Use of heat capacities for the estimation of cubic equation-of-state parameters—application to the prediction of very low vapor pressures of heavy hydrocarbons

Fluid Phase Equilibria 200 (2002) 375–398

Use of heat capacities for the estimation of cubic equation-of-state parameters—application to the prediction of very low vapor pressures of heavy hydrocarbons Lucie Coniglio a,∗ , Evelyne Rauzy b , André Péneloux b , Evelyne Neau b a

Département de Chimie Physique des Réactions, Institut National Polytechnique de Lorraine, Ecole Nationale Supérieure des Industries Chimiques de Nancy, 1 rue Grandville, 54000 Nancy, France b Laboratoire de Chimie Physique, Université Aix-Marseille II, Aix-Marseille, France Received 9 August 2001; received in revised form 3 January 2002; accepted 12 January 2002

Abstract Improvement in the prediction of very low vapor pressures is checked by introducing heat capacity data into the estimation of cubic equation-of-state (EOS) parameters. As the key parameter is the temperature-dependent parameter a, several expressions (mainly of exponential form) were investigated. All of them were chosen in order to show a consistent behavior for the two considered properties (vapor pressures and heat capacities). The cubic EOS used as an illustration is of the Peng–Robinson type applied to heavy hydrocarbons. No satisfactory refinement in the prediction of the very low vapor pressures was observed in comparison with the results obtained by extrapolating the EOS from medium to very low pressures. This work has, however, the following benefits: (1) to point out the changes that should be made to improve these predictions; (2) to inform on the accuracy that may be obtained if vapor pressures of heavy organic compounds are predicted from heat capacity data as the sole alternative for estimating the temperature-dependent parameter a of a cubic EOS; (3) to confirm the reliability of the cubic group-contribution (GC)-based EOS proposed by Coniglio et al. [Ind. Eng. Chem. Res. 39 (2000) 5037] when extrapolated for modeling crude oils or gas condensates encountered in the petroleum industry. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Very low vapor pressures; Heat capacities; Cubic; Equation-of-state; Formulation of the temperature-dependent parameter a; Heavy hydrocarbons

1. Introduction Accurate prediction of pure component vapor pressures is essential for phase equilibrium calculations in multicomponent mixtures. These predictions with cubic equation-of-state (EOS) depend on the estimation ∗

Corresponding author. Tel.: +33-3-83-17-50-25; fax: +33-3-83-37-81-20. E-mail address: [email protected] (L. Coniglio). 0378-3812/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 2 ) 0 0 0 4 6 - 8

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of the co-volume b and the temperature-dependent parameter a. The latter is traditionally determined by regression on experimental vapor pressure data. Organic compounds with high molecular weight (>250) have, however, very low vapor pressures (9 × 10−6 bar at 340 K for n-octadecane) whose measurement is a difficult task. As a result, experimental data are quite scarce, sometimes subject to major systematic errors and in a relatively limited temperature range. Under such circumstances, it may be difficult to ensure that the analytical expression chosen for modeling the EOS parameter a behaves reliably in terms of temperature. On the other hand, measurement of saturated liquid heat capacities is perfectly mastered nowadays, even at very low pressures (corresponding to very low temperatures). As a result, experimental data of saturated liquid heat capacities are more abundant and more reliable than the very low vapor pressure measurements performed in the same temperature range. Therefore, an initial problem can be raised. “Can experimental data of saturated liquid heat capacities (coupled with data of ideal gas heat capacities) add complementary information and sufficient refinement in the estimation of the cubic EOS parameter a to increase the reliability of the very low vapor pressure prediction?” Encouraging results were obtained by Ruzicka and Majer [1,2] who achieved a simultaneous correlation of vapor pressures and related thermal data (enthalpies of vaporization and heat capacities) between the triple point and the normal boiling temperature. From their procedure, the authors were able to recommend mutually consistent vapor pressures and thermal data, with a particular refinement at low pressures. Simultaneous correlation between vapor pressures in the medium pressure range and related thermal data available at lower temperatures was also investigated in order to propose a solution for extrapolating vapor pressures toward lower pressure range [3]. Here again, satisfactory results were obtained by the authors. In addition, various classes of organic substances (1-alkanols and normal alkanes [1–3], compounds of petroleum interest [4,5], chlorobenzenes [6], and dimethylphtalate [7]) were selected as illustrations by the authors and their collaborators. Their approach is however, somewhat different from ours since they used the three or four parameter Cox equation (instead of a cubic EOS) for correlating the various investigated properties simultaneously as a function of temperature. The second problem that can be raised concerns organic compounds of very high molecular weight and/or high normal boiling temperature. For this class of compounds, vapor pressures can never be determined experimentally for technical reasons (thermal decomposition of the compound far below its normal boiling temperature, too low values of the vapor pressure at low temperature). On the other hand, measurements of saturated liquid heat capacities could be achieved, if necessary, at low temperatures. Aware that this type of problem is often encountered in petroleum/petrochemical industry (as an example, in modeling heavy cut crude oils with waxes or asphalthenes), Coniglio et al. [8] and Coniglio and Nouviaire [9] proposed a cubic group-contribution (GC)-based EOS suitable for the estimation of thermophysical properties of heavy hydrocarbons (with molecular weights higher than 250 and normal boiling temperatures ranging from 280 to 750 K). In this cubic EOS, the temperature-dependent parameter a is expressed in terms of GCs which were determined by regression on experimental vapor pressure data exclusively. The very good estimates (within the experimental uncertainties) obtained by the authors from the triple point to 7 bar for various thermophysical properties (vapor pressures, saturated liquid densities, enthalpies of vaporization, saturated liquid heat capacities, speeds of sound in saturated liquids), confirm the internal consistency of the GC-based EOS developed by Coniglio et al. [8]. Also, the predictive abilities of this model with respect to vapor pressures were checked by the authors by considering hydrocarbons of higher molecular weight than those selected to regress the GCs involved in the parameter a. Here again, the GC-based EOS gave a very good performance yielding vapor pressure predictions with an error smaller than 2%. Although the objective of a GC method is to use it for extrapolation (i.e. to predict as reliably

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as possible properties for which no verification can be made due to the lack of experimental data), the question is: what degree of accuracy in vapor pressure prediction can be obtained for hydrocarbons of even higher molecular weight than those investigated by Coniglio et al. [8] and Coniglio and Nouviaire [9] ? It would be very hazardous to attempt to give any answer to this question. The only solution is to rely on the results of the prediction given by the authors unless another alternative can reasonably be adopted. Indeed, the “real” question related to the second problem raised can be put as follows: “for heavy organic compounds, is it possible to predict the vapor pressures reasonably from heat capacity data by using a cubic EOS, or is it better to extrapolate towards low temperatures a cubic GC-based EOS whose parameters were determined from vapor pressures of lighter organic compounds?” With regard to the work developed by Ruzicka and Majer [3], heat capacity data should be used only as supporting information (i.e. together with vapor pressures available in the medium pressure range) to extrapolate vapor pressures below the lower limit of experimental vapor pressure data. However, as it has been said previously (first problem raised), the authors developed their work in another context than that of cubic EOSs. As a result, the accuracy that can be obtained in the prediction of vapor pressures of very heavy organic compounds by using their heat capacity data within the cubic EOS’s framework still remains to be evaluated. The purpose of this paper is to bring partial answers to the two problems raised above which both aim at improving extrapolation abilities of a cubic EOS, in terms of temperature range (first problem), and in terms of classes of compounds of different homologous series (second problem). For this work, the thermodynamic model used has to be internally consistent, i.e. to be able to yield accurate estimates of both vapor pressures and heat capacities. Since the cubic GC-based EOS proposed by Coniglio et al. [8] satisfies this requirement, it was adopted as a starting point, and therefore, this work was focused on heavy hydrocarbons. 2. Thermodynamic model This section presents the main qualitative features of the cubic GC-based EOS of Coniglio et al. [8] and states the changes made to this model in order to fulfill the purpose of this work. The cubic EOS is of the Peng–Robinson [10] type and uses the combination cubic EOS-consistent volume correction introduced by Péneloux and Rauzy [11] and Rauzy [12]: P =

RT a(T ) − 2 v − b v + 2bv − b2

with

vcorrected = v − c(T )

(1)

The three EOS parameters a, b, and c, have been modified by Coniglio et al. [8] to obtain reliable estimations of heavy hydrocarbon properties. Therefore, the approach used by the authors for calculating them is not based on the three-parameter corresponding state principle which involves the use of the critical temperature, the critical pressure, and the acentric factor, because all these properties are not known experimentally for most high molecular weight/high normal boiling point substances. Instead, the following relations were used: b=

bCH4 Vw Vw,CH4

(2)




T a(T ) = a(Tb ) α , C1 , C2 , m Tb

(3)

378


c(T ) = c(Tb ) ξ

L. Coniglio et al. / Fluid Phase Equilibria 200 (2002) 375–398

 T , α0 , α1 , Tb

c(Tb ) = ξTb (Tb , m, αb , βb )

(4)

where C1 , C2 , α 0 , α 1 , α b , and β b are universal constants (i.e. they apply to all hydrocarbons), while Vw , m and c(Tb ) are parameters which are characteristic for a given hydrocarbon and are calculated according to the GC concept. The co-volume b is related to the van der Waals volume (Vw ) calculated from Bondi’s GC method [13] by taking methane as the reference for evaluating the proportionality between the two properties (b and Vw ). The two temperature-dependent parameters of the EOS, i.e. the parameter a and the volume correction c, are expressed in terms of the normal boiling temperature Tb and the shape parameter m. The former property (Tb ) is taken as the reference temperature instead of the conventional critical temperature, while the second property (m) has a role similar to that of the acentric factor. Furthermore, the temperature-dependence of the two EOS parameters a and c (α and ξ , respectively) are expressed using exponential forms. It is also important to emphasize that the temperature-dependent volume correction c is an “external” correction which has to be applied to the volume v and not to the co-volume b. This feature has two results: (i) The contribution of the temperature-dependent volume correction c to the thermophysical properties which can be derived from the EOS (prior to volume correction) is deduced from the thermodynamic relationships. For example, ln ϕcor (T , P ) = ln ϕ(T , P ) − P c(T )/RT, CPcor (T , P ) = CP (T , P ) + P T (∂ 2 c/∂T 2 ), where ϕ cor and CPcor are the corrected fugacity coefficient and the corrected heat capacity at constant pressure, respectively. (ii) The co-volume b is temperature independent. In that sense, the approach used by Trebble and Bishnoi [14] who used a temperature-dependent co-volume b is different from the approach used by Coniglio et al. [8]. Also, the conclusions of Salim and Trebble [15] which pointed out the dangerous implications of temperature dependence in the co-volume b of the Trebble and Bishnoi [14] EOS, and more generally of any cubic EOS, i.e. prediction of negative values of isochoric heat capacity and imaginary values for speed of sound at extremely low temperatures or high pressures, should not be applied to the Coniglio et al. [8] EOS. Particularly, in a temperature range from the triple point to 1.1 T /Tb , predictions obtained with the Coniglio et al. EOS led to positive values of the saturated liquid heat capacities (CPS L and CvS L which show respectively, the variation in enthalpy and in internal energy of a saturated liquid with temperature) and realistic values of the speeds of sound in saturated liquids. Concerning property predictions at high pressures, this would tend to push the Coniglio et al. EOS out of its applicability range. In the cubic GC-based EOS of Coniglio et al. [8], the co-volume b, the shape parameter m, and the value of the volume correction at the normal boiling temperature c(Tb ) are calculated using homogeneous GC methods like that of Bondi [13]. Indeed, the same calculation procedure is used for the three parameters (b, m, and c(Tb )) according to which the property estimation of a compound from its molecular structure is considered to be a collection of two types of contributions: (a) the contributions of basic functional groups, and (b) the contributions of structural corrections that account for various intramolecular effects. According to the purpose of this work, the shape parameter m will not be estimated here by using the GC method of Coniglio et al. [8], but will be specifically determined by regression on the experimental data of each hydrocarbon.

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From the cubic EOS (Eq. (1)), it is possible to calculate the δ C difference between the heat capacity of the saturated liquid CPS L and that of the ideal gas CPid by δC (T ) = CPS L (T ) − CPid (T )   √
2  1 (∂P /∂T )2v ∂ c(T ) 1 + (1 + 2)(b/v) T ∂ 2 a(T ) = −R + √ ln −T + PT √ (∂P /∂v)T ∂T 2 2 2 1 + (1 − 2)(b/v) b ∂T 2

(5)

The contribution of the temperature-dependent volume correction c to the calculation of the δ C difference (i.e. PT(∂ 2 c(T )/∂T 2 ), Eq. (5)) is very small: from 10−12 at the triple point to 10−2 J mol−1 K−1 at the normal boiling temperature. Consequently, within the use of the cubic EOS of the Peng–Robinson type (Eq. (1)), the key parameter that plays an important role for the estimation of vapor pressures and δ C differences (Eq. (5)) between the heat capacities of saturated liquids and those of the ideal gases is the temperature-dependent parameter a. However, as it can be observed in Eq. (5), the temperature-dependent parameter a appears only as its second derivative in the expression for the δ C difference. Therefore, in addition to the three basic requirements that any formulation of the function a(T) must satisfy, i.e. (i) to be finite and positive for all temperatures; (ii) to equal a(Tref ) at the temperature Tref taken as reference; (iii) to approach a finite value (close to zero) as the temperature approaches infinity, a fourth requirement has also to be satisfied: (iv) the second derivative of the function a(T) must exist so that the δ C difference (Eq. (5)) can be calculated. Also, it is important that the second derivative should contain all the parameters that appear in the expression for a(T) in order that no information be lost during derivation. This requirement is not necessarily satisfied if a polynomial form is used for the function a(T). Table 1 Description of the seven formulations investigated for the function a(T) Coding (source)

Equation

Exponent (x, y) value

1 (Melhem et al. [16]; Coniglio et al. [8])

a(Tb ) exp[R1 (m) Fx (Tr ) − R2 (m) Fy (Tr )]

x = 0.4, y = 1/x

2 (Carrier et al. [17]; Coniglio et al. [18])

a(Tb )[1 + R1 (m) Fx (Tr ) − R2 (m) Fy (Tr )]

x = 0.05, y = 1.00

3 (Trebble and Bishnoi [14]; this work)

a(Tb ) exp[R1 (m) Fx (Tr )]

x = 0.5

4 (this work)

a(Tb ) exp{[ R1 (m) Tr + R2 (m)] Fx (Tr )}

x = 0.5

5 (this work)

a(Tb ) exp{[I1 (m) Tr (1+x) + I2 (m) Trx ] Fy (Tr )}

x = −0.1, y = 0.5

6 (this work)

a(Tb ){1 + I1 (m) [1 − Ex (Tr )] − I2 (m) [1 − Ex (Tr )]2 }

x = 0.1

7 (this work)

a(Tb ){1 + R1 (m) [1 − Ex (Tr )] − R2 (m) [1 − Ey (Tr )]}

x = 0.05, y = 5.00

Tr =

T , Tb

Fx (Tr ) = [1 − Trx ],

Ex (Tr ) = exp[Fx (Tr )],

C2 1 yC1 xC1 m− , R2 (m) = m− , xC2 − y xC2 − y xC2 − y xC2 − y m C1 C2 C1 1 R1 = , R1 (m) = m+ , R2 (m) = m− , x x(C2 + 1) C2 + 1 x(C2 + 1) C2 + 1 C2 C1 m+ , I1 (m) = y(C2 + 1) C2 + 1 C1 1 m 1 C1 I2 (m) = m− , I1 (m) = , I2 (m) = m− (C1 and C2 are universal constants). y(C2 + 1) C2 + 1 x xC2 C2

R1 (m) =

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Given the important role of the function a(T) in this work, seven different a(T) formulations have been investigated. They are listed in Table 1. The a(T) formulation proposed by Coniglio et al. [8] has also been included, and is designated as form 1. All these a(T) formulations satisfy the four aforementioned requirements, and are of exponential form (with the exception of the a(T) formulation designated as form 2 which is of polynomial form). The values of powers x and y involved in the various a(T) formulations have been optimized to fit the two thermophysical properties considered in this work: vapor pressures and δ C differences between the heat capacity of the saturated liquid and that of the ideal gas (Eq. (5)). 3. Experimental databases This section presents details of the experimental databases of vapor pressures and heat capacities which were selected. These databases should be divided into two classes: (i) the first covers the databases selected to verify the consistency of the thermodynamic model with respect to vapor pressures and heat capacities. This class of databases will be designated as class A; (ii) the second refers to the databases used to check the prediction of the very low vapor pressures obtained by considering medium to low vapor pressure and/or heat capacity data in the regression of the temperature-dependent parameter a of the EOS. This class of databases will be designated as class B. The databases of class A the main features of which are given in the following sections are the same as the databases used by Coniglio et al. [8]. The hydrocarbons which belong to the databases of class B, and literature sources of the related data, are listed in the Appendix A. 3.1. Databases selected to verify the consistency of the thermodynamic model (class A) The data of vapor pressures and heat capacities cover a wide pressure range (from 4 × 10−6 to 7 bar), and a large number of classes of hydrocarbons commonly found in crude oils or gas condensates (alkanes, naphthenes, alkylbenzenes, and condensed polynuclear aromatics). Also, most of these hydrocarbons are of medium to high molecular weight and have high normal boiling temperatures (from 300 to 600 K). The values of the experimental uncertainties in the various classes of data given in the following sections should be considered as minimum average values. Larger experimental uncertainties may be encountered for some data relating to the hydrocarbons with the highest normal boiling temperatures. 3.1.1. Vapor pressures The selected database comprises 128 hydrocarbons corresponding to 3385 vapor pressure data points. As the vapor pressure measurements are of very different orders of magnitude and experimental accuracy, there were divided into three classes: (i) The first class contains hydrocarbons with very low vapor pressures ranging from 4 × 10−6 to 0.01 bar. Most of them are from laboratories specialized in this kind of measurements [20–26]. The experimental uncertainties typically given by the authors are, for the temperature ±0.001 [23] to ±0.02 K [20,21,27], and for the pressure ±1 [23] to ±2% [20,21,27]. (ii) The second class contains hydrocarbons with low vapor pressures ranging from 0.01 to 0.06 bar. The data are from various sources, and the experimental uncertainties in the temperature and pressure vary widely from one author to another.

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(iii) The third class contains hydrocarbons with medium vapor pressures ranging from 0.06 to 2 bar (7 bar for some heavy alkylbenzenes and condensed polynuclear aromatics). These data are ebulliometric data of great accuracy and most of them were made around 1945 by the NBS/American Petroleum Institute. The experimental uncertainties typically given by the authors are ±0.003 K for the temperature and ±0.06 mmHg for the pressure. The relative proportion of each class of measurements in the whole database is 11, 7, and 82%, respectively. To build a vapor pressure database that could be consistent and reliable, the measurements were also screened with the graphical method proposed by Wilsak and Thodos [43] and modified by Coniglio [44] for application to very low vapor pressures. During this step of screening, two features were noticed, confirming the arguments stated in the Introduction: (i) for class I, two sets of measurements made by two different authors which show a very small internal scattering, can nevertheless, exhibit a discrepancy of 2–12% between them, depending on the authors and the location of these measurements in the vapor pressure range. In these cases, all data were retained, as it was impossible to decide which author made the most consistent measurements; (ii) overall, there is a disparity of quality between the vapor pressure measurements in classes (i)–(iii). 3.1.2. Heat capacities Among the 128 hydrocarbons taken as references to build the vapor pressure database, data of saturated liquid heat capacities (CPS L ) could be collected for only 69 compounds (including alkanes, naphthenes, alkylbenzenes, and condensed polynuclear aromatics). The CPS L database is mainly from the TRC tables [31] and the Stephan and Hildwein [33] compilation. However, for naphthenes, no data were available in these compilations, and thus, some measurements from various sources were also considered [28,36,39,41]. The overall experimental uncertainties of the CPS L data typically given by the authors range from 0.1 to 0.2%. The temperature range of the selected CPS L data as a whole (i.e. 1454 data points) is from 120 to 460 K. The data of ideal gas heat capacities (CPid ) of the 69 hydrocarbons considered previously all stem from the TRC tables [31]. The whole of the CPid database comprises 1357 data points ranging from 100 to 1500 K. In order to calculate the δ C differences (Eq. (5)) at the temperatures relating to the CPS L data, the CPid data have been regressed by using a semi-empirical correlation proposed by Coniglio et al. [18] which is derived from statistical mechanics. This correlation led to a mean deviation of 0.05% when compared with the CPid tabulated values [31]. This deviation is substantially less than the expected errors in CPid data which is 0.1–0.2% for light hydrocarbons (having up to eight carbon atoms), and up to 1% for heavier compounds [2]. 3.2. Databases selected to check the various approaches of vapor pressure prediction (class B) The hydrocarbons considered here all stem from the databases of class A described previously with the particularity that both vapor pressure and CPS L data are available. According to Section 3.1.2, 69 hydrocarbons are considered for which 1454 data points of CPS L (ranging from the triple point to an upper temperature limit corresponding to a vapor pressure of 2 bar), and 1865 data points of medium to low vapor pressures are available. For only 24 of these 69 hydrocarbons, very low vapor pressure data are available (248 data points). This subclass of hydrocarbons (designated as class B∗ in the Appendix A) is quite small compared to the initial databases (see Section 3.1). However, it comprises various classes of hydrocarbons (alkanes, naphthenes, alkylbenzenes, and condensed polynuclear aromatics) and very low

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vapor pressure data from various sources. Therefore, it can be considered as a representative sample to fulfill the purpose of this work.

4. Methodology The constants C1 and C2 (Eq. (3) and Table 1) are characteristic of the function a(T), and were estimated by regression on the values of the temperature-dependent parameter a that fit the saturation curve exactly. As explained in Section 2, however, the shape parameter m (Eq. (3) and Table 1) is characteristic of each compound. Therefore, it was decided to keep C1 and C2 constant and equal to their initial values, and to calculate the very low vapor pressures after successive regression of the shape parameter m on: (i) the medium and low vapor pressure data, and the δ C differences (Eq. (5)), simultaneously; (ii) the δ C differences (Eq. (5)); (iii) the medium and low vapor pressure data. The comparison of the percent average relative deviations δr (PS )% (definition given in the list of symbols) between experimental data and values obtained in each aforementioned case of the very low vapor pressures should help to determine: (i) whether the heat capacity data could add complementary information to the medium vapor pressure data, and sufficient refinement in the estimation of the cubic EOS parameter a to improve the reliability of the very low vapor pressure prediction (first problem raised); (ii) whether the heat capacity data could be used as an alternative for the vapor pressure data for the estimation of the cubic EOS parameters, which would be particularly interesting in the case of very heavy organic compounds (second problem raised); (iii) whether the very low vapor pressures of an organic compound could be predicted reasonably with a cubic GC-based EOS for which the GCs involved in the function a(T) would be estimated from vapor pressure data of lighter organic compounds (second problem raised). The various objective functions minimized during the successive regressions of the shape parameter m 1 are described in Table 2. The objective function Fob chosen for the simultaneous regression of the shape parameter m on the medium and low vapor pressure data and the δ C differences (Eq. (5)) requires some comments. Since the regression is performed by considering two different classes of properties (vapor pressures and heat capacities), the corresponding data points have been weighted by their respective overall errors: σ (PS ) for the vapor pressures, and σ (δ C ) for the heat capacities. Strictly speaking, these overall errors should be determined from the experimental uncertainties in the measurements which, in the case of vapor pressures, are not always clearly specified in the literature. Therefore, the variances σ 2 (PS ) relating to each point of the vapor pressure database of class B have been estimated by using a statistical procedure based on the “observed deviation method” developed by Péneloux et al. [45,46]. Typical overall errors in vapor pressures that were found were: 0.1% for the very low vapor pressures, and 0.001% for the medium vapor pressures. These values are lower than the experimental uncertainties stated by the authors of the measurements (see Section 3.1.1). However, one should keep in mind that the vapor pressure data were previously screened graphically [43,44]. After this treatment, the data retained for each hydrocarbon and each source exhibit very little internal scattering which is shown by small values of the statistically calculated overall errors in vapor pressures. Concerning heat capacities, the variances

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Table 2 Objective functions minimized for the estimation of the shape parameter m (Eq. (3) and Table 1) by considering vapor pressure and/or heat capacity data Data used for regression of the shape parameter m Medium and low PS + heat capacities

Objective function useda 2 2   NP PSexp,k − PScal,k NC δCexp,k − δCcal,k + k=1 , k=1 σ (PSk ) σ (δCk ) exp,k

Heat capacities

Medium and low PS

N P

k=1

exp,k



CP S L exp,k

PS

1 Fob

exp,k

− CPid,k where δC = CPS L − CPid,k and δCcal,k = CPcal,k SL  exp,k  cal,k 2 NC CPS L − CPS L k=1

Coding

− PScal,k exp,k

PS

2 Fob

2 3 Fob

a

NP and NC are the number of data points of vapor pressures and of heat capacities relating to each investigated compound, respectively.

of the δ C differences (Eq. (5)) are given by σ 2 (δC ) = σ 2 (CPS L ) + σ 2 (CPid )

(6)

According to the error propagation law of Gauss and by neglecting the error in temperature, Eq. (6) can be rewritten as σ 2 (δC ) = σe2 (CPS L ) + σe2 (CPid )

(7)

where σe (CPS L ) and σe (CPid ) are the experimental uncertainties in CPS L and CPid data, respectively. Since the selected database of saturated liquid heat capacities comprises both experimental data and values which are from the TRC tables [31] (i.e. values which were smoothed in order to be presented in a tabulated form), it was decided to consider only the term σe2 (CPid ). This assumption was also guided by the differences in the experimental uncertainties of the CPS L and CPid data observed particularly for heavy hydrocarbons (σe (CPS L ) = 0.2%CPS L , while σe (CPid ) = 1%CPid , see Section 3.1.2). Therefore, (i) σ (δC ) = σe (CPid ) = 0.2%CPid was set for light hydrocarbons having up to eight carbon atoms, and (ii) σ (δC ) = σe (CPid ) = 1%CPid for heavier compounds. 5. Results and discussion Results relative to the different steps described in the previous section are reported in Tables 3–5 for the seven formulations of the function a(T) that were investigated (Table 1). All results are expressed in terms of percent average relative deviations in vapor pressures δr (PS )% and in saturated liquid heat capacities δr (CPS L )%. Tables 3 and 4 confirm the consistency of the various a(T) formulations with respect to vapor pressure and heat capacity, in a large pressure/temperature range (databases of class A). Indeed, regression on vapor

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Table 3 Verification of the consistency of the investigated a(T) formulations with respect to vapor pressures (database of class A)a,b δr (PS )% obtained with the a(T) formulation

Overall PS database (128 comp., 3385 pts.) Medium and low PS data Very low PS data

(∈ CPS L ) (69 comp., 1865 pts.) (∈ / CPS L ) (59 comp., 1148 pts.) (∈ CPS L ) (24 comp., 248 pts.) (∈ / CPS L ) (13 comp., 124 pts.)

1

2

3

4

5

6

7

0.30

0.31

0.60

0.30

0.30

0.31

0.31

0.16 0.21 1.37 1.03

0.19 0.20 1.45 0.86

0.46 0.40 1.92 2.01

0.16 0.21 1.38 1.01

0.17 0.21 1.38 1.01

0.19 0.19 1.43 0.88

0.19 0.19 1.46 0.85

Regression of the a(T) function parameters (C1 , C2 , and m) on the whole vapor pressure data (including very low, low and medium vapor pressure data). Results are given here by distinguishing between compounds for which CPS L data are available (∈CPS L ) from those for which no CPS L data could be found in the literature (∈ / CPS L ). a Results are expressed in terms of percent average relative deviations in vapor pressures (δr (PS )%). b Abbreviations: comp. for compounds and pts. for data points.

pressures of the parameters C1 , C2 , and m involved in the function a(T) yields an average relative deviation δr (PS ) = 0.3% for the whole of the very low, low and medium vapor pressure data (δr (PS ) = 0.6% for a(T) formulation 3, the only adjustable parameter involved being the shape parameter m). With the values of C1 , C2 , and m obtained thereby, saturated liquid heat capacities are estimated within 1.5–2.9% depending on the a(T) formulation concerned, which is quite satisfactory according to the expected errors in the ideal gas heat capacities (from 0.2 to 1%). Analysis of Table 5 which summarizes the results obtained for the very low vapor pressure prediction by considering medium to low vapor pressure data and/or heat capacity data (databases of class B) demonstrates the two following findings. (i) Concerning the simultaneous regression of the shape parameter m on the medium and low vapor pressure data and the δ C differences (Eq. (5)), the very low vapor pressures are predicted with slightly larger errors than when adjusting the shape parameter m on only the medium and low vapor pressure data (see steps B1 and B3). In both cases, the deviations δr (PS ) range from 1.8 to 2.0% depending on the a(T) formulation used (with the exception of the a(T) formulation 3 for which δr (PS ) = 7.2% in step B1 and 5.4% in step B3). It is worth noting that this finding depends mainly on the choice of the 1 1 objective function Fob (Table 2) that is used here. Indeed, using the objective function Fob , where the overall errors in the vapor pressures and in the δ C differences are of different orders of magnitude, Table 4 Verification of the consistency of the investigated a(T) formulations with respect to heat capacities (database of class A)a,b δr (CPS L )% obtained with the a(T) formulation

Overall CPS L database (69 comp., 1454 pts.)

1

2

3

4

5

6

7

1.75

2.09

2.19

2.91

1.47

2.33

1.80

Estimation of the saturated liquid heat capacities with the values of a(T) function parameters (C1 , C2 , and m) obtained in Table 3. a Results are expressed in terms of percent average relative deviations in saturated liquid heat capacities (δr (CPS L )%). b Abbreviations: comp. for compounds and pts. for data points.

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causes the vapor pressure data to be taken into account more than the heat capacity data during the parameter regression (see steps A1–A3). (ii) Much better predictions of the very low vapor pressures are obtained by considering in the regression of the shape parameter m the medium and low vapor pressure data (δr (PS ) ranging from 1.8 to 2.0%, 5.4% for the a(T) formulation 3, step B3) rather than the heat capacity data (δr (PS ) ranging from 7 to 19%, step B2). Three other important points, useful for the continuation of this work, should be made. (i) The various a(T) formulations that have been checked (with the exception of the a(T) formulation 3) can be extrapolated with confidence towards low temperatures; the errors relating to the very low vapor pressure prediction are overall 0.5% larger than the errors relating to the very low vapor pressure regression (see Table 5, step B3, and Table 3, very low PS data (∈ CPS L )). (ii) The saturated liquid heat capacities are calculated with errors of different orders of magnitude when the shape parameter m is regressed on the δ C differences (Eq. (5)) or on the vapor pressures (see Table 5, step A2 and Table 4). (iii) The a(T) formulation that gave the best performance at all steps of the present study is the one proposed by Coniglio et al. [8] (a(T) formulation 1).

6. Parametric sensitivity In order to better understand why very low vapor pressures are predicted with such different orders of magnitude when either heat capacity data or medium to low vapor pressure data are used in the regression of the shape parameter m, two issues have been raised. (i) What is the degree of sensitivity of the shape parameter m with respect to the estimation of vapor pressures or heat capacities? The answer to this question should provide comparative data on the quality of the information acquired by the shape parameter m when it is estimated by successive regression on vapor pressures or heat capacities. (ii) When the heat capacities are taken as input data, what is the effect of uncertainty in the heat capacities on the vapor pressure estimates? To answer these questions, an approach should be adopted in which deviations due to the thermodynamic model used can be avoided. Therefore, it was decided to generate, using the thermodynamic model described in Section 2 combined with a single a(T) formulation (chosen from Table 1), sets of pseudo-databases of vapor pressures (PS ) and saturated liquid heat capacities (CPS L ) that would be mutually consistent. The a(T) formulation used corresponds to the formulation 3, i.e. an exponential form involving a single adjustable parameter: the shape parameter m. Thus, the two constants C1 and C2 characteristic of the function a(T) need not be mentioned in the discussion. Furthermore, the vapor pressure and heat capacity pseudo-data were generated by reproducing the temperature ranges of the “real” experimental data. Therefore, the sample of hydrocarbons selected for this part of the work is the same as that used to build the databases of class B∗ , and so has the same number of very low, low and medium vapor pressure and heat capacity data points (see Section 3.2).

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6.1. Sensitivity of the shape parameter m with respect to the estimation of vapor pressures or heat capacities 6.1.1. Method Step 1: An initial set of pseudo-databases is generated from values of the shape parameter m obtained by prior regression on the “real” data of experimental vapor pressures relating to each hydrocarbon. This first set of pseudo-databases will be designated as PS1 and CP1 L . Step 2: A second set of pseudo-databases S is generated from the values of the shape parameter m as obtained in step 1, but shifted by 0.1%. This second set of pseudo-databases will be designated as PS2 and CP2 S L . Step 3: Deviations in vapor pressures δr (PS )% and in saturated liquid heat capacities δr (CPS L )% between the pseudo-data generated in each step are evaluated and compared. These will only be the result of the errors in the shape parameter m estimates (errors of 0.1%). 6.1.2. Results For the 24 considered hydrocarbons, the shift of 0.1% in the shape parameter m induces a deviation δr (PS ) = 0.15% in vapor pressures, whereas a deviation δr (CPS L ) = 0.05% is observed for the saturated liquid heat capacities. Furthermore, these two results correspond to average deviations, and according to Fig. 1 and Table 6 which show deviations in vapor pressures and saturated liquid heat capacities as a function of temperature or pressure range, one important observation can be made: deviations in vapor pressures δr (PS )% increase strongly with decreasing temperature (for the medium and very low vapor pressure pseudo-data, δr (PS ) are 0.05 and 0.4%, respectively) while deviations in saturated liquid heat capacities δr (CPS L )% are low, and almost independent of temperature (around 0.05%). To sum up, the shape parameter m is a much more sensitive parameter with respect to the vapor pressure estimation than to the heat capacity estimation, particularly in the very low pressure/temperature region. This result leads to the conclusion that the value of the shape parameter m is more reliable when estimated by regression on vapor pressure data than when it is estimated by regression on heat capacity data. 6.2. Effect of uncertainty in the heat capacity data on the vapor pressure estimates As stated previously in Section 3.1.2, estimates of ideal gas heat capacities (CPid ) in this work are all based on the data published in the TRC tables [31]. The CPid data relating to light hydrocarbons (with up

Table 6 Deviations in vapor pressures (δr (PS )%) as a function of the pressure range for a shift of 0.1% in the shape parameter m (a(T) formulation used: formulation 3, Table 1)a Pressure range (bar) 4 × 10−6 –0.01 (very low PS ) 0.01–0.06 (low PS ) 0.06–7 (medium PS ) Overall a

Number of hydrocarbons: 24.

Number of data points

δr (PS )%

247 147 643

0.39 0.17 0.05

1037

0.15

388

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Fig. 1. Sensitivity of the shape parameter m with respect to the estimation of vapor pressures PS or saturated liquid heat capacities CPS L : deviations δr (PS )% and δr (CPS L )% in terms of temperature for an error of 0.1% in the shape parameter m (a(T) formulation used: formulation 3, Table 1). (+) δ r (PS )%; (䊊) δr (CPS L )%. (a) Toluene: δr (PS ) = 0.13% for 82 pseudo-data, PS (bar) : 0.00001 → 1.7; δr (CPS L ) = 0.05% for 36 pseudo-data, T (K) : 178.15 → 400. (b) n-Decane: δr (PS ) = 0.15% for 94 pseudo-data, PS (bar) : 0.00001 → 2.7; δr (CPS L ) = 0.05% for nine pseudo-data, T (K) : 250 → 320.

to eight carbon atoms) were found to be in excellent agreement with experimental data available in the literature [47–49]. The deviations found are δr (CPid ) = 0.1–0.2% for 18 hydrocarbons (97 data points). However, for heavier hydrocarbons, no CPid experimental data are available in the literature. Furthermore, for this class of compounds, it is well known that the CPid data from the TRC tables [31] are estimated by a GC method from which the expected errors in CPid estimates can not be inferior to at most 1%. Also, the use

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of heat capacities in the regression of the a(T) function parameters (i.e. the shape parameter m here) is fully justified in the particular case of very heavy hydrocarbons for which no measurement of vapor pressure would be available. For this class of compounds, the experimental uncertainty in the saturated liquid heat capacities (CPS L ) is, however, very likely to be greater than 0.2% (5% for n-eicosane as found by Ruzicka and Majer [2]). Therefore, it was necessary to evaluate the effect of uncertainty in the heat capacity data on the vapor pressure estimates, and particularly those relating to the very low vapor pressures. 6.2.1. Method Step 1: The two pseudo-databases of vapor pressures (PS1 ) and of saturated liquid heat capacities (CP1 S L ) built previously in Section 6.1.1 were considered again. Step 2: The CP1 S L pseudo-data were then shifted

Fig. 2. Effect of uncertainty in the heat capacity data (1%) on the vapor pressure estimates: deviations in vapor pressures (δ r (PS )%) in terms of temperature (a(T) formulation used: formulation 3, Table 1). (a) Toluene: δr (PS ) = 2.14% for 82 pseudo-data, PS (bar) : 0.00001 → 1.7. (b) n-Decane: δr (PS ) = 3.03% for 94 pseudo-data, PS (bar) : 0.00001 → 2.7.

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Table 7 Effect of uncertainty in the heat capacity data (1%) on the vapor pressure estimates: deviations in vapor pressures (δr (PS )%) as a function of the pressure range (a(T) formulation used: formulation 3, Table 1)a Pressure range (bar) 4 × 10−6 –0.01 (very low PS ) 0.01−0.06 (low PS ) 0.06–7 (medium PS ) Overall a

Number of data points

δr (PS )%

247 147 643

7.33 3.04 0.88

1037

2.72

Number of hydrocarbons: 24.

by 1%. The shifted pseudo-data were designated as CP2 S L . Step 3: The shape parameter m was regressed on the pseudo-data CP2 S L (new values of the shape parameter m designated as mS ). Step 4: With the values of the shape parameter m obtained in step 3 (mS ), a third set of pseudo-databases of saturated liquid heat capacities (designated as CP3 S L ) and vapor pressures (designated as PS3 ) were generated. Step 5: Deviations δr (PS )% between the PS1 and PS3 pseudo-databases were evaluated. It is worth noting that a comparison of the CP1 S L (step 1) and CP3 S L (step 4) pseudo-data yielded a deviation δr (CPS L ) = 0.96% (close to 1%) for the whole pseudo-databases. As a result, it can be said that there is no significant loss of information during the regression of the shape parameter m on the CP2S L pseudo-data (step 3). Therefore, the δr (PS )% deviations between the PS1 and PS3 pseudo-databases can only be the result of the errors in the saturated liquid heat capacity pseudo-data (1%). 6.2.2. Results For all of the 24 hydrocarbons investigated, a shift of 1% in the saturated liquid heat capacities (which can be considered as the overall error in the measurement of this property, and in the estimation of the ideal gas heat capacities) induced a deviation δr (PS ) in the vapor pressures of 2.72% for the pseudo-database as a whole, i.e. 7.33–0.88% for the very low and medium vapor pressure pseudo-data, respectively (Table 7). To illustrate this, the deviations δr (PS )% were plotted in terms of temperature for two typical hydrocarbons (Fig. 2). As had been found previously in Section 6.1.2, the largest deviations δr (PS )% were again observed at the lowest temperatures.

7. Conclusion The purpose of this work was to suggest the use of experimental saturated liquid heat capacities, which are available from the triple point to higher temperatures, for the estimation of the temperature-dependent parameter a of a cubic EOS. The goals of this technique were firstly to improve the extrapolation abilities of a cubic EOS towards very low temperatures and pressures, and secondly to calculate thermophysical properties of very heavy hydrocarbons for which no experimentally determined vapor pressure is available. Seven a(T) formulations were used to express analytically the temperature-dependence of the parameter a. For all seven a(T) formulations, the specificity of the parameter a with respect to the nature of compound was expressed through the normal boiling temperature (Tb ) and a shape parameter m which, respectively,

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fulfill the same role as the critical temperature and the acentric factor which are traditionally used in cubic EOSs. Tb was considered as an experimentally known property, and therefore, the shape parameter m had the key role in this study. In this work, two important points have been demonstrated. (i) The shape parameter m is a much more sensitive parameter with respect to the vapor pressure estimation than to the heat capacity estimation; this is particularly true in the very low pressure/temperature region. Consequently, it may be deduced from this result, that within the context of this work, the heat capacity data provide information of lesser importance than the vapor pressure data, particularly in the very low temperature region. This may be due to a mathematical problem that has already been touched on in this work (Section 2). Indeed, the expression derived from the cubic EOS for the δ C difference (Eq. (5)) between the heat capacity of the saturated liquid (CPS L ) and that of the ideal gas (CPid ), contains the temperature-dependent parameter a only under its second derivative. This mathematical problem may not have been solved by using a(T) formulations of exponential forms. (ii) If the shape parameter m is regressed on heat capacity data which are available with an uncertainty of 1% (which may be considered as the overall error in the CPS L measurements and in the CPid estimates) very low vapor pressures are predicted with an error of 7%. On the other hand, the same very low vapor pressures are predicted with an error of 2% when extrapolating towards very low temperatures (pressures) using an a(T) formulation with parameters previously derived from medium vapor pressures. However, the CPid of very heavy hydrocarbons can only be estimated by GC methods with an expected accuracy of at most 1%. To sum up, in order to predict very low vapor pressures reliably, it seems more reasonable to estimate the temperature-dependent parameter a from medium vapor pressure experimental data rather than from heat capacity data, even if the latter are available at very low temperatures. Moreover, this conclusion confirms the reliability of the property predictions obtained with the cubic GC-based EOS proposed by Coniglio et al. [8] (i.e. when the model is used for hydrocarbons of much higher molecular weights/higher normal boiling points than those investigated by the authors). It is nevertheless, worth noting that all these conclusions are related to the EOS used (of cubic type) combined with the a(T) formulations investigated (one of polynomial form, and six of exponential form). List of symbols a temperature-dependent energy parameter of the cubic EOS a(T) function taken as model for the temperature-dependent parameter a a(Tb ) value of the function a(T) at the normal boiling temperature b co-volume of the cubic EOS bCH4 methane co-volume calculated from critical properties c volume correction of the cubic EOS c(T) temperature-dependent volume correction of the cubic EOS c(Tb ) value of the temperature-dependent volume correction c(T) at the normal boiling temperature C1 , C2 universal constants which are characteristic of the function a(T) CPid molar ideal gas heat capacity

392

CPS L Fob m NC NP P Pmin Pmax PS R T Tb Tmin Tmax v Vw Vw,CH4 x, y

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molar saturated liquid heat capacity objective function minimized for parameter regression shape parameter number of heat capacity data points number of vapor pressure measurements pressure lower pressure limit of the whole vapor pressure data of a given compound upper pressure limit of the whole vapor pressure data of a given compound vapor pressure ideal gas constant temperature normal boiling temperature (normal boiling point) lower temperature limit of the whole vapor pressure (or saturated liquid heat capacity) data of a given compound upper temperature limit of the whole vapor pressure (or saturated liquid heat capacity) data of a given compound molar volume van der Waals volume of a molecule van der Waals volume of methane powers of function a(T)

Greek symbols α(T) temperature-dependent form of the parameter a δC difference (calculated by the cubic EOS) between the heat capacity of the saturated liquid and that of the ideal gas δ r (X)% percent average relative deviation in the property X  P exp exp δr (X)% = (100/Np ) N − Xical )/Xi | i=1 |(Xi σ (X) estimated overall error in property X estimated variance relating to the property X σ 2 (X) σ e (X) experimental uncertainty in property X ξ (T) temperature-dependent form of the volume correction c(T) Superscripts cal calculated exp experimental Subscripts corrected property calculated by a cubic EOS corrected with a volume correction L relative to the saturated liquid min minimum max maximum ref reference S saturation

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