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Predictive Tait equation for non-polar and weakly polar fluids: Applications to liquids and liquid mixtures
Accepted Manuscript Predictive Tait equation for non-polar and weakly polar fluids: Applications to liquids and liquid mixtures Hai Hoang, Guillaume Galliero PII:
S0378-3812(16)30259-X
DOI:
10.1016/j.fluid.2016.05.026
Reference:
FLUID 11111
To appear in:
Fluid Phase Equilibria
Received Date: 10 April 2016 Revised Date:
20 May 2016
Accepted Date: 25 May 2016
Please cite this article as: H. Hoang, G. Galliero, Predictive Tait equation for non-polar and weakly polar fluids: Applications to liquids and liquid mixtures, Fluid Phase Equilibria (2016), doi: 10.1016/ j.fluid.2016.05.026. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Predictive Tait equation for non-polar and weakly polar fluids: Applications to liquids and liquid mixtures Hai Hoang1,2, Guillaume Galliero1* 1
Laboratoire des Fluides Complexes et leurs Réservoirs (UMR-5150 with CNRS and TOTAL),
2
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Université de Pau et des Pays de l'Adour, BP 1155, F-64013 Pau Cedex, France Institute of Research and Development, Duy Tan University, Da Nang, Viet Nam *[email protected]
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Abstract
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In this study, we describe a method based on the Tait equation which allows accurate estimation of density and isothermal compressibility of non-polar and weakly polar liquids and liquid mixtures over a wide range of pressures. We have applied the approach to various species in conditions ranging from 0.4 to 0.99 of the critical temperature and for pressures up to 150 MPa. If a reference experimental point of density is known and used as an input for
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this method, the approach yields average absolute deviations around 0.15 % on density and around 2 % on isothermal compressibility. In its fully predictive version, i.e. without
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experimental data point on density, this approach yields deviations on density around 0.3 %, while deviations on isothermal compressibility are kept around 2 %. Interestingly, when
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applied to compounds with a dipole moment larger than one Debye this approach is still able to provide reasonable results. In addition, this approach has been applied on binary liquid mixtures, using classical mixing rules. On three different hydrocarbon mixtures it has been found that the proposed approached yields results as good as on pure liquids emphasizing the consistency of the proposed methodology. All these results compare very favorably to those obtained from other approaches of the literature. Keywords Density; Isothermal compressibility; Tait Equation; Liquids; Mixtures; 1
ACCEPTED MANUSCRIPT 1. Introduction High-pressures densities and isothermal compressibilities of liquids are key quantities in the oil and gas industry [1-2]. To estimate these properties various methods, both correlative and theoretical ones, have been developed and reported in the literature [1-4].
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Among them, the so-called Tait equation [5], a simple heuristic approach has shown its efficiency to deal with compressed liquids [5-16]. This approach has attracted a lot of attention as it only requires a value of density at a reference pressure and two numerical
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parameters, often noted B and C, to provide the density and the isothermal compressibility of a given liquid for pressures up to several MPa.
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Rather surprisingly, the Tait equation has mostly been employed in a purely correlative manner and not as a predictive tool, i.e. B and C parameters are regressed for each fluid of interest using experimental data. This is probably due to the difficulties in physically interpreting these two numerical parameters [16-18], even if B is viewed as a measure of the
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cohesive energy density of the liquid and is closely linked to the repulsive force between molecules [7, 16-17]. Thus, to the best of our knowledge, only two schemes based on a Tait equation are truly predictive for a large variety of species, those of Thomson et al. [8] and of
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Eslami and Azin [14]. However, these approaches usually yield non-negligible deviations compared to experimental data, their average absolute deviations (AAD) on density being of
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the order of 1% [8, 14] which leads to AAD on isothermal compressibility of the order of 10% or even more.
In a recent work [15], we have recast the Tait equation in the framework of the
extended corresponding states (CS) which allowed to us to develop a “universal” function for the parameter B. This approach has shown to be accurate, with AAD compared to experimental data of the order of 0.1% and 1% on density and isothermal compressibility respectively [15]. However, this method requires an accurate value of the density and the
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ACCEPTED MANUSCRIPT isothermal compressibility at the reference point, the last being a quantity which is not always available even if it can be deduced, in some cases, from experimental compression at high pressures [19]. Thus, we propose in this article an extension of the developments of our previous work [15], which removes the need of experimental values of the density and the
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isothermal compressibility at the reference point. In addition, by combining this new model with mixing rules, we provide a methodology easily applicable to a large class of liquids and liquids mixtures in a predictive manner.
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The structure of the article is as follow. In section 2, we describe the theoretical framework used to develop the proposed approach. Then, in section 3, we present and discuss
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the results obtained. Finally, the main conclusions are given in section 4.
2. Theoretical framework 2.1 Tait equation
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Along a given isotherm, the so-called Tait equation describing the evolution of density with pressure is given by [5]:
(1)
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=
where is the density, P is the pressure, and are the density and the pressure,
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respectively, at a reference point. B and C are numerical parameters that depend both on the chosen isotherm and on the studied materials. From Eq. (1), the isothermal compressibility, , is straightforwardly deduced:
= =
(2)
Interestingly, when using the Tait equation, the estimation of density of a liquid at a target pressure depends on the values of , B and C, whereas its isothermal compressibility only depends on B and C. 3
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2.2 Corresponding states formulation In a recent study [15], we have reformulated the Tait equation in the extended corresponding states framework, i.e. using critical pressure, critical temperature and acentric
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factor as scaling parameters. This allowed us to build a generic function describing the B parameter. This function is applicable to a large variety of not-highly-polar compounds over a wide range of temperature and pressure and writes for a given liquid: 0
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!"# = $ %−2 + )1 + +,1 − -" ./ 1
(3)
where, $ is the critical pressure, " = , where " is the critical temperature, and + is a
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2
parameter function of the acentric factor given by:
+!3# = −0.88330 + 14.343 + 10.17
(4)
It should be pointed out that, to develop Eqs. (3-4), we have used a reference pressure in the Tait equation, Eq. (1), taken at 1.5 [15]. In the following, and , taken at 1.5 will
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3. Results
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be noted .; 2 and .; 2 , respectively.
3.1 Development of the P-Tait scheme
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Once the parameter B is obtained thanks to Eqs. (3-4), the application of the Tait
equation to determine density and isothermal compressibility only requires the knowledge of parameter C and .; 2 . To determine these quantities, a general framework is described below. The so-developed scheme, applicable to liquids and liquid mixtures, is noted as P-Tait in the following and its corresponding flowchart is provided in Fig. 1 for sake of clarity.
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ACCEPTED MANUSCRIPT 3.1.1. C parameter function To define a generic function describing the C parameter we have used an extended CS framework [20-21]. More precisely, we have first studied the variations of the parameter C with the reduced temperature, " , when Eqs. (3-4) are used to compute the B parameter. To do
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so, using the accurate NIST database [22], we have adjusted the values of C with " for
sixteen non-polar and weakly-polar pure compounds over a wide range of temperature and pressure, in which the adjustments are carried out based on the isothermal compressibility
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data [15], i.e. using Eq. (2), see Table 1 for the studied systems and the range of thermodynamic conditions. We recall here that the reference point has been taken at =
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1.5 consistently with our previous work [15]. However, as it will be shown later on, the scheme proposed in this work can be straightforwardly extended to any reference point (e.g. the saturation pressure).
Results shown in Fig. 2 indicate that the so-fitted parameter C depends both on the
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studied compounds and on the reduced temperature. This means that the usual assumption of a parameter C independent on the temperature could lead to non-negligible deviations when dealing with a wide range of temperatures. This would be particularly noticeable on the
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isothermal compressibility as this property is strongly dependent on C, see Eq. (2). In
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addition, Figure 2 shows that compounds possessing a similar acentric factor do not yield superimposing C versus Tr curves, e.g. butane and sulfur hexafluoride or heptane and octafluorocyclobutane. This indicates that the parameter C does not depend only on Tr and
ω as in a classical extended corresponding states framework. However, as shown in Fig. 2, the deviations between the curves associated to compounds possessing similar acentric factors occur when the ratios between the triple point temperature, "< , and the critical temperature, "$ , differs noticeably. Hence, to take into account this effect and so to go beyond the classical
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extended corresponding states framework, we have used the "< ⁄"$ ratio as an extra variable to correlate the parameter C. In addition, similarly to what obtained by Eslami and Azin [14] when they proposed a “universal” C parameter function, we have noticed that the C parameter, when B is given by
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Eqs. (3-4), is correlated to the choice of the reference point and more precisely on the compressibility factor, Z=P/(ρRT), at the reference point.
Thus, as a starting point to build the C parameter correlation, we have used the fitted C
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values obtained on argon and n-alkanes from methane to dodecane, i.e. compounds with relatively small "< ⁄"$ ratios ("< ⁄"$ smaller than 0.55). For these compounds, it has been
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found that C can be well correlated by the following relation:
= exp,−)A .; 2 " B /.
(5)
= 0.06030 − 0.0543 + 0.1205
(6a)
where, A .; 2 is the compressibility factor at the reference point = 1.5 , and
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and
0 = −1.8743 + 1.040
(6b)
Then, to be applicable to compounds possessing a relatively high "< ⁄"$ ratio, Eq. (5)
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should be extended by adding two terms D and 0D that are functions of "< ⁄"$ , as:
(7)
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= E + D Fexp,−)A .; 2 " B + 0D /.
To determine the correlation describing D and 0D , we have used the fitted C values obtained on the compounds of the fitting database, see Table 1. This has lead us to the following correlations: D = 0.17exp G−!0.255⁄3#H − 20-!3 − 0.255#0 − 250!"< ⁄"$ − 0.7#0 I and
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(8a)
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0
O
0D = 0.15,1 − ",< . K L + 1.5exp,— 800!3 − 0.2#0 . L P M
M
(8b)
with ",< = L . M
L
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As shown in Fig. 2, Eq. (7) is able to describe very reasonably, with absolute deviations (AD) around 1-2%, the parameter C for all compounds included in the fitting database. In the following Eq. (7) is used to compute the parameter C for all compounds
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studied in this work whatever their "< ⁄"$ ratio as indicated in Fig. 1. 3.1.2. Reference point
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If the reference density, , of the studied compound is known at the reference
pressure = 1.5 , then the P-Tait scheme is straightforwardly applied by simply using Eqs. (1-8). However, for some liquids .; 2 may be unavailable. In such situations, if the, R ,
is known at another pressure Px, the proposed approach is still applicable with a similar
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accuracy. To do so, one has to calculate the value of .; 2 that yields a computed R,ST<
which equals the exact value of R , where
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R,ST< =
U.V2
Z [P U.V2
K WXY
(9)
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and the parameters B and C are defined by Eqs. (3-4) and Eqs. (6-8), respectively. This can be easily achieved using a Newton-Raphson numerical scheme [23] allowing the determination of the zero of the non linear function: \) .; 2 / = R,ST< ) .; 2 / − R
(10)
Once the value of .; 2 is estimated using this approach, the P-Tait method can be straightforwardly applied using Eqs. (1-8).
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ACCEPTED MANUSCRIPT When no liquid density data point is available, the scheme is still applicable in a purely predictive manner but yields a slightly deteriorated accuracy as it will be shown latter. To do so, one has to use a correlation yielding estimate of liquid density. In this work, we have used the correlations developed by Hankinson and Thomson (HT) [24] which yields
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saturation liquid density, ] , and vapor pressure, ^ , with a reasonable accuracy. Then, by using Eqs. (9-10), it is possible to retrieve an estimation of .; 2 , which, in turn, allows to
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use Eqs. (1-8).
The different possibilities, regarding the availability of the density at the reference
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pressure .; 2 , and the corresponding flowcharts, are summarized in Fig. 1. 3.1.3. Mixtures
To extend the P-Tait scheme to a mixture, we have used a set of mixing rules similar
parameters as:
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to those proposed by Thomson et al. [8] in order to define a pseudo-pure fluid that has its own
3 = ∑T `T 3T
(11a) ⁄
⁄
a$ = O ,∑T `T a$T + 3)∑T `T a$T0 b /)∑T `T a$T b /.
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"$ = ^ G∑T ∑c `T `c )a$T "$T a$c "$c /
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M
D.;
I
(11b) (11c)
"< = ∑T `T "
(11d)
A$ =
Md
(11e)
fM g M
(11f)
Z ∑dY d [fMd e
$ =
Z ∑d d
eMd
^M
where, `T and a$T are the molar fraction and the critical molar volume of species h in the
mixture, respectively, and i is the gas constant. Then, the density and the isothermal
compressibility of the mixture are equal to the ones of the pseudo-pure fluid that are straightforwardly estimated as presented above. 8
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3.2 Applications of the P-Tait scheme To quantify the accuracy of the P-Tait approach, average absolute deviations (AAD) between P-Tait and experimental values have been computed using: nopqrstu nvwx
m
(12)
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AAD = l ∑l Ty 100 m1 −
where X is the density or the isothermal compressibility and N is the number of data points. In
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addition, as an illustration, are also displayed the absolute deviations of some fluids to observe their variations with the thermodynamics conditions, see Fig. 3.
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3.2.1. Application on the species included in the fitting database
As a first test of the P-Tait approach, we have applied it on the database (16 species) used to build the function describing the C parameter, see Table 1. For each species, between 500 and 3000 points generated using the NIST database [22], and uniformly distributed over
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the studied range of thermodynamic conditions, have been considered to compute the AAD. When the P-Tait scheme is used in combination with the experimental value .; 2 ,
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the AAD is, on average, equal to 0.1 % on density and 1.7 % on isothermal compressibility as shown in Tables 2-3. Despite being pure correlation, this can be considered as excellent
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keeping in mind the uncertainties of the database itself [22]. However, it should be noticed that the quality of the P-Tait scheme slightly deteriorates when applied to species possessing a high ratio of the triple point temperature over the critical temperature (e.g. carbon dioxide for which "< ⁄"$ = 0.7121).
Interestingly, the results are still very good when the P-Tait scheme is used with the Hankinson and Thomson correlations [24] (when applicable) and Eqs. (9-10) to determine .; 2 , see Tables 2-3 and Fig. 3. The AAD on density slightly increases to 0.27% on average, 9
ACCEPTED MANUSCRIPT which is consistent with deviations induced by the Hankinson and Thomson correlation [24], whereas the AAD on isothermal compressibility is kept constant on average slightly below 1.7 %.
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In addition, we have compared our results with those coming from the other schemes based on the Tait equation [8, 10, 14] and the non-Tait equation [25-26], see Tables 2-3. It can be noticed that our approach, even when the HT correlation [24] combined with Eqs. (9-10) is used, yields AADs on density and isothermal compressibility which are, by far, much smaller
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than those provided by the other existing approaches including both Tait and non-Tait
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equations.
3.2.2. Predictions on pure fluids
To quantify the predictive capabilities of the P-Tait scheme, the method has been
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applied to 21 additional weakly polar compounds (i.e. with a dipole moment, µ, smaller than 1 Debye), including model and real ones [22, 27-30], which were not included in the fitting database, see Table 4.
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In a first step, we have applied the P-Tait scheme with experimental values of ρref. For compounds whose densities at 1.5 were not available, we have retrieved .; 2 by using a
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known density at a point the closest to 1.5 combined with the use of Eq. (10). Tables 5-6 and Fig. 3 clearly show that the P-Tait approach provides excellent predictions of density and isothermal compressibility of the studied species. The AADs obtained are similar to those obtained on the species used in the fitting database. On average, the AAD on density equals to 0.16 % and to 2.35 % on isothermal compressibility. These predictions validate the reliability of the approach even when dealing with species possessing molecular characteristics out of the boundaries of those used to develop the C function.
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ACCEPTED MANUSCRIPT In a second step, we have tested the P-Tait scheme combined with the HT correlations [24] (when applicable) and Eqs. (9-10) to determine .; 2 . By doing so, it is worth noticing that the proposed scheme is purely predictive. Results given in Tables 5-6 and Fig. 3 indicate that the results are still very good. The AADs on density are slightly larger than the ones
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provided from the P-Tait scheme using an experimental data point, while the AAD on isothermal compressibility is globally unchanged. As shown in Tables 5-6, this approach yields predictions which are much better than what provided by other schemes from the
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literature, based [8, 10, 14] or not [25-26] on the Tait Equation.
In addition, to better quantify the limits of the proposed scheme, we have applied it to
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compounds which were not dedicated for, i.e. highly polar species such as refrigerants and ethanol, see Table 7, using experimental values of ρref. Interestingly, see Table 8, it has been found that the proposed scheme is still able to yield reasonable AADs when dealing with the tested refrigerants, i.e. 0.2-1.0% on the density and 3.0-12.0% on the isothermal
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compressibility. However, for a hydrogen bonding compound such as ethanol the results noticeably deteriorate as shown in Table 8. Thus, this indicates the consistency of the proposed scheme, and seems to be applicable to a wider kind of compounds, including highly
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polar species, than it was dedicated for.
3.2.3. Application on binary liquid mixtures As a further test of the P-Tait scheme, we have applied it on three binary hydrocarbons
liquid mixtures for which both density and isothermal compressibility data were available: Octane-Hexadecane, Decane-Hexadecane and Cyclohexane-Hexadecane [31-33]. As on pure fluids the P-Tait scheme has been applied with an experimental reference density (+ Eqs. (9-10) to retrieve .; o ) and with a predicted reference density using the HT correlation [24] combined with Eqs. (9-10). 11
ACCEPTED MANUSCRIPT Results shown in Tables 9-10 and Fig. 3 indicate that the P-Tait approach, combined with the mixing rules given by Eq. (11), yields a very good prediction of the density and of the isothermal compressibility of the tested liquid mixtures even when using the purely predictive scheme (i.e. combined with the HT correlation to generate the reference point).
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Quantitatively, the results are as good as on pure fluids. Keeping in mind the claimed uncertainties on the experimental database itself (approximately 0.1% on and 1% on ) [31-33], these results are very encouraging. The only case for which results deteriorate is on
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the isothermal compressibility of the cyclohexane-hexadecane mixture whereas the density is well estimated, see Tables 9-10. This slight deterioration is possibly related to the way the
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isothermal compressibilities of this mixture have been deduced, as they have been computed using a cubic equation of state fitted on the density data [33]. In addition, we have compared our results with those provided by the Tait and non-Tait schemes of Thomson et al. [8] and of Nasrifar et al. [25], see Tables 9-10. As found on pure fluids, the P-Tait predictions are
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generally noticeably better than those provided by the other approaches. In addition, we have applied the proposed scheme to two refrigerant mixtures, R507A and R508B, that contain relatively high-polar refrigerants, see Table 11. As found on pure
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fluids, the results are not as good as those on non/weakly polar compounds, but they are still
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reasonable see Table 11.
4. Conclusions
In this study, we have developed a method based on the Tait equation, named P-Tait,
which allows accurate estimations of density and isothermal compressibility of various compressed liquids and liquids mixtures typical of the oil and gas industry. More precisely, following a previous work [15] providing the B parameter of the Tait equation, we have developed here a correlation for the parameter C using an extended corresponding state
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ACCEPTED MANUSCRIPT framework and fitted on a database of 16 species. To be applicable, this approach requires a value of the density at a reference point as well as the critical and triple point temperature, the critical pressure and the acentric factor. If one point of density is known experimentally, the P-Tait yields average absolute
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deviations of non polar weakly polar liquids (noble gases, normal alkanes, alkenes, aromatic, refrigerants …) of the order of 0.15% on the density and of the order of 2% on the isothermal compressibility over a wide range of subcritical thermodynamic conditions (from 0.4 to 0.99
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in reduced temperature and for pressures up to 150 MPa). Very interestingly, the results on the species not included in the fitting database (21 compounds) have been found as good as
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those on the species included in the fitting database (16 compounds). In addition it has been found that, when applied to non-hydrogen bonding compounds with a dipole moment larger than one Debye this approach is still able to provide reasonable results If no data is available on the density, this approach is still applicable by combining it
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with a correlation providing a value for the reference density. As a test, we applied the P-Tait on the same species database in combination with the liquid density at saturation (as the reference point) provided the Hankinson and Thomson correlation [24]. Interestingly, this
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predictive approach yields deviations that are still very good. Deviations on density are two times higher than with an experimental density value (i.e. around 0.3 %), consistently with
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deviations induced by the Hankinson and Thomson correlation, while deviations on isothermal compressibility are globally unchanged. The extension of the P-Tait to mixtures is relatively straightforward because of its
corresponding states nature. Thus, we have applied it to binary liquid mixtures by using mixing rules similar to the ones proposed in the work of Thomson et al. [8]. Very interestingly, the P-Tait combined with these mixing rules yields results that are as good as
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ACCEPTED MANUSCRIPT those on pure fluids on both density and isothermal compressibility. All these results confirm the interest of the proposed scheme. Finally, as a test of its efficiency, the proposed approach has been compared with existing schemes available in the literature (based on Tait equation or not). In all cases, it has
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been found that the P-Tait outperforms the other existing approaches, in particular on
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isothermal compressibility estimation.
Acknowledgements:
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The authors express their warm thanks to Dr. J. Bickert and Dr. F. Montel for the continuous stimulating discussions. We gratefully acknowledge TOTAL S.A. for the post-
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doctoral grant awarded to one of us (HH) and for letting us publish these results.
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ACCEPTED MANUSCRIPT References [1] A. Danesh, PVT and Phase Behaviour of Petroleum Reservoir Fluids, Elsevier Science (1998). [2] A. Firoozabadi, Thermodynamics of Hydrocarbon Reservoirs, McGraw-Hill (1999).
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[3] B. E. Poling, M. Prausnitz and J.P. O’Connell, The Properties of Gases and Liquids, 5th edition, McGraw-Hill(2004).
[4] W.D. McCain, J. P. Spivey and C.P. Lenn, Petroleum Reservoir Fluid Property
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Correlations, PennWellCorporation (2011).
Thermophysics, 9 (1988), 941-951.
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[5] J. H. Dymond and R. Malhotra, The Tait Equation: 100 Years On, International Journal of
[6] R. E. Gibson and O. H. Loeffler, Pressure-Volume-Temperature relation in solution. I. Observations on the Behavior of Solutions of Benzene and some of its derivatives, Journal of Physical Chemistry, 43 (1939), 207-217.
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[7] G. A. Neece and D. R. Squire, On the Tait and Related Empirical Equations of State, The Journal of Physical Chemistry, 72 (1968), 128-136. [8] G.H. Thomson, K.R. Brobst and R.W. Hankinson, An improved Correlation for Densities
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of Compressed Liquids and Liquid Mixtures, AIChE Journal, 28 (1982), 671-676. [9] J. H. Dymond and R. Malhotra, Densities of n-Alkanes and Their Mixtures at Elevated
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Pressures, International Journal of Thermophysics, 8 (1987), 541-555. [10] M. J. Assael, J. H. Dymond and D. Exadaktilou, An Improved Representation for nAlkane Liquid Densities, International Journal of Thermophysics, 15 (1994), 155-164. [11] I. Cibulka and L. Hnedkovsky, Liquid Densities at Elevated Pressures of n-Alkanes from C5 to C16: A Critical Evaluation of Experimental Data, Journal of. Chemical & Engineering Data, 41(1996), 657-668.
15
ACCEPTED MANUSCRIPT [12] I. A. Fakhretdinov, and E. R. Zhdanov, Tait’s Equation of State for Liquid Mixtures, High Temperature, 42 (2004), 396–400. [13] M. Ramos-Estrada, G.A. Iglesias-Silva and K. R. Hall, Experimental measurements and prediction of liquid densities for n-alkane mixtures, The Journal of Chemical
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Thermodynamics, 38 (2006), 337–347.
[14] H. Eslami and R. Azin, Corresponding-states correlation for compressed liquid densities, Fluid Phase Equilibria, 209 (2003), 245-254.
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[15] H. Hoang, G. Galliero, F. Montel, J. Bickert, Tait equation in the extended corresponding
11.
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states frame: Application to liquids and liquid mixtures, Fluid Phase Equilibria, 387 (2015), 5-
[16] V.N. Verveyko, M.V. Verveyko, Y.F. Melikhov, G.A. Melnikov, Perspectives of Applicability and Validity of Tait Equation for Liquids, International Journal of
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Thermophysics, 35 (2014), 2158-2165.
[17] V.S. Nanda, R. Simha, Theoretical interpretation of Tait Equation Parameters, Journal of Chemical Physics, 41 (1964), 1884-1885.
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[18] E.B. Postnikov, AL. Goncharov, V.V. Melent’ev, Tait equation revisited from the entropic and fluctuational points of view, International Journal of Thermophysics, 35 (2014),
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2115-2123.
[19] V. D. Kiselev, A. V. Bolotov, A. Satonin, I. Shakirova, H. A. Kashaeva and A. I. Konovalov, Compressibility of Liquids. Rule of Noncrossing V−P Curvatures, J. Phys. Chem. B, 2008, 112 (21), pp 6674–6682 [20] H. W. Xiang, The corresponding-States Principle and its Practice, Elsevier (2005). [21] M. J. Assael, J. P. M. Trusler and T. F. Tsolakis, An Introduction to Their Prediction: Thermophysical Properties of Fluids, Imperial College Press (1996) 16
ACCEPTED MANUSCRIPT [22] E. W. Lemmon, M. L. Huber, and M. O. McLinden, Reference Fluid Thermodynamic and Transport Properties, NIST Standard Reference Database 23, Version 8.0, (2007). [23] C.T. Kelley, Solving Nonlinear Equations with Newton’s Method (Fundamental of Algorithms), SIAM, Philadelphia (2003).
Liquids and Their Mixtures, AIChE J., 25 (1979), 653
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[24] R. W. Hankinson and G. H. Thomson, A New Correlation for Saturated Densities of
[25] Kh. Nasrifar, Sh. Ayatollahi, M. Moshfeghian, A compressed liquid density correlation,
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Fluid Phase Equilibria, 168 (2000) 149–163
[26] M. Aalto, K. I. Keskinen, J. Aittamaa and S. Liukkonen, An improved correlation for
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compressed liquid densities of hydrocarbons. Part 1. Pure compounds, Fluid Phase Equilibria, 114 (1996) 1-19
[27] J. Kolafa and I. Nezbeda, The Lennard-Jones fluid: an accurate analytic and theoretically-based equation of state, Fluid Phase Equilibria, 100 (1994), 1-34.
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[28] T. S. Khasanshin and A. P. Shchemelev, The Thermodynamic Properties of nTetradecane in Liquid State, High Temperature, 40 (2002), 207–211 [29] T. S. Khasanshin, V. S. Samuilov, and A. P. Shchemelev, Determination of The
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Thermodynamic properties of Liquid n-Hexadecane from The Measurements of The Velocity of Sound, Journal of Engineering Physics and Thermophysics, 82 (2009), 149-156
AC C
[30] S. Dutour, J. L. Daridon and B. Lagourette, Pressure and Temperature Dependence of the Speed of Sound and Related Properties in Normal Octadecane and Nonadecane, International Journal of Thermophysics, 21 (2000), 173-184 [31] T. S. Khasanshin, V. S. Samuilov, and A. P. Shchamialiou, Thermodynamic properties of binary liquid mixtures of n-octane + n-hexadecane, High Temperatures-High Pressures, 39 (2010), 321–338 [32] T. S. Khasanshin, V. S. Samuilov, and A. P. Shchemelev, The Thermodynamic
17
ACCEPTED MANUSCRIPT Properties of Liquid Binary Mixtures of n-Alkanes: n-Decane + n-Hexadecane, High Temperature, 48 (2010), 665-672 [33] J. A. Amorim, O.Chiavone-Filho, M. L. L. Paredes, and K.Rajagopal, High-Pressure Density Measurements for the Binary System Cyclohexane + n-Hexadecane in the
RI PT
Temperature Range of (318.15 to 413.15) K, Journal of Chemical & Engineering Data, 52
AC C
EP
TE D
M AN U
SC
(2007), 613-618
18
ACCEPTED MANUSCRIPT
" , , 3, "<
No
z{|.
} .; 2 Yes
Yes
No
Estimate ^ and ] from Ref. [24]
SC
z{|.
y .; 2
RI PT
Compute B, , D , 0 , 0D using Eqs. (3), (6) and (8)
TE D
Compute and using Eqs. (1) and (2)
M AN U
Solve Eq. (10) to obtain y .; 2
Calculate C using Eq. (7)
Figure 1: Flowchart of the P-Tait scheme proposed in this work. "< is the triple point
AC C
EP
temperature, ^ is the vapor pressure and ] is the saturation liquid density .
19
ACCEPTED MANUSCRIPT 3
0.105
0.09 0.085 0.08 0.3
2 1.5 1 0.5
0.4
0.5
0.6
Tr
0.7
0.8
0.9
1
0 0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
RI PT
C
0.095
2.5
Ar (ω=-0.00219) Kr (ω=-0.00089) Xe (ω=0.00363) CH4 (ω=0.0114) C4H10 (ω=0.201) C7H16 (ω=0.349) C6H6 (ω=0.209) SF6 (ω=0.210) C4F8 (ω=0.355)
|CCorr-Cfit|/Cfit×100
0.1
Tr
Figure 2: Left: Dependence of the C parameter with the reduced temperature " for some
SC
compounds. Right: Absolute deviation on the C parameter obtained from Eq. (7). The same
AC C
EP
TE D
M AN U
symbol is employed for compounds possessing similar acentric factor.
20
ACCEPTED MANUSCRIPT 0.8 0.6 0.4 0.2 0
0
20
40
60
80
P (MPa)
100
120
2 1 0
20
40
0.1
0
20
40
60
80
P (MPa)
100
120
120
140
2
1
0
20
40
60
80
P (MPa)
100
120
140
6
0.6
Octane-Hexadecane
0.2
20
40
60
P (MPa)
80
100
EP
0
TE D
0.4
0
100
Hexadecane 3
0
140
80
P (MPa)
SC
abs(βT,Exp-βT,Pred)/βT,Exp×100%
0.2
60
M AN U
abs(ρExp-ρPred)/ρExp×100%
3
4
Hexadecane
abs(ρExp-ρPred)/ρExp×100%
Argon
4
0
140
0.3
0
5
RI PT
abs(βT,Exp-βT,Pred)/βT,Exp×100%
Argon
abs(βT,Exp-βT,Pred)/βT,Exp×100%
abs(ρExp-ρPred)/ρExp×100%
1
Octane-Hexadecane
5 4 3 2 1 0
0
20
40
60
P (MPa)
80
100
AC C
Figure 3. Absolute deviation on density (left figures) and on isothermal compressibility (right figures) yielded by the proposed scheme on some systems studied in this work. Solid circles correspond to the results using an experimental reference point. Open diamonds correspond to the results using a reference point computed from the HT correlation [24].
21
ACCEPTED MANUSCRIPT Table 1: Triple point temperature, critical temperature, critical pressure and acentric factor [22] of the compounds included in the fitting database and range of temperature and pressure used to fit the C parameter in Eqs. (5-8). Tt(K)
Tc(K)
Pc (MPa)
ω
Tmin-Tmax(K)
Pmax(MPa)
Methane
90.694
190.564
4.5992
0.01142
100-190
150
Ethane
90.368
305.330
4.8718
0.09930
100-290
150
Propane
85.53
369.890
4.2512
0.15210
90-360
150
Butane
134.9
425.125
3.7960
0.20100
140-420
70
Pentane
143.47
469.700
3.3700
Hexane
177.83
507.820
3.0340
Heptane
182.55
540.13
2.7360
Octane
216.37
569.32
Nonane
219.7
Decane
SC
RI PT
Species
150-450
100
0.29900
180-450
100
0.349
190-450
100
2.4970
0.393
220-450
100
594.55
2.2810
0.4433
220-450
150
243.5
617.7
2.1030
0.488
250-450
150
Dodecane
263.6
658.1
1.8170
0.574
270-450
150
Argon
83.806
150.69
4.863
-0.00219
90-150
150
Benzene
278.7
562.05
4.894
0.2092
280-450
80
Sulfur Hexafluoride
223.56
318.72
3.755
0.21
240-310
80
216.58
304.13
7.3773
0.22394
220-300
150
233.35
388.38
2.7775
0.3553
240-380
60
TE D
AC C
Octafluorocyclobutane
EP
Carbon dioxide
M AN U
0.25100
22
ACCEPTED MANUSCRIPT Table 2: AAD on the density yielded by the P-Tait scheme and some other Tait and non-Tait schemes from literature on the species included in the fitting database.
AAD on ρ P-Tait + [24]
Ref. [10]
Ref. [14]
Methane
0.12
0.26
1.37
0.63
0.86
Ethane
0.08
0.21
0.54
1.19
1.22
Propane
0.10
0.34
0.87
1.26
1.14
Butane
0.10
0.28
1.59
2.64
Pentane
0.17
0.24
1.03
2.16
Hexane
0.05
0.16
Heptane
0.12
0.33
Octane
0.05
0.21
Nonane
0.06
0.28
Decane
0.04
0.08
Dodecane
0.01
0.14
Argon
0.14
Benzene
0.01
Ref. [25]
Ref. [26]
1.40
4.12
[22]
0.62
2.74
[22]
0.66
2.64
[22]
SC
P-Tait + Exp.
0.44
0.48
1.23
[22]
0.64
0.35
1.80
[22]
M AN U
Ref. [8]
Database
RI PT
Species
1.05
0.75
0.37
1.44
[22]
0.77
0.40
0.69
0.21
1.62
[22]
0.48
0.24
0.55
0.43
1.23
[22]
0.67
0.12
0.77
0.48
1.65
[22]
0.48
0.09
0.66
0.40
1.67
[22]
0.28
0.06
---
0.41
1.75
[22]
0.59
5.30
---
1.71
1.64
4.87
[22]
0.15
0.50
---
0.19
0.17
0.69
[22]
EP
TE D
0.62
0.17
---
1.25
---
0.72
---
---
[22]
Carbon dioxide
0.27
0.45
1.58
---
0.84
0.77
2.22
[22]
---
0.88
---
---
---
---
[22]
AC C
Sulfur Hexafluoride
Octafluorocyclobutane 0.15
23
ACCEPTED MANUSCRIPT Table 3: AAD on the isothermal compressibility yielded by the P-Tait scheme and some other Tait and non-Tait schemes from literature on the species included in the fitting database. AAD on ~ P-Tait + [24]
Ref. [10]
Methane
2.52
2.50
9.06
7.23
15.24
Ethane
1.75
1.73
9.77
8.20
32.69
Propane
1.15
1.17
24.56
7.68
43.77
Butane
1.36
1.40
18.00
11.50
Pentane
1.84
1.84
18.71
8.51
Hexane
1.85
1.82
15.82
Heptane
1.14
1.10
Octane
1.52
Nonane
1.48
Decane
0.96
0.95
Dodecane
0.79
Argon Benzene
Ref. [25]
Ref. [26]
50.13
[22]
19.04
56.28
[22]
19.65
63.36
[22]
15.18
9.35
39.18
[22]
23.75
10.33
45.24
[22]
6.51
22.75
9.76
44.76
[22]
20.14
3.16
29.23
8.60
44.53
[22]
1.49
15.22
2.91
26.14
9.08
43.32
[22]
1.39
18.08
1.65
33.09
9.95
49.41
[22]
15.62
0.91
30.77
9.59
47.04
[22]
0.80
10.90
1.14
---
11.36
46.69
[22]
1.79
1.83
13.29
---
13.21
17.91
46.74
[22]
1.07
0.84
13.50
---
11.56
6.05
24.48
[22]
1.51
---
11.96
---
8.66
---
---
[22]
4.24
5.31
---
9.11
15.20
32.06
[22]
---
12.12
---
---
---
---
[22]
EP
TE D
M AN U
19.14
AC C
Sulfur Hexafluoride
Ref. [14]
RI PT
P-Tait + Exp.
Carbon dioxide
Ref. [8]
Database
SC
Species
4.21
Octafluorocyclobutane 1.94
24
ACCEPTED MANUSCRIPT Table 4: Triple point temperature, critical temperature, critical pressure and acentric factor [22] of the compounds not included in the fitting database and range of temperature and pressure explored. Tt(K)
Tc(K)
Pc (MPa)
ω
Tmin-Tmax(K)
Pmax(MPa)
Krypton
115.78
209.48
5.525
-0.00089
120-200
150
Xenon
161.41
289.73
5.842
0.00363
170-280
150
Nitrogen
63.151
126.19
3.3958
0.0372
70-120
150
Neopentane
256.6
433.74
3.196
0.1961
320-430
131
Tetradecane§
279.02
693.00
1.600
Hexadecane§
291.3
723.00
1.400
Octadecane§
301.0
747.00
1.29
Nonadecane§
304.0
755.00
Oxygen
54.361
Ethylene
SC
RI PT
Species
293-433
140
0.7180
298-433
140
0.800
313-383
150
1.20
0.845
313-383
150
154.58
5.043
0.0222
60-150
82
103.99
282.35
5.0418
0.0866
110-280
150
Isopentane
112.65
460.35
3.378
0.2274
120-450
150
Cyclohexane
279.47
553.64
4.075
0.20926
290-450
80
Toluene
178.0
591.75
4.1263
0.2657
180-450
150
Lennard-Jones*§
0.657
1.301
0.121
-0.0220
0.75-1.25
8.33
87.953
364.21
4.555
0.146
90-350
150
87.80
419.29
4.0051
0.192
100-400
70
R12§
116.1
385.12
4.1361
0.17948
125-375
150
R14§
89.54
227.51
3.750
0.1785
120-220
51
R113§
236.93
487.21
3.3922
0.25253
280-450
150
R115§
173.75
353.1
3.129
0.2500
180-350
60
R116§
173.1
293.03
3.048
0.2566
180-290
50
TE D
AC C
Butene§
EP
Propene§
M AN U
0.644
(*) Values are expressed in Lennard-Jones (LJ) units [22] (§) Compounds for which .; 2 was not available. 25
ACCEPTED MANUSCRIPT Table 5: AAD on the density yielded by the P-Tait scheme and some other Tait and non-Tait schemes from literature on the species that were not included in the fitting database.
AAD on ρ P-Tait + [24]
Krypton
0.12
---
1.17
---
1.40
Xenon
0.07
---
0.70
---
0.90
Nitrogen
0.34
0.41
1.12
---
1.75
Neopentane
0.22
---
4.16
---
Lennard-Jones*§
0.25
---
1.86
---
Tetradecane§
0.03
0.08
0.46
0.14
Hexadecane§
0.08
0.14
0.63
Octadecane§
0.09
0.06
Nonadecane§
0.06
Oxygen
0.12
Ethylene
0.25
Isopentane
0.18
Cyclohexane
0.14
Ref. [14]
Ref. [25]
Ref. [26]
---
---
[22]
---
---
[22]
1.07
5.50
[22]
---
---
[22]
---
---
---
[27]
---
0.70
2.63
[28]
0.05
---
0.69
1.91
[29]
0.47
---
---
1.23
2.50
[30]
0.05
0.36
---
---
0.82
2.51
[30]
0.34
0.45
---
0.62
0.91
2.38
[22]
TE D
M AN U
---
9.30
---
1.18
1.33
3.99
[22]
0.40
1.29
---
---
0.57
2.82
[22]
0.29
0.83
---
---
0.69
0.64
[22]
EP
0.56
0.09
0.28
1.01
---
---
0.39
1.54
[22]
0.13
0.65
0.85
---
---
1.26
3.66
[22]
AC C
Propene§
Ref. [10]
RI PT
P-Tait + Exp.
Toluene
Ref. [8]
Database
SC
Species
Butene§
0.08
0.90
0.66
---
0.63
0.88
1.96
[22]
R12§
0.25
0.37
1.29
---
---
0.52
3.27
[22]
R14§
0.08
---
0.57
---
---
---
---
[22]
R113§
0.15
0.29
1.06
---
0.96
2.31
[22]
R115§
0.30
---
0.99
---
---
---
---
[22]
R116§
0.29
---
2.63
---
---
---
---
[22]
(*) Values are expressed in Lennard-Jones (LJ) units [22] (§) Compounds for which .; 2 was not available. 26
ACCEPTED MANUSCRIPT Table 6: AAD on the isothermal compressibility yielded by the P-Tait scheme and some other Tait and non-Tait schemes from literature on the species that were not included in the fitting database. AAD on ~ Ref. [14]
RI PT
P-Tait + [24]
Ref. [8]
Ref. [25]
Krypton
1.68
---
8.28
---
12.32
---
---
[22]
Xenon
0.88
---
6.55
---
12.03
---
---
[22]
Nitrogen
3.67
3.69
8.70
---
17.82
10.09
51.88
[22]
Neopentane
1.37
---
13.54
---
---
---
---
[22]
LennardJones*§
2.77
---
13.09
---
---
---
---
[27]
Tetradecane§
1.28
1.31
7.52
1.00
---
11.71
57.37
[28]
Hexadecane§
1.50
1.53
9.38
0.86
---
12.68
51.48
[29]
Octadecane§
2.72
2.69
6.44
---
---
14.60
60.95
[30]
Nonadecane§
1.65
1.64
6.36
---
---
11.34
61.01
[30]
Oxygen
2.95
2.93
6.61
---
15.44
17.54
44.46
[22]
Ethylene
3.12
3.18
23.04
---
20.27
19.22
51.83
[22]
Isopentane
2.31
2.33
25.88
---
---
12.58
60.70
[22]
Cyclohexane
3.88
3.91
17.39
---
---
2.62
25.62
[22]
EP
M AN U
SC
P-Tait + Exp.
AC C
Ref. [10]
Database
TE D
Species
Ref. [26]
Toluene
1.73
1.61
30.33
---
---
17.01
52.36
[22]
Propene§
1.34
1.42
13.27
---
---
20.41
56.84
[22]
Butene§
1.27
1.26
20.24
---
29.68
12.32
44.03
[22]
R12§
3.17
3.11
14.27
---
---
8.65
49.44
[22]
R14§
1.83
---
7.62
---
---
---
---
[22]
R113§
2.34
2.35
10.00
---
---
7.15
43.23
[22]
R115§
4.18
---
16.40
---
---
---
---
[22]
R116§
3.88
---
7.60
---
---
---
---
[22]
27
ACCEPTED MANUSCRIPT (*) Values are expressed in Lennard-Jones (LJ) units [22]
AC C
EP
TE D
M AN U
SC
RI PT
(§) Compounds for which .; 2 was not available.
28
ACCEPTED MANUSCRIPT Table 7: Triple point temperature, critical temperature, critical pressure, acentric factor and dipole moment [22] of the highly polar compounds tested in this work and range of temperature and pressure explored. Tt(K)
Tc(K)
Pc (MPa)
ω
µ (D)
R125
172.52
339.17
3.6177
0.3052
1.563
R141b
169.68
477.50
4.212
0.2195
2.014
R143a
161.34
345.86
3.761
0.2615
2.340
R21
142.80
451.48
5.1812
0.2061
1.370
R23
118.02
299.29
4.832
0.2630
R32
136.34
351.26
5.782
0.2769
Ethanol
159.00
513.90
6.148
0.6440
TminTmax(K)
60
200-450
150
190-340
100
250-440
137
1.649
210-280
120
1.978
150-134
70
1.691
250-425
150
SC
190-330
M AN U
TE D EP AC C 29
Pmax(MPa)
RI PT
Species
ACCEPTED MANUSCRIPT Table 8: AAD on the density and isothermal compressibility yielded by the P-Tait scheme when applied highly polar compounds. AAD on ρ
AAD on ~
Database
R125
0.21
2.85
[22]
R141b
0.15
2.38
[22]
R143a
0.57
4.84
R21
0.20
4.69
R23
0.83
7.09
R32
0.67
12.45
[22]
Ethanol
2.38
36.91
[22]
RI PT
Species
[22]
[22]
AC C
EP
TE D
M AN U
SC
[22]
30
ACCEPTED MANUSCRIPT Table 9: AAD on the density yielded by the P-Tait scheme in this work and other Tait and non-Tait schemes reported in the literature when applied to three different binary mixtures. TminTmax(K)
AAD on ρ
Pmax(MPa) P-Tait + Exp.
x1
P-Tait + [24]
Ref. [8]
Ref. [25]
RI PT
Mole Fraction
Mixture of Octane (1) and Hexadecane (2)§ [31] 313.15393.15
100
0.07
0.07
0.50
313.15393.15
100
0.10
0.14
0.75
313.15393.15
100
0.13
0.27
0.28
0.33
0.29
SC
0.25
M AN U
0.13
0.42
0.41
Mixture of Decane (1) and Hexadecane (2)§ [32] 298.15433.15
100
0.50
298.15433.15
100
0.75
298.15433.15
100
0.08
0.08
0.45
0.33
0.09
0.16
0.43
0.28
0.10
0.16
0.43
0.26
TE D
0.25
Mixture of Cyclohexane (1) and Hexadecane (2)§ [33] 318.15413.15
0.30
318.15413.15
63
0.09
0.15
0.42
0.27
63
0.03
0.13
0.32
0.60
AC C
EP
0.10
0.50
318.15413.15
63
0.06
0.24
0.69
1.04
0.70
318.15413.15
63
0.18
0.49
1.15
1.32
0.90
318.15413.15
63
0.24
0.41
1.12
1.02
(§) Compounds for which .; 2 was not available.
31
ACCEPTED MANUSCRIPT Table 10: AAD on the isothermal compressibility yielded by the P-Tait scheme in this work and other Tait and non-Tait schemes reported in the literature when applied to three different binary mixtures. TminTmax(K)
AAD on ~
Pmax(MPa) P-Tait + Exp.
x1
P-Tait + [24]
Ref. [8]
Ref. [25]
7.12
15.15
RI PT
Mole Fraction
Mixture of Octane (1) and Hexadecane (2)§ [31] 100
1.84
0.50
313.15393.15
100
2.75
0.75
313.15393.15
100
1.82
SC
313.15393.15
2.70
M AN U
0.25
3.18
3.14
10.41
13.29
12.97
11.59
Mixture of Decane (1) and Hexadecane (2)§ [32] 298.15433.15
100
0.50
298.15433.15
100
0.75
298.15433.15
1.21
1.19
7.70
18.47
1.27
1.23
9.13
16.02
1.19
10.79
14.23
TE D
0.25
100
1.24
318.15413.15
AC C
0.10
EP
Mixture of Cyclohexane (1) and Hexadecane (2)§ [33] 63
2.42
2.42
7.48
14.97
0.30
318.15413.15
63
1.05
1.05
9.87
10.56
0.50
318.15413.15
63
2.27
2.30
16.75
8.74
0.70
318.15413.15
63
5.49
5.55
21.85
9.53
0.90
318.15413.15
63
6.58
6.61
20.98
6.96
(§) Compounds for which .; 2 was not available.
32
ACCEPTED MANUSCRIPT Table 11: AAD on the density and isothermal compressibility yielded by the P-Tait scheme when applied to two mixtures of polar compounds. Mole Fraction Tmin-Tmax(K)
Pmax(MPa)
AAD on ρ
AAD on ~
x1
0.41184
190-320
83
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[R507A]: Mixture of R125 (1) and R143a (2)
0.41
5.94
[R508B]: Mixture of R23 (1) and R116 (2) 150-260
50
0.96
AC C
EP
TE D
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0.46
33
16.14
S0378-3812(16)30259-X
DOI:
10.1016/j.fluid.2016.05.026
Reference:
FLUID 11111
To appear in:
Fluid Phase Equilibria
Received Date: 10 April 2016 Revised Date:
20 May 2016
Accepted Date: 25 May 2016
Please cite this article as: H. Hoang, G. Galliero, Predictive Tait equation for non-polar and weakly polar fluids: Applications to liquids and liquid mixtures, Fluid Phase Equilibria (2016), doi: 10.1016/ j.fluid.2016.05.026. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Predictive Tait equation for non-polar and weakly polar fluids: Applications to liquids and liquid mixtures Hai Hoang1,2, Guillaume Galliero1* 1
Laboratoire des Fluides Complexes et leurs Réservoirs (UMR-5150 with CNRS and TOTAL),
2
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Université de Pau et des Pays de l'Adour, BP 1155, F-64013 Pau Cedex, France Institute of Research and Development, Duy Tan University, Da Nang, Viet Nam *[email protected]
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Abstract
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In this study, we describe a method based on the Tait equation which allows accurate estimation of density and isothermal compressibility of non-polar and weakly polar liquids and liquid mixtures over a wide range of pressures. We have applied the approach to various species in conditions ranging from 0.4 to 0.99 of the critical temperature and for pressures up to 150 MPa. If a reference experimental point of density is known and used as an input for
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this method, the approach yields average absolute deviations around 0.15 % on density and around 2 % on isothermal compressibility. In its fully predictive version, i.e. without
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experimental data point on density, this approach yields deviations on density around 0.3 %, while deviations on isothermal compressibility are kept around 2 %. Interestingly, when
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applied to compounds with a dipole moment larger than one Debye this approach is still able to provide reasonable results. In addition, this approach has been applied on binary liquid mixtures, using classical mixing rules. On three different hydrocarbon mixtures it has been found that the proposed approached yields results as good as on pure liquids emphasizing the consistency of the proposed methodology. All these results compare very favorably to those obtained from other approaches of the literature. Keywords Density; Isothermal compressibility; Tait Equation; Liquids; Mixtures; 1
ACCEPTED MANUSCRIPT 1. Introduction High-pressures densities and isothermal compressibilities of liquids are key quantities in the oil and gas industry [1-2]. To estimate these properties various methods, both correlative and theoretical ones, have been developed and reported in the literature [1-4].
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Among them, the so-called Tait equation [5], a simple heuristic approach has shown its efficiency to deal with compressed liquids [5-16]. This approach has attracted a lot of attention as it only requires a value of density at a reference pressure and two numerical
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parameters, often noted B and C, to provide the density and the isothermal compressibility of a given liquid for pressures up to several MPa.
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Rather surprisingly, the Tait equation has mostly been employed in a purely correlative manner and not as a predictive tool, i.e. B and C parameters are regressed for each fluid of interest using experimental data. This is probably due to the difficulties in physically interpreting these two numerical parameters [16-18], even if B is viewed as a measure of the
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cohesive energy density of the liquid and is closely linked to the repulsive force between molecules [7, 16-17]. Thus, to the best of our knowledge, only two schemes based on a Tait equation are truly predictive for a large variety of species, those of Thomson et al. [8] and of
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Eslami and Azin [14]. However, these approaches usually yield non-negligible deviations compared to experimental data, their average absolute deviations (AAD) on density being of
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the order of 1% [8, 14] which leads to AAD on isothermal compressibility of the order of 10% or even more.
In a recent work [15], we have recast the Tait equation in the framework of the
extended corresponding states (CS) which allowed to us to develop a “universal” function for the parameter B. This approach has shown to be accurate, with AAD compared to experimental data of the order of 0.1% and 1% on density and isothermal compressibility respectively [15]. However, this method requires an accurate value of the density and the
2
ACCEPTED MANUSCRIPT isothermal compressibility at the reference point, the last being a quantity which is not always available even if it can be deduced, in some cases, from experimental compression at high pressures [19]. Thus, we propose in this article an extension of the developments of our previous work [15], which removes the need of experimental values of the density and the
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isothermal compressibility at the reference point. In addition, by combining this new model with mixing rules, we provide a methodology easily applicable to a large class of liquids and liquids mixtures in a predictive manner.
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The structure of the article is as follow. In section 2, we describe the theoretical framework used to develop the proposed approach. Then, in section 3, we present and discuss
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the results obtained. Finally, the main conclusions are given in section 4.
2. Theoretical framework 2.1 Tait equation
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Along a given isotherm, the so-called Tait equation describing the evolution of density with pressure is given by [5]:
(1)
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=
where is the density, P is the pressure, and are the density and the pressure,
AC C
respectively, at a reference point. B and C are numerical parameters that depend both on the chosen isotherm and on the studied materials. From Eq. (1), the isothermal compressibility, , is straightforwardly deduced:
= =
(2)
Interestingly, when using the Tait equation, the estimation of density of a liquid at a target pressure depends on the values of , B and C, whereas its isothermal compressibility only depends on B and C. 3
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2.2 Corresponding states formulation In a recent study [15], we have reformulated the Tait equation in the extended corresponding states framework, i.e. using critical pressure, critical temperature and acentric
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factor as scaling parameters. This allowed us to build a generic function describing the B parameter. This function is applicable to a large variety of not-highly-polar compounds over a wide range of temperature and pressure and writes for a given liquid: 0
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!"# = $ %−2 + )1 + +,1 − -" ./ 1
(3)
where, $ is the critical pressure, " = , where " is the critical temperature, and + is a
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2
parameter function of the acentric factor given by:
+!3# = −0.88330 + 14.343 + 10.17
(4)
It should be pointed out that, to develop Eqs. (3-4), we have used a reference pressure in the Tait equation, Eq. (1), taken at 1.5 [15]. In the following, and , taken at 1.5 will
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3. Results
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be noted .; 2 and .; 2 , respectively.
3.1 Development of the P-Tait scheme
AC C
Once the parameter B is obtained thanks to Eqs. (3-4), the application of the Tait
equation to determine density and isothermal compressibility only requires the knowledge of parameter C and .; 2 . To determine these quantities, a general framework is described below. The so-developed scheme, applicable to liquids and liquid mixtures, is noted as P-Tait in the following and its corresponding flowchart is provided in Fig. 1 for sake of clarity.
4
ACCEPTED MANUSCRIPT 3.1.1. C parameter function To define a generic function describing the C parameter we have used an extended CS framework [20-21]. More precisely, we have first studied the variations of the parameter C with the reduced temperature, " , when Eqs. (3-4) are used to compute the B parameter. To do
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so, using the accurate NIST database [22], we have adjusted the values of C with " for
sixteen non-polar and weakly-polar pure compounds over a wide range of temperature and pressure, in which the adjustments are carried out based on the isothermal compressibility
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data [15], i.e. using Eq. (2), see Table 1 for the studied systems and the range of thermodynamic conditions. We recall here that the reference point has been taken at =
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1.5 consistently with our previous work [15]. However, as it will be shown later on, the scheme proposed in this work can be straightforwardly extended to any reference point (e.g. the saturation pressure).
Results shown in Fig. 2 indicate that the so-fitted parameter C depends both on the
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studied compounds and on the reduced temperature. This means that the usual assumption of a parameter C independent on the temperature could lead to non-negligible deviations when dealing with a wide range of temperatures. This would be particularly noticeable on the
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isothermal compressibility as this property is strongly dependent on C, see Eq. (2). In
AC C
addition, Figure 2 shows that compounds possessing a similar acentric factor do not yield superimposing C versus Tr curves, e.g. butane and sulfur hexafluoride or heptane and octafluorocyclobutane. This indicates that the parameter C does not depend only on Tr and
ω as in a classical extended corresponding states framework. However, as shown in Fig. 2, the deviations between the curves associated to compounds possessing similar acentric factors occur when the ratios between the triple point temperature, "< , and the critical temperature, "$ , differs noticeably. Hence, to take into account this effect and so to go beyond the classical
5
ACCEPTED MANUSCRIPT
extended corresponding states framework, we have used the "< ⁄"$ ratio as an extra variable to correlate the parameter C. In addition, similarly to what obtained by Eslami and Azin [14] when they proposed a “universal” C parameter function, we have noticed that the C parameter, when B is given by
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Eqs. (3-4), is correlated to the choice of the reference point and more precisely on the compressibility factor, Z=P/(ρRT), at the reference point.
Thus, as a starting point to build the C parameter correlation, we have used the fitted C
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values obtained on argon and n-alkanes from methane to dodecane, i.e. compounds with relatively small "< ⁄"$ ratios ("< ⁄"$ smaller than 0.55). For these compounds, it has been
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found that C can be well correlated by the following relation:
= exp,−)A .; 2 " B /.
(5)
= 0.06030 − 0.0543 + 0.1205
(6a)
where, A .; 2 is the compressibility factor at the reference point = 1.5 , and
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and
0 = −1.8743 + 1.040
(6b)
Then, to be applicable to compounds possessing a relatively high "< ⁄"$ ratio, Eq. (5)
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should be extended by adding two terms D and 0D that are functions of "< ⁄"$ , as:
(7)
AC C
= E + D Fexp,−)A .; 2 " B + 0D /.
To determine the correlation describing D and 0D , we have used the fitted C values obtained on the compounds of the fitting database, see Table 1. This has lead us to the following correlations: D = 0.17exp G−!0.255⁄3#H − 20-!3 − 0.255#0 − 250!"< ⁄"$ − 0.7#0 I and
6
(8a)
ACCEPTED MANUSCRIPT
0
O
0D = 0.15,1 − ",< . K L + 1.5exp,— 800!3 − 0.2#0 . L P M
M
(8b)
with ",< = L . M
L
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As shown in Fig. 2, Eq. (7) is able to describe very reasonably, with absolute deviations (AD) around 1-2%, the parameter C for all compounds included in the fitting database. In the following Eq. (7) is used to compute the parameter C for all compounds
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studied in this work whatever their "< ⁄"$ ratio as indicated in Fig. 1. 3.1.2. Reference point
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If the reference density, , of the studied compound is known at the reference
pressure = 1.5 , then the P-Tait scheme is straightforwardly applied by simply using Eqs. (1-8). However, for some liquids .; 2 may be unavailable. In such situations, if the, R ,
is known at another pressure Px, the proposed approach is still applicable with a similar
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accuracy. To do so, one has to calculate the value of .; 2 that yields a computed R,ST<
which equals the exact value of R , where
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R,ST< =
U.V2
Z [P U.V2
K WXY
(9)
AC C
and the parameters B and C are defined by Eqs. (3-4) and Eqs. (6-8), respectively. This can be easily achieved using a Newton-Raphson numerical scheme [23] allowing the determination of the zero of the non linear function: \) .; 2 / = R,ST< ) .; 2 / − R
(10)
Once the value of .; 2 is estimated using this approach, the P-Tait method can be straightforwardly applied using Eqs. (1-8).
7
ACCEPTED MANUSCRIPT When no liquid density data point is available, the scheme is still applicable in a purely predictive manner but yields a slightly deteriorated accuracy as it will be shown latter. To do so, one has to use a correlation yielding estimate of liquid density. In this work, we have used the correlations developed by Hankinson and Thomson (HT) [24] which yields
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saturation liquid density, ] , and vapor pressure, ^ , with a reasonable accuracy. Then, by using Eqs. (9-10), it is possible to retrieve an estimation of .; 2 , which, in turn, allows to
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use Eqs. (1-8).
The different possibilities, regarding the availability of the density at the reference
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pressure .; 2 , and the corresponding flowcharts, are summarized in Fig. 1. 3.1.3. Mixtures
To extend the P-Tait scheme to a mixture, we have used a set of mixing rules similar
parameters as:
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to those proposed by Thomson et al. [8] in order to define a pseudo-pure fluid that has its own
3 = ∑T `T 3T
(11a) ⁄
⁄
a$ = O ,∑T `T a$T + 3)∑T `T a$T0 b /)∑T `T a$T b /.
EP
"$ = ^ G∑T ∑c `T `c )a$T "$T a$c "$c /
AC C
M
D.;
I
(11b) (11c)
"< = ∑T `T "
(11d)
A$ =
Md
(11e)
fM g M
(11f)
Z ∑dY d [fMd e
$ =
Z ∑d d
eMd
^M
where, `T and a$T are the molar fraction and the critical molar volume of species h in the
mixture, respectively, and i is the gas constant. Then, the density and the isothermal
compressibility of the mixture are equal to the ones of the pseudo-pure fluid that are straightforwardly estimated as presented above. 8
ACCEPTED MANUSCRIPT
3.2 Applications of the P-Tait scheme To quantify the accuracy of the P-Tait approach, average absolute deviations (AAD) between P-Tait and experimental values have been computed using: nopqrstu nvwx
m
(12)
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AAD = l ∑l Ty 100 m1 −
where X is the density or the isothermal compressibility and N is the number of data points. In
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addition, as an illustration, are also displayed the absolute deviations of some fluids to observe their variations with the thermodynamics conditions, see Fig. 3.
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3.2.1. Application on the species included in the fitting database
As a first test of the P-Tait approach, we have applied it on the database (16 species) used to build the function describing the C parameter, see Table 1. For each species, between 500 and 3000 points generated using the NIST database [22], and uniformly distributed over
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the studied range of thermodynamic conditions, have been considered to compute the AAD. When the P-Tait scheme is used in combination with the experimental value .; 2 ,
EP
the AAD is, on average, equal to 0.1 % on density and 1.7 % on isothermal compressibility as shown in Tables 2-3. Despite being pure correlation, this can be considered as excellent
AC C
keeping in mind the uncertainties of the database itself [22]. However, it should be noticed that the quality of the P-Tait scheme slightly deteriorates when applied to species possessing a high ratio of the triple point temperature over the critical temperature (e.g. carbon dioxide for which "< ⁄"$ = 0.7121).
Interestingly, the results are still very good when the P-Tait scheme is used with the Hankinson and Thomson correlations [24] (when applicable) and Eqs. (9-10) to determine .; 2 , see Tables 2-3 and Fig. 3. The AAD on density slightly increases to 0.27% on average, 9
ACCEPTED MANUSCRIPT which is consistent with deviations induced by the Hankinson and Thomson correlation [24], whereas the AAD on isothermal compressibility is kept constant on average slightly below 1.7 %.
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In addition, we have compared our results with those coming from the other schemes based on the Tait equation [8, 10, 14] and the non-Tait equation [25-26], see Tables 2-3. It can be noticed that our approach, even when the HT correlation [24] combined with Eqs. (9-10) is used, yields AADs on density and isothermal compressibility which are, by far, much smaller
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than those provided by the other existing approaches including both Tait and non-Tait
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equations.
3.2.2. Predictions on pure fluids
To quantify the predictive capabilities of the P-Tait scheme, the method has been
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applied to 21 additional weakly polar compounds (i.e. with a dipole moment, µ, smaller than 1 Debye), including model and real ones [22, 27-30], which were not included in the fitting database, see Table 4.
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In a first step, we have applied the P-Tait scheme with experimental values of ρref. For compounds whose densities at 1.5 were not available, we have retrieved .; 2 by using a
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known density at a point the closest to 1.5 combined with the use of Eq. (10). Tables 5-6 and Fig. 3 clearly show that the P-Tait approach provides excellent predictions of density and isothermal compressibility of the studied species. The AADs obtained are similar to those obtained on the species used in the fitting database. On average, the AAD on density equals to 0.16 % and to 2.35 % on isothermal compressibility. These predictions validate the reliability of the approach even when dealing with species possessing molecular characteristics out of the boundaries of those used to develop the C function.
10
ACCEPTED MANUSCRIPT In a second step, we have tested the P-Tait scheme combined with the HT correlations [24] (when applicable) and Eqs. (9-10) to determine .; 2 . By doing so, it is worth noticing that the proposed scheme is purely predictive. Results given in Tables 5-6 and Fig. 3 indicate that the results are still very good. The AADs on density are slightly larger than the ones
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provided from the P-Tait scheme using an experimental data point, while the AAD on isothermal compressibility is globally unchanged. As shown in Tables 5-6, this approach yields predictions which are much better than what provided by other schemes from the
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literature, based [8, 10, 14] or not [25-26] on the Tait Equation.
In addition, to better quantify the limits of the proposed scheme, we have applied it to
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compounds which were not dedicated for, i.e. highly polar species such as refrigerants and ethanol, see Table 7, using experimental values of ρref. Interestingly, see Table 8, it has been found that the proposed scheme is still able to yield reasonable AADs when dealing with the tested refrigerants, i.e. 0.2-1.0% on the density and 3.0-12.0% on the isothermal
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compressibility. However, for a hydrogen bonding compound such as ethanol the results noticeably deteriorate as shown in Table 8. Thus, this indicates the consistency of the proposed scheme, and seems to be applicable to a wider kind of compounds, including highly
AC C
EP
polar species, than it was dedicated for.
3.2.3. Application on binary liquid mixtures As a further test of the P-Tait scheme, we have applied it on three binary hydrocarbons
liquid mixtures for which both density and isothermal compressibility data were available: Octane-Hexadecane, Decane-Hexadecane and Cyclohexane-Hexadecane [31-33]. As on pure fluids the P-Tait scheme has been applied with an experimental reference density (+ Eqs. (9-10) to retrieve .; o ) and with a predicted reference density using the HT correlation [24] combined with Eqs. (9-10). 11
ACCEPTED MANUSCRIPT Results shown in Tables 9-10 and Fig. 3 indicate that the P-Tait approach, combined with the mixing rules given by Eq. (11), yields a very good prediction of the density and of the isothermal compressibility of the tested liquid mixtures even when using the purely predictive scheme (i.e. combined with the HT correlation to generate the reference point).
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Quantitatively, the results are as good as on pure fluids. Keeping in mind the claimed uncertainties on the experimental database itself (approximately 0.1% on and 1% on ) [31-33], these results are very encouraging. The only case for which results deteriorate is on
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the isothermal compressibility of the cyclohexane-hexadecane mixture whereas the density is well estimated, see Tables 9-10. This slight deterioration is possibly related to the way the
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isothermal compressibilities of this mixture have been deduced, as they have been computed using a cubic equation of state fitted on the density data [33]. In addition, we have compared our results with those provided by the Tait and non-Tait schemes of Thomson et al. [8] and of Nasrifar et al. [25], see Tables 9-10. As found on pure fluids, the P-Tait predictions are
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generally noticeably better than those provided by the other approaches. In addition, we have applied the proposed scheme to two refrigerant mixtures, R507A and R508B, that contain relatively high-polar refrigerants, see Table 11. As found on pure
EP
fluids, the results are not as good as those on non/weakly polar compounds, but they are still
AC C
reasonable see Table 11.
4. Conclusions
In this study, we have developed a method based on the Tait equation, named P-Tait,
which allows accurate estimations of density and isothermal compressibility of various compressed liquids and liquids mixtures typical of the oil and gas industry. More precisely, following a previous work [15] providing the B parameter of the Tait equation, we have developed here a correlation for the parameter C using an extended corresponding state
12
ACCEPTED MANUSCRIPT framework and fitted on a database of 16 species. To be applicable, this approach requires a value of the density at a reference point as well as the critical and triple point temperature, the critical pressure and the acentric factor. If one point of density is known experimentally, the P-Tait yields average absolute
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deviations of non polar weakly polar liquids (noble gases, normal alkanes, alkenes, aromatic, refrigerants …) of the order of 0.15% on the density and of the order of 2% on the isothermal compressibility over a wide range of subcritical thermodynamic conditions (from 0.4 to 0.99
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in reduced temperature and for pressures up to 150 MPa). Very interestingly, the results on the species not included in the fitting database (21 compounds) have been found as good as
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those on the species included in the fitting database (16 compounds). In addition it has been found that, when applied to non-hydrogen bonding compounds with a dipole moment larger than one Debye this approach is still able to provide reasonable results If no data is available on the density, this approach is still applicable by combining it
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with a correlation providing a value for the reference density. As a test, we applied the P-Tait on the same species database in combination with the liquid density at saturation (as the reference point) provided the Hankinson and Thomson correlation [24]. Interestingly, this
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predictive approach yields deviations that are still very good. Deviations on density are two times higher than with an experimental density value (i.e. around 0.3 %), consistently with
AC C
deviations induced by the Hankinson and Thomson correlation, while deviations on isothermal compressibility are globally unchanged. The extension of the P-Tait to mixtures is relatively straightforward because of its
corresponding states nature. Thus, we have applied it to binary liquid mixtures by using mixing rules similar to the ones proposed in the work of Thomson et al. [8]. Very interestingly, the P-Tait combined with these mixing rules yields results that are as good as
13
ACCEPTED MANUSCRIPT those on pure fluids on both density and isothermal compressibility. All these results confirm the interest of the proposed scheme. Finally, as a test of its efficiency, the proposed approach has been compared with existing schemes available in the literature (based on Tait equation or not). In all cases, it has
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been found that the P-Tait outperforms the other existing approaches, in particular on
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isothermal compressibility estimation.
Acknowledgements:
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The authors express their warm thanks to Dr. J. Bickert and Dr. F. Montel for the continuous stimulating discussions. We gratefully acknowledge TOTAL S.A. for the post-
AC C
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doctoral grant awarded to one of us (HH) and for letting us publish these results.
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ACCEPTED MANUSCRIPT References [1] A. Danesh, PVT and Phase Behaviour of Petroleum Reservoir Fluids, Elsevier Science (1998). [2] A. Firoozabadi, Thermodynamics of Hydrocarbon Reservoirs, McGraw-Hill (1999).
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[3] B. E. Poling, M. Prausnitz and J.P. O’Connell, The Properties of Gases and Liquids, 5th edition, McGraw-Hill(2004).
[4] W.D. McCain, J. P. Spivey and C.P. Lenn, Petroleum Reservoir Fluid Property
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Correlations, PennWellCorporation (2011).
Thermophysics, 9 (1988), 941-951.
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[5] J. H. Dymond and R. Malhotra, The Tait Equation: 100 Years On, International Journal of
[6] R. E. Gibson and O. H. Loeffler, Pressure-Volume-Temperature relation in solution. I. Observations on the Behavior of Solutions of Benzene and some of its derivatives, Journal of Physical Chemistry, 43 (1939), 207-217.
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[7] G. A. Neece and D. R. Squire, On the Tait and Related Empirical Equations of State, The Journal of Physical Chemistry, 72 (1968), 128-136. [8] G.H. Thomson, K.R. Brobst and R.W. Hankinson, An improved Correlation for Densities
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of Compressed Liquids and Liquid Mixtures, AIChE Journal, 28 (1982), 671-676. [9] J. H. Dymond and R. Malhotra, Densities of n-Alkanes and Their Mixtures at Elevated
AC C
Pressures, International Journal of Thermophysics, 8 (1987), 541-555. [10] M. J. Assael, J. H. Dymond and D. Exadaktilou, An Improved Representation for nAlkane Liquid Densities, International Journal of Thermophysics, 15 (1994), 155-164. [11] I. Cibulka and L. Hnedkovsky, Liquid Densities at Elevated Pressures of n-Alkanes from C5 to C16: A Critical Evaluation of Experimental Data, Journal of. Chemical & Engineering Data, 41(1996), 657-668.
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ACCEPTED MANUSCRIPT [12] I. A. Fakhretdinov, and E. R. Zhdanov, Tait’s Equation of State for Liquid Mixtures, High Temperature, 42 (2004), 396–400. [13] M. Ramos-Estrada, G.A. Iglesias-Silva and K. R. Hall, Experimental measurements and prediction of liquid densities for n-alkane mixtures, The Journal of Chemical
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Thermodynamics, 38 (2006), 337–347.
[14] H. Eslami and R. Azin, Corresponding-states correlation for compressed liquid densities, Fluid Phase Equilibria, 209 (2003), 245-254.
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[15] H. Hoang, G. Galliero, F. Montel, J. Bickert, Tait equation in the extended corresponding
11.
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states frame: Application to liquids and liquid mixtures, Fluid Phase Equilibria, 387 (2015), 5-
[16] V.N. Verveyko, M.V. Verveyko, Y.F. Melikhov, G.A. Melnikov, Perspectives of Applicability and Validity of Tait Equation for Liquids, International Journal of
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Thermophysics, 35 (2014), 2158-2165.
[17] V.S. Nanda, R. Simha, Theoretical interpretation of Tait Equation Parameters, Journal of Chemical Physics, 41 (1964), 1884-1885.
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[18] E.B. Postnikov, AL. Goncharov, V.V. Melent’ev, Tait equation revisited from the entropic and fluctuational points of view, International Journal of Thermophysics, 35 (2014),
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2115-2123.
[19] V. D. Kiselev, A. V. Bolotov, A. Satonin, I. Shakirova, H. A. Kashaeva and A. I. Konovalov, Compressibility of Liquids. Rule of Noncrossing V−P Curvatures, J. Phys. Chem. B, 2008, 112 (21), pp 6674–6682 [20] H. W. Xiang, The corresponding-States Principle and its Practice, Elsevier (2005). [21] M. J. Assael, J. P. M. Trusler and T. F. Tsolakis, An Introduction to Their Prediction: Thermophysical Properties of Fluids, Imperial College Press (1996) 16
ACCEPTED MANUSCRIPT [22] E. W. Lemmon, M. L. Huber, and M. O. McLinden, Reference Fluid Thermodynamic and Transport Properties, NIST Standard Reference Database 23, Version 8.0, (2007). [23] C.T. Kelley, Solving Nonlinear Equations with Newton’s Method (Fundamental of Algorithms), SIAM, Philadelphia (2003).
Liquids and Their Mixtures, AIChE J., 25 (1979), 653
RI PT
[24] R. W. Hankinson and G. H. Thomson, A New Correlation for Saturated Densities of
[25] Kh. Nasrifar, Sh. Ayatollahi, M. Moshfeghian, A compressed liquid density correlation,
SC
Fluid Phase Equilibria, 168 (2000) 149–163
[26] M. Aalto, K. I. Keskinen, J. Aittamaa and S. Liukkonen, An improved correlation for
M AN U
compressed liquid densities of hydrocarbons. Part 1. Pure compounds, Fluid Phase Equilibria, 114 (1996) 1-19
[27] J. Kolafa and I. Nezbeda, The Lennard-Jones fluid: an accurate analytic and theoretically-based equation of state, Fluid Phase Equilibria, 100 (1994), 1-34.
TE D
[28] T. S. Khasanshin and A. P. Shchemelev, The Thermodynamic Properties of nTetradecane in Liquid State, High Temperature, 40 (2002), 207–211 [29] T. S. Khasanshin, V. S. Samuilov, and A. P. Shchemelev, Determination of The
EP
Thermodynamic properties of Liquid n-Hexadecane from The Measurements of The Velocity of Sound, Journal of Engineering Physics and Thermophysics, 82 (2009), 149-156
AC C
[30] S. Dutour, J. L. Daridon and B. Lagourette, Pressure and Temperature Dependence of the Speed of Sound and Related Properties in Normal Octadecane and Nonadecane, International Journal of Thermophysics, 21 (2000), 173-184 [31] T. S. Khasanshin, V. S. Samuilov, and A. P. Shchamialiou, Thermodynamic properties of binary liquid mixtures of n-octane + n-hexadecane, High Temperatures-High Pressures, 39 (2010), 321–338 [32] T. S. Khasanshin, V. S. Samuilov, and A. P. Shchemelev, The Thermodynamic
17
ACCEPTED MANUSCRIPT Properties of Liquid Binary Mixtures of n-Alkanes: n-Decane + n-Hexadecane, High Temperature, 48 (2010), 665-672 [33] J. A. Amorim, O.Chiavone-Filho, M. L. L. Paredes, and K.Rajagopal, High-Pressure Density Measurements for the Binary System Cyclohexane + n-Hexadecane in the
RI PT
Temperature Range of (318.15 to 413.15) K, Journal of Chemical & Engineering Data, 52
AC C
EP
TE D
M AN U
SC
(2007), 613-618
18
ACCEPTED MANUSCRIPT
" , , 3, "<
No
z{|.
} .; 2 Yes
Yes
No
Estimate ^ and ] from Ref. [24]
SC
z{|.
y .; 2
RI PT
Compute B, , D , 0 , 0D using Eqs. (3), (6) and (8)
TE D
Compute and using Eqs. (1) and (2)
M AN U
Solve Eq. (10) to obtain y .; 2
Calculate C using Eq. (7)
Figure 1: Flowchart of the P-Tait scheme proposed in this work. "< is the triple point
AC C
EP
temperature, ^ is the vapor pressure and ] is the saturation liquid density .
19
ACCEPTED MANUSCRIPT 3
0.105
0.09 0.085 0.08 0.3
2 1.5 1 0.5
0.4
0.5
0.6
Tr
0.7
0.8
0.9
1
0 0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
RI PT
C
0.095
2.5
Ar (ω=-0.00219) Kr (ω=-0.00089) Xe (ω=0.00363) CH4 (ω=0.0114) C4H10 (ω=0.201) C7H16 (ω=0.349) C6H6 (ω=0.209) SF6 (ω=0.210) C4F8 (ω=0.355)
|CCorr-Cfit|/Cfit×100
0.1
Tr
Figure 2: Left: Dependence of the C parameter with the reduced temperature " for some
SC
compounds. Right: Absolute deviation on the C parameter obtained from Eq. (7). The same
AC C
EP
TE D
M AN U
symbol is employed for compounds possessing similar acentric factor.
20
ACCEPTED MANUSCRIPT 0.8 0.6 0.4 0.2 0
0
20
40
60
80
P (MPa)
100
120
2 1 0
20
40
0.1
0
20
40
60
80
P (MPa)
100
120
120
140
2
1
0
20
40
60
80
P (MPa)
100
120
140
6
0.6
Octane-Hexadecane
0.2
20
40
60
P (MPa)
80
100
EP
0
TE D
0.4
0
100
Hexadecane 3
0
140
80
P (MPa)
SC
abs(βT,Exp-βT,Pred)/βT,Exp×100%
0.2
60
M AN U
abs(ρExp-ρPred)/ρExp×100%
3
4
Hexadecane
abs(ρExp-ρPred)/ρExp×100%
Argon
4
0
140
0.3
0
5
RI PT
abs(βT,Exp-βT,Pred)/βT,Exp×100%
Argon
abs(βT,Exp-βT,Pred)/βT,Exp×100%
abs(ρExp-ρPred)/ρExp×100%
1
Octane-Hexadecane
5 4 3 2 1 0
0
20
40
60
P (MPa)
80
100
AC C
Figure 3. Absolute deviation on density (left figures) and on isothermal compressibility (right figures) yielded by the proposed scheme on some systems studied in this work. Solid circles correspond to the results using an experimental reference point. Open diamonds correspond to the results using a reference point computed from the HT correlation [24].
21
ACCEPTED MANUSCRIPT Table 1: Triple point temperature, critical temperature, critical pressure and acentric factor [22] of the compounds included in the fitting database and range of temperature and pressure used to fit the C parameter in Eqs. (5-8). Tt(K)
Tc(K)
Pc (MPa)
ω
Tmin-Tmax(K)
Pmax(MPa)
Methane
90.694
190.564
4.5992
0.01142
100-190
150
Ethane
90.368
305.330
4.8718
0.09930
100-290
150
Propane
85.53
369.890
4.2512
0.15210
90-360
150
Butane
134.9
425.125
3.7960
0.20100
140-420
70
Pentane
143.47
469.700
3.3700
Hexane
177.83
507.820
3.0340
Heptane
182.55
540.13
2.7360
Octane
216.37
569.32
Nonane
219.7
Decane
SC
RI PT
Species
150-450
100
0.29900
180-450
100
0.349
190-450
100
2.4970
0.393
220-450
100
594.55
2.2810
0.4433
220-450
150
243.5
617.7
2.1030
0.488
250-450
150
Dodecane
263.6
658.1
1.8170
0.574
270-450
150
Argon
83.806
150.69
4.863
-0.00219
90-150
150
Benzene
278.7
562.05
4.894
0.2092
280-450
80
Sulfur Hexafluoride
223.56
318.72
3.755
0.21
240-310
80
216.58
304.13
7.3773
0.22394
220-300
150
233.35
388.38
2.7775
0.3553
240-380
60
TE D
AC C
Octafluorocyclobutane
EP
Carbon dioxide
M AN U
0.25100
22
ACCEPTED MANUSCRIPT Table 2: AAD on the density yielded by the P-Tait scheme and some other Tait and non-Tait schemes from literature on the species included in the fitting database.
AAD on ρ P-Tait + [24]
Ref. [10]
Ref. [14]
Methane
0.12
0.26
1.37
0.63
0.86
Ethane
0.08
0.21
0.54
1.19
1.22
Propane
0.10
0.34
0.87
1.26
1.14
Butane
0.10
0.28
1.59
2.64
Pentane
0.17
0.24
1.03
2.16
Hexane
0.05
0.16
Heptane
0.12
0.33
Octane
0.05
0.21
Nonane
0.06
0.28
Decane
0.04
0.08
Dodecane
0.01
0.14
Argon
0.14
Benzene
0.01
Ref. [25]
Ref. [26]
1.40
4.12
[22]
0.62
2.74
[22]
0.66
2.64
[22]
SC
P-Tait + Exp.
0.44
0.48
1.23
[22]
0.64
0.35
1.80
[22]
M AN U
Ref. [8]
Database
RI PT
Species
1.05
0.75
0.37
1.44
[22]
0.77
0.40
0.69
0.21
1.62
[22]
0.48
0.24
0.55
0.43
1.23
[22]
0.67
0.12
0.77
0.48
1.65
[22]
0.48
0.09
0.66
0.40
1.67
[22]
0.28
0.06
---
0.41
1.75
[22]
0.59
5.30
---
1.71
1.64
4.87
[22]
0.15
0.50
---
0.19
0.17
0.69
[22]
EP
TE D
0.62
0.17
---
1.25
---
0.72
---
---
[22]
Carbon dioxide
0.27
0.45
1.58
---
0.84
0.77
2.22
[22]
---
0.88
---
---
---
---
[22]
AC C
Sulfur Hexafluoride
Octafluorocyclobutane 0.15
23
ACCEPTED MANUSCRIPT Table 3: AAD on the isothermal compressibility yielded by the P-Tait scheme and some other Tait and non-Tait schemes from literature on the species included in the fitting database. AAD on ~ P-Tait + [24]
Ref. [10]
Methane
2.52
2.50
9.06
7.23
15.24
Ethane
1.75
1.73
9.77
8.20
32.69
Propane
1.15
1.17
24.56
7.68
43.77
Butane
1.36
1.40
18.00
11.50
Pentane
1.84
1.84
18.71
8.51
Hexane
1.85
1.82
15.82
Heptane
1.14
1.10
Octane
1.52
Nonane
1.48
Decane
0.96
0.95
Dodecane
0.79
Argon Benzene
Ref. [25]
Ref. [26]
50.13
[22]
19.04
56.28
[22]
19.65
63.36
[22]
15.18
9.35
39.18
[22]
23.75
10.33
45.24
[22]
6.51
22.75
9.76
44.76
[22]
20.14
3.16
29.23
8.60
44.53
[22]
1.49
15.22
2.91
26.14
9.08
43.32
[22]
1.39
18.08
1.65
33.09
9.95
49.41
[22]
15.62
0.91
30.77
9.59
47.04
[22]
0.80
10.90
1.14
---
11.36
46.69
[22]
1.79
1.83
13.29
---
13.21
17.91
46.74
[22]
1.07
0.84
13.50
---
11.56
6.05
24.48
[22]
1.51
---
11.96
---
8.66
---
---
[22]
4.24
5.31
---
9.11
15.20
32.06
[22]
---
12.12
---
---
---
---
[22]
EP
TE D
M AN U
19.14
AC C
Sulfur Hexafluoride
Ref. [14]
RI PT
P-Tait + Exp.
Carbon dioxide
Ref. [8]
Database
SC
Species
4.21
Octafluorocyclobutane 1.94
24
ACCEPTED MANUSCRIPT Table 4: Triple point temperature, critical temperature, critical pressure and acentric factor [22] of the compounds not included in the fitting database and range of temperature and pressure explored. Tt(K)
Tc(K)
Pc (MPa)
ω
Tmin-Tmax(K)
Pmax(MPa)
Krypton
115.78
209.48
5.525
-0.00089
120-200
150
Xenon
161.41
289.73
5.842
0.00363
170-280
150
Nitrogen
63.151
126.19
3.3958
0.0372
70-120
150
Neopentane
256.6
433.74
3.196
0.1961
320-430
131
Tetradecane§
279.02
693.00
1.600
Hexadecane§
291.3
723.00
1.400
Octadecane§
301.0
747.00
1.29
Nonadecane§
304.0
755.00
Oxygen
54.361
Ethylene
SC
RI PT
Species
293-433
140
0.7180
298-433
140
0.800
313-383
150
1.20
0.845
313-383
150
154.58
5.043
0.0222
60-150
82
103.99
282.35
5.0418
0.0866
110-280
150
Isopentane
112.65
460.35
3.378
0.2274
120-450
150
Cyclohexane
279.47
553.64
4.075
0.20926
290-450
80
Toluene
178.0
591.75
4.1263
0.2657
180-450
150
Lennard-Jones*§
0.657
1.301
0.121
-0.0220
0.75-1.25
8.33
87.953
364.21
4.555
0.146
90-350
150
87.80
419.29
4.0051
0.192
100-400
70
R12§
116.1
385.12
4.1361
0.17948
125-375
150
R14§
89.54
227.51
3.750
0.1785
120-220
51
R113§
236.93
487.21
3.3922
0.25253
280-450
150
R115§
173.75
353.1
3.129
0.2500
180-350
60
R116§
173.1
293.03
3.048
0.2566
180-290
50
TE D
AC C
Butene§
EP
Propene§
M AN U
0.644
(*) Values are expressed in Lennard-Jones (LJ) units [22] (§) Compounds for which .; 2 was not available. 25
ACCEPTED MANUSCRIPT Table 5: AAD on the density yielded by the P-Tait scheme and some other Tait and non-Tait schemes from literature on the species that were not included in the fitting database.
AAD on ρ P-Tait + [24]
Krypton
0.12
---
1.17
---
1.40
Xenon
0.07
---
0.70
---
0.90
Nitrogen
0.34
0.41
1.12
---
1.75
Neopentane
0.22
---
4.16
---
Lennard-Jones*§
0.25
---
1.86
---
Tetradecane§
0.03
0.08
0.46
0.14
Hexadecane§
0.08
0.14
0.63
Octadecane§
0.09
0.06
Nonadecane§
0.06
Oxygen
0.12
Ethylene
0.25
Isopentane
0.18
Cyclohexane
0.14
Ref. [14]
Ref. [25]
Ref. [26]
---
---
[22]
---
---
[22]
1.07
5.50
[22]
---
---
[22]
---
---
---
[27]
---
0.70
2.63
[28]
0.05
---
0.69
1.91
[29]
0.47
---
---
1.23
2.50
[30]
0.05
0.36
---
---
0.82
2.51
[30]
0.34
0.45
---
0.62
0.91
2.38
[22]
TE D
M AN U
---
9.30
---
1.18
1.33
3.99
[22]
0.40
1.29
---
---
0.57
2.82
[22]
0.29
0.83
---
---
0.69
0.64
[22]
EP
0.56
0.09
0.28
1.01
---
---
0.39
1.54
[22]
0.13
0.65
0.85
---
---
1.26
3.66
[22]
AC C
Propene§
Ref. [10]
RI PT
P-Tait + Exp.
Toluene
Ref. [8]
Database
SC
Species
Butene§
0.08
0.90
0.66
---
0.63
0.88
1.96
[22]
R12§
0.25
0.37
1.29
---
---
0.52
3.27
[22]
R14§
0.08
---
0.57
---
---
---
---
[22]
R113§
0.15
0.29
1.06
---
0.96
2.31
[22]
R115§
0.30
---
0.99
---
---
---
---
[22]
R116§
0.29
---
2.63
---
---
---
---
[22]
(*) Values are expressed in Lennard-Jones (LJ) units [22] (§) Compounds for which .; 2 was not available. 26
ACCEPTED MANUSCRIPT Table 6: AAD on the isothermal compressibility yielded by the P-Tait scheme and some other Tait and non-Tait schemes from literature on the species that were not included in the fitting database. AAD on ~ Ref. [14]
RI PT
P-Tait + [24]
Ref. [8]
Ref. [25]
Krypton
1.68
---
8.28
---
12.32
---
---
[22]
Xenon
0.88
---
6.55
---
12.03
---
---
[22]
Nitrogen
3.67
3.69
8.70
---
17.82
10.09
51.88
[22]
Neopentane
1.37
---
13.54
---
---
---
---
[22]
LennardJones*§
2.77
---
13.09
---
---
---
---
[27]
Tetradecane§
1.28
1.31
7.52
1.00
---
11.71
57.37
[28]
Hexadecane§
1.50
1.53
9.38
0.86
---
12.68
51.48
[29]
Octadecane§
2.72
2.69
6.44
---
---
14.60
60.95
[30]
Nonadecane§
1.65
1.64
6.36
---
---
11.34
61.01
[30]
Oxygen
2.95
2.93
6.61
---
15.44
17.54
44.46
[22]
Ethylene
3.12
3.18
23.04
---
20.27
19.22
51.83
[22]
Isopentane
2.31
2.33
25.88
---
---
12.58
60.70
[22]
Cyclohexane
3.88
3.91
17.39
---
---
2.62
25.62
[22]
EP
M AN U
SC
P-Tait + Exp.
AC C
Ref. [10]
Database
TE D
Species
Ref. [26]
Toluene
1.73
1.61
30.33
---
---
17.01
52.36
[22]
Propene§
1.34
1.42
13.27
---
---
20.41
56.84
[22]
Butene§
1.27
1.26
20.24
---
29.68
12.32
44.03
[22]
R12§
3.17
3.11
14.27
---
---
8.65
49.44
[22]
R14§
1.83
---
7.62
---
---
---
---
[22]
R113§
2.34
2.35
10.00
---
---
7.15
43.23
[22]
R115§
4.18
---
16.40
---
---
---
---
[22]
R116§
3.88
---
7.60
---
---
---
---
[22]
27
ACCEPTED MANUSCRIPT (*) Values are expressed in Lennard-Jones (LJ) units [22]
AC C
EP
TE D
M AN U
SC
RI PT
(§) Compounds for which .; 2 was not available.
28
ACCEPTED MANUSCRIPT Table 7: Triple point temperature, critical temperature, critical pressure, acentric factor and dipole moment [22] of the highly polar compounds tested in this work and range of temperature and pressure explored. Tt(K)
Tc(K)
Pc (MPa)
ω
µ (D)
R125
172.52
339.17
3.6177
0.3052
1.563
R141b
169.68
477.50
4.212
0.2195
2.014
R143a
161.34
345.86
3.761
0.2615
2.340
R21
142.80
451.48
5.1812
0.2061
1.370
R23
118.02
299.29
4.832
0.2630
R32
136.34
351.26
5.782
0.2769
Ethanol
159.00
513.90
6.148
0.6440
TminTmax(K)
60
200-450
150
190-340
100
250-440
137
1.649
210-280
120
1.978
150-134
70
1.691
250-425
150
SC
190-330
M AN U
TE D EP AC C 29
Pmax(MPa)
RI PT
Species
ACCEPTED MANUSCRIPT Table 8: AAD on the density and isothermal compressibility yielded by the P-Tait scheme when applied highly polar compounds. AAD on ρ
AAD on ~
Database
R125
0.21
2.85
[22]
R141b
0.15
2.38
[22]
R143a
0.57
4.84
R21
0.20
4.69
R23
0.83
7.09
R32
0.67
12.45
[22]
Ethanol
2.38
36.91
[22]
RI PT
Species
[22]
[22]
AC C
EP
TE D
M AN U
SC
[22]
30
ACCEPTED MANUSCRIPT Table 9: AAD on the density yielded by the P-Tait scheme in this work and other Tait and non-Tait schemes reported in the literature when applied to three different binary mixtures. TminTmax(K)
AAD on ρ
Pmax(MPa) P-Tait + Exp.
x1
P-Tait + [24]
Ref. [8]
Ref. [25]
RI PT
Mole Fraction
Mixture of Octane (1) and Hexadecane (2)§ [31] 313.15393.15
100
0.07
0.07
0.50
313.15393.15
100
0.10
0.14
0.75
313.15393.15
100
0.13
0.27
0.28
0.33
0.29
SC
0.25
M AN U
0.13
0.42
0.41
Mixture of Decane (1) and Hexadecane (2)§ [32] 298.15433.15
100
0.50
298.15433.15
100
0.75
298.15433.15
100
0.08
0.08
0.45
0.33
0.09
0.16
0.43
0.28
0.10
0.16
0.43
0.26
TE D
0.25
Mixture of Cyclohexane (1) and Hexadecane (2)§ [33] 318.15413.15
0.30
318.15413.15
63
0.09
0.15
0.42
0.27
63
0.03
0.13
0.32
0.60
AC C
EP
0.10
0.50
318.15413.15
63
0.06
0.24
0.69
1.04
0.70
318.15413.15
63
0.18
0.49
1.15
1.32
0.90
318.15413.15
63
0.24
0.41
1.12
1.02
(§) Compounds for which .; 2 was not available.
31
ACCEPTED MANUSCRIPT Table 10: AAD on the isothermal compressibility yielded by the P-Tait scheme in this work and other Tait and non-Tait schemes reported in the literature when applied to three different binary mixtures. TminTmax(K)
AAD on ~
Pmax(MPa) P-Tait + Exp.
x1
P-Tait + [24]
Ref. [8]
Ref. [25]
7.12
15.15
RI PT
Mole Fraction
Mixture of Octane (1) and Hexadecane (2)§ [31] 100
1.84
0.50
313.15393.15
100
2.75
0.75
313.15393.15
100
1.82
SC
313.15393.15
2.70
M AN U
0.25
3.18
3.14
10.41
13.29
12.97
11.59
Mixture of Decane (1) and Hexadecane (2)§ [32] 298.15433.15
100
0.50
298.15433.15
100
0.75
298.15433.15
1.21
1.19
7.70
18.47
1.27
1.23
9.13
16.02
1.19
10.79
14.23
TE D
0.25
100
1.24
318.15413.15
AC C
0.10
EP
Mixture of Cyclohexane (1) and Hexadecane (2)§ [33] 63
2.42
2.42
7.48
14.97
0.30
318.15413.15
63
1.05
1.05
9.87
10.56
0.50
318.15413.15
63
2.27
2.30
16.75
8.74
0.70
318.15413.15
63
5.49
5.55
21.85
9.53
0.90
318.15413.15
63
6.58
6.61
20.98
6.96
(§) Compounds for which .; 2 was not available.
32
ACCEPTED MANUSCRIPT Table 11: AAD on the density and isothermal compressibility yielded by the P-Tait scheme when applied to two mixtures of polar compounds. Mole Fraction Tmin-Tmax(K)
Pmax(MPa)
AAD on ρ
AAD on ~
x1
0.41184
190-320
83
RI PT
[R507A]: Mixture of R125 (1) and R143a (2)
0.41
5.94
[R508B]: Mixture of R23 (1) and R116 (2) 150-260
50
0.96
AC C
EP
TE D
M AN U
SC
0.46
33
16.14