Analysis of the Combinations of Property Data That Are Suitable for a

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Cite This: J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Analysis of the Combinations of Property Data That Are Suitable for a Safe Estimation of Consistent Twu α‑Function Parameters: Updated Parameter Values for the Translated-Consistent tc-PR and tc-RK Cubic Equations of State Andreś Pina-Martinez,† Yohann Le Guennec,† Romain Privat,*,† Jean-Noël Jaubert,*,† and Paul M. Mathias‡ Université de Lorraine, É cole Nationale Supérieure des Industries Chimiques, Laboratoire Réactions et Génie des Procédés (UMR CNRS 7274), 1 rue Grandville, 54000, Nancy, France ‡ Fluor Corporation, 3 Polaris Way, Aliso Viejo, California 92698, United States

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†

S Supporting Information *

ABSTRACT: The suitability of different combinations of property data among vapor-pressure (Psat), enthalpy of vaporization (ΔvapH) and saturated-liquid heat-capacity (csat P,liq) data for fitting consistent Twu α-function parameters is evaluated. A given combination is declared to be suitable if the obtained fitted parameters produce acceptable predictions for the three considered properties for the 783 molecules included in the reference database. Calculations are carried out using the translated-consistent versions of the Peng−Robinson (tc-PR) and Redlich−Kwong (tc-RK) cubic equations of state (CEoS). It is demonstrated that reliable parameters are obtained on the condition that they are, at least, fitted to vapor-pressure data. It is strongly advised to not exclusively fit α-function parameters to enthalpy of vaporization and/or saturated-liquid heat capacity data. Finally, Twu α-function parameters (L, M, N) suitable for the tc-PR and tc-RK CEoS are determined for the 1721 molecules, for which at least accurate vapor-pressure data are available in the DIPPR database. The volume-translation parameter c has also been determined but for only 1489 pure fluids. In addition, the generalized versions of the tc-PR and tc-RK CEoS (“generalized” means that the input parameters of these models are the experimental critical temperature, critical pressure, and acentric factor) are updated.

1. INTRODUCTION Most of the two-parameter cubic equations of state (CEoS) obey the general formulation: P(T , v) =

a(T ) RT − (v − r1b)(v − r2b) v−b

determined using correlations that only depend on the acentric factor (ω), while the parameters of the component-dependent α-functions must be regressed, component by component, from experimental data. It is worth noting that the generalized α-functions, although less accurate than the componentdependent α-functions, are extremely useful when experimental data to fit the specific parameters of a given α-function are not available. Moreover, generalized α-functions are easy to implement because the acentric factor is known for thousands of pure components or can be estimated by well-established correlations or group-contribution methods.1 As recently highlighted by Le Guennec et al.,2,3 α-functions cannot be freely chosen. The temperature dependence of αfunctions must indeed obey the following list of constraints to

(1)

where parameter a is a measure of the attractive forces between molecules; parameter b is the so-called covolume, which attempts to correct the perfect-gas law for the fact that molecules have a finite volume; and r1 and r2 are two universal constants that depend on the selected EoS. The temperaturedependent a parameter of any CEoS is classically written as a(T ) = acα(T )

(2)

that is, as the product of the value of the attractive parameter at the critical temperature (ac) multiplied by a so-called αfunction, which is dimensionless. It is possible to distinguish two types of α-functions: generalized and component-dependent. The coefficients of the generalized α-functions are usually © XXXX American Chemical Society

Received: July 22, 2018 Accepted: September 13, 2018

A

DOI: 10.1021/acs.jced.8b00640 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

Article

tc-PR:

guarantee safe property predictions in both subcritical and supercritical domains: l α ≥ 0 and α(T ) continuous o r o o o o o dα o dα o o ≤ 0 and continuous o o o dTr Tr d o o o o 2 o α(Tc) = 1 and for all T :m dα d2α o o ≥ 0 and continuous o 2 2 o o d T d T o r r o o o o o o d3α o o ≤0 o o o dTr3 n

P(T , v) =

l o α(Tr) = TrN(M − 1) exp[L(1 − TrMN )] o o o o o o sat,u − PR sat o o (Tr = 0.8) − v liq,exp (Tr = 0.8) c = v liq o o o o o o o o ηc = [1 + 3 4 − 2 2 + 3 4 + 2 2 ]−1 o o o o o o ≈ 0.25308 with: m o o 2 2 o R2Tc,exp R2Tc,exp o o o a = 40ηc + 8 o 0.45724 ≈ o c o o 49 − 37ηc Pc,exp Pc,exp o o o o o o o RTc,exp ηc RTc,exp o o o − c ≈ 0.07780 −c ob = o o Pc,exp ηc + 3 Pc,exp o n

(3)

An α-function that fulfills all of the conditions reported in eq 3 was described as consistent by Le Guennec et al. These authors also demonstrated that the highly flexible 3-parameter αfunction developed by Twu et al. in 19914 (denoted hereafter as Twu91): α(Tr) = TrN(M − 1) exp[L(1 − TrMN )]

or

P(T , v) =

lδ < 0 o o o o o o Lγ > 0 o o o m o o γ < 1 − 2δ + 2 δ(δ − 1) o o o o o o o 4Y 3 + 4ZX3 + 27Z2 − 18XYZ − X2Y 2 > 0 n

By working with 1116 pure fluids, Le Guennec et al. not only published a library of consistent parameters that are suitable for the tc-PR and tc-RK EoS but also developed a generalized version of the Twu α-function for both CEoS (RK and PR). Recently, Bell et al.11 also published consistent Twu parameters for more than 2500 pure fluids. While the overall aims of these studies were similar, a noticeable difference should be highlighted: Bell et al.11 directly employed experimental data from the ThermoDataEngine (TDE) database of NIST, while Le Guennec et al.10 fitted the three parameters of the α-function on pseudoexperimental data generated from correlations available in the DIPPR database. Indeed, the DIPPR database provides a set of temperaturedependent correlations, the coefficients of which are fit to actual experimental data, making it possible to estimate, among other properties, vapor pressures, enthalpies of vaporization, and saturated-liquid heat capacities of pure components. Note that for each component and each property, an experimental error is provided. The benefit of using DIPPR correlations instead of raw experimental data is the ability to carry out fitting procedures on pseudoexperimental data that are regularly distributed between the triple point and critical point. By doing so, the optimization results (i.e., α-function parameters) are not biased by experimental data points that are concentrated in a specific temperature range. The approach also eliminates the opportunity for independent evaluation of the raw data. It is worth noting that Le Guennec et al.10 determined αfunction parameters for 1116 pure compounds by minimizing an objective function accounting for deviations between calculated and pseudoexperimental vapor pressure (Psat) and/ or enthalpy of vaporization (ΔvapH) and/or saturated-liquid heat capacity (csat P,liq) data. For many components, thanks to the

(5)

with Tr =

(8) 10

Le Guennec et al. coupled this noteworthy α-function to the volume translation concept5,6 discovered by Péneloux et al.7 in 1982 to define the translated-consistent versions of the Peng− Robinson8 (PR) and Redlich−Kwong9 (RK) EoS. The volume correction, denoted hereafter as c, was selected to exactly reproduce the experimental saturated liquid volume at a reduced temperature of 0.8 [vsat liq,exp(Tr = 0.8)], that is c = vLsat,u − CEoS(Tr = 0.8) − vLsat,exp(Tr = 0.8)

acα(T ) RT − v−b (v + c)(v + b + 2c)

l o α(Tr) = TrN(M − 1) exp[L(1 − TrMN )] o o o o o o sat,u − RK sat o o c = v liq (Tr = 0.8) − v liq,exp (Tr = 0.8) o o o o o o 2 o R2Tc,exp o 1 with: m ac = 3 o o o 9( 2 − 1) Pc,exp o o o o o o o 3 o 2 − 1 RTc,exp o o o b= −c o o 3 Pc,exp o n

(4)

l δ = N (M − 1) o o o o o o γ = MN o o o o o o X = − 3(γ + δ − 1) with: o m o o o o o o Y = γ 2 + 3δγ − 3γ + 3δ 2 − 6δ + 2 o o o o o o Z = − δ(δ 2 − 3δ + 2) o n

(7)

tc-RK:

became consistent if the following constraints were added to the three parameters: l δ 0 m o o o o oγ < 1 − δ n

acα(Tr) RT − (v + c)(v + b + 2c) + (b + c)(v − b) v−b

T Tc (6)

where vsat,u−CEoS (Tr = 0.8) is the molar volume calculated with L the original (untranslated) CEoS at Tr = 0.8. These EoS, for which safe property predictions are guaranteed in both subcritical and supercritical domains because the α-function passes the consistency test proposed by Le Guennec et al. are also almost certainly the most accurate three-parameter CEoS ever published. They can be expressed as B



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Twu91 α-function parameters on vapor pressure (Psat), enthalpy of vaporization (ΔvapH), and saturated-liquid heat capacity (csat P,liq) data. It is thus decided to draw up the list of pure components for which the DIPPR database has reasonable experimental error for at least one out of the three aforementioned properties. At this step, it should be noted that only the residual part of the isobaric heat capacity can be estimated from a CEoS. The cP of the fluid is obtained by adding the heat capacity of the corresponding perfect gas to the residual contribution expressed from the EoS:

DIPPR correlations, it was possible to generate accurate pseudoexperimental data for all three properties, whereas for other components, only 2 or even 1 accurate property could be generated. Therefore, each set of α-function parameters was fitted on one out of the seven different possible combinations of the pseudoexperimental data properties shown in Table 1. Table 1. Possible Combinations of Accurate Pseudoexperimental Data Available in the DIPPR Database for a Given Pure Component type of pure fluida 1 2 3 4 5 6 7

properties for which accurate data are available (that were considered in the fitting procedure of α-function parameters by Le Guennec et al.10)

cP(T , v) = cP ,perfectgas(T ) + cP ,residual(T , v)

sat

(P ) (ΔvapH) (csat P,liq) (Psat + ΔvapH) (Psat + csat P,liq) (ΔvapH + csat P,liq) (Psat + ΔvapH + csat P,liq)

(9)

The heat capacities of perfect gases can be estimated from DIPPR correlations. Consequently, the error of four correlations per compound must be taken into account when evaluating a pure component: Psat, ΔvapH, csat P,liq and cP,perfect gas. Henceforth, in this work, when it is stated that csat P,liq data can be accurately generated, it is implied that both the csat P,liq and cP,perfect gas pseudoexperimental data can be accurately generated. In this study, the 11.1.0 version (May 2016) of the DIPPR database is used. The following should be noted for the DIPPR database:

a The pure fluid type is an arbitrary index that indicates which properties among Psat, ΔvapH and csat P,liq are associated with accurate data and can thus be used to fit α-function parameters.

In view of this situation, key questions are addressed in this paper: • Are all of the possible arrangements of the properties listed in Table 1 suitable for parameter fitting? If not, some of the parameters we published in 201610 should not be used. • Which property combinations are the most appropriate for parameter fitting? As previously explained, Bell et al.11 published Twu parameters for more than 2500 pure fluids, while the library published by Le Guennec10 contains parameters for less than half the number of pure components. This is because Le Guennec et al. only considered a DIPPR correlation to be acceptable if its experimental error (reported in the DIPPR database) was lower or equal to 5%. However, it is believed that, in many cases, this error is overestimated. Thus, another objective of this paper is to review the uncertainty of the correlations reported in the DIPPR database and to develop a new procedure to evaluate the error of DIPPR correlations. This new procedure makes it possible to notably increase the number of compounds for which the Twu parameters can be determined. In the first section of this study, an error evaluation of DIPPR correlations is performed in order to draw up a list of compounds for which accurate data of interest (Psat and/or ΔvapH and/or csat P,liq) are available in the DIPPR database. The next section presents the combinations of the property data that are suitable for parameter regression. Finally, an update of the Twu91 α-function and volume-translation parameters with respect to Le Guennec et al.’s initial study is proposed. Component-dependent parameters are determined for 1721 pure fluids. The updated versions of the generalized Twu88 αfunctions and volume-translation correlations for both the PR and RK EoS are also provided.

• For a series of experimental data, the single attribute ERROR that is reported is the experimental uncertainty of the measured values. This uncertainty can either be provided by the source or calculated by the DIPPR staff. • For a temperature-dependent correlation, the way that the attribute ERROR is calculated is not known with certainty. Generally, ERROR is equal to the maximum error among those of the data series used to fit correlation parameters. For certain compounds it is found that the ERROR reported in the DIPPR was overestimated. Indeed, for such cases, the DIPPR reported uncertainty (henceforth called type I error), set to the maximum uncertainty among the data series, was much higher than the average uncertainty value and did not reflect the confidence we could have on the experimental data. Therefore, the actual average uncertainty is calculated (henceforth called type II error) for each of the four sat temperature-dependent properties (Psat,ΔvapH, cP,liq and cP,perfect gas) and for all pure compounds in the DIPPR database. For a given property and a given pure fluid, a type II error is calculated as n

type II error =

∑ j =series × errorj N 1 data, j n

∑ j =series N 1 data, j

(10)

where Ndata,j is the number of experimental data points available in the series j and errorj the corresponding experimental error. A third strategy to evaluate the correlation errors is to compare pseudoexperimental data with calculated values obtained using the generalized version of the tc-PR model, which was developed by Le Guennec et al.10 On the basis of the predictive performance of the model, it is possible to state that if a correlation is well represented by the generalized model, it is implied that there is no reason to reject the pseudoexperimental data, and the correlation is considered to be accurate.

2. OVERVIEW OF THE DIPPR DATABASE 2.1. Error Evaluation of the DIPPR Correlations. As previously stated, this paper addresses the fitting of consistent C



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For type III errors, the threshold of acceptance (noted TH) for each property is defined as follows: • For Psat: the mean absolute percentage error (MAPE) on Psat calculated by the generalized tc-PR EoS (see Table 6 in Le Guennec et al.10) from a database including 980 compounds + one standard deviation given by THPtypeIII = 2.0% + 1.2% = 3.2% sat

removed from the database, leaving 1881 compounds. In comparison to the work by Le Guennec et al., who only considered type I errors, the definition of two new types of error (type II and type III) makes it possible to significantly increase the number of potentially suitable components for parameter fitting. Let us recall that 1116 pure fluids were considered by Le Guennec et al., whereas 1881 pure fluids are considered in this study. In Table 3, these 1881 molecules are distributed along the seven fluid types defined in Table 1 (i.e., according to the

(11)

• For ΔvapH: the mean absolute percentage error (MAPE) on ΔvapH calculated by the generalized tc-PR EoS (see Table 6 in Le Guennec et al.10) from a database including 805 compounds + one standard deviation is given by THΔtypeIII = 2.8% + 1.5% = 4.3% vapH

•

Table 3. Comparison of the Number of Molecules Included in the Database by Le Guennec et al.3 and This Work

(12)

For csat P,liq: the mean absolute percentage error on csat P,liq is calculated by the generalized tc-PR 10

(MAPE) EoS (see Table 6 in Le Guennec et al. ) from a database including 529 compounds + one standard deviation − 5% (maximum allowed error for cP,perfect gas). The maximum error of the perfect gas heat capacity was subtracted to focus on the deviation on the residual heat capacity, which is the sole property calculable from the EoS: THctypeIII = 6.7% + 4.3% − 5% = 6% sat

The thresholds of the three types of error are reported in Table 2. Table 2. Acceptance Thresholds for the Different Property Correlations and Types of Error acceptance thresholds for each property error type type I (error reported in the DIPPR database) type II (average exp. error calculated by eq 10) type III (deviation calculated by the gen. tc-PR EoS)

ΔvapH

csat P,liq

cP,perfect gas

5%

5%

5%

5%

5%

5%

5%

5%

3.2%

4.3%

6%

Psat

accurate property data available

Le Guennec et al.

this work

1 2 3 4 5 6 7 total

(Psat) (ΔvapH) (csat P,liq) (Psat + ΔvapH) (Psat + csat P,liq) (ΔvapH + csat P,liq) (Psat + ΔvapH + csat P,liq)

166 20 98 401 47 18 366 1116

217 25 53 705 80 18 783 1881

number and type of accurate properties that can be generated for each of them), and a comparison is made with the previous work of Le Guennec et al. Among the 1116 molecules considered by Le Guennec et al., 684 kept the same fluid type, whereas 432 moved to a new fluid type. For instance, if type I error is solely considered to evaluate the uncertainty of the correlations (as in Le Guennec et al.), only csat P,liq can be estimated for 1-hexene, and consequently, it belongs to type 3 (as defined by Tables 1 and 3). If type II and III errors are also considered, all three of the temperature-dependent properties, Psat, ΔvapH and csat P,liq, can be estimated and 1-hexene moves to type 7.

(13)

P,liq

type of pure fluid

3. SUITABLE COMBINATIONS OF PROPERTY DATA FOR THE SAFE ESTIMATION OF CONSISTENT TWU91 α-FUNCTION PARAMETERS Table 1 indicates that it was possible to define 7 types of pure fluids depending on the availability of accurate pseudoexperimental data. As explained in the introduction, our key concern is to determine which of these seven types are suitable for safely fitting consistent Twu91 α-function parameters and to order these types. To answer this question, the approach proposed in this section is aimed at fitting Twu91 α-function parameters for type 7 pure-compounds, such that all three temperature-dependent properties, Psat, ΔvapH and csat P,liq, can be accurately estimated from DIPPR correlations. The interest in choosing these pure-compounds is that if it is decided to fit the (L, M, N) parameters using not three but one or two of the selected properties, it becomes possible to predict the remaining ones and to compare them with pseudoexperimental values. It is thus possible to determine whether taking specific combinations of 1 or 2 properties into account for parameter fitting may lead to nonreliable sets of (L, M, N) parameters. Consequently, for the 783 type 7 fluids (see Table 3), seven different fittings were carried out, as shown in Table 4. In the second column, the properties that were taken into account for each fitting are presented. In the third column, the properties

Finally, it is considered that a property can be accurately estimated by a DIPPR correlation if at least one out of the three types of error is less than or equal to its respective threshold. 2.2. Drawing up of a List of Compounds Potentially Suitable for Parameter Fitting. First, the use of cubic equations of state requires critical temperature (Tc) and critical pressure (Pc) data. Whether a generalized α-function is used, the acentric factor (ω) is also needed. Therefore, the first criterion that suitable molecules must meet is that the three experimental constants, Tc, Pc and ω, are simultaneously reported in the DIPPR database. Additionally, molecules for which none of the three saturation properties of interest (Psat, sat ΔvapH, cP,liq ) can be accurately generated are removed. Application of these criteria results in 1887 suitable molecules among 2162 available in the DIPPR database. The properties of some compounds, especially metals with a critical temperature higher than 2000 K (Li, K, Ca, LiI) or quantum fluids (He-3 and p-H2) that are not of industrial interest, were D


ÅÄÅ sat sat 50 ÅÅ ÅÅ 100 P sat,DIPPR(TiP ) − P sat,EoS(TiP ) ∑ OFα = ÅÅÅ sat ÅÅ 50 i = 1 P sat,DIPPR(TiP ) ÅÅ ÅÇ

Journal of Chemical & Engineering Data Table 4. Arrangements of the Properties for Twu91 αFunction Parameter Fitting fitting procedure index

property data included in the Twu parameter fitting procedure

properties predicted with the optimized Twu parameter set

sat

1 2 3 4 5 6 7

(P ) (ΔvapH) (csat P,liq) (Psat + ΔvapH) (Psat + csat P,liq) (ΔvapH + csat P,liq) (Psat + ΔvapH + csat P,liq)

Δ vapH

csat P,liq)

+

(ΔvapH + (Psat + csat P,liq) (Psat + ΔvapH) (csat P,liq) (ΔvapH) (Psat)

Δ vapHDIPPR (Ti

Δ vapH

) − Δ vapHEoS(Ti

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)

ÉÑ ÑÑ cPsat,liq cPsat,liq sat,EoS ÑÑ cPsat,DIPPR T c T − ( ) ( ) i i ,liq P ,liq ÑÑ +WF ÑÑ sat cP ,liq ÑÑ sat,DIPPR cP ,liq (Ti ) ÑÑ ÑÖ Δ vapH

Δ vapHDIPPR (Ti

)

(14)

For fitting procedure 6, the weighting factor on csat P,liq (denoted as WF in eq 14) is set to 1. In turn, for fitting procedures 5 and 7, special attention is devoted to the estimation of WF; as an initial choice, it is set to 1. Nevertheless, for certain compounds, deviations on csat P,liq are found to be much smaller than those on Psat. In these cases, the deviation on csat P,liq is reduced to better correlate Psat (i.e., to obtain smaller deviations on Psat). This goal is made possible by carrying out successive minimizations by changing the value of WF from 1 to 0.1, with a step of 0.1. In practice, this procedure is run for a considered compound if (i) the MAPE on Psat is larger than 2% (which corresponds to the average MAPE plus one standard deviation calculated for the 782 considered compounds when WF = 1) and the MAPE on csat P,liq is smaller than that on Psat; (ii) the MAPE on csat is considerably smaller P,liq (two times) than that on Psat and the MAPE on csat P,liq is larger than 1%. These conditions are summarized in eq 15:

predicted by the CEoS with the fitted (L, M, N) parameters are provided. 3.1. Fitting Procedure. In this study, generation of the pseudoexperimental data obeys the following rules: 1. Regardless of the considered property among Psat, ΔvapH and csat P,liq, 50 equidistant pseudoexperimental data points are generated in their valid temperature range, [Tmin;Tmax]. 2. For the ΔvapH and csat P,liq correlations, in the case that Tmax exceeds 0.98Tc, Tmax = 0.98Tc is set. The reason for this choice is simple: ΔvapH goes to zero at the critical temperature and the calculation of the relative deviation with the experimental data induces a division by zero. Similarly, csat P,liq goes to infinity at the critical temperature so that, once again, the relative deviation with the experimental value cannot be calculated.

MAPE on P sat > 2% and MAPE on cPsat,liq < MAPE on P sat 2 × MAPE on cPsat,liq < MAPE on P sat and MAPE on cPsat,liq > 1%

sat

3. For the P correlations, Tmax = Tc is set. However, in the case where Psat(Tmin) < 0.1 bar, the value of Tmin is increased to enforce that P sat (T min ) = 0.1 bar (considering that most industrial processes operate at higher values of pressure and that Psat is highly uncertain at low pressures, say less than 1 Torr).

(15)

In a second step, the weighing factor is selected on a case by case basis to obtain what we consider to be the best compromise between the MAPE on the vapor pressures and MAPE on the saturated-liquid heat capacities. 3.2. Results. The MAPE on Psat, ΔvapH, and csat P,liq is reported in Table 5 for the seven fitting procedures described in Table 4. Procedure 7, for which the three properties are simultaneously considered, serves as a reference. It is observed that fitting procedures that are carried out when only considering csat P,liq (procedure 3) yield extremely high deviations: 24.8% and 16.0% for the predicted properties Psat and ΔvapH, respectively. Conversely, it is crucial to include experimental sat csat P,liq data in the fitting procedure to obtain a MAPE on cP,liq lower than 5% (see procedures 1, 2, and 4 in Table 5). Procedures 2 and 6, for which no vapor pressure data are considered, seem to yield acceptable average deviations for the

The optimal set of (L, M, N) parameters is the one which minimizes the generic objective function depicted in eq 14, which is a ponderation of the mean absolute percentage error (MAPE) on Psat, ΔvapH, and csat P,liq. The objective function is obviously adapted according to the fitting procedure index defined in Table 4. As an example, for the fitting procedure number 1, the objective function only considers the MAPE on Psat. The fitting procedure is conducted by imposing the mathematical constraints necessary to obtain consistent parameters, as explained by Le Guennec et al. and recalled from the introduction.

Table 5. MAPE on Psat, ΔvapH, and csat P,liq According to the Property Considered for Parameter Fitting fitting no. procedure

property taken into account for fitting

MAPE on Psat (783 compounds)

MAPE on ΔvapH (783 compounds)

MAPE on csat P,liq (783 compounds)

1 2 3 4 5 6 7

(Psat) (ΔvapH) (csat P,liq) (Psat + ΔvapH) (Psat + csat P,liq) (ΔvapH + csat P,liq) (Psat + ΔvapH + csat P,liq)

0.7% 2.1% 24.8% 0.9% 1.3% 3.1% 1.1%

2.7% 1.4% 16.0% 1.6% 2.8% 2.3% 2.2%

6.8% 5.1% 0.6% 5.1% 1.7% 1.7% 2.3%

E



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predicted properties. However, the deviations on Psat are higher than the average deviation plus one standard deviation of the reference case (procedure 7). Moreover, for certain molecules, the average deviation on Psat unfortunately reaches 20%. We thus strongly recommend against fitting α-function parameters when experimental Psat data are not available. For procedure 5, very accurate predictions are achieved. To sum up, if we accept deviations on csat P,liq higher than 5%, it is recommended to only estimate consistent Twu91 α-function parameters for molecules for which at least the pseudoexperimental data of Psat can be accurately generated (procedures 1, 4, 5, and 7 in Table 6).

Table 7. Number of Nonquantum Compounds for Which the Experimental Properties Can Be Accurately Estimated property (P ) (ΔvapH) (csat P,liq)

1 4 5 7 total

accurate property data available sat

(P ) (Psat + ΔvapH) (Psat + csat P,liq) (Psat + ΔvapH + csat P,liq)

1781 1487 863

4.2. Results. Among the 1781 studied molecules, it is found that by working with the tc-PR EoS, 61 reach a mean deviation on at least one property among Psat, ΔvapH, or csat P,liq greater than the average MAPE (x̅) calculated for the 1781 compounds plus 3 times the standard deviation (σ). These thresholds are provided in Table 8.

Table 6. Number of Molecules for Which at Least the Pseudoexperimental Data of Psat Can Be Accurately Generated type of pure fluid

number of compounds

sat

Table 8. Rejection Thresholds for Molecule Selection

this work

property

217 705 80 783 1785

(Psat) (ΔvapH) (csat P,liq)

average MAPE over 1781 fluids (x̅) 1.2% 2.0% 2.3%

standard deviation over 1781 fluids (σ)

threshold (x̅+3σ)

1.6% 2.0% 3.4%

6% 8% 13%

These molecules are considered to be outliers, and it is decided not to publish the corresponding values of the L, M, N parameters (i.e., to remove them from the consolidated database) except for mercury, which is considered to be a molecule of interest as shown by a number of recent research articles. Among these 60 molecules, 14 were unfortunately considered as reliable by Le Guennec et al. and α-function parameters should not have been published.10 These large deviations are justified as follows: 1. Some of these 60 molecules do not obey the 3parameter corresponding-state theorem, so that their properties cannot be modeled with an EoS from the mere knowledge of Tc, Pc and α(T). This behavior is detected when the acentric factor does not reflect at all the size of the molecule. As an example, the acentric factor of trimethylolpropane (C6H14O3) is 1.6; by comparison, a similar value is reached in the alkane family for n-hexatriacontane (n-C36) (its acentric factor is equal to 1.5). 2. There is a problem with the experimental data that was not detected by the correlation selection procedure. The regressed parameters of the 1721 remaining molecules (Table 9) are reported in the Supporting Information for both

After the reference case for which the three experimental properties are available, the most favorable case is that for which Psat and csat P,liq are available (procedure 5), followed by procedure 4, for which Psat and ΔvapH are experimentally known, and procedure 1, for which only Psat is known. The grading of the four procedures is as follows: procedure 7 > procedure 5 > procedure 4 > procedure 1. The number of molecules belonging to each of these four categories of fluids for which vapor pressure data are available are summarized in Table 6. Among the 1116 molecules studied by Le Guennec et al.10 for which L,M,N parameters are published, it is recommended not to use these values for 34 molecules (see the Supporting Information) since they were estimated by only taking experimental ΔvapH data or experimental csat P,liq data, or both of them, into account (fluids of types 2, 3, or 6 in Table 3).

4. UPDATED CONSISTENT Twu91 α-FUNCTION PARAMETERS Following the conclusion of the previous section, a safe estimation of consistent Twu91 α-functions parameters is carried out only for molecules for which at least P sat pseudoexperimental data can be accurately generated from DIPPR correlations. The objective function, imposed constraints, and data generation rules used in this section are identical to those used previously. 4.1. Constitution of a Pure-Compound Database. As highlighted in Table 6, among the 2162 molecules available in the DIPPR database, it is possible to accurately generate at least experimental Psat data for 1785 of them. It should be noted that among these 1785 molecules, there are four quantum fluids (He, H2, Ne, and deuterium), which require special treatment.10 The (L, M, N) parameters for these fluids are presented in the Supporting Information section, but are identical to those published by Le Guennec et al.10 In this paper, deviations are thus calculated for 1781 nonquantum fluids for which accurate Psat and/or ΔvapH and/ or csat P,liq data are available (see Table 7).

Table 9. Distribution of the 1721 Retained Molecules According to the Available Accurate Property Data type of pure fluid


this work

1 4 5 7 total

(Psat) (Psat + ΔvapH) (Psat + csat P,liq) (Psat + ΔvapH + csat P,liq)

198 694 70 759 1721

the PR and the RK EoS. The MAPE on Psat, ΔvapH, and csat P,liq are reported in Table 10 for both consistent-PR and consistentRK EoS. It can be stated that the PR or RK EoS, coupled with a consistent Twu91 α-function, is capable of accurately reproducing the properties of interest. The MAPE on vapor F



Article

Table 10. MAPE on Psat, ΔvapH, and csat P,liq as Predicted from the Updated Consistent PR and RK EoS source

EoS

this work (1721 fluids) consistentPR consistentRK Le Guennec et al.10 (1116 fluids)

consistentPR consistentRK

11

Bell et al. (2571 fluids) consistentPR

MAPE on Psat

MAPE on ΔvapH

MAPE on csat P,liq

(1721 compounds) 1.0%



1.2%

1.9%

2.2%




1.1%

2.1%

2.2%




(i.e., error reported in the DIPPR database or average experimental error calculated by eq 10 is lower than 5%). As highlighted in Table 11, among the 1721 compounds included in the database, 1489 have accurate correlations for the liquid molar volume data. As previously done for the Psat, ΔvapH, and csat P,liq properties, for each of these 1489 compounds, the DIPPR correlation was used to generate 50 equidistant liquid molar volume data points between Tmin and Tmax = 0.9Tc. Here, Tmin is the minimum temperature at which vsat liq can be estimated from the DIPPR correlation, whereas Tmax = 0.9Tc is the highest temperature at which the volume-translation concept has beneficial effect. By denoting vsat,t‑CEoS , the molar volume liq calculated with the translated EoS, the MAPE on vsat liq is calculated as v sat

liq

i

(17)

pressures for more than 1700 compounds is only 1%, while the deviations on ΔvapH and csat P,liq are approximately 2%. It is possible to observe that the addition of up to 600 compounds with respect to that used in Le Guennec et al.3 did not affect the mean deviations on the studied properties. Furthermore, it is possible to compare the MAPE on Psat, ΔvapH, and csat P,liq obtained in this work with those calculated by Bell et al.11 It is observed that the average deviation on Psat in this work is 5-fold lower than that obtained by Bell et al. (see Table 10). Conversely, the MAPE on csat P,liq is 2-fold greater than that obtained by Bell et al.11 This difference is probably due to the mathematical expressions of the objective functions, which are different in the two studies and to the fact that vapor pressures considered in this work were cut off at 0.1 bar, whereas Bell et al. acceptedas part of the fitting exercise much lower vapor pressures for which it is not scarce to get large relative deviations.

These MAPEs are reported in Table 12 for both the PR and RK EoS in their translated and original versions. The Table 12. MAPE on vsat liq As Predicted by the consistent PR and RK EoS on an Updated Database. Effect of the Volume Translation MAPE on vsat liq (Tr < 0.9) (1489 compounds) EoS

untranslated EoS

translated EoS

consistent-PR consistent-RK

8.7% 19.2%

2.2% 3.7%

calculated volume-translation parameters are reported in the Supporting Information for both the PR and RK EoS. It is observed that the reproduction of the vsat liq data is dramatically improved for both models. In addition, the average deviations on the molar critical volume are reported in Table 13. At the critical point, the

5. UPDATED VOLUME-TRANSLATION PARAMETERS (c) To calculate the volume-translation parameter, it is necessary to exactly reproduce the pseudoexperimental saturated-liquid molar volume returned by the DIPPR correlation at a reduced temperature of 0.8, as described by Le Guennec et al.10 The volume translation parameter is thus calculated as sat,u ‐ CEoS sat c = v liq (Tr = 0.8) − v liq,exp (Tr = 0.8)

v sat

sat,DIPPR sat,t ‐ CEoS 50 v liq (Ti liq ) − v liq (Ti liq ) 100 MAPE = ∑ v sat 50 i = 1 v sat,DIPPR (T liq )

Table 13. MAPE on vc As Predicted by the Consistent PR and RK EoS. Effect of the Selected Volume Translation MAPE on vc (1489 compounds)

(16)

where vsat,u‑CEoS (Tr = 0.8) is the molar volume calculated with liq the original (untranslated) CEoS at Tr = 0.8. 5.1. Results. To estimate the MAPE on the liquid molar volumes, it is necessary to identify the molecules for which the pseudoexperimental liquid densities can be accurately generated following the procedure described in section 2.1

EoS

untranslated EoS

translated EoS

consistent-PR consistent-RK

21% 31%

20% 24%

volume-translation parameter c has too weak an influence on the calculated critical volume (vc) to obtain a good match with the experimental value. Although the influence of the volume translation concept is more visible on the RK EoS, the PR CEoS yields better predictions of vc.

Table 11. Distribution of the 1489 Molecules for Which at Least Psat and vsat liq Can Be Accurately Generated from the DIPPR Database type of pure fluid


this work

1 4 5 7 total

(Psat) (Psat + ΔvapH) (Psat + csat P,liq) (Psat + ΔvapH + csat P,liq)

198 694 70 759 1721 G

accurate property data available (Psat (Psat (Psat (Psat

+ + + +

vsat liq ) ΔvapH + vsat liq ) sat csat + v ) P,liq liq sat ΔvapH + csat P,liq + vliq )

this work 164 543 65 717 1489



Article

sat Table 14. MAPE on Psat, ΔvapH, csat P,liq, vliq , vc, Tc, and Pc As Predicted by the Translated-Consistent PR and RK EoS with the Updated Parameters Determined in This Study

EoS tcPR tcRK

MAPE on Psat(1721 fluids)

MAPE on ΔvapH(1453 fluids)

MAPE on csat P,liq(829 fluids)

MAPE on vsat liq (1489 fluids) (Tr < 0.9)

MAPE on vc(1489 fluids)

MAPE on Tc(1721 fluids)

MAPE on Pc (1721 fluids)

1.0%

1.9%

2.0%

2.2%

20%

0%

0%

1.2%

2.0%

2.2%

3.7%

24%

0%

0%

In Table 14, the accurate results obtained with the tc-PR and tc-RK CEoS using the updated parameters determined in this study are summarized. This table highlights that the accuracies of both EoS are very similar, with the exception of the molar volumes, which are better correlated with the PR EoS. In Table 14, the MAPEs on the critical temperature and critical pressure, which are necessarily zero, are reported to remind the reader that these errors are not null with EoS that do not satisfy the critical constraints, for example, the SAFTtype EoS. A proper prediction of the pure-component critical coordinates is, however, a prerequisite to accurately predict the global phase equilibrium diagrams (GPED) of binary systems.12,13

Table 15. Comparison of the MAPEs Calculated with the 3Parameter Twu91, Generalized Twu88 and Soave αFunctions for the Translated PR and RK EoS Peng−Robinson

6. UPDATED GENERALIZED TWU88 α-FUNCTIONS SUITABLE FOR THE tc-PR AND tc-RK EOS This section aims to update the generalized formulations for the tc-PR and tc-RK EoS. It is recalled that the interest of these formulations is that they can be applied to components for which consistent Twu91 α-function parameters cannot be determined from experimental data since no vapor pressure data are available in the literature. As explained in our previous paper,10 parameter N is set to 2 in order to work with the so-called Twu8814 α-function, whereas L and M are correlated as second order polynomials with respect to the acentric factor ω (see eq 18): l o α(Tr) = Tr2(M − 1) exp[L(1 − Tr2M )] o o o o o o m L(ω) = L1ω 2 + L 2ω + L3 o o o o o 2 o o (18) n M(ω) = M1ω + M 2ω + M3 To update the polynomial expressions for parameters L and M, the 759 molecules that have simultaneously accurate correlations for the Psat, ΔvapH, and csat P,liq properties are used (see Table 11). The resulting expressions, for both the PR and RK CEoS, are l o L(ω) = 0.0925ω 2 + 0.6693ω + 0.0728 o PR CEoS: o m o 2 o o n M(ω) = 0.1695ω − 0.2258ω + 0.8788 2 l o o L(ω) = 0.0611ω + 0.7535ω + 0.1359 RK CEoS: o m o 2 o o n M(ω) = 0.1709ω − 0.2063ω + 0.8787

Redlich−Kwong

α-function

Twu91

gen. Twu88

Soave

Twu91

gen. Twu88

MAPE on Psat (1721 compounds) MAPE on ΔvapH (1453 compounds) MAPE on csat P,liq (829 compounds) MAPE on vsat liq (1489 compounds) (Tr < 0.9)

1.0%

1.8%

2.8%

1.2%

2.1%

Soave 2.3%

1.9%

2.7%

3.1%

1.9%

2.9%

2.9%

2.0%

4.1%

7.1%

2.2%

4.3%

5.3%

2.2%

2.2%

2.2%

3.7%

3.8%

3.8%

with, however, an advantage to the first one, especially when predicting Psat and csat P,liq with the PR EoS. Conversely, it is observed that the results are strongly improved if the deviations calculated for the generalized Twu88 α-function are compared with those calculated for the componentdependent Twu91 α-function. The conclusions are identical to those drawn previously and with the exception of molar volumes which are better correlated with the PR EoS, Table 15 does not make it possible to conclude which of the PR or RK EoS is the most accurate. Working with the Soave α-function, the RK EoS is the best-suited but the reverse situation occurs when considering the Twu88 or Twu91 α-functions. Many process simulation software programs however express cPc/(RTc) as a linear function of the Rackett compressibility factor (zRA), appearing in the improvement of the original Rackett equation15 proposed by Spencer and Danner.16 In this study, it was decided to work with the 1489 compounds, for which vsat liq can be accurately estimated. The obtained correlations are l o o o cPR = o o o o m o o o o o cRK = o o o n

(19)

(20)

As proposed by Le Guennec et al., a comparison of the MAPEs obtained with the following three α-functions is performed: (i) the component-dependent Twu91 α-function with the updated parameters determined in this paper, (ii) the updated generalized Twu88 α-function, and (iii) the Soave α-function. Table 15 highlights that the two generalized α-functions, namely “generalized Twu88” and Soave have similar accuracy

RTc (0.1975 − 0.7325z RA ) Pc RTc (0.2150 − 0.7314z RA) Pc

(21)

With such correlations, MAPEs on vsat liq data, over 1489 compounds modeled with the generalized Twu88 α-function, in a temperature range so that Tr ≤ 0.9, coupled with the PR and RK EoS, are 3.0% and 4.0%, respectively. To conclude this section, the updated correlations of volume-translation parameters to ω are presented below: H


Journal of Chemical & Engineering Data l o o o cPR = o o o o m o o o o o o cRK = o o n

Article

ORCID

RTc (0.0096 + 0.0048ω) Pc RTc (0.0227 + 0.0093ω) Pc

Romain Privat: 0000-0001-6174-9160 Jean-Noël Jaubert: 0000-0001-7831-5684 Paul M. Mathias: 0000-0001-5781-9525 Funding

(22)

The French Petroleum Company TOTAL, and particularly Dr. Laurent Avaullée and Freddy Garcia (experts in thermodynamics), are gratefully acknowledged for sponsoring this research.

We know by experience that these correlations are not very accurate but can be useful in the case where Tc, Pc, and ω are the unique data available for a given compound, which is often the case for the pseudocomponents encountered in the oil and gas industry.

Notes

The authors declare no competing financial interest.

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7. CONCLUSION In this study, the suitable combinations of properties among vapor pressures, enthalpies of vaporization, and saturated liquid heat capacities for fitting Twu91 α-function parameters were investigated. The study reveals that reliable (L, M, N) parameters are obtained if the fitting includes at least vapor pressure data. Fitting with only the enthalpies of vaporization and/or saturated liquid heat capacities are wholly inadvisable. One thousand seven hundred twenty-one compounds, for which at least accurate vapor-pressure data could be generated, were found in the DIPPR database, and the 3 (L, M, N) parameters were determined for these fluids. Using these parameters, the tc-PR EoS leads to the following very small average deviations: ΔPsat = 1%, ΔvapH = Δcsat P,liq=2% and Δvsat (T < 0.9) = 2.2%. Moreover, for the tc-PR and tc-RK liq r CEoS to be applied to components not included in the DIPPR database, a generalized version of these models was developed. The 3-parameter Twu91 α-function was substituted by the 2parameter Twu88 function, and the parameters L and M were correlated to the acentric factor. In comparison with the previous work of Le Guennec et al. and as explained notably in sections 3.2 and 4.2, 52 molecules that were considered to be suitable by these authors for fitting α-function parameters were declared nonreliable in this study. The reject of these 52 molecules is justified as follows: • for 34 molecules, no vapor pressure data were available (see section 3.2). • For 14 molecules, the deviations between experimental and calculated properties were abnormally high so that such molecules were considered to be outliers (see section 4.2) • Four molecules (Ca, K, Li, and LiI) are metals with critical temperature higher than 2000 K which we believe can not be seriously modeled by a CEoS. In return, 661 new molecules were added to the library developed in this work.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jced.8b00640.

■

REFERENCES

(1) Poling, B. E.; Prausnitz, J. M.; O’Connell, J. P. The Properties of Gases and Liquids, 5th ed.; McGraw-Hill: New York, 2001. (2) Le Guennec, Y.; Lasala, S.; Privat, R.; Jaubert, J.-N. A Consistency Test for α-Functions of Cubic Equations of State. Fluid Phase Equilib. 2016, 427, 513−538. (3) Le Guennec, Y.; Privat, R.; Lasala, S.; Jaubert, J.-N. On the Imperative Need to Use a Consistent α-Function for the Prediction of Pure-Compound Supercritical Properties with a Cubic Equation of State. Fluid Phase Equilib. 2017, 445, 45−53. (4) Twu, C. H.; Bluck, D.; Cunningham, J. R.; Coon, J. E. A Cubic Equation of State with a New Alpha Function and a New Mixing Rule. Fluid Phase Equilib. 1991, 69, 33−50. (5) Jaubert, J.-N.; Privat, R.; Le Guennec, Y.; Coniglio, L. Note on the Properties Altered by Application of a Péneloux-Type Volume Translation to an Equation of State. Fluid Phase Equilib. 2016, 419, 88−95. (6) Privat, R.; Jaubert, J.-N.; Le Guennec, Y. Incorporation of a Volume Translation in an Equation of State for Fluid Mixtures: Which Combining Rule? Which Effect on Properties of Mixing? Fluid Phase Equilib. 2016, 427, 414−420. (7) Péneloux, A.; Rauzy, E.; Freze, R. A Consistent Correction for Redlich-Kwong-Soave Volumes. Fluid Phase Equilib. 1982, 8, 7−23. (8) Peng, D.-Y.; Robinson, D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15, 59−64. (9) Redlich, O.; Kwong, J. N. S. On the Thermodynamics of Solutions. V. An Equation of State. Fugacities of Gaseous Solutions. Chem. Rev. 1949, 44, 233−244. (10) Le Guennec, Y.; Privat, R.; Jaubert, J.-N. Development of the Translated-Consistent tc-PR and tc-RK Cubic Equations of State for a Safe and Accurate Prediction of Volumetric, Energetic and Saturation Properties of Pure Compounds in the Sub- and Super-Critical Domains. Fluid Phase Equilib. 2016, 429, 301−312. (11) Bell, I. H.; Satyro, M.; Lemmon, E. W. Consistent Twu Parameters for More than 2500 Pure Fluids from Critically Evaluated Experimental Data. J. Chem. Eng. Data 2018, 63, 2402−2409. (12) Privat, R.; Jaubert, J.-N. Classification of Global Fluid-Phase Equilibrium Behaviors in Binary Systems. Chem. Eng. Res. Des. 2013, 91, 1807−1839. (13) Qian, J.-W.; Privat, R.; Jaubert, J.-N. Predicting the Phase Equilibria, Critical Phenomena, and Mixing Enthalpies of Binary Aqueous Systems Containing Alkanes, Cycloalkanes, Aromatics, Alkenes, and Gases (N2, CO2, H2S, H2) with the PPR78 Equation of State. Ind. Eng. Chem. Res. 2013, 52, 16457−16490. (14) Twu, C. H. A Modified Redlich-Kwong Equation of State for Highly Polar, Supercritical Systems. Proceedings of the International Symposium on Thermodynamics in Chemical Engineering and Industry, Beijing, China, 30 May-2 June, 1988; pp 148−169. (15) Rackett, H. G. Equation of State for Saturated Liquids. J. Chem. Eng. Data 1970, 15, 514−517. (16) Spencer, C. F.; Danner, R. P. Improved Equation for Prediction of Saturated Liquid Density. J. Chem. Eng. Data 1972, 17, 236−241.

Updated parameter values for the translated-consistent tc-PR and tc-RK cubic equations of state (PDF)

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Analysis of the Combinations of Property Data That Are Suitable for a

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