Optimization of the Reduced Temperature ... - ACS Publications

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Optimization of the Reduced Temperature Associated with Peng− Robinson Equation of State and Soave−Redlich−Kwong Equation of State To Improve Vapor Pressure Prediction for Heavy Hydrocarbon Compounds Zehua Chen and Daoyong Yang* Petroleum Systems Engineering, Faculty of Engineering and Applied Science, University of Regina, Regina, Saskatchewan S4S 0A2, Canada ABSTRACT: A pragmatic technique has been developed to optimize the reduced temperature for the acentric factor associated with the Peng−Robinson equation of state (PR-EOS) and the Soave−Redlich−Kwong equation of state (SRK-EOS) by minimizing the deviation between the measured and calculated vapor pressures for nonhydrocarbon compounds and hydrocarbon compounds including heavy alkanes up to n-tritetracontane (n-C43H88) under different conditions. All the compounds are divided into four categories, that is, lightsaturated hydrocarbons, heavy-saturated hydrocarbons, aromatic compounds, and other compounds, among which the first three categories are used to examine their effects on the optimum reduced temperature for the entire database. By redefining the reduced temperature, three existing alpha functions together with the newly developed alpha functions for the PR-EOS as well as one existing alpha function and the newly developed alpha functions for the SRK-EOS are then used to evaluate their respective accuracy of predicting vapor pressures for pure substances. As for the newly expanded database with 1880 data points, the reduced temperature has its optimum value of 0.59 for the acentric factor for both the PR-EOS and SRK-EOS corresponding to the minimum absolute average relative deviations (AARDs) of 4.04% and 4.08%, respectively. Therefore, it is recommended that a reduced temperature of 0.60 be used for predicting the vapor pressures of heavy hydrocarbon compounds and their mixtures, yielding AARDs of 4.08% and 4.12% and maximum absolute relative deviations (MARDs) of 77.20% and 79.74% for the PR-EOS and SRK-EOS, respectively. Among the three subdivided categories, the heavysaturated hydrocarbons impose the largest effect on the optimum reduced temperature for the entire database, while the aromatic compounds take the second place, and the light-saturated hydrocarbons have the smallest effect. The sensitivity of the calculated alpha functions reduces with an increase in the reduced temperature, while it remains no change as the acentric factor varies. Finally, the newly developed alpha functions all lead to the minimum AARDs for the corresponding compound categories or for the entire database compared with existing alpha functions except for the light-saturated hydrocarbons. Also, the newly developed alpha function leads to the most accurate predictions of vaporization enthalpy with an AARD of 1.93% and MARD of 8.03% for the PR-EOS as well as an AARD of 2.02% and MARD of 7.76% for the SRK-EOS compared with the existing alpha functions.

1. INTRODUCTION

efforts have been made to examine the effect of the reduced temperature for the acentric factor on the prediction of vapor pressure for a pure substance. It is of practical and fundamental importance, therefore, to develop an accurate representation of the vapor pressures for pure substances pertaining to heavy hydrocarbon compounds, allowing for more accurate modeling of the phase behavior for their mixtures under various conditions. Because of their simplicity and good accuracy, the twoparameter CEOSs have been extensively used to predict phase behavior for the petroleum and chemical industries. In particular, the Peng−Robinson equation of state (PR-EOS)6

The cubic equation of state (CEOS) plays a significant role in simulating phase behavior pertaining to enhanced oil recovery (EOR) processes, such as CO2 huff and puff,1 vapor extraction (VAPEX),2 and solvent expanded-steam assisted gravity drainage (ES-SAGD).3 Such EOR processes are directly associated with the mixture of the injected gas/liquid (e.g., CO2, propane, butane, hexane, and diluents) and crude oil.4 As for heavy oil, its components are mainly composed of heavysaturated hydrocarbons and aromatic compounds,5 the vapor pressures of which cannot be accurately reproduced by using the existing CEOS due mainly to the limitations of the alpha function which is used to characterize the deviation of a pure substance from its ideal behavior. Physically, the alpha function depends on the acentric factor which is a measure of the nonsphericity or centricity of a molecule. So far, no explicit © 2017 American Chemical Society

Received: June 2, 2017 Accepted: August 18, 2017 Published: August 31, 2017 3488

DOI: 10.1021/acs.jced.7b00496 J. Chem. Eng. Data 2017, 62, 3488−3500

Journal of Chemical & Engineering Data

Article

and the Soave−Redlich−Kwong equation of state (SRK-EOS)7 have been widely used in the petroleum industries, while numerous efforts have been made to modify the original alpha function in the attractive term for more accurate vapor pressure prediction for either pure substances or their mixtures.8−17 All modifications on the alpha function pertain to changes in either the alpha function formulation or databases used for developing the alpha functions. The Soave-type alpha function has been found to be the most widely used one for the two-parameter CEOSs; however, its original version does not provide accurate vapor pressure prediction for heavy hydrocarbon compounds because limited data were made available for both critical properties and vapor pressures of heavy hydrocarbon compounds; that is, only those properties and vapor pressures of hydrocarbon compounds lighter than C10 were used for developing the original alpha function pertaining to both the SRK-EOS and PR-EOS.15 On the basis of the 6000 vapor pressure data points of 237 diverse hydrocarbon compounds, including heavy alkanes up to nhexacosane (n-C26H54), Nji et al.16 modified the Soave-type alpha function for the PR-EOS, allowing for more accurate prediction of vapor pressures of heavy hydrocarbon compounds. Although the Soave-type alpha function is the most popular one, it fails to yield a finite and positive value as temperature approaches infinity,15 which is one of the three basic criteria for the alpha function.9,12 Alternatively, a logarithm-type alpha function was developed by Heyen9 and later modified by other researchers.11−14 Twu et al.13 pointed out that the relationship between alpha function and acentric factor is a linear function instead of a high-order polynomial one. Subsequently, Gasem15 confirmed that the linear acentric factor dependence of the alpha function for n-paraffins can be extended to C28. This is another reason why the logarithmictype alpha functions are used to replace the Soave-type alpha functions in the literature. In addition to expanding the database by adding heavy alkanes up to C28H58, Gasem15 proposed a modified logarithmic-type alpha function to reproduce vapor pressures of 1100 data points with an absolute average relative deviation (AARD) of 1.10%, which is more accurate than other alpha functions. Recently, Li and Yang17 have developed a modified alpha function which combines the advantages of both the Soave-type alpha function and the logarithm-type alpha function by using a comprehensive database comprising 59 hydrocarbon and nonhydrocarbon compounds, including heavy alkanes up to n-tritetracontane (nC43H88). This modified alpha function has been successfully used to more accurately predict the phase behavior of both pure substances17 and alkane solvent(s)−CO2−heavy oil systems by treating heavy oil as one pseudocomponent18−22 and multiple pseudocomponents.23−35 By definition, the alpha function includes two important parameters: acentric factor and reduced temperature. The acentric factor is originally defined at the reduced temperature of 0.70 by Pitzer et al.,36,37 and such a definition is still the most widely used one. However, a generalized vapor pressure equation with this definition yields a good accuracy at reduced temperatures from 0.70 and 1.00, but a lower accuracy at a lower reduced temperature.13,14,38 Thus, the reduced temperature at which the acentric factor is defined can impose a significant impact on the vapor pressure prediction for a pure substance, especially for heavy hydrocarbon compounds. To improve its prediction accuracy, Twu et al.38 empirically redefined the acentric factor for the alpha function at a reduced

temperature of 0.50 which leads to a more accurate vapor pressure prediction compared with the original acentric factor, especially at low reduced temperatures; however, such an improvement was made by generalizing vapor pressure correlations based on the Pitzer’s corresponding states rather than by regressing a new alpha function. Recently, Li and Yang17 empirically redefined the acentric factor at the reduced temperature of 0.60 for their modified alpha function, resulting in a maximum absolute relative deviation (MARD) of 21.22% for vapor pressure predictions for the 1163 data points, compared with a MARD of 40.56% resulting from the original acentric factor. Although the reduced temperature for the acentric factor shows a great impact on vapor pressure prediction, few attempts have been made to systematically examine the effects of reduced temperature for the acentric factor on the vapor pressure prediction for pure substances by CEOSs, especially for substances associated with heavy hydrocarbon compounds. In this study, a pragmatic technique, which is found to be able to conveniently and accurately develop alpha functions and predict vapor pressure for pure substances based on the experimentally measured vapor pressure data, has been developed to optimize the reduced temperature for the acentric factor for both the PR-EOS and SRK-EOS by minimizing the deviation between the measured and calculated vapor pressures for hydrocarbon and nonhydrocarbon compounds including heavy alkanes up to n-tritetracontane (n-C43H88). As for the existing database of 1163 data points,17 an expansion has been made by adding an extra 717 data points, which include vapor pressures of heavy alkanes at low reduced temperatures and those of aromatic compounds. By redefining the reduced temperature, different alpha functions are obtained to predict the vapor pressure for a given hydrocarbon compound. Three existing alpha functions and the newly developed alpha functions for the PR-EOS as well as one existing alpha function and the newly developed alpha functions for the SRK-EOS are then used to evaluate their respective accuracy of predicting vapor pressures for pure substances. Also, the effects of different compounds on the prediction of vapor pressure are examined, while sensitivity analysis is performed to identify the influence of different parameters on the calculated alpha functions. Finally, the respective prediction accuracy of vaporization enthalpy by the newly developed alpha function and the existing alpha functions is examined and compared.

2. MATHEMATICAL FORMULATIONS 2.1. EOS. Both PR-EOS6 and SRK-EOS7 have been widely used in the petroleum and chemical industries; they can be respectively expressed as follows: PR-EOS: P=

RT a − V−b V (V + b) + b(V − b)

a = acα(Tr , ω)

3489

(1) (2a)

ac =

0.457235R2Tc 2 Pc

(2b)

b=

0.0777969RTc Pc

(2c) DOI: 10.1021/acs.jced.7b00496 J. Chem. Eng. Data 2017, 62, 3488−3500


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Table 1. Vapor Pressure Database

a

name

formula

Tr

Pexp v (kPa)

NPTSa

ref

pyridine y-picoline toluene thianaphthene ethylbenzene o-xylene quinoline cumene 2-ethyltoluene 3-ethyltoluene 1,2,3-trimethylbenzene pseudocumene naphthalene tetralin tetralin n-butylbenzene sec-butylbenzene l, 2-diethylbenzene cis-deca-hydronaphthalene trans-Deca-hydronaphthalene 1-methylnaphtalene 1:3:5-trimethyl-ethylbenzene cyclohexylbenzene 5-tert-butyl-m-xylene n-hexylbenzene bicyclohexyl fluorene diphenylmethane phenanthrene n-nonylbenzene n-undecylbenzene 1,3,5-triter-butyl-benzene n-eicosane n-heneicosane n-docosane n-tricosane n-tetracosane n-pentacosane n-octacosane n-nonacosane n-triacontane n-hentriacontane n-tetratriacontane n-hexatriacontane

C5H5N C6H7N C7H8 C8H6S C8H10 C8H10 C9H7N C9H12 C9H12 C9H12 C9H12 C9H12 C10H8 C10H12 C10H12 C10H14 C10H14 C10H14 C10H18 C10H18 C11H10 C11H16 C12H16 C12H18 C12H18 C12H22 C13H10 C13H12 C14H10 C15H24 C17H28 C18H30 n-C20H42 n-C21H44 n-C22H46 n-C23H48 n-C24H50 n-C25H52 n-C28H58 n-C29H60 n-C30H62 n-C31H64 n-C34H70 n-C36H74

0.557−0.585 0.593−0.633 0.497−0.881 0.563−0.836 0.534−0.665 0.496−0.728 0.605−0.861 0.544−0.663 0.543−0.661 0.539−0.655 0.547−0.678 0.493−0.637 0.472−0.819 0.436−0.478 0.626−0.809 0.444−0.693 0.442−0.562 0.546−0.675 0.530−0.667 0.532−0.671 0.380−0.670 0.516−0.900 0.465−0.531 0.463−0.654 0.479−0.664 0.594−0.808 0.464−0.518 0.513−0.852 0.430−0.543 0.502−0.630 0.512−0.565 0.497−0.651 0.462−0.550 0.478−0.595 0.478−0.522 0.519−0.582 0.489−0.566 0.496−0.570 0.512−0.575 0.517−0.541 0.515−0.578 0.518−0.557 0.528−0.576 0.537−0.592

23.890−44.260 26.340−59.380 3.090−1683.700 15.380−1059.550 6.413−103.990 2.006−270.000 42.120−1489.000 6.402−83.753 6.402−83.753 6.402−83.753 6.402−103.980 0.933−45.543 1.001−854.300 0.153−0.868 48.250−684.500 0.107−103.980 0.186−10.790 6.402−103.980 5.529−102.730 5.529−102.730 0.006−102.760 1.573−1059.550 0.250−2.733 0.112−37.782 0.185−40.695 9.591−346.750 0.213−1.683 0.879−670.230 0.028−2.912 0.155−10.100 0.117−0.908 0.150−18.825 0.001−0.135 0.002−0.611 0.001−0.030 0.013−0.242 0.001−0.089 0.002−0.094 0.001−0.044 0.002−0.008 0.001−0.080 0.001−0.019 0.001−0.031 0.001−0.052

10 7 24 30 20 33 19 15 15 15 20 16 45 4 8 13 9 20 19 19 27 45 6 14 14 23 7 35 9 12 6 14 10 9 12 7 18 12 15 5 20 16 16 14

39 39 40−42 43 44 45 46, 47 44 44 44 44 48 41, 49, 50 40 46 40, 44 40 44 50 50 50, 51 52 40 40 40 43 51 43, 53 54 40 40 40 55, 56 57 55 57 55, 56 57 55, 56 57 55 55 55 55

NTPS denotes the number of experimental data points.

the universal gas constant, Tc is the critical temperature, and Pc is the critical pressure. 2.2. Vapor Pressure Database. In this study, the existing database,17 which is composed of 59 hydrocarbon and nonhydrocarbon compounds, including heavy alkanes up to n-tritetracontane (n-C43H88), has been expanded with the available data collected from literature39−57 (see Table 1). As can be seen, heavy oil components are closely associated with heavy alkanes and aromatic compounds according to the SARA analysis.5 Thus, the existing database has been expanded by adding vapor pressures of heavy alkanes at lower reduced temperatures and vapor pressures of aromatic compounds. In total, there are 90 compounds in the newly expanded database. It should be noted that 7 aromatic compounds and 16 heavy

SRK-EOS: P=

RT a − V−b V (V + b)

ac =

0.42748R2Tc 2 Pc

(4a)

b=

0.08664RTc Pc

(4b)

(3)

where α(Tr, ω) is the alpha function, which is the function of both the reduced temperature Tr and the acentric factor ω, T is the temperature, P is the pressure, V is the molar volume, R is 3490



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Table 2. Alpha Functions model no.

functions

EOS 2

1

α = [1 + (0.37464 + 1.54226ω − 0.26992ω )(1 −

2

2 α = [1 + (0.388187 + 1.56132ω − 0.29014ω2 + 0.0617513ω3)(1 − T0.5 r )]

3 4 5

α = exp[(2 + 0.836Tr)(1 −

2 Tr0.134 + 0.508ω − 0.0467ω )]

α = exp{k1(1 − Tr) + l ln[1 + k 2(1 − 2

2

Tr )] }

α = [1 + (0.48 + 1.574ω − 0.176ω )(1 −

Tr0.5)]2

ω = −log(Pr)Tr − 1

AARD =

PR

6

PR

16

PR

15

PR and SRK

17 and this work

SRK

7

1 n

n

∑ 1

Pv cal − Pv exp Pv exp

(7)

where n is the number of data points, Pcal v is the calculated vapor exp pressure, and Pv is the measured vapor pressure. To compare the superiority of the newly developed alpha functions, three existing alpha functions for the PR-EOS and one existing alpha function for the SRK-EOS are used to predict vapor pressures for pure substances, respectively. The alpha functions used for evaluation are listed in Table 2. 2.4. Enthalpy of Vaporization. As an important thermodynamic property, the enthalpy of vaporization is also used to further validate the alpha function developed in this study with corresponding vaporization enthalpy data available in the literature. Specific expressions of the fundamental enthalpy departure equation by using PR-EOS and SRK-EOS can be respectively written as follows:7,17

(5)

where Pr is the reduced pressure at the certain Tr and is calculated as follows:

Pv Pc

ref

substances, provided that Pc, Tr, and the alpha function are already known. The direct calculation of alpha function and vapor pressure makes the development of a new alpha function and prediction of vapor pressure both expedient and accurate. All the regression is performed using the SigmaPlot software (version 2013). In this study, the newly developed alpha function takes the form as defined earlier17 because all the three required criteria can be satisfied simultaneously. It should be noted that all the new alpha functions are regressed using the expanded database. The absolute average relative deviation (AARD) is used as the objective function to evaluate the prediction accuracy of each alpha function, that is,

alkanes (carbon number is larger than 9) are included in the existing database17 and that the reduced temperatures of the heavy alkanes are in the range of 0.58−0.74. More specifically, the Tr range of the C10−C20, C20−C30, and C30+ are 0.55−0.82, 0.59−0.71, and 0.63−0.66, respectively. As for the expanded database, there are 34 aromatic compounds and 20 heavy alkanes and the reduced temperatures of the heavy alkanes are in the range of 0.53−0.71 (i.e., the Tr range of C10−C20, C20− C30, and C30+ are 0.55−0.82, 0.50−0.64, and 0.57−0.66, respectively). In addition, some polycyclic aromatic hydrocarbons have been added in the expanded database, such as naphthalene, 1-methylnaphtalene, fluorine, phenanthrene. Such aromatic compounds can be more closely related to resin and asphaltene which are found to be composed of condensed aromatic rings.58 To facilitate the corresponding analysis, all the compounds are divided into four categories, that is, light-saturated hydrocarbons (alkanes with carbon number 0 (12)

where n is the total number of the compound types included in the database. These constraints can ensure that the coefficient k1 is always positive. Other types of constraints can also be applied to achieve the same purpose according to the properties of the quadratic function. For example, if a3 > 0, the constraint can be set as follows:

a1 −

a2 2 >0 4a3

(13)

This constraint ensures k1 > 0 no matter what the value of the acentric factor is.17 In other cases, if a3 < 0, there is only a need to exert the constraint that the k1 value is positive at the minimum acentric factor and the maximum acentric factor for all the compounds in the database. It should be noted that the coefficients in the alpha function vary with the change of constraints for the same acentric factor. The coefficients may affect some properties of the alpha function. However, the AARDs for vapor pressure prediction using different alpha functions (due to different constraints) at the same acentric factor are almost the same. 3.4.2. AARDs for Different Alpha Functions. Table 5 shows the AARDs for vapor pressure prediction for different categories with different alpha functions for the PR-EOS. It is found that the original alpha function (i.e., alpha function a1 in Table 5, which corresponds to model no. 1 in Table 2) yields a small AARD (i.e., 0.49%) for the light-saturated hydrocarbons. Owing to the limited database for regressing this alpha function,6,15 however, it does not obtain a small AARD for the heavy-saturated hydrocarbons (i.e., 47.42%) and aromatic compounds (i.e., 8.79%). Thus, this alpha function yields a high AARD (i.e., 13.38%) for the entire database. With the addition of heavy hydrocarbons and other compounds to the original database,15,16 the subsequently developed alpha functions b and 3495



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greater the coefficient number is, the higher the prediction accuracy of the alpha functions will be. Table 5 also shows the MARDs of vapor pressure predictions for different categories with different alpha functions for the PR-EOS. As can be seen in Table 5, similar to the trend of the AARDs for the PR-EOS, the original alpha function yields a small MARD (i.e., 2.39%) for the light-saturated hydrocarbons; however, it leads to very high MARDs for the heavy-saturated hydrocarbons (i.e., 223.67%) and aromatic compounds (i.e., 99.45%). Thus, this alpha function yields a high MARD (i.e., 223.67%) for the entire database. The subsequently developed alpha functions b and c yield much smaller MARDs for the heavy-saturated hydrocarbons and entire database compared with the original alpha function, respectively, though all these three alpha functions (i.e., a1, b, c) yield high MARDs (all the three MARDs are higher than 76.72%) for the aromatic compounds. In contrast, alpha function d4 which uses the acentric factor defined at a reduced temperature of 0.59 yields smaller MARDs for the heavy-saturated hydrocarbons (i.e., 84.73%), entire database (i.e., 84.73%), and especially aromatic compounds (i.e., 26.31%) compared with previous alpha functions. It should be noted that the newly developed alpha functions using the optimized reduced temperatures for the acentric factor all yield relatively low MARDs for the corresponding categories (except for the light-saturated hydrocarbons) or the entire database, though they do not necessarily yield the lowest MARDs due to the fact that the alpha regression is based on AARD rather than MARD. It is expected that the prediction accuracy of an alpha function is also related to the database used for its development. Thus, the four different function formats in Table 2 are used to develop alpha functions using acentric factors defined at different reduced temperatures and the same database (i.e., the expanded database in this study) to make a comparison. The corresponding AARDs of different function formats with reduced temperature for the acentric factor are listed in Table 6 and plotted in Figure 6. As can be seen, as a whole, by using the same database, the four function formats

Figure 6. Variation of AARDs of different alpha functions regressed using different function formats and the same database.

yield a similar prediction accuracy, indicating that the database used for developing the alpha functions is also an important factor to improve their prediction accuracy. However, there are still differences between the four function formats. Among the three existing alpha function formats (i.e., models nos. 1, 2, and 3), model no. 2 yields the highest accuracy at reduced temperatures smaller than 0.59, while model no. 3 yields the highest accuracy at reduced temperatures larger than 0.59. In summary, model no. 1 yields the highest AARD among the three alpha function formats because it has the lowest number of coefficients. Because the function format in this study (i.e., model no. 4) has the highest number of coefficients and also because it combines the advantages of both the Soave-type and logarithm-type alpha functions, it yields the highest prediction accuracy among the four function formats when no constraints are exerted on the regression. However, the prediction accuracy can be compromised slightly when constraints are exerted on the regression to ensure that the developed alpha function meets the third criterion. Still, model no. 4 with constraints yields the similar prediction accuracy as those of model nos. 2 and 3. It should be noted that, physically, the third criterion (i.e., the alpha function approaches a finite value as the temperature approaches infinity9,17) cannot be satisfied by the Soave-type alpha functions, indicating that these alpha functions do not yield correct results under certain conditions. Take model no. 1 as an example, when Tr approaches infinity, the term (1 − T0.5 r ) will approach negative infinity. Then, the alpha function will approach infinity, while ω is a constant value for a given pure substance. Also, the logarithm-type alpha function, that is, model no. 3 in Table 2, can meet the third criterion, provided that the term 0.134 + 0.508ω − 0.0467ω2 is forced to be positive. Thus, constraints are still needed (for the expanded database in this study) to ensure the logarithm-type alpha function meets the third criterion, though the prediction accuracy will be slightly compromised. A higher accuracy for vapor pressure prediction for aromatic compounds is very significant for vapor pressure prediction for heavy oil components because all the three important categories, that is, the aromatics, resin, and asphaltene, are related to aromatic compounds.5 Another important category of heavy oil components is the heavy-saturated hydrocarbons.18−24 Since the critical temperatures of heavy-saturated hydrocarbons are relatively high, the temperature range of the

Table 6. AARDs for Alpha Functions Regressed with Different Function Formats and the Same Database AARDs by using different function formats (%) model

model no. 4

reduced temp

no. 1

no. 2

no. 3

with constraints

without constraints

0.45 0.50 0.52 0.54 0.56 0.58 0.59 0.60 0.61 0.62 0.64 0.66 0.68 0.70 0.75 0.80

17.20 9.44 7.40 5.85 4.85 4.26 4.13 4.13 4.23 4.40 4.91 5.61 6.49 7.56 12.03 23.63

16.77 8.67 6.68 5.26 4.47 4.09 4.07 4.12 4.25 4.41 4.89 5.50 6.29 7.38 11.95 23.27

17.33 9.45 7.42 5.83 4.82 4.19 4.04 4.01 4.07 4.23 4.71 5.41 6.27 7.34 11.75 23.05

17.45 8.81 6.87 5.40 4.52 4.09 4.04 4.07 4.19 4.36 4.86 5.51 6.33 7.24 11.59 22.57

16.37 8.61 6.64 5.22 4.42 3.97 3.91 3.94 4.04 4.20 4.65 5.27 6.11 7.21 11.23 22.25 3496



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data points of the heavy-saturated hydrocarbons is also high.17 For instance, the average temperature of the data points of the heavy-saturated hydrocarbons in the existing database is 252.20 °C, which is higher than the operating temperature of regular solvent recovery processes for enhancing heavy-oil recovery.1−4 By adding vapor pressures of heavy-saturated hydrocarbons at average temperature of 166.80 °C to the newly expanded database, the data points can be more related to the operating temperature range of the relevant solvent recovery processes. Also, it is found that the optimized alpha functions for the entire database, heavy-saturated hydrocarbons, and aromatic compounds, have almost the same optimum reduced temperature for the acentric factor, which is around 0.59 (see Table 4). Thus, one optimum alpha function can be used for all the categories and entire database, so 0.60 is recommended as the reduced temperature for the acentric factor for the pragmatic purpose. Table 7 shows the AARDs and MARDs for different categories with different alpha functions for the SRK-EOS.

3.4.3. Optimum Alpha Functions. The coefficients in the optimum alpha functions for different categories for both the PR-EOS and SRK-EOS are listed in Table 8, including the recommended alpha functions for the entire database which use the acentric factor defined at the reduced temperature of 0.60. Also, all the alpha functions can be expressed as equation 14 together with 7 coefficients in different alpha functions tabulated in Table 8, respectively. α = exp{(C1 + C2ω + C3ω 2)(1 − Tr) + C4 ln[1 + (C5 + C6ω + C7ω 2)(1 −

(14)

3.5. Enthalpy of Vaporization. Table 9 shows the AARD of vaporization enthalpy prediction for different compounds by using the existing alpha functions and the newly developed alpha function. As can be seen, for the PR-EOS, the original alpha function (i.e., alpha function a1) yields the largest predictive error among the four alpha functions, while the other three alpha functions yield similar predictive accuracy due to the fact that the latter three alpha functions can predict the vaporization enthalpy for heavy alkanes much more accurately. Furthermore, the alpha function developed in this study yields the lowest AARD. As for the SRK-EOS, the alpha function developed in this study yields lower AARD of vaporization enthalpy prediction than the original alpha function (i.e., alpha function a2). Also, for both the PR-EOS and SRK-EOR, the newly developed alpha functions yield the lowest MARDs among the evaluated alpha functions. These results indicate that the newly developed alpha functions can both accurately predict the vapor pressure and vaporization enthalpy for heavy hydrocarbon compounds.

Table 7. AARDs and MARDs for Different Categories with Different Alpha Functions for the SRK-EOSa alpha functions category

a2

A B C D

0.93 15.00 8.10 7.30

A B C D

7.34 120.46 75.69 120.46

e1

e2

AARDs (%) 0.68 0.90 11.46 9.64 5.99 3.76 5.44 4.12 MARDs (%) 3.68 1.70 54.90 79.74 47.13 29.69 54.90 79.74

e3

e4

0.96 10.51 3.46 4.30

0.92 9.66 3.57 4.08

1.88 104.88 25.82 104.88

1.70 86.85 25.48 86.85

Tr )]2 }

4. CONCLUSIONS A pragmatic technique has been developed to optimize the reduced temperature for acentric factor for both the PR-EOS and SRK-EOS by minimizing the deviation between the measured and calculated vapor pressures for hydrocarbon and nonhydrocarbon compounds, including heavy alkanes up to ntritetracontane (n-C43H88). Obviously, the optimum reduced temperature 0.59 for the acentric factors for both the PR-EOS and SRK-EOS are determined, though the optimum value of 0.60 shall be used for pragmatic purposes. The optimum reduced temperature for the acentric factor is found to be related to the reduced temperatures of the data points. The higher the reduced temperature range of the data points is, the higher the optimum reduced temperature for the acentric factor will be. Among the three subdivided categories, the heavy-

a

Note: a2 is the alpha function no. 5 in Table 2, it uses acentric factor defined at reduced temperature of 0.70; e1, e2, e3, and e4 are optimum alpha functions for the light-saturated hydrocarbons, heavy-saturated hydrocarbons, aromatic compounds, and entire database, respectively, and they use acentric factors defined at reduced temperatures of 0.66, 0.60, 0.57, and 0.59, respectively.

These results follow the similar trend for the comparison between the existing alpha functions and the newly developed alpha functions as those for the PR-EOS (see Table 5). For the SRK-EOS, all the optimum alpha functions yield the minimum AARDs for the targeted categories or entire database.

Table 8. Coefficients of the Optimum Alpha Functions for Different Categories for Both the PR-EOS and SRK-EOS coefficients categories entire database

heavy-saturated hydrocarbons aromatic compounds

a

EOS

optimum Tr

C1

C2

C3

C4

C5

C6

C7

PR SRK PR SRK PR SRK PR SRK

0.59 0.59 0.60a 0.60a 0.59 0.60 0.57 0.57

0.40692 0.22914 0.33730 0.20463 0.40692 0.20463 0.55328 0.32072

−0.33004 −0.13707 −0.26346 −0.04711 −0.33004 −0.04711 −0.45839 −0.33137

0.06861 0.04946 0.05297 0.02697 0.06861 0.02697 0.09668 0.08856

1.01311 0.73559 1.01711 0.70851 1.01311 0.70851 0.99019 0.79412

−0.52578 −0.44853 −0.42587 −0.36525 −0.52578 −0.36525 −0.74457 −0.64298

1.19103 1.53047 1.16423 1.50270 1.19103 1.50270 1.27231 1.59192

−0.11862 −0.15521 −0.10857 −0.13084 −0.11862 −0.13084 −0.14148 −0.18838

Recommended value. 3497



■

Table 9. AARDs of the Predicted Vaporization Enthalpy Using Different Alpha Functions for PR-EOS and SRKEOSa PR-EOS 5

name

a

b

c

d

methane n-pentane n-hexane n-heptane n-octane n-nonane n-decane n-dodecane n-tetradecane n-hexadecane n-eicosane n-tetracosane n-octacosane n-triacontane n-hentriacontane n-tetratriacontane n-pentatriacontane n-hexatriacontane n-tritetracontane benzene toluene ethylbenzene tetralin naphthalene overall MARD (%)

2.23 0.14 1.39 1.01 1.48 1.88 0.61 4.04 6.40 5.91 5.56 4.81 2.33 4.80 7.55 6.63 7.17 9.34 17.23 2.54 1.26 1.17 0.84 0.85 4.05 17.23

2.79 1.23 1.65 0.62 0.61 0.62 2.55 2.06 4.24 7.76 2.47 0.45 3.20 1.08 1.35 0.53 0.85 1.39 8.47 1.30 2.57 0.53 1.62 1.98 2.16 8.47

2.64 0.17 1.31 0.29 0.37 0.50 2.82 1.57 3.58 8.02 2.19 0.08 3.63 1.49 0.97 0.88 1.08 1.16 8.42 1.85 1.06 0.26 0.63 1.11 2.01 8.42

2.19 0.18 1.38 0.57 0.68 0.74 1.30 2.30 3.96 8.03 1.88 0.27 3.78 1.46 0.85 0.66 1.94 3.10 6.00 1.66 0.63 0.53 1.66 0.73 1.93 8.03

AUTHOR INFORMATION

Corresponding Author

*Tel.: 1-306-337-2660. Fax: 1-306-585-4855. E-mail: tony. [email protected].

AARDs (%) 1

Article

SRK-EOS

ORCID

2

Daoyong Yang: 0000-0001-8820-6625

2

a

e

3.42 1.58 2.05 0.99 0.78 0.74 3.75 0.77 2.80 8.80 1.21 0.73 4.21 1.96 0.62 1.02 1.08 1.22 8.88 0.82 2.32 1.03 1.15 2.41 2.26 8.88

2.69 0.49 1.71 0.47 0.66 0.70 1.19 2.60 4.47 7.76 2.10 0.41 3.71 1.40 0.90 0.63 1.95 3.10 6.04 1.10 0.98 0.46 2.14 1.02 2.02 7.76

Funding

The authors acknowledge a Discovery Development Grant, a Discovery Grant, and a Collaborative Research and Development (CRD) Grant from the Natural Sciences and Engineering Research Council of Canada (NSERC) to D. Yang and EHR Enhanced Hydrocarbon Recovery Inc. for financial support. Notes

The authors declare no competing financial interest.

■

REFERENCES

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a

Note: a1, b, c, and a2 are alpha functions nos. 1, 2, 3, and 5 in Table 2, they all use acentric factor defined at reduced temperature of 0.70; d5 and e2are the recommended alpha functions for the PR-EOS and SRKEOS, respectively, both of which use the acentric factor defined at reduced temperature of 0.60.

saturated hydrocarbons have the largest effect on the optimum reduced temperature for the entire database, the aromatic compounds take the second place, and the light-saturated hydrocarbons have the smallest effect. The sensitivity of alpha function reduces with an increase in reduced temperature, while it remains unchanged as acentric factor is varied. By adding data points of vapor pressures of aromatic compounds and heavysaturated hydrocarbons at low reduced temperatures, the newly expanded database is more related to the heavy oil components and the relevant solvent recovery processes. Accordingly, corresponding alpha functions at the optimum reduced temperatures have been regressed for the entire database and various compound categories. All of the newly developed alpha functions lead to the minimum AARDs either for the corresponding compound categories or for the entire database except for the light-saturated hydrocarbons. This is due mainly to the fact that the AARDs for the light-saturated hydrocarbons are small and not sensitive to the reduced temperature for the acentric factor and the fact that the newly developed alpha functions must account more for the newly added compounds which are quite deviated from the light-saturated hydrocarbons. Also, the newly developed alpha functions yield good prediction accuracy of vaporization enthalpy for heavy hydrocarbon compounds. 3498



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