 0.336M-0.006S0.894
(29)
I20 ) 2.3435 × 10-2 exp(7.029 × 10-4Tb + 2.468S 10.267 × 10-4TbS)Tb0.0572S-0.72 (30) I20 ) 0.328ν-0.003S0.915
(31)
where Tb is the normal boiling point in degrees Kelvin.
Ind. Eng. Chem. Res., Vol. 40, No. 8, 2001 1981
for M > 300 I20 ) 0.01102 exp(-8.61126 × 10-4M + 3.22861S + 9.07171 × 10-4MS)M0.02426S-2.25051 (32) -2
I20 ) 1.843 × 10
To calculate the refractive index of mixtures of known composition, one of the following two mixing rules may be applied:
Imix )
-3
exp(1.1635 × 10 Tb + 5.144S -
5.92 × 10-4TbS)Tb-0.4077S-3.333 (33) The above equations predict the refractive index with an AAD of less than 0.2% for pure compounds. Relations in terms of molecular weight M (eqs 29 and 32) are recommendeded only when Tb is not available. For light fractions (C5-C20) in a case when neither Tb nor M is available, eq 31 in terms of the kinematic viscosity at 37.8 °C (100 °F) may be used. For pure homologous hydrocarbons and single carbon number fractions, the following relation is recommended in terms of M:
I20 ) I∞ - exp(a2 - b2Mc2)
(34)
where constants I∞, a2, b2, and c2 are given in Table 6. This equation is slightly modified for the n-alkylbenzene group as indicated in Table 6. Because for aromatics the refractive index decreases with an increase in the molecular weight, in the right-hand side of eq 34 the two terms are added together. In the above equation, I∞ is the value of parameter I for compounds of extremely high molecular weights (M f ∞). Relations given by eqs 29-34 can estimate I20; then the refractive index, n20, can be estimated from definition of parameter I (eq 4).
n20 )
(
)
1 + 2I20 1 - I20
1/2
(35)
Once the refractive index at 20 °C (n20) is known, the refractive index at any other temperature (n) can be determined by two methods. One approach is based on the assumption that the value of the molar refraction (Rm) for each compound is independent of the temperature. Therefore, once Rm is determined at 20 °C from eq 12, then eq 2 may be used to estimate I at any temperature using the value of Rm at 20 °C and the density (d) at the temperature of interest. The density at any temperature may be estimated through the Rackett equation2 based on the density at 20 °C. However, this method is somewhat inconvenient because of density calculations. Another approach is to use the linear relation between the refractive index and the temperature as follows:29
n ) n20 - 0.0004(T - 293)
(36)
where n20 is the sodium D line refractive index of liquid at 293 °K and 1 atm. T is the temperature in degress Kelvin, and n is the refractive index at temperature T. Both approaches predict values of the refractive index at various temperatures with the same accuracy. Equation 36 can predict the refractive index of pure compounds with an average error of about 0.3%. Equation 36 is the basis for the calculation of parameter I through eq 4 at various temperatures, which is needed to be used in eqs 15-17. The only data required for any compound in eq 36 is the refractive index at 20 °C (n20), which can be calculated through eqs 29-35.
1 Imix
)
∑i xviIi xwi
∑i I
(37)
(38)
i
where xvi is the volume fraction of component i in the mixture and Ii is the corresponding refractive index parameter for component i. In eq 38, xwi is the weight fraction of component i and Imix is the value of I for the mixture. Both eqs 37 and 38 have nearly the same accuracy. Application of these equations in the calculation of the refractive index of wide-boiling-range petroleum fractions and crude oils has been shown in previous publications.12-14 Application to Undefined Petroleum Fractions. In this part we demonstrate the application of proposed methods to estimate some physical properties and the quality of some narrow-boiling-range petroleum fractions. Experimental data on the molecular weight (M), ASTM-D86 temperature at 50% volume vaporized (Tb), specific gravity (S), refractive index (n20), liquid density at 20 °C (d20), and sulfur weight percent (S %) of some petroleum fractions34,35 are given in Table 7. Equation 5 in terms of Tb and I20 was used to estimate the specific gravity and molecular weight of the fractions. Then eq 27 was used to estimate the density at 20 °C. Using estimated values of M, n20, and d20, parameters RI and m were calculated from eqs 18 and 19. Finally the sulfur content of these fractions was estimated from eq 26, using parameters RI and m and estimated values of S. The results of calculations are also given in Table 7 for these samples. It should be noted that in using eq 26 for the estimation of the sulfur contents, estimated values of M, S, and d20 from eq 5 with Tb and I20 as the only available parameters have been used. If, instead of estimated parameters, available experimental parameters for M, S, n20, and d20 were used, more accurate results for the sulfur contents could be obtained. Another set of data on the composition of some 12 petroleum products from crude oils around the world36 were used to show the application of the proposed methods for estimation of the composition of undefined petroleum fractions. Experimental data on Tb, S, and the volume percents of paraffins (XP), naphthenes, (XN), and aromatics (XA) of these fractions as well as calculated parameters for M, m, and the composition are given in Table 8. The calculation procedure for the input parameters is similar to that of Table 7. In these calculations, parameter I20 was calculated from eq 30 from Tb and S, and then M was calculated from eq 5 using Tb and estimated I20. Parameter n20 was calculated from eq 35. As shown in Table 8, the estimated composition for most fractions is close to the experimental composition. Again, if experimental values of M and n20 were used instead of estimated values, better results can be obtained. It should be noted that data for the fractions presented in Tables 7 and 8 were not used in the development of the proposed equations. These are simply random fractions used to demonstrate application of the proposed methods for undefined
1982
Ind. Eng. Chem. Res., Vol. 40, No. 8, 2001
Table 6. Constants in Equation 34 for the Estimation of the Refractive Index hydrocarbon type
carbon no. range
I∞
a2
b2
c2
AAD %
n-alkanes n-alkylcyclopentanes n-alkylcyclohexanes n-alkylbenzenea single carbon number
C5-C40 C5-C41 C6-C20 C6-C42 C6-C50
0.2833 0.283 0.277 0.2829 0.34
87.6593 87.552 38 -2.455 12 137.0918 2.308 84
86.621 67 86.975 56 0.056 36 135.433 2.965 08
0.01 0.01 0.7 0.01 0.1
0.002 0.003 0.06 0.008 0.1
a
For n-alkylbenzene group, eq 34 should be revised as I20 ) I∞ + exp(a2 - b2Mc2).
Table 7. Estimation of Basic Parameters from the Refractive Index for Some Undefined Petroleum Fractions S
d20
fraction
exp Tb, K
exp n20
exp
calc
Kuwaiti kerosene Kuwaiti diesel oil U.S. jet naphtha U.S. high boiling naphtha U.S. kerosene U.S. fuel oil
468 583 434 435 480 559
1.441 1.480 1.444 1.426 1.444 1.478
0.791 0.860 0.805 0.762 0.808 0.862
0.795 0.860 0.804 0.769 0.800 0.856
Mexp
Mcalc
144 142.4 162.3 227.5
153.3 228.9 130 135.0 160.7 210.2
sulfur %
exp
calc
calc m
calc RI
exp
calc
0.801 0.759 0.804 0.858
0.792 0.856 0.800 0.765 0.796 0.852
-5.214 1.052 -4.030 -6.615 -4.982 0.631
1.045 1.144 1.044 1.0435 1.046 1.052
0.01 1.3 1.0 0.0 0.0 1.3
0.7 1.4 1.3 0.6 0.6 1.2
a Experimental data for the first two Kuwaiti fractions are taken from Riazi et al.,34 and those for the U.S. fractions are taken from Lenoir and Hipkin data set.35
Table 8. Estimation of the Composition of Petroleum Fractions from Proposed Correlationsa exp composition
estimated composition
no.
fraction
exp Tb, K
exp S
n20
M
m
XP
XN
XA
XP
XN
XA
1 2 3 4 5 6 7 8 9 10 11 12
China heavy naphtha Malaysia light naphtha Indonesia heavy naphtha Venezuela kerosene heavy Iranian gasoline Qatar gasoline Sharjah gasoline American gasoline Libya kerosene U.K. North Sea kerosene U.K. North Sea gas oil Mexico naphtha
444.1 326.6 405.2 463.6 323.5 309.4 337.7 317.2 465.5 464.9 574.7 324.7
0.791 0.666 0.738 0.806 0.647 0.649 0.693 0.653 0.794 0.798 0.855 0.677
1.441 1.373 1.412 1.449 1.364 1.364 1.386 1.366 1.442 1.444 1.475 1.377
137.5 83.1 119.8 148.2 83.2 75.8 86.3 79.3 151.1 150.3 223.6 81.1
-4.767 -8.495 -7.608 -3.832 -9.204 -8.411 -7.713 -8.609 -4.920 -4.596 -0.029 -7.915
48.9 83.0 62.0 39.8 93.5 95.0 78.4 92.0 51.2 42.5 34.3 81.9
30.9 17.0 30.0 41.1 5.7 3.9 14.4 7.3 34.7 36.4 39.8 13.9
20.2 0.0 8.0 19.0 0.8 1.1 7.2 0.7 14.1 21.1 25.9 4.2
44.8 84.0 59.9 40.7 89.8 90.0 75.6 88.3 43.6 42.7 28.0 80.9
31.9 13.9 29.4 32.3 10.2 9.1 17.8 11.0 33.4 33.1 30.0 14.4
23.3 2.1 10.7 27.1 0.0 0.9 6.6 0.7 23.0 24.2 42.0 4.7
a Experimental data on T , S, and the composition are taken from the Oil & Gas Journal Data Book.36 Values of n b 20 are calculated from eqs 30 and 35 using Tb and S. Values of M are calculated from eq 5 using Tb and I20 as input parameters. Values of XP, XN, and XA are calculated from eqs 20-22.
petroleum fractions. The calculation procedure for one of the samples in Table 8 is shown in the following example. Example 1. A heavy Iranian gasoline36 has an average boiling point of 50.5 °C and a density at 15.5 °C of 0.647 g/cm3 (sample 5 in Table 8). Determine the composition of this fraction. Solution. Available input parameters are Tb ) 323.7 K and S ) 0.647. Assuming M e 300, from eq 30, I20 ) 0.223, and from eq 35, n20 ) 1.364. Using eq 5 with constants given in Table 1 for estimating M from Tb and I20, the molecular weight can be estimated as M ) 83.2. So, the assumption of M e 300 for the use of eq 30 was correct. Using estimated values of M and n20, parameter m can be calculated from eq 19; m ) -9.204. Because M e 250, eqs 20-22 can be used to estimate the composition. Substituting values of m and S in eqs 20 and 21 gives XP ) 89.7% and XN ) 11.9%, and from eq 22, one obtains XA ) -1.6%. Because XA e 0, it is set equal to zero (XA ) 0), and XN should be adjusted in a way that XP + XN + XA ) 100. This means XN ) 100 - XP or XN ) 10.2%. Therefore, the estimated composition for this fraction is 89.8 vol % paraffins, 10.2 vol % naphthenes, and 0% aromatics. Experimental compositions as reported in the Oil & Gas Journal Data Book36 are 93.5% for paraffins, 5.7% for naphthenes, and 0.8% for aromatics. The difference between the estimated and actual compositions is within experimental uncertainty.
The aromatic content of the fraction may also be calculated from eqs 23-25 using parameters RI and m. From eq 27, the density is calculated as d20 ) 0.642 g/cm3. Using values of n20 and d20 for this fraction, parameter RI is calculated from eq 18; RI ) 1.043. Substituting the values of RI and m into eqs 23 and 24 results in negative values for both XMA and XPA, and hence both are set equal to zero. Therefore, from eq 25, XA ) 0 + 0 ) 0, which is consistent with the results obtained from eqs 20-22. Also, substituting values of RI, m, and S into eq 26 gives XS ) -0.3%, which must be set as zero (XS ) 0) and is the same as the experimental value. Example 2. A petroleum fraction has an average boiling point of 401 °C and a specific gravity of 0.8686. Estimate the liquid density of this hydrocarbon mixture at 373.3 bar and 20 °C. Solution. Available parameters are Tb ) 674 K and S ) 0.8686. Assuming M e 300, from eq 30, I20 ) 0.283, and from eq 35, n20 ) 1.477. Using eq 5 with constants given in Table 1 for estimating Tc and M from Tb and I20, the critical temperature (Tc) and molecular weight (M) can be estimated as Tc ) 832.9 K and M ) 311.8. The calculated value of M is not too far from the limit of 300, and the application of eq 30 is acceptable. Equation 5 with parameters in Table 1 for the estimation of the critical pressure (Pc) from M and I20 can be used to get Pc ) 11.2 bar. Substituting values of Tb and
Ind. Eng. Chem. Res., Vol. 40, No. 8, 2001 1983
S into eq 27 results in d20 ) 0.866 g/cm3. Using estimated values of M, I20, and d20 in eq 12 yields Rm ) 101.9 cm3/mol. Parameter r is calculated from eq 11: r ) 101.9/6.987 ) 14.58. At 20 °C the reduced temperature is Tr ) 0.352. Substituting values of Tr and r in eq 10 gives β ) 0.637. Using values of Tc and Pc in eqs 7 and 8, the RK-EOS parameters are calculated as a ) 528.2 Pa (m3/mol)2 and bRK ) 5.36 × 10-4 m3/mol. Parameter b for eq 6 is calculated from eq 9 using the correction factor of β: b ) 0.637 × 5.36 × 10-4 ) 3.48 × 10-4 m3/mol. Substituting values of a and b in the RK-EOS (eq 6) and solving the cubic equation for the molar volume results in V ) 3.54 × 10-4 m3/mol. The absolute density is then d ) M/V ) 311.8/354 ) 0.881 g/cm3. The vigorous procedure designed for the estimation of densities of liquid petroleum fractions at high pressures (procedure 6A3.10 in the API Technical Data Book1) gives a density value of 0.883 g/cm3. To summarize, in this work it is shown that the refractive index can be used to accurately estimate basic thermodynamic and physical properties of hydrocarbons and their mixtures as well as the quality and composition of undefined petroleum fractions. The above few examples show how one can use laboratory data combined with the proposed methods in this paper to estimate basic thermophysical properties of undefined petroleum fractions. Initial investigations show that the refractive index is also capable of characterizing polar and non-hydrogen compounds as well.29 However, to show the application of the refractive index to such systems, further studies are required. Polar compounds cannot be characterized by only two parameters, and they need a third parameter to define the intermolecular forces.6 However, one of these parameters can be refractive index parameter I or molar refraction Rm. Therefore, it seems that the refractive index is also a suitable parameter for the estimation of properties of nonhydrocarbon systems. This could be a subject of future studies. Conclusions In this paper use of the refractive index in the estimation of thermodynamic and physical properties of pure hydrocarbons and petroleum fluids is demonstrated. The refractive index is also used to estimate EOS parameters which result in greater accuracy in density calculations especially for heavier compounds. The application of the refractive index in the prediction of the composition and sulfur content of undefined petroleum products is demonstrated. Finally, methods of the estimation of the refractive index for petroleum mixtures and pure compounds are presented. Because the refractive index is easily measurable, it is recommended that petroleum-related laboratories report this property as one of the basic characteristic parameters in order to determine the quality of a petroleum product. In this paper we have shown that the properties calculated through the refractive index have a good accuracy and that they are reliable used for the design and operation of petroleum-related equipment. Acknowledgment A part of this paper was presented at the 1999 AIChE Spring Meeting, Houston, TX (Mar 14-18, 1999). It was also presented at the Department of Physical Chemsitry of Institut Francais du Petrole (IFP, Paris) on Nov 8,
1999, as an invited lecture. Part of the calculations were made by Sami Alkandari. The funding for this research was provided by the Research Administration (RA) of Kuwait University under Grant EC-092, which is appreciated. Nomenclature A ) constant in eq 15 A1 ) constant in eq 16 a ) constant in eq 6 a0 ) constant in eq 5 a1 ) constant in eq 17 a2 ) constant in eq 34 B ) constant in eq 15 B1 ) constant in eq 16 B0 ) constant in eq 13 b ) constant in eq 6 b0 ) constant in eq 5 b1 ) constant in eq 17 b2 ) constant in eq 34 bRK ) constant in eq 8 c0 ) constant in eq 5 c1 ) constant in eq 17 c2 ) constant in eq 34 Cp ) heat capacity in eqs 16 and 17, kJ/kg‚K C0 ) constant in eq 14 D ) diffusion coefficient (self or at infinite dilution), 109 m2/s d ) liquid density at temperature T and 1 atm, kg/dL (g/ cm3) d0 ) constant in eq 5 d1 ) constant in eq 17 and Table 3 d20 ) liquid density at temperature 20 °C and 1 atm, kg/ dL (g/cm3) e0 ) constant in eq 5 f0 ) constant in eq 5 I ) refractive index parameter defined in eq 4 Ii ) refractive index parameter for component i I20 ) refractive index parameter at 20 °C I∞ ) refractive index parameter at M f ∞ k ) thermal conductivity, W/mK M ) molecular weight m ) parameter defined in eq 19 n ) sodium D line refractive index of liquid at temperature T and 1 atm n20 ) sodium D line refractive index of liquid at 20 °C and 1 atm P ) pressure in eq 6 Pc ) critical pressure, bar RI ) refractivity intercept defined in eq 18 Rm ) molar refraction defined in eq 2 Rm,ref ) molar refraction for the reference compound of methane, 6.987 cm3/g‚mol r ) parameter defined in eq 11 T ) absolute temperature, K Tb ) normal boiling point, K Tc ) critical temperature, K Tr ) reduced temperature ()T/Tc) S ) specific gravity at 15.5 °C (60 °F) V ) molar volume Vc ) critical molar volume, m3/kg Vo ) molar volume at melting point XA ) volume percentage of aromatics in a petroleum fraction XN ) volume percentage of naphthenes in a petroleum fraction XP ) volume percentage of paraffins in a petroleum fraction XMA ) volume percentage of monoaromatics in a petroleum fraction
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Ind. Eng. Chem. Res., Vol. 40, No. 8, 2001
XPA ) volume percentage of polyaromatics in a petroleum fraction XS ) weight percentage of sulfur in a petroleum fraction xiv ) volume fraction of component i in a mixture xiw ) weight fraction of component i in a mixture Greek Symbols R ) polarizability defined in eq 1 β ) parameter defined in eqs 9 and 10 θ ) property given in eq 5 and Table 1 θtr ) transport property (1/µ, 1/k, D) given in eq 15 µ ) absolute viscosity in eqs 13 and 15, mPa‚s (cP) ν ) kinematic viscosity in eqs 5 and 31 and Table 1, mm2/s (cSt) ∆H ) heat of vaporization at normal boiling point, kJ/kmol
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Received for review April 21, 2000 Accepted January 7, 2001 IE000419Y
