Article pubs.acs.org/jced
Calculation of Energies and Entropies from Isochoric and Isothermal Experimental Data Andrea del Pilar Tibaduiza, Diego E. Cristancho,† Diego Ortiz-Vega,† Ivan D Mantilla,‡ Martin A. Gomez-Osorio, Robert A. Browne, James C. Holste,* and Kenneth R. Hall§ Artie McFerrin Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843-3122, United States S Supporting Information *
ABSTRACT: This paper presents residual values of Helmholtz energy (Ar/(RT)), entropy (Sr/R), and internal energy (Ur/(RT)) for a ternary mixture that resembles a distribution natural gas between 223.15 K and 303.15 K up to 20 MPa. The methodology uses isochoric and isothermal density measurements to apply corrections to the residuals for Helmholtz energy (δAr/(RT)), entropy (δSr/R), and internal energy (δUr/(RT)) calculated using an equation of state. The method is demonstrated for three representative equations of state: REFPROP, Peng−Robinson, and Redlich−Kwong. Accurate (p−ρ−T) isochoric and isothermal data provide experimental values of the compression factor (Z) and the derivative of pressure with respect to temperature at constant density (∂p/∂T)ρ which are used to calculate residual entropies and energies. The results obtained by using different equations of state at the same conditions have slight differences in the residual values. It is possible to represent those differences by equivalent changes in temperature for entropy (δTS) and internal energy (δTU) that are ≤ 0.5 K for temperatures above 225 K. For this mixture, the values of δAr/(RT), δSr/R, and δUr/(RT) determined from REFPROP are sufficiently small that no corrections are required. For the Peng−Robinson and Redlich−Kwong equations of state, a global fit describing the residual corrections presents a practical application. The residual deviations of the plots lie within ± 0.0005, ± 0.001, and ± 0.002 in the residuals for δAr/(RT), δSr/(RT), and δUr/(RT), respectively.
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INTRODUCTION
tion directly determined from experimental data is useful for further development of such equations. Recently Cristancho et al.4 published experimental densities and saturation properties for a synthetic natural gas mixture with mole fractions of 0.95039 (methane), 0.03961 (ethane), and 0.01000 (propane). The experimental apparatus included a magnetic suspension densimeter (MSD) and an isochoric apparatus. Experimental details and uncertainties for these systems appear elsewhere.4−7 In this paper, the experimental data provide isochoric slopes (∂p/∂T)ρ,x, that enable determination of the residual Helmholtz energy (Ar), the residual entropy (Sr) and the residual internal energy (Ur). This paper provides the equations needed to determine the energy properties (presented in the theory section). It also presents a practical methodology to calculate residual energy and entropy functions from isochoric data in conjunction with an equation of state. The methodology is demonstrated for several categories of EoS. One group is highly accurate but sufficiently complex to discourage some design applications. This group is illustrated in this work using equations contained
The experimental determination of caloric properties for natural gas mixtures and associated mixtures is essential information for the gas processing industry. This information is useful for developing new equations of state (EoS), for empirical correlations needed to determine the energy content of hydrocarbon gas streams, and it can be used in conjunction with other sets of experimental data, such as volumetric data (p−ρ−T) for developing thermodynamic models. Unfortunately, the caloric properties for gas mixtures determined by calorimetric methods have more experimental scatter than volumetric data, which impacts the thermodynamic consistency of multiproperty thermodynamic models.1,2 This particular behavior of gaseous systems in calorimetric measurements is a result of the low density of the gas mixtures, which makes the analysis of their thermal responses difficult. On the other hand knowledge of the experimental derivatives such as (∂p/∂T)ρ,x helps to evaluate the thermodynamic consistency of equations of state and the impact of using numerical and analytical derivatives during its construction and application. The modern multiparameter equations of state,2,3 which use an explicit form of the Helmholtz energy function, have proven to provide the most accurate representation of the experimental data. These expressions also have excellent extrapolation capabilities. Having Helmholtz energy informa© XXXX American Chemical Society
Special Issue: In Honor of Grant Wilson Received: August 15, 2013 Accepted: December 19, 2013
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in the NIST Standard Reference Data Base 23 (REFPROP).3 Other groups include cubic EoS optimized for the prediction of phase equilibrium, illustrated by Peng−Robinson (PR),8 or of density, represented by Redlich−Kwong (RK).9 The purpose of these comparisons is to determine the level of accuracy achieved by using experimental data to adjust numbers calculated from both the complex and the cubic EoS.
δS r = R
Sr = R
ρ
∫0
(Z − 1)
∫0
(3)
∂Z ⎞ dρ ⎜ ⎟ ⎝ ∂(1/T ) ⎠ ρ ρ
ρ⎛
(4)
in which the integrations follow isothermal paths, and p Z= ρRT
(5)
The required changes in pressures with respect to temperature at constant density are available from isochoric measurements. Following eq 1, a residual value calculated using an EoS is r XEoS = XEoS − X ig
(6)
Because the EoS is not perfect, Xr ≠ XrEoS, so that r
X =
r XEoS
+
r δXEoS
correlation factor = ρMSD / ρREF (7)
(8)
Hr Ur = +Z−1 RT RT
(9)
Gr Ar = +Z−1 RT RT
(10)
The correction terms in eq 7 for the Helmholtz energy, entropy, and internal energy residuals are δAr = RT
∫0
ρ
(Z − Z EoS)
dρ = ρ
∫0
ρ
δIA
dρ ρ
dρ ρ
⎤ ⎥ dρ ⎥ ρ ⎥ EoS ⎦ (13)
(14)
that compares densities measured in the MSD to densities calculated for the same conditions using an accurate EoS (REFPROP in this work). Experimental isothermal densities are used to determine the correlation factors for each measured pressure at 300 K. For an accurate EoS these correlations vary extremely slowly with temperature and pressure, so the values at 300 K can be used without loss of accuracy at 303.15 K (the reference temperature for the isochores in this work). The reference density (ρ*) for each isochore then is calculated using the correlation factor and densities calculated using the EoS in eq 14. 2. Calculation of Densities in the Cell at Any Point (p− T) along Each Isochore. Two factors cause the density of the fluid in the cell to vary slightly with temperature along the experimental path. First, the volume of the cell varies with temperature, and second, part of the volume occupied by the fluid is external to the cell at a different temperature. The first is true of all isochoric apparatus, and the second is true for many, including the apparatus used to collect the data analyzed in this work. The calculation of densities from pseudoisochoric data relies upon the fact that the total number of moles is constant for each pseudoisochore. The total number of moles is the sum
Any EoS used with eq 7 can calculate accurate residual properties when δXrEoS is calculated from experimental data using the procedure described below. Only two of the five residual functions of interest must be calculated from experimental data because the following definitions allow calculations of the other three
Ur Ar Sr = + RT RT R
δIS
METHODOLOGY The procedure used to calculate energies and entropies from density data involves a number of steps discussed in detail below. A complication is the fact that apparatus effects cause the experimental “isochore” not to be a path of constant density, rather it is a pseudoisochore of nearly constant density. This work suggests an approach that uses densities measured in a magnetic suspension densimeter (MSD), and isochoric measurements (p−T) to determine densities, useful derivatives, and residual values of energies and entropies. This paper applies the methodology to a set of seven isochores, and isothermal data at 300 K measured at Texas A&M University and published by Cristancho et al.4 1. Determination of the Reference Density (ρ*) for Each Isochore. Densities measured at constant temperature in the MSD are used to determine the density of each isochore at the reference temperature. Interpolation between the measured isothermal densities uses a correlation factor
ρ⎡
Ur 1 = RT T
ρ
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(2)
⎛ ⎞ ⎤ ⎢1 − 1 ⎜ ∂p ⎟ ⎥ dρ ⎢⎣ ρR ⎝ ∂T ⎠ ρ⎥⎦ ρ
∫0
The dimensionless factors in the integrands are denoted by δI to simplify the notation, and they represent directly the shortcomings of the EoS. The calculation requires only one of eq 12 or 13, but both are included here to compare the accuracies obtained when using entropy or energy as the second choice. Eq 12 is preferred because the partial derivative of pressure with respect to temperature is related more directly to the measured variables (p,T) in the isochoric experiment.
(1)
dρ ρ
∫
ρ
∫
where X represents any thermodynamic property. The relationships between the observable properties p, T, ρ and the dimensionless forms for the Helmholtz energy, entropy, and internal energy density residuals respectively are
∫0
EoS
⎤ ⎛ ∂p ⎞ ⎥ dρ −⎜ ⎟ ⎥ = ⎝ ∂T ⎠ ρ ρ ⎦
⎡ ⎛ ∂Z ⎞ 1 ⎢⎛ ∂Z ⎞ ⎜ ⎟ −⎜ ⎟ ⎢ 0 T ⎝ ∂(1/ T ) ⎠ ⎝ ∂(1/T ) ⎠ ρ ⎢⎣ ρ ρ dρ δIU = 0 ρ
δU r = RT
THEORY The density residual function is the difference between the value of a property for a real fluid and the value of that same property for an ideal gas at the same temperature and density:
Ar = RT
⎡ 1 ⎢⎛ ∂p ⎞ ⎜ ⎟ ρR ⎢⎝ ∂T ⎠ ρ ⎣
(12)
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X r(T , ρ) = X(T , ρ) − X ig(T , ρ)
∫0
ρ
(11) B
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A quadratic equation provides a sufficient fit for the experimental data from 223.15 K to 303.15 K. A polynomial in inverse temperature did not improve the fits. No reasonable model could fit the data below 223.15 K. 5. Determination of the Dimensionless Factors δIA, δIS, and δIU. Eq 11 requires experimental compression factors (Z*), calculated using eqs 5 and 21. Eqs 12 and 13 require the experimental derivative of p* as function of T at constant density, which results from the derivative of eq 21. The corresponding EoS values come directly from that EoS. 6. Isothermal Integration for Determining the Residual Adjustment Term. Because the integrations in eqs 11 to 13 follow isothermal paths, it is necessary to reorganize the isochoric data into isothermal data sets, and then fit each δI with respect to density. Each isochore provides one value for the fit. When using a polynomial in density with zero intercept,
of the moles in the cell and the moles in the external volume (primarily the pressure transducer). In the following equations, the subscripts “e” and “c” denote the external and cell volumes, respectively. The “r” superscript denotes the respective property at reference conditions (Tr = 303.15 K and isochoric pressure at Tr). Because the total number of moles remains constant, nc + ne = ncr + ner
(15)
which may be rearranged to obtain nc (ner − ne) = 1 + ncr ncr
(16)
or in terms of density: ρc Vc ρcr Vcr
ρer Ver − ρe Ve
=1+
ρcr Vcr
(17)
δI =
Three terms are sufficient for these data. Also, with the polynomials, the general form for the integrals is
∫0
(18)
0
0
V (T , p) = V [1 + α(T − T ) + κ(P − P )]
∑ aiT i i=0
dρ = ρ
r
∑ i=1
bi (ρ ∗)i i
(23)
r
(24) r
δH δU δU = + δZ = + δIA RT RT RT
(25)
δGr δAr δAr = + δZ = + δIA (26) RT RT RT so that only two variables must be calculated from the data. 7. Determination of Residual Properties Based upon Experimental Measurements. The values of the residual properties based upon experimental measurements come from eq 7, in which the EoS value is calculated directly and the adjustment term is calculated as described above. This paper uses a polynomial of the form:
(19)
δX r = RT
(20)
m
n
∑ ∑ cXijT i(ρ*) j i=0 j=1
(27)
to describe temperature and density dependence of the residual adjustment factors within the limits of the data set. For RK and PR, the binary interaction parameters are zero. Nonzero interaction parameters could improve the accuracy, but part of the purpose here is to compare the performance of less accurate but more practical EoS with a more accurate EoS.
provides a sufficiently accurate adjustment when using values of (∂p/∂ρ)T from an accurate equation of state such as those contained in REFPROP. 4. Fitting the Isochoric Pressures As a Function of Temperature. Here it is necessary to select the best model that correlates the pressure to temperature with the smallest deviation using a regression model that produces the standard error for each coefficient. For purposes of energy and entropy determination only p*−T pairs in the single phase region should be included in the fit of p* vs T. The isochores are nearly straight, so a simple polynomial is sufficient to describe them:
pfit* =
δI
δU r δAr δS r = + RT RT R
Supporting Information, Table 1S contains the parameters for the isochoric apparatus. The fluid density in the cell, (ρc) in eq 18, corresponds to the isochoric density and from this point forward appears as ρ. 3. Adjustment of the Experimental Pressures (p) to Isochoric Pressures (p*). The pressure derivatives required for the energy and entropy calculations are isochoric; therefore, the measured pressures must be converted to truly isochoric values. Because the changes in fluid density are small, ⎛ ∂p ⎞ p* = p + ⎜ ⎟ (ρ * − ρ ) ⎝ ∂ρ ⎠T
ρ
The residual adjustment factors obey eqs 8 through 10 in the forms
The volume terms in eq 18 are functions of temperature and pressure. In the following equations the “0” superscripts refer to the calibration conditions of the cell, α is the volumetric thermal expansion coefficient, and κ is the pressure distortion parameter for the material of the isochoric cell, which is stainless steel. The volume is given as a function of temperature and pressure by 0
(22)
i=1
The temperature of the external volume remains essentially constant throughout the experiment, so assuming that Vre = Ve and solving for ρc: ⎛Vr ⎞ ⎛V ⎞ ρc = ⎜ c ⎟ρcr + ⎜ e ⎟(ρer − ρe ) ⎝ Vc ⎠ ⎝ Vc ⎠
∑ bi(ρ*)i
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RESULTS AND ANALYSIS The numerical results appear in the order of the steps described in the Methodology Section. Figure 1 illustrates the distribution of the experimental measurements. Figure 2 shows the correlation factor as a function of pressure (p ≤ 50 MPa) for the isothermal data at 300 K. This figure shows that a value of 0.9997 for the correlation factor represents all data below 20 MPa within ± 0.01%. Supporting Information, Table 2S also shows values of ρ* calculated for each isochore.
(21) C
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Figure 3. Isochoric pressure adjustments: ●, isochore 1; ○, isochore 2; ▼, isochore 3; △, isochore 4; ■, isochore 5; □, isochore 6; ◆, isochore 7.
Figure 1. p−T and isothermal data: ●, isochore 1; ○, isochore 2;▼, isochore 3; △, isochore 4; ■, isochore 5; □, isochore 6; ◆, isochore 7; ×, 300 K isotherm; , phase boundary predicted by Peng−Robinson.
Figure 2. Correlation factor at T = 300 K: ▲, calculated using ρexp from the 300 K MSD isotherm reported by Cristancho et al.4 and ρref predicted by REFPROP.
Figure 4. Residuals for the fits of eq 21 to the isochoric pressures (p*): ●, isochore 1; ○, isochore 2; ▼, isochore 3; △, isochore 4; ■, isochore 5; □, isochore 6; ◆, isochore 7.
Supporting Information, Table 3S contains the experimental densities (ρ) calculated using eq 18 and reference conditions T = 303.15 K and ρ*. Because of different methodology, these densities are slightly, but not significantly, different than those reported by Cristancho et al.4 Supporting Information, Table 3S also presents the relative differences between the experimental densities and those calculated using REFPROP. Values of the isochoric pressures, p*, calculated using eq 20 are in Supporting Information, Table 3S. Figure 3 shows the adjustments (p* − p) made to the experimental pressures. The largest pressure adjustments are only about 20 times larger than the accuracy of the pressure measurement (± 0.002 MPa10) so the accuracy requirements for the adjustments are not stringent. The contributions from the volume change of the cell and the effects of the external volume (first and second terms respectively in eq 18) are approximately equal in magnitude. Supporting Information, Table 3S and Figure 4 present the differences between the p* values and a fit of the quadratic version of eq 21 to the p* values for T ≥ 220 K. No version of eq 21 provided a satisfactory fit to data sets that included those at lower pressures, so this analysis was limited to temperatures above 220 K. All data are included in Supporting Information,
Table 3S and Figure 4; the residuals for data not included in the fit appear in italics in Supporting Information, Table 3S. These results show that the distribution of experimental temperature and pressures must be chosen carefully to maximize the amount of information obtained. This includes not only the phase envelope determination11 but also the derivative values that have tremendous importance in the determination of caloric properties. Supporting Information, Table 4S reports the coefficients, the standard errors of the coefficients, and the standard deviations of the fits for the seven isochores. Coefficients in Supporting Information, Table 4S contain a sufficient number of significant figures to ensure a repeatability of six digits in p* and five digits in dp*/dT. The residual deviations of the fit (p* − pfit*) also appear in Supporting Information, Table 3S and Figure 4. For temperatures above 220 K, the scatter of the residuals is less than ± 0.01 MPa for the quadratic model, which is sufficient for the purpose of this work. Experimental values of Z* and dp*/dT and values predicted by REFPROP, PR, and RK appear in Supporting Information, Tables 5S and 6S, respectively. Figure 5 illustrates the relative differences between the measured and predicted values of Z*, and Figure 6 illustrates the relative differences between the D
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Figure 7. Integrand of Helmholtz energy residual (δIA) for T = 303.15 K: ●, REFPROP; △, Peng−Robinson; ☆, Redlich−Kwong.
Figure 5. Relative deviations of experimental Z* from values calculated using EoS: ●, REFPROP; △, Peng−Robinson; ☆, Redlich−Kwong.
Figure 8. Integrand of entropy residual (δIS) for T = 303.15 K: ●, REFPROP; △, Peng−Robinson; ☆, Redlich−Kwong.
Figure 6. Relative deviations of experimental dp*/dT from values calculated using EoS. ●, REFPROP;Δ, Peng−Robinson; ☆, Redlich− Kwong.
columns of Supporting Information, Tables 7S, 8S, and 9S respectively. The values of the residual properties calculated using different EoS show only slight differences among them that can be attributed to the error in the fit of the integrands. To provide insight into the practical implications of these differences, they can be translated into an equivalent changes in temperature for entropy (δTS) and internal energy (δTU) using
measured and predicted values of dp*/dT. Finally the values of δIA, δIS, and δIU appear in Supporting Information, Tables 7S, 8S, and 9S, respectively. The cubic version of eq 23 is the best choice for the isothermal integration of δIA and δIs with respect to ρ*. The values of coefficients, their standard errors, and the standard deviations of the fits appear in Supporting Information, Tables 10S and 11S. Also, Figures 7 and 8 illustrate the behavior of δIA and δIS as a function of density for T = 303.15 K. The behavior in these figures is similar for all the isotherms in this work. These figures indicate that the integrands determined using REFPROP are significantly smaller than those determined using PR or RK, and that the residual adjustment factors δAr/ (RT) and δSr/R are significantly smaller for REFPROP. Values of δAr/(RT) and δSr/R resulting from the integration appear in Supporting Information, Tables 7S and 8S. For determining δUr/(RT) two methods exist as expressed in eqs 13 and 24; both methods were used to test for consistency between them. Values of δUr/(RT) determined using eq 24 appear in Supporting Information, Table 9S. Values of the residual properties for Helmholtz energy (Ar/ (RT)), entropy (Sr/R), and internal energy (Ur/(RT)) based upon experimental data calculated using eq 7 appear in the last
δS r T ⎛ δS r ⎞ ⎜ ⎟ = Cv (Cv /R ) ⎝ R ⎠
(28)
T ⎛ δU r ⎞ δU r ⎜ ⎟ = Cv (Cv /R ) ⎝ RT ⎠
(29)
δTS = T δTU =
Figures 9 and 10 show the differences between the residual values for Sr/R and Ur/(RT) using different EoS. The equivalent differences in temperature were calculated according to eqs 28 and 29 finding that δTS and δTU are ≤ 0.5 K for temperatures above 225 K, which suggests that the small differences in the values of the residual properties calculated using different EoS are acceptable for most engineering applications. The equivalent temperature differences δTS and δTU appear as error bars in Figures 9 and 10. Fits of eq 27 to the residual adjustment factors (δXr/(RT)) presented in Supporting Information, Tables 7S, 8S, and 9S E
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Figure 9. Differences between residual values of entropy (Sr/R) using different equation of state: ▲, Sr/RREFPROP − Sr/RPR; ○, Sr/RREFPROP − Sr/RRK.
Figure 11. Residual deviations of δAr/(RT) predicted by the fit for different EoS: ○, (δAr/(RT) − δAr/(RT)fit)PR; ●, (δAr/(RT) − δAr/ (RT)fit)RK.
Figure 10. Differences between residual values of internal energy (Ur/ (RT)) for different EoS: ▲, Ur/(RT)REFPROP − Ur/(RT)PR; ○, Ur/ (RT)REFPROP − Ur/(RT)RK..
Figure 12. Residual deviations of δSr/R predicted by the fit for different EoS: ○, (δSr/R − δSr/Rfit)PR; ●, (δSr/R − δSr/Rfit)RK.
provide correlations for extending the calculations to the entire temperature and pressure range of the isochoric data.4 The values of (δXr/(RT)) determined using REFPROP are sufficiently small that it is not necessary to use, for practical applications, a residual adjustment factor when calculating properties for this mixture using REFPROP. Supporting Information, Tables 12S to 14S contain the coefficients, standard errors of the coefficients, and standard errors of the fits of eq 27 to calculated δAr/(RT), δSr/R, and δUr/(RT) respectively for the PR and RK. Figures 11 through 13 show residuals for the fits of δAr/(RT), δSr/R and δUr/(RT), respectively.
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CONCLUSIONS This work presents a new methodology for determining residual energies and entropies for fluid samples from isochoric and isothermal (p−ρ−T) measurements. For the ternary mixture used here as an example, the residual adjustment factors are about 2 orders of magnitude smaller than the residual properties predicted by REFPROP. On the other hand, PR and RK yield residual adjustment factors approximately 1 order of magnitude smaller than the respective residual properties. This implies that for this gas sample, the value of residual properties can be obtained with high confidence
Figure 13. Residual deviations of δUr/R predicted by the fit using different equations of state: ○, (δUr/(RT) − δUr/(RT)fit)PR; ●, (δUr/ (RT) − δUr/(RT)fit)RK.
directly from REFPROP, and it is useful to use the global fits to calculate corrections for the residual properties predicted by PR and RK. However, even though the residual adjustment factors are much larger for the PR and RK than for REFPROP, the residuals calculated using each of the three equations agree F
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Apparatus from (250 to 450) K with Pressures up to 150 MPa: Part II. J. Chem. Eng. Data 2011, 56 (10), 3766−3774. (7) Zhou, J.; Patil, P.; Ejaz, S.; Atilhan, M.; Holste, J. C.; Hall, K. R. (p, V, m, T) and Phase Equilibrium Measurements for a Natural Gaslike Mixture Using an Automated Isochoric Apparatus. J. Chem. Thermodyn. 2006, 38 (11), 1489−1494. (8) Peng, D.-Y.; Robinson, D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15 (1), 59−64. (9) Redlich, O.; Kwong, J. N. S. On the Thermodynamics of Solutions. V. An Equation of State. Fugacities of Gaseous Solutions. Chem. Rev. 1949, 44 (1), 233−244. (10) Zhou, J. Automated Isochoric Apparatus for PVT and Phase Equilibrium Studies of Natural Gas Mixtures. Ph.D. Dissertation, Texas A&M University, College Station, TX, USA, 2005. (11) Acosta-Perez, P. L.; Cristancho, D. E.; Mantilla, I. D.; Hall, K. R.; Iglesias-Silva, G. A. Method and Uncertainties to Determine Phase Boundaries from Isochoric Data. Fluid Phase Equilib. 2009, 283 (1−2), 17−21.
within acceptable practical limits. This means that the procedure may be used with any EoS in practical applications to utilize isochoric (p−ρ−T) measurements to improve the accuracy of energy and entropy calculations. This work also provides valuable insight into experimental design. For example, when determining the isochoric reference densities, it is highly desirable to have isochoric measurements at the same temperature as the isothermal measurements in the MSD (or other isothermal device). This procedure assumes that the amount of sample in the isochoric apparatus is not determined directly. If the amount of material is measured directly, isothermal density measurements are not necessary. Also, accurate determinations of (∂p/∂T)ρ require a sufficient number of measurements, especially near the ends of the temperature and pressure ranges, to establish the derivatives as accurately as possible from the experiment. The derivatives may be used as higher order constraints when fitting complex equations of state, and they are involved directly in the calculations of energies and entropies.
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ASSOCIATED CONTENT
S Supporting Information *
Additional tables as described in the text. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Phone: +1.979.845.3384. Fax: +1.979.845.6446. Present Addresses †
The Dow Chemical Company, Freeport TX, USA. Bryan Research & Engineering, Bryan, TX, USA. § Currently on assignment at Texas A&M University at Qatar, Doha QATAR. ‡
Funding
The authors gratefully acknowledge financial support from the Jack E. & Frances Brown Chair Endowment and the Texas A&M Engineering Experiment Station. Notes
The authors declare no competing financial interest.
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REFERENCES
(1) Kunz, O., Klimeck, R., Wagner, W., and Jaeschke, M., The GERG-2004 wide-range equation of state for natural gases and other mixtures. Fortschr.-Ber. VDI: 2007; Vol. 6. (2) Kunz, O.; Wagner, W. J. Chem. Eng. Data 2012, 57 (11), 3032− 3091. (3) Lemmon, E. W., Huber, M. L., McLinden, M. O., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP, version 9.1; Standard Reference Data Program; National Institute of Standards and Technology: Gaithersburg, Md, 2013. (4) Cristancho, D. E.; Mantilla, I. D.; Coy, L. A.; Tibaduiza, A.; OrtizVega, D. O.; Hall, K. R.; Iglesias-Silva, G. A. Accurate P−ρ−T Data and Phase Boundary Determination for a Synthetic Residual Natural Gas Mixture. J. Chem. Eng. Data 2010, 56 (4), 826−832. (5) Atilhan, M.; Aparicio, S.; Ejaz, S.; Cristancho, D.; Hall, K. R. P−ρ−T Behavior of a Lean Synthetic Natural Gas Mixture Using Magnetic Suspension Densimeters and an Isochoric Apparatus: Part I. J. Chem. Eng. Data 2011, 56 (2), 212−221. (6) Atilhan, M.; Aparicio, S.; Ejaz, S.; Cristancho, D.; Mantilla, I.; Hall, K. R. p−ρ−T Behavior of Three Lean Synthetic Natural Gas Mixtures Using a Magnetic Suspension Densimeter and Isochoric G
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