Determination of Volumetric Properties Using Refractive Index

Feb 27, 2017 - A novel method to evaluate volumetric properties, namely the thermal expansivity (αP) and the isothermal compressibility (κT) for non...
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Determination of Volumetric Properties Using Refractive Index Measurements for Nonpolar Hydrocarbons and Crude Oils Fei Wang, Renato F. Evangelista, Timothy J. Threatt, Mohammad Tavakkoli, and Francisco M. Vargas* Department of Chemical and Biomolecular Engineering, Rice University, Houston, Texas 77005, United States S Supporting Information *

ABSTRACT: A novel method to evaluate volumetric properties, namely the thermal expansivity (αP) and the isothermal compressibility (κT) for nonpolar hydrocarbon systems using refractive index measurements, is presented in this work. New expressions for αP and κT are derived from the Lorentz−Lorenz equation and the One-Third rule, respectively. A further simplified expression for αP is proposed requiring only refractive index data and molecular weight for calculation. Densities and refractive indices of 12 pure nonpolar hydrocarbons, 6 hydrocarbon mixtures, and 3 crude oils are measured at temperatures from 283.15 K up to 343.15 K and at 0.1 MPa. The measured refractive indices are used to calculate αP for a wide range of temperatures using the proposed method, and the measured densities are used to calculate αP for comparison. Reported densities and refractive indices of benzene at 298.15 K and at pressures up to 90 MPa are used for κT evaluations with the proposed method. Values of αP and κT calculated from refractive index measurements are in good agreement with experimental data and those determined from densities. This work aims to establish the foundation for experimental methods to determine volumetric properties of nonpolar hydrocarbon systems based on refractive index measurements. A high temperature and high pressure refractometer is expected to have multiple advantages over conventional techniques for density measurements, which include but are not limited to smaller amounts of sample needed, simpler calibration, faster measurement, and cells that are corrosion-resistant (i.e., sapphire glass).

1. INTRODUCTION Thermal expansivity (αP) and isothermal compressibility (κT) are crucial thermophysical volumetric properties that quantify the volume change of a substance with temperature and pressure variations, respectively. The expansivity and compressibility of a material are helpful properties for identifying possible applications and are widely used to test new materials, such as polymers,1−3 carbon materials,4 ionic liquids5,6 and biophysical materials.7,8 For petroleum fluids, αP and κT are used to describe the volumetric behavior of complex mixtures at different conditions through production, transportation, and storage. Additionally, αP and κT have been extensively used to evaluate other properties through fundamental thermodynamic relations. Singh et al.9,10 evaluated the excess enthalpy of mixing and the excess Gibbs free energy of binary mixtures using αP and κT. Verdier et al.11,12 calculated the solubility parameters of crude oils and pure hydrocarbons from αP and κT measurements. The significance of solubility parameter evaluations at high temperatures and high pressures was introduced by Vargas and Chapman,13 proposing rigorous methods to calculate the solubility parameters of nonpolar hydrocarbons as a function of αP and κT. Therefore, accurate and easy determination of thermal expansivity and isothermal compressibility is advantageous under many circumstances. Different experimental techniques have been developed to determine thermal expansivity and isothermal compressibility at various temperatures and pressures. Davis and Gordon14 originally used acoustic-wave-velocities to generate αP and κT © XXXX American Chemical Society

values for mercury at three different temperatures and pressures up to 130 MPa. Biswas et al.15−17 improved the acoustic velocity method by using the pulse-echo overlap technique between two refractors for several pure hydrocarbons. A piezothermal technique was established later using a high pressure calorimeter.18−20 This method was followed by Randzio et al.21,22 who designed an isothermal scanning calorimeter with high accuracy by relating the linear change in pressure to αP and κT. The calorimetric technique was widely adopted for αP and κT evaluations and was extended to calculate solubility parameters as well.23,24 Another popular approach is to calculate αP and κT from densities based on their thermophysical definitions.25−30 An equation of state type of expression is needed to correlate the measured densities with temperature and pressure variations31−35 in order to obtain the necessary derivatives. Calado et al.26 measured densities of liquid ethylene at temperatures from 110 to 280 K and pressures up to 130 MPa. The data were fit to the Strobridge equation of state to calculate other thermophysical properties of ethylene including αP and κT. For nonpolar hydrocarbons, densities and refractive indices are related by the Lorentz−Lorenz equation and other proposed correlations,36−40 such as the One-Third rule,13 Received: Revised: Accepted: Published: A

December 9, 2016 February 18, 2017 February 27, 2017 February 27, 2017 DOI: 10.1021/acs.iecr.6b04773 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research allowing volumetric property evaluations using refractive indices instead of densities. In this work, a novel method is proposed to determine thermal expansivity and isothermal compressibility using refractive index measurements for nonpolar hydrocarbons, their mixtures, and crude oils at a wide range of temperatures and pressures.

κT =

(2)

where αP is the thermal expansivity, κT is the isothermal compressibility, ρ is the mass density, T is the temperature, and P is the pressure. The subscript “ref ” will be included in this work when referring to values calculated using eqs 1 and 2 and considering all acquired density data within the reported temperature ranges. For nonpolar systems, the intermolecular interactions are dominated by the dispersion forces associated with polarizability and refractive index.42 The relationship between those thermophysical properties is described by the Lorentz−Lorenz equation, given in eq 3

2. EXPERIMENTAL SECTION 2.1. Materials. All pure hydrocarbon materials used in this work are listed in the Supporting Information, Table S1, with their supplier information and purities. The purchased chemicals are of high purity and used without further purification. The measured density and refractive index of each chemical at 298.15 K and 0.1 MPa are also listed and compared against reported data. Binary and ternary hydrocarbon mixtures, B1,B2, B3, T1, T2, and T3, were prepared using the pure chemicals. Their compositions and molecular weights are provided in the Supporting Information, Table S2, together with the measured density and refractive index at 298.15 K and 0.1 MPa. The molecular weight, mass density, and the saturate, aromatic, resin, and asphaltene (SARA) compositional analysis of the crude oils used in this work are listed in the Supporting Information, Table S3. The SARA analysis was performed following the standard ASTM method.41 2.2. Apparatus. Density Measurement. Density measurements were performed using the Anton Paar DMA 4500 digital vibrating U-tube densitometer with a measuring range of 0 to 3 g/cm3. The resolution was 0.00001 g/cm3 for the measured density. The temperature in the measuring cell was measured by two integrated Pt-100 platinum thermometers and precisely controlled by a built-in thermostat with an uncertainty of ±0.01 K. The measuring temperature range was from 273.15 to 368.15 K. The apparatus was calibrated with air and water. The uncertainty of the measured densities in this work was 0.0004 g/cm3 for pure components, 0.001 g/cm3 for their mixtures, and 0.002 g/cm3 for crude oils with a 95% confidence interval. Refractive Index Measurement. Refractive index measurements were performed using the automatic Anton Paar WR refractometer under the sodium D wavelength, 589.3 nm, with a measuring range of 1.30 to 1.72. The resolution was 0.000001 for the measured refractive index. The temperature was measured by two internal Pt-1000 platinum resistance temperature sensors. A built-in internal solid state Peltier thermostat with an accuracy of ±0.03 K was used for more precise control of the temperature due to the high sensitivity of refractive index to temperature variations for organic compounds. The working temperature range was from 283.15 to 343.15 K. The uncertainty of the measured refractive indices was 0.0002 for pure components, 0.0005 for their mixtures, and 0.001 for crude oils with a 95% confidence interval.

Rm F = RI Mw ρ

(3)

where Rm is the molar refractivity, Mw is the molecular weight, FRI = (n2D − 1)/(n2D + 2) is a function of the refractive index, and nD is the refractive index measured at the sodium-D line. Vargas and Chapman13 have reported that the ratio of FRI over ρ is a constant characteristic value for nonpolar hydrocarbons and their mixtures and is a weak function of temperature and pressure. This characteristic value represents the specific volume of molecules without considering the void space between the molecules, approximately equal to 1/3 for nonpolar hydrocarbon systems. This One-Third rule is expressed by eq 4 Rm F 1 1 = RI = 0 ≅ Mw ρ 3 ρ

(4)

where ρ0 is the mass density of molecules without considering the void space between the molecules. This simple relation works well at a wide range of temperatures and pressures because both Mw and Rm are almost constant. New expressions for αP and κT are derived from the One-Third rule and given by eqs 5 and 6 αP , OTR = −

κT , OTR =

1 ⎛ ∂FRI ⎞ ⎜ ⎟ FRI ⎝ ∂T ⎠ P

1 ⎛ ∂FRI ⎞ ⎜ ⎟ FRI ⎝ ∂P ⎠T

(5)

(6)

where the subscript “OTR” indicates properties derived from the One Third rule. Similar equations were derived from the Lorentz−Lorenz equation assuming constant Rm.43−45 However, there are discrepancies between αP and κT calculated according to their thermodynamic definition and the respective values from the One-Third rule, demonstrating that treating Rm as a constant is not appropriate for accurate αP and κT evaluations. In addition, the polarizability, which is directly related to Rm, was observed to be temperature dependent with a low increasing rate of 1%/1000 K for diatomic molecules, where this effect can be explained by the occupation of higher rotational and vibrational levels46−48 at higher temperatures. A more rigorous approach to calculate αP and κT is proposed based on the Lorentz−Lorenz equation but without assuming a constant Rm. The derived expressions are shown as eqs 7 and 8.

3. MATHEMATICAL EXPRESSIONS FOR αP AND κT The thermal expansivity and the isothermal compressibility are defined as the rate of volume or density change with temperature and pressure, respectively, per unit volume or density of a certain material, expressed by eqs 1 and 2 1 ⎛ ∂ρ ⎞ αP = − ⎜ ⎟ ρ ⎝ ∂T ⎠ P

1 ⎛ ∂ρ ⎞ ⎜ ⎟ ρ ⎝ ∂P ⎠T

αP = αP , OTR − (1) B

1 ⎛ ∂ρ0 ⎞ ⎟ ⎜ ρ0 ⎝ ∂T ⎠ P

(7) DOI: 10.1021/acs.iecr.6b04773 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research κT = κT , OTR +

1 ⎛ ∂ρ0 ⎞ ⎟ ⎜ ρ0 ⎝ ∂P ⎠T

αP , m =

(8)

The additional terms in eqs 7 and 8 represent the effects of temperature and pressure on the volume of a single molecule without considering the space between molecules. According to the similarity to αP and κT definitions, the second terms in eqs 7 and 8 are defined as molecular thermal expansivity and molecular isothermal compressibility as proposed by eqs 9 and 10, respectively αP , m = −

κT , m

1 ⎛ ∂ρ0 ⎞ ⎟ ⎜ ρ0 ⎝ ∂T ⎠ P

1 ⎛ ∂ρ0 ⎞ = 0⎜ ⎟ ρ ⎝ ∂P ⎠T

( ) + ln( )

ln

FRI , a

ρb

FRI , b

ρa

Ta − Tb

(11)

( ) + ln( )

ln κT , m =

FRI , b

ρa

FRI , a

ρb

Pa − Pb

(12)

where the subscripts “a” and “b” indicate properties at fixed points. With the new expressions for αP,m and κT,m, αP and κT can be calculated using refractive index data and two densities by eqs 13 and 14

(9)

αP ,2p

κT ,2p

ln 1 ⎛ ∂FRI ⎞ ⎜ ⎟ + = FRI ⎝ ∂P ⎠T

(10)

where the subscript “m” indicates molecular properties. The bulk volumetric properties have two contributions based on the derived equations. One is from the molecules themselves related to the electron configuration. The other one arises from the void space between the molecules, αP,OTR and κT,OTR, which largely accounts for the behavior of the bulk material. Difficulties are expected when evaluating αP,m and κT,m using eqs 9 and 10 because ρ0 and its derivatives with respect to temperature and pressure are not easily accessible. The temperature effect on αP,m is shown in Figure 1. αP,m at various

( ) + ln( )

ln 1 ⎛ ∂F ⎞ = − ⎜ RI ⎟ + FRI ⎝ ∂T ⎠ P

FRI , a

ρb

FRI , b

ρa

Ta − Tb

(13)

( ) + ln( ) FRI , b

ρa

FRI , a

ρb

Pa − Pb

(14)

where the subscript “2p” indicates properties calculated including the molecular contribution term, which is obtained based on two fixed points. For cases without any available density measurements, critical properties-based correlations for saturated liquid densities can be applied for the calculation of αP. Among a variety of correlations currently available in this category,49−51 Hankinson and Thomson’s correlation33 (the HT correlation) yields the best results for the considered components and has shown the lowest dependence on the selected temperatures for the αP calculation. The H-T correlation is given in eqs 15−17 ρc ρs = (0) V R [1 − ωSRK V R(1)] (15) V R(0) = 1 − 1.52816(1 − Tr )1/3 + 1.43907(1 − Tr )2/3 − 0.81446(1 − Tr ) + 0.190454(1 − Tr )4/3 V R(1) =

Figure 1. Molecular thermal expansivity at temperatures from (283.15 up to 343.15) K compares with the molecular thermal expansivity at 293.15 K for pure nonpolar hydrocarbons, mixtures, and crude oils at 0.1 MPa.

(16)

− 0.296123 + 0.386914Tr − 0.0427258Tr2 − 0.0480645Tr3 Tr − 1.00001 (17)

where ρs is the saturated liquid density; ρc is the characteristic density fit using experimental data; ωSRK is the acentric factor which provides best fit of SRK equation of state for vapor pressure data; Tr is the reduced temperature, Tr = T/Tc, and Tc is the critical temperature. An additional advantage of expressing αP,m as a function of the density ratio is that the H-T component specific parameter ρc is no longer required. The new expression for αP incorporating the H-T correlation is given by eq 18.

temperatures was calculated by its defining equation using ρ0 calculated from eq 4 using the molecular weight, measured density, and refractive index. ρ0 was fit as a linear function of temperature. The calculated αP,m of tested samples at temperatures from 283.15 up to 343.15 K is plotted against αP,m calculated at 293.15 K in Figure 1. All data points lie very close to the illustrative line Y = X. From Figure 1, one may conclude that the molecular thermal expansivity is a weak function of temperature. The variations in αP,m at different temperatures are fairly small, thus it is reasonable to neglect temperature dependence. Appropriate expressions for the molecular expansivity and compressibility, eqs 11 and 12, are derived by integrating eqs 9 and 10 assuming that αP,m and κT,m are independent of the temperature and pressure

αP , RI = −

+

⎛ FRI , a ⎞ 1 ⎛ ∂FRI ⎞ 1 ⎟⎟ ⎜ ⎟ + ln⎜⎜ FRI ⎝ ∂T ⎠ P Ta − Tb ⎝ FRI , b ⎠ ⎡ V (0)(1 − ω V (1) ) ⎤ 1 R ,a SRK R , a ⎥ ln⎢ (0) Ta − Tb ⎢⎣ V R , b(1 − ωSRK V R(1), b) ⎥⎦

(18)

The density ratio in eq 13 is replaced by the saturated density ratio in eq 18. The subscript “RI” is used in this specific case where the property is calculated solely based on refractive index measurements. Extended correlations for liquid densities C

DOI: 10.1021/acs.iecr.6b04773 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research considering the pressure effect34,35,52 are suitable to use for the calculation. However, more parameters are needed to correct the liquid density from the saturated condition. The HBT correlation proposed by Thomson et al.52 allows predictions of liquid densities for nonsaturation conditions by incorporating the Tait equation constants. Compared to the results from the H-T correlation, the difference between the density ratios is within 0.01%, and the difference between the calculated αP is within 0.4% for the tested pure hydrocarbons. The H-T correlation was chosen in this work for its relative simplicity. However, advanced density correlations including pressure effects are necessary for κT evaluations using only refractive index data. The H-T correlation needs critical properties in order to predict the density ratio, which are widely reported for pure substances. Mixing rules are available for critical properties of mixtures with known compositions. For complex mixtures, such as crude oils, the correlation is hard to apply. Correlations for ωSRK, Tc, and Pc of nonpolar hydrocarbons and their mixtures as functions of FRI,293K and Mw are proposed and expressed by eqs 19−21

Information, Table S11, for the tested samples. The predicted densities and refractive index functions from the polynomial fittings are highly accurate with a maximum deviation of 0.18% from the experimental data. The quadratic fitting for densities is appropriate for αP evaluations at atmospheric pressure as performed in the literature.5 Further validations are provided in Figure 2. Values of αP,ref calculated by combining eqs 1 and 22

Figure 2. Experimental and calculated thermal expansivity of 7 pure hydrocarbons at temperatures from (283.15 up to 343.15) K or its normal boiling point and at 0.1 MPa; symbols represent the experimental data,9,12,16,17,54−56 and continuous curves represent αP,ref calculated using densities.

ωSRK = 0.0009438(Mw/FRI ,293K ) − 0.04892, 300 ≤ Mw/FRI ,293K ≤ 900

(19)

2 0.2961 Tc = 324.5859(FRI ,293K Mw)

(20)

5/3 Pc = 28724.6707(FRI ,293K / Mw) + 0.7629

(21)

are in good agreement with experimental data at a wide range of temperatures, with deviations within the 2% uncertainty of the thermal expansivity measurements. Therefore, α P,ref calculated from the measured densities is used as “true values” for comparison with αP,RI, calculated by the proposed method using refractive index measurements as shown in eq 18. 4.2. Acentric Factor and Critical Property Correlations. The SRK acentric factors were taken from the H-T correlation paper,33 and the critical temperatures and critical pressures were taken from the DIPPR database57 to generate and validate the proposed correlations, eqs 19−21. These data are provided in the Supporting Information (Tables S7 and S8) together with predicted results. For most cases, the predictions are in good agreement with reported values. The correlation results for the SRK acentric factor, the critical temperature, and the critical pressure are shown in Figure 3 −Figure 5,

where FRI,293K is the value of the refractive index function at 293.15 K and 0.1 MPa, and Pc is the critical pressure in bar. The correlations are generated using data of the tested pure hydrocarbons and further validated by another 17 nonpolar hydrocarbons. The critical properties of the tested hydrocarbon mixtures and crude oils are calculated by the proposed correlations in eqs 19−21 with satisfactory αP results because the crude oils are mainly composed by nonpolar and slightly polar hydrocarbons. Combining eqs 16−20, a new method for αP evaluations at a wide range of temperatures is achieved for nonpolar hydrocarbon systems requiring only refractive indices and the molecular weight.

4. RESULTS AND DISCUSSION 4.1. Density and Refractive Index Temperature Dependence. The densities and refractive indices of the tested pure hydrocarbons, their mixtures, and crude oil samples were measured at 0.1 MPa and at temperatures from 283.15 to 343.15 K or to their normal boiling points if lower than 343.15 K. The data are tabulated and provided in the Supporting Information (Tables S4−S10) together with deviations from NIST database.53 The measured densities are in good agreement with reported values. In order to calculate derivative properties, the measured densities and refractive indices were fit as functions of temperature by eqs 22 and 23 ρ = a0 + a1T + a 2T 2

(22)

FRI = b0 + b1T + b2T 2

(23)

Figure 3. SRK acentric factor correlation for pure nonpolar hydrocarbons as a function of molecular weight and ambient refractive index function.

respectively. The acentric factor correlation shows satisfactory predictions for nonpolar hydrocarbons with Mw/FRI,293K in the range 300−900 as shown in Figure 3. The proposed eq 20 relates the critical temperature with F2RI,293KMw very well for all the tested nonpolar hydrocarbons as shown in Figure 4. Though not required for this work, a correlation for estimating

where a0, a1, a2, b0, b1, and b2 are adjustable coefficients generated from the least-squares fitting method. Both a2 and b2 are restricted to be negative values. The adjustable coefficients with units and the fitting errors are provided in the Supporting D

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ωSRK and Tc is not substantial, with a difference less than 7 × 10−5 K−1. 4.3. The Thermal Expansivity. Comparisons for the calculated thermal expansivity at 293.15 K and at 0.1 MPa using the various equations presented in this work are presented in Table 1. The calculated αP,RI by the proposed method using measured refractive indices and correlated ωSRK and Tc is compared with αP,ref calculated using densities. The two temperatures used in eq 18 were 293.15 and 303.15 K for pure hydrocarbons and 293.15 and 343.15 K for mixtures and crude oils. The calculated αP,OTR is also provided for comparison. The average absolute relative difference (AARD) over all tested samples is decreased from 5.4% to 2.4% by introducing the molecular contribution term, calculated as proposed by eq 18, to the One-Third rule. The results indicate the One-Third rule is a satisfactory correlation between density and refractive index but not optimal for derivative property calculations. The αP predictions can be further improved by using eq 13 with two measured densities instead of the H-T correlation. To validate the proposed approach, αP,NIST of several pure hydrocarbons are acquired from the NIST database53 and plotted against those calculated using the refractive index data in Figure 6. The relative difference is within 5%, and they are in good agreement with each other. The comparison between αP,RI and αP,ref, calculated from measured densities, at a wide range of temperatures is shown in Figure 7 for all tested samples. For the experimental temperature range, the deviation of αP,RI from αP,ref is less than 4.8% for all tested samples except for benzene and the T3 mixture, which includes benzene as a component. The poor prediction for benzene originates from the density ratio calculation using the H-T correlation. The predicted density ratio for benzene shows a 0.11% difference from the experimental value compared to the maximum deviation of 0.04% for the other components. If the calculation is performed

Figure 4. Critical temperature correlation for pure nonpolar hydrocarbons as a function of molecular weight and ambient refractive index function.

Figure 5. Critical pressure correlation for pure nonpolar hydrocarbons as a function of molecular weight and ambient refractive index function.

critical pressure has been proposed, and its predictability is displayed in Figure 5. Caution should be paid to cyclic hydrocarbons when using the critical pressure correlation. For the interest of αP evaluations in this work, the difference between the calculated results using the correlated and reported

Table 1. Comparison between the Thermal Expansivity at 293.15 K and at 0.1 MPa Calculated from Different Methods component pentane n-hexane heptane n-octane decane dodecane n-pentadecane cyclohexane benzene toluene p-xylene 1-MN B1 B2 B3 T1 T2 T3 crude A crude C crude S average a

αP,ref (K−1) 1.58 1.36 1.23 1.14 1.03 9.66 9.09 1.20 1.21 1.07 1.01 7.29 1.09 1.15 1.04 1.06 1.08 1.08 7.57 7.69 7.86

× × × × × × × × × × × × × × × × × × × × ×

−03

10 10−03 10−03 10−03 10−03 10−04 10−04 10−03 10−03 10−03 10−03 10−04 10−03 10−03 10−03 10−03 10−03 10−03 10−04 10−04 10−04

αP,OTRa (K−1)

αP,RIa (K−1)

αP,2pa (K−1)

−03

−03

−03

1.47 1.31 1.18 1.10 9.86 9.24 8.68 1.14 1.11 9.97 9.30 6.52 1.04 1.10 9.62 1.03 1.03 1.00 7.29 7.32 7.53

× × × × × × × × × × × × × × × × × × × × ×

10 10−03 10−03 10−03 10−04 10−04 10−04 10−03 10−03 10−04 10−04 10−04 10−03 10−03 10−04 10−03 10−03 10−03 10−04 10−04 10−04

1.62 1.38 1.27 1.17 1.05 9.69 9.01 1.24 1.11 1.03 9.71 7.48 1.10 1.15 1.01 1.10 1.09 1.03 7.34 7.68 7.89

× × × × × × × × × × × × × × × × × × × × ×

10 10−03 10−03 10−03 10−03 10−04 10−04 10−03 10−03 10−03 10−04 10−04 10−03 10−03 10−03 10−03 10−03 10−03 10−04 10−04 10−04

1.58 1.35 1.25 1.15 1.03 9.64 9.06 1.21 1.23 1.07 9.98 7.33 1.08 1.16 1.04 1.08 1.09 1.08 7.65 7.77 8.04

× × × × × × × × × × × × × × × × × × × × ×

10 10−03 10−03 10−03 10−03 10−04 10−04 10−03 10−03 10−03 10−04 10−04 10−03 10−03 10−03 10−03 10−03 10−03 10−04 10−04 10−04

ARDOTRb (%)

ARDRIb (%)

ARD2pb (%)

6.7 3.7 4.2 3.5 4.4 4.4 4.5 5.0 8.0 6.7 7.5 10.6 4.6 4.7 7.2 2.8 5.0 7.5 3.8 4.7 4.3 5.4

2.6 1.4 3.1 3.0 1.6 0.3 0.9 3.1 7.7 3.6 3.4 2.6 0.7 0.2 2.5 3.5 1.2 4.7 3.1 0.1 0.3 2.4

0.5 0.6 1.1 1.1 0.4 0.2 0.3 0.3 1.6 0.1 0.7 0.5 1.0 0.3 0.1 1.8 0.5 0.6 1.0 1.1 2.2 0.8

αP,OTR is calculated by eq 7, αP,RI is calculated by eq 18, and αP,2p is calculated by eq 13. bARDX = |αP,X − αP,ref |/αP,ref × 100%. E

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Figure 8. Isothermal compressibility of benzene at 298.15 K and at pressures up to 90 MPa; symbols represent the experimental data;59 the black continuous curve represents those calculated using densities; the red continuous curve represents those calculated using refractive index data and two densities; the black dashed curve represents those calculated using the equation derived from the One-Third rule.

Figure 6. Thermal expansivity reported by NIST database,53 αP,NIST, compared with those calculated using the proposed method, αP,RI, at temperatures from 283.15 up to 343.15 K and at 0.1 MPa for pure hydrocarbons; open symbols represent the calculated data; the black dashed line represents Y = X, and the red dashed lines represent ±5% deviation from Y = X.

derivative properties. From Figure 8, the calculated κT,RI is in good agreement with κT,ref. The calculated κT,OTR shows a constant difference compared to κT,2p and κT,ref and a higher deviation from the experimental value. The absolute relative difference between the experimental value and κT,2p from the proposed method is 4.9%. The deviation likely comes from the uncertainty of the high pressure refractive index data used and the error induced by the derivative calculation with limited data. However, the prediction is significantly improved compared to κT,OTR, which deviates from the experimental value by 14.8%. An expression for κT,RI using only refractive index data can be achieved incorporating a density correlation that accounts for the pressure effect on liquid densities, such as that given by the Tait equation. More refractive index data at high pressures are needed to further evaluate the proposed method for isothermal compressibility calculations.

Figure 7. Thermal expansivity calculated using densities, αP,ref, compared with those calculated using the proposed method, αP,RI, at temperatures from 283.15 up to 343.15 K and at 0.1 MPa for pure hydrocarbons, their mixtures, and crude oils; open symbols represent the calculated data; the black dashed line represents Y = X, and the red dashed lines represent ±5% deviation from Y = X.

5. ADVANTAGES OF THE REFRACTIVE INDEX-BASED APPROACH The proposed method established the framework for the determination of volumetric properties, such as thermal expansivity and isothermal compressibility, for nonpolar hydrocarbons in a wide range of temperatures and pressures using refractive index measurements, especially for complex mixtures such as crude oils. Some important advantages of the proposed method over conventional density-based methods include but are not limited to 1. Reduced volume of sample needed (less than 5% of the volume needed in the densitometer), 2. Simple and infrequent calibration of the instrument (no recalibration needed after location change of the instrument), 3. Sample is not in contact with metal surfaces, which makes this method suitable for analyzing hydrocarbon blends containing CO2 or H2S, 4. Measurements are much faster (temperature can be more easily adjusted and controlled in a short period of time), 5. Risk of instrument fouling is significantly reduced (the sapphire prism is the only part of the instrument contaminated with the sample).

using eq 13 with two measured densities for the ratio term, the αP,2p matches better with the αP,ref as shown in the last column of Table 1. The proposed method also demonstrates great ability to predict thermal expansivity of complex mixtures such as crude oils with unknown compositions, as displayed in Figure 7. The thermal expansivity of such complex systems at a wide range of temperatures can be evaluated using solely refractive index data and molecular weight, given the proposed correlations for ωSRK and Tc. 4.4. The Isothermal Compressibility. The isothermal compressibility, κT, can be evaluated following the same methodology as that for αP using the proposed eq 14. Due to limited refractive indices reported in the literature at high pressures, a preliminary implementation of the proposed method is provided for pure benzene as an example using density and refractive index data taken from the literature.58 The calculated κT,2p at 298.15 K and pressures up to 90 MPa is shown in Figure 8 along with κT,ref and κT,OTR. The experimental κT measured by the piezometer-filling technique59 at 298.15 K and at 0.1 MPa is plotted for comparison. The two densities used for calculation were at 0.1 MPa and at 86.8 MPa. The densities and refractive indices were fit to second order polynomials as a function of pressure in order to calculate the

6. CONCLUSION In this work, a new method to evaluate the thermal expansivity using refractive index measurements and the molecular weight is proposed and validated for nonpolar hydrocarbon systems. The results from the proposed method are in good agreement with those calculated from densities, which are consistent with thermal expansivity measured by experimental techniques. The F

DOI: 10.1021/acs.iecr.6b04773 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research proposed method shows significant improvement on thermal expansivity and isothermal compressibility predictions compared to the equations derived from the One-Third rule. There are two contributions to the volumetric properties of a substance as discussed, one due to molecular configuration and the other due to void space between molecules. The proposed method takes both into consideration and requires the refractive index data and the molecular weight of the substance. The thermal expansivity calculated by the equation derived from the One-Third rule includes only the void space contribution, which inevitably results in large discrepancies. The conclusions are drawn through the thermal expansivity evaluations for 12 pure nonpolar hydrocarbons, 3 binary mixtures, 3 ternary mixtures, and 3 crude oils over a wide range of temperatures and at atmospheric pressure. Benzene is studied as an example to show the feasibility of the proposed method for isothermal compressibility evaluations using refractive index measurements and two densities. However, refractive index data acquisition at a wide range of pressures is currently impractical due to the lack of commercial high temperature high pressure refractometers. Therefore, this work expects to stimulate potential advances in this topic and presumes that, with larger data availability, the proposed equations can be further inspected for volumetric property evaluations at high pressures. Along these lines, a variety of advantages of using the proposed approach over the conventional density-based method have been discussed, including but not limited to easier and faster experimental procedure, better equipment preservation, and significantly less sample required.

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ASSOCIATED CONTENT

S Supporting Information *

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The Supporting Information is available free of charge via the Internet at The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/ acs.iecr.6b04773. Density and refractive index data measured in this work together with the correlated ωSRK, Tc, and Pc (PDF)

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αP,OTR = thermal expansivity calculated using the equation derived from the One-Third rule, K−1 αP,2p = thermal expansivity calculated using refractive index data and two densities, K−1 αP,RI = thermal expansivity calculated using refractive index data and the H-T correlation, K−1 αP,m = molecular thermal expansivity, K−1 κT = isothermal compressibility, MPa−1 κT,ref = isothermal compressibility calculated using densities, MPa−1 κT,OTR = isothermal compressibility calculated using the equation derived from the One-Third rule, MPa−1 κT,2p = isothermal compressibility calculated using refractive index data and two densities, MPa−1 κT,m = molecular isothermal compressibility, MPa−1 ρ = density, g/cm3 ρ0 = molecular density, g/cm3 ρs = saturated liquid density, g/cm3 ρc = characteristic density for the H-T correlation, g/cm3 ωSRK = acentric factor which provides best fit for SRK equation of state for vapor pressure data FRI = function of refractive index, FRI = (n2D − 1)/(n2D + 2) Mw = molecular weight, g/mol nD = refractive index measured at sodium-D line Rm = molar refractivity, cm3/mol P = pressure, MPa Pc = critical pressure, bar T = temperature, K Tr = reduced temperature, Tr = T/Tc Tc = critical temperature, K V(0) R = spherical molecule function for the H-T correlation V(1) R = deviation function for the H-T correlation

REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*Phone: +1 (713) 348-2384. E-mail: [email protected]. Corresponding author address: 6100 Main MS-362, Houston, Texas 77005-1827, USA. ORCID

Fei Wang: 0000-0001-8475-9097 Francisco M. Vargas: 0000-0001-5686-5140 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The financial support from Rice University, through a faculty start-up fund, is greatly appreciated. R.E. thanks CAPES Foundation for the financial support (grant no, 18644/12-0). The authors are also grateful to Xiaoyu Liu, Mohan Boggara, Sara Rezaee, and Caleb Sisco for their contributions to this work.

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NOMENCLATURE αP = thermal expansivity, K−1 αP,ref = thermal expansivity calculated using densities, K−1 G

DOI: 10.1021/acs.iecr.6b04773 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

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DOI: 10.1021/acs.iecr.6b04773 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX