Consistent Twu parameters for more than 2500 pure uids ...

Consistent Twu parameters for more than 2500pure fluids from critically evaluated experimental

data†

Ian H. Bell,∗,‡ Marco Satyro,¶ and Eric W. Lemmon‡

‡National Institute of Standards and Technology, Boulder, CO, USA¶Formerly of Virtual Materials Group Inc.

E-mail: [email protected]

Abstract

The construction of the standard cubic equa-tions of state such as Peng-Robinson or Soave-Redlich-Kwong does not automatically yieldphysically reasonable values when the equa-tion of state is extrapolated beyond the rangewhere experimental data are available. A multi-property fitting exercise was carried out inwhich we obtained a consistent set of Twu αfunction parameters for 2570 pure fluids basedon the experimental data contained in the Ther-moDataEngine (TDE) database developed atNIST. We have applied the consistency checksof Le Guennec et al. to the Twu α func-tion of the Peng-Robinson equation of state.The experimental data stored in TDE passedthrough a critical evaluation, and we used onlythe data that were determined to be thermo-dynamically reliable. Over all the consideredfluids, the mean average percentage error is ap-proximately 7% for vapor pressure, 1% for la-tent heat of vaporization, and 1% for saturationspecific heat. Comprehensive supplemental ma-

†Commercial equipment, instruments, or materialsare identified only in order to adequately specify cer-tain procedures. In no case does such identification im-ply recommendation or endorsement by the NationalInstitute of Standards and Technology, nor does it im-ply that the products identified are necessarily the bestavailable for the purpose. Contribution of the NationalInstitute of Standards and Technology, not subject tocopyright in the US

terials with the complete set of analytic deriva-tives, the obtained parameters, and the fittingcode in C++, is provided.

1 Introduction

Cubic equations of state (EOS) have a pedigreedating back to the work of van der Waals in1873.1 In spite of their well-documented defi-ciencies described by for instance Trebble andBishnoi2 or Boshkova and Deiters,3 cubic EOSretain a prominent place in chemical engineer-ing (and many other fields) thanks to theirsimplicity. Though much more accurate mul-tiparameter fundamental equations of state areavailable in thermophysical property libraries,cubic equations of state show no signs of fallinginto disuse.

One of the well-documented challenges withcubic equations of state is their relatively poorpredictions of thermodynamic properties (e.g.,the vapor pressures of polar fluids like wa-ter) when generalized estimation schemes areemployed for the attractive parameters in theequation of state, such as in the conventionalPeng-Robinson equation of state.4,5 When theattractive parameters are fit to experimentaldata, as in this work, the equation of statecan yield much better predictions of thermo-dynamic properties. Figure 1 gives a graph-ical representation of the problem. This fig-ure shows that at low reduced temperatures,

1

the errors in vapor pressure prediction from theconventional Peng-Robinson equation are morethan 20% as compared to the reference equa-tion of state of Wagner and Pruß6! The com-mon “solution” to this problem is to introduceempiricism, or adjustable parameters that mustbe fit by the correlator.

0.4 0.5 0.6 0.7 0.8 0.9 1.0T/Tc (-)

505

10152025

(pσ,r

ef/p

σ,c

ubic−

1)×

100 (

%)

ConventionalPeng-Robinson (Soave-type α function)

This work (consistent Twu α function)

Figure 1: Error in the prediction of the va-por pressure of water from the default Peng-Robinson α function with the α function fromthis work. The subscript “ref” refers to thereference multiparameter equation of state ofWagner and Pruß6

In the literature there have been a few groupsthat have published large libraries of these ad-justable parameters for cubic equations of state.The most significant recent contributions havebeen those of Horstmann et al.7 and Le Guen-nec et al.8 In particular the work of Le Guennecet al. is important here because their library ofadjustable parameters were obtained accordingto a set of rigorous consistency checks, the samechecks applied here.

While the overall aim of this work is similarto that of Le Guennec et al., we made someimprovements to the fitting procedure:

• More comprehensive set of pure fluids : InLe Guennec et al., the DIPPR databasewas used to generate pseudo-experimentaldata points for 1197 pure fluids for whichvapor pressure, latent heat, or satura-tion specific heat data were available. Inthis work we used the ThermoDataEngine

(TDE) database of NIST, which providesdata coverage for more than 23,000 purefluids (though the data coverage is veryheterogeneous; see below). Of those flu-ids, 2570 were included in our fit becausethey had sufficient experimental data cov-erage.

• Direct fitting of experimental data:Rather than fitting curve fitted pseudo-experimental data as in Le Guennec etal., we have directly employed the exper-imental data of NIST TDE in our fitter.

• Algorithm: We provide comprehensivedescription of the fitting process and pro-vide the source code used in our fit-ter. All parts of the fitting process useopen-source tools and packages for cross-platform replicability.

2 Thermodynamic model-

ing

2.1 Cubic equations of state

The development of new cubic equations ofstate remains an active field of research,9 as wellas the extension of cubic equations of state withactivity coefficient models.7,10–14 One of the pri-mary motivations for this work is to serve as areference for the properties of the pure fluids incubic + activity coefficient models such as thoseof the group-contribution volume-translatedPeng-Robinson (VTPR)12,15–19 equation or pre-dictive Soave-Redlich-Kwong (PSRK).7 More-over, as demonstrated by Le Guennec et al.,20

it is absolutely imperative to use a consistent αfunction when fitting only subcritical data andapplying the model to supercritical states.

While a comprehensive analysis of the multi-tude of equations of state that are either directdescendants (or distant offspring) of the vander Waals equation of state is beyond the scopeof this paper, we refer the interested reader tothe literature for a further review.9,21,22 In thisstudy we focus on one of the cubic equationsof state with the most significant present-day

2

influence – the Peng-Robinson4,5 equation ofstate.

The most industrially relevant cubic equa-tions of state can be given in the form23

p =RT

v − b− a(T )

(v + ∆1b)(v + ∆2b)(1)

where ∆1 and ∆2 are constants that are set toyield the desired equation of state (see Bell andJager24 or Michelsen23 for more information).In the case of Peng-Robinson, ∆1 = 1+

√2 and

∆2 = 1−√

2, and in the case of Soave-Redlich-Kwong, ∆1 = 1 and ∆2 = 0. In addition, b isthe co-volume, a is the attractive term, and vis the molar volume.

In the standard implementations of a cubicEOS, the attractive term a generally takes theform

a = a0α(Tr), (2)

where a0 is an constant of the form a0 =c0R

2T 2c /pc, and where c0 is EOS-dependent.

In the Peng-Robinson or Soave-Redlich-KwongEOS the α function is given by the form

α =[1 +m

(1−

√T/Tc

)]2(3)

as proposed by Soave and co-workers.25 The pa-rameter m is a function of the acentric factorof the fluid. The attractive term of Mathias &Copeman26 is an extension of this general form.An alternative (and now preferred) form for theattractive function is that of Twu:27

α =

(T

Tc

)C2(C1−1)

exp

[C0

(1−

(T

Tc

)C1C2)]

.

(4)This form is now preferred28 because the pa-rameters C0, C1, and C2 can be selected to meetsome important consistency conditions that aredifficult or impossible to enforce with otherfunctional forms. These conditions are furtherdescribed in Section 4.4.

A number of authors have attempted to de-velop generalized approaches for the attractiveparameters for cubic EOS, with varying levelsof success.29 There have also been attempts todetermine physical constraints on the terms in

the cubic equation of state.8,20,28,30,31

2.2 Helmholtz transformations

In the state-of-the-art thermophysical propertylibraries, the equation of state is expressed interms of non-dimensionalized Helmholtz energyαe = ae/(RT ) rather than in a pressure-explicitform. The Helmholtz energy ae is a thermody-namic potential from which all other thermo-dynamic properties can be obtained. Thereforethe EOS can be expressed as

αe(τ, δ) = α0e(τ, δ) + αr

e(τ, δ), (5)

where τ = Tr/T and δ = ρ/ρr are the reciprocalreduced temperature and the reduced density,respectively. The reducing temperature Tr andthe reducing density ρr are usually, but not al-ways, their values at the critical point.

The generalized non-dimensionalized residualHelmholtz energy contribution αr

e can be givenby24

αre = − ln(1− bδρr)−

τa

RTr

ln

(∆1bρrδ + 1

∆2bρrδ + 1

)b(∆1 −∆2)

.

(6)As in Bell and Jager,24 Eq. (6) can be factoredinto the form

αre = ψ(−)(δ)− τa(τ)

RTrψ(+)(δ). (7)

The ideal-gas contribution α0e of the equation

of state (see for instance Lemmon et al.32) isgiven by

α0e =

h00τ

RTc−s

00

R−1+ln

δτ0δ0τ− τR

∫ τ

τ0

c0pτ 2

dτ+1

R

∫ τ

τ0

c0pτ

dτ

(8)where δ0 = ρ0/ρc, τ0 = Tc/T0, and c0p is theideal-gas specific heat capacity as a functionof temperature (or reciprocal reduced temper-ature τ). The subscript “0” indicates that theproperty is for the reference state of the EOS.The ideal-gas contribution can also be writtenin the form α0

e = ln δ + f(T ) to highlight theseparability of the temperature and density de-pendence of α0

e .

3

3 Data curation and prepa-

ration

In this work we used the experimental datacontained in the ThermoData Engine database(TDE).33–36 This database contains a largebody of experimental data (for nearly 24,000pure fluids), and includes methods for criticallyevaluating the uncertainty and thermodynamicconsistency of experimental data. We includein this study only experimental data that passthe critical evaluation checks of TDE.

The data were prepared by selecting all thepure fluids for which, at a minimum:

• Critical temperature data were available• Saturation pressure data were available

with at least 10 data points passing thecritical evaluation tests; saturation pres-sures below pc/106 or saturation temper-atures above 0.8Tc were not included1.

For some fluids, additional useful data wereavailable in TDE, including critical pressuredata, latent heat of vaporization data, triple-point temperature data, etc. For many fluidsthere is a significant paucity of experimentaldata; for some fluids, there are as few as onedata point among all data types.

While the critical temperature is usually mea-sured directly, the “experimentally measured”critical pressure given in literature is often eval-uated by extrapolating the saturation pres-sure data to the measured critical temperature,for instance by fitting an Antoine-type equa-tion to the saturation pressure versus tempera-ture data and extrapolating to the given criti-cal temperature to obtain the critical pressure.The critical pressure value calculated by TDEis used without modification.

The acentric factor ω appearing in the base-line equation of state forms for the Peng-Robinson and SRK equations is never mea-sured directly, rather it is obtained as a post-

1These points were rejected due to the numerical dif-ficulties of carrying out vapor-liquid-equilibrium calcu-lations at these states; derived vapor pressures for heavylinear alkanes are available at pressures below 10−20

Pa37

processing step applied to the experimentaldata. The definition of the acentric factor is21

ω = − log10

(pσ(0.7Tc)

pc

)− 1. (9)

Determination of the acentric factor from ex-perimental data is a multi-step process:

1. Determine the critical temperature andpressure as described above.

2. Obtain a saturation pressure curve (froma functional form such as the Antoineequation) for the saturation pressuredata, with the functional dependencypσ = f(T ).

3. Evaluate the saturation pressure curve at0.7Tc, and evaluate the acentric factorfrom Eq. (9).

Figure 2 shows the data distribution for thepure fluids included in TDE that were used inthis study. There are comparatively few fluidsthat have both a significant number of vaporpressure measurements and well as latent heatmeasurements. The bulk of the fluids are foundin the domain with only a few vapor pressuremeasurements, and for many of those fluids, few(or no) latent heat measurements.

The metadata associated with each fluid isbased on the InChI key of the fluid, a uniqueidentifier based on the molecular connectivityinformation2. The InChI key is broadly under-stood by cheminformatics systems, and is unen-cumbered by intellectual property restrictions.CAS registry numbers, on the other hand, areproprietary information.

4 Algorithmic approach

4.1 Objective function

The objective function for this problem is thesum of the squared residues in the residue vec-tor. Mathematically, our objective function is

2Though useful for generating unique identifiers fornearly all compounds based on molecular connectivityinformation, the InChI string/key is not able to dis-ambiguate spin isomers like ortho-, para-, or normal-hydrogen.

4

0 500 1000 1500 2000 2500100101102103104

Npσ

0 20 40 60 80 100 120100101102103104

NL

0 100 200 300 400 500 600Data points per fluid

100101102103104

Nc σ

Figure 2: Distributions of the experimentaldata available for each data type for pure fluidsin the NIST ThermoDataEngine. Npσ : num-ber of fluids with this many experimental va-por pressure data points, NL: number of fluidswith this many latent heat of vaporization datapoints, Ncσ : number of fluids with this many ex-perimental saturation specific heat data points.

given by:

O(~C) =∑i

(wiri(~C))2 (10)

where each of the residue contributions ri =ymodel,i−yexp,i corresponds to a given data point(saturation pressure, latent heat, or saturationspecific heat), as is described in the followingsections, and as described in detail in the sup-plemental material. The parameters wi weightthe different property types, where a weight ofone is used for the vapor pressure and latentheat of vaporization data, and a weight of 0.5is used for the saturation specific heat data.These weights were obtained by experimenta-tion to enforce the desired balance between thedifferent properties. Within the properties, nor-malized weights as specified by the Thermody-namics Research Center were used to weight thedata points.

The evaluation of the residues is an embarass-ingly parallel problem; each row in the residuevector can be evaluated entirely independentlyof the other rows. This creates a problem that isperfectly suited to parallel evaluation over sev-eral threads. In this case, we use the nativelymultithreaded C++11 library NISTfit38 devel-oped by the authors to evaluate the residues inparallel, yielding a nearly linear speedup versuspurely serial evaluation. The NISTfit libraryalso includes a thread-parallel implementationof the Levenberg-Marquardt sum of squaresminimizer governed by a derivative-based trustregion minimization. At the beginning of thefitting campaign, the Levenberg-Marquardt op-timizer was used, and for that reason, analyticderivatives of each of the residues with respectto the coefficients were developed. These an-alytic derivatives are mathematically complex,and for that reason are presented in the sup-plemental material, where they are covered indetail.

One of the major disadvantages of Levenberg-Marquardt minimization is that there is no easymeans of integrating nonlinear inequality con-straints such as the consistency checks imple-mented in this paper. One of the most straight-forward means of implementing inequality con-

5

straints in fitting is to use a nature-inspiredevolutionary optimization technique in concertwith penalty functions for the inequality con-straints. There are numerous evolutionary-likeoptimization methods available in the litera-ture, and the differential evolution39 algorithmas implemented in the Python scipy.optimize

package was used. Differential evolution oper-ates by generating a large population of indi-viduals (an individual is a set of Twu coeffi-cients), and, for each individual, evaluating thecost function. Varying hybridization schemesare described in the literature for finding thelowest cost individual (best set of Twu coeffi-cients).

The cost function for differential evolution isthen given by

COST(~C) = O(~C) + PENALTY(~C) (11)

where PENALTY(~C) is the sum of the penal-ties for each constraint that is not satisfied.Each unfulfilled constraint added a large num-ber (here, 1000) to the cost function. The dif-ferential evolution optimizer was then able tosuccessfully carry out the global optimizationproblem within the domain specified by the con-straints. Solutions not meeting the constraintswere implicitly rejected by the optimizer. Dif-ferential evolution is a non-derivative-based op-timization method and is therefore able to han-dle the discontinuities in the objective functioncaused by the constraints.

4.2 Constraints

As is described in Le Guennec et al.,8,28 thereare constraints on the Twu attractive parame-ters ~C that should be enforced in order to en-sure reasonable outputs and extrapolation be-havior from the equation of state. These con-straints are that:

• α should be 1 at Tr = 1 .

• α should always be greater than zero.

• The derivative dα/dTr should always beless than or equal to zero.

• The derivative d2α/dT2r should always be

greater than or equal to zero.

• The derivative d3α/dT3r should always be

less than or equal to zero.

Many authors that have developed sets ofTwu attractive parameters have not enforcedthese constraints, yielding highly suspect ex-trapolation behavior outside the domain inwhich the parameters were fit.

Le Guennec et al.28 gives the following twoconstraints, both of which MUST be enforced:Constraint 1:

−∆ ≥ 0 (12)

Constraint 2:

C0γ ≥ 0 (13)

where ∆ = C2(C1 − 1) and γ = C1C2.In this case we express each constraint as the

expression being greater than or equal to zero- this is the general form of the constraints re-quired in many optimization routines. In eachconstraint, the left hand side is only a functionof the attractive parameters ~C.

There are furthermore two constraints (3aand 3b), at least one of which must be sat-isfied to ensure consistency.Constraint 3a:

1−∆− γ ≥ 0 (14)

Constraint 3b (both conditions must bemet):{

1− 2∆ + 2√

∆(∆− 1)− γ ≥ 04Y 3 + 4ZX3 + 27Z2 − 18XY Z −X2Y 2 ≥ 0

(15)where

X = −3(γ + δ − 1) (16)

Y = γ2 + 3δγ − 3γ + 3δ2 − 6δ + 2 (17)

Z = −δ(δ2 − 3δ + 2) (18)

A consistent set of Twu parameters is thereforeone that satisfies constraints 1 and 2 and atleast one of constraints 3a and 3b.

6

4.3 Residues

The residues used in this work are of three fun-damental types:

1. Saturation pressure: Equality of the ex-perimental saturation pressure with themodel prediction is thermodynamicallyequivalent to the model-predicted Gibbsenergy being the same in both phases forthe experimental pressure. Therefore, thedifference in Gibbs energy between theliquid and vapor phases, each evaluated atthe experimental temperature and pres-sure, is driven to zero.

2. Latent heat of vaporization: The dif-ference in latent heat of vaporizationwith the experimentally-measured valueis driven to zero.

3. Saturation specific heat : The saturationspecific heat can be experimentally mea-sured at states where the measurement ofthe vapor pressure is difficult or impossi-ble. Therefore, saturation specific heatdata can provide useful information onthe shape of the thermodynamic surface.As noted by Le Guennec et al.,20 the in-clusion of cσ data is imperative to morefully constrain the behavior of the equa-tion of state. One disadvantage of the useof saturation specific heat data is that amodel for the ideal-gas specific heat of thefluid must be available. In this case, weused the fitted values for c0p provided bythe Wilhoit correlation coefficients avail-able in TDE.

The residues are added together as describedin Eq. (10). The analytic form of each residue ispresented in the supplemental material, alongwith additional derivations required to imple-ment each of the residue terms.

4.4 Implementation

For a given fluid, the following approach is em-ployed:

1. The experimental data for a given fluid isretrieved from cached data obtained fromNIST ThermoDataEngine.

2. If sufficient data are not available, thefluid is not included.

3. If sufficient data are available, the acen-tric factor is obtained by fitting an An-toine curve to the saturated pressure dataover the entire temperature range, andis then evaluated according to Eq. (9).These values are only used to provideguess values for the saturation calls; thus,extremely precise acentric factors are notrequired.

4. Differential evolution is used to carry outthe optimization in two parts:

(a) Constraints 1, 2, and 3a are imposed,and the optimization is carried out.If the optimization terminates suc-cessfully, the result is stored.

(b) Constraints 1, 2, and both parts of3b are imposed, and the optimiza-tion is carried out. If the optimiza-tion terminates successfully, the re-sult is stored.

5. The best individual from the optimizationis retained. The output of this step isthe final optimized value of the consistentTwu vector of coefficients ~C.

The final minimization algorithm was imple-mented by hybridizing a number of open-sourcetools. The implementation of the generalizedHelmholtz energy transformations of the cu-bic equations of state of Bell and Jager24 wereused to evaluate the Helmholtz energy contri-butions found in the residues and the Jacobianmatrix. Data input and output uses the stan-dardized JSON (javascript object notation) fileformat for native interoperability between C++and Python (or other high-level languages). InC++, the rapidjson library is used for JSONfile input/output, and the Eigen library is usedfor the matrix math operations. Some com-putational routines (e.g., generic C++ routines

7

for nonlinear equation solving) have been takenfrom the CoolProp library.40

The C++ interface was wrapped into amodule in the Python programming languagethrough the use of the pybind1141 package.The Python module retains the convenience ofa high-level programming language while alsoachieving computational speeds commensuratewith a low-level programming language (C++in this case). The amount of shim code to con-struct the interface between C++ and Pythonis minimal; the pybind11 templates carry outmost of the datatype conversions.

The source code used to carry out the mini-mization is given in the supplemental material,as well as some artificial “experimental” datagenerated from the multiparameter equation ofstate of n-hexane42 for testing purposes.

5 Results

We applied the fitting methodology describedabove to fit a consistent set of Twu attractiveparameters for 2570 fluids and obtained the pa-rameters given in the supplemental material.For each property, we define the average ab-solute deviation (AAD) as a percentage givenas

AADY =100

N

N∑i=1

∣∣∣∣1− Ycalc,iYexp,i

∣∣∣∣ (19)

where Y is the property of interest (vapor pres-sure, latent heat, or saturation specific heat).If a calculated value is unable to be evaluatedby the model (most especially at extremely lowpressures), it is not included in the AAD.

Figures 3 to 5 show the coverage and errordistributions for the saturation pressure, latentheat of vaporization, and saturation specificheat data for the fluids included in this fit-ting exercise. These figures give a high-leveloverview of the representation of the experi-mental data by the Peng-Robinson equation ofstate augmented by the Twu attractive functionparameters obtained in this work. In each fig-ure, a two-dimensional density plot is shown,and histograms show for the distributions ineach of the two plotted variables, here the num-

ber of data points and the AAD in the repre-sentation of the given variable.

The average of the AAD of a property Y forall the fluids that have this property is definedby

AADY = mean(−−−−→AADY ) (20)

results in the following values:

• AADpσ = 4.9%

• AADcσ = 0.9%

• AADL = 1.5%

The AAD of the properties can vary over afew orders of magnitude. Therefore, the morerelevant metric is the logarithm of the AAD.The log-average of the AAD of a property Yfor all the fluids that have this property,

logAADY = exp(mean(log(−−−−→AADY ))), (21)

results in the following values for the propertiesunder consideration:

• logAADpσ = 2.05%

• logAADcσ = 0.39%

• logAADL = 0.34%

These values correspond to the peaks of each ofthe histograms in AAD in Figures 3 to 5.

8

10 100 1000Npσ

10−3

10−2

10−1

100

101

102

103

AA

Dpσ (%

)

Figure 3: Distribution of the AAD error of sat-uration pressure from fitting Twu parametersand data point availability. The darker the hex,the more data points fall within it.

1 10 100NL

10−3

10−2

10−1

100

101

102

103

AA

DL (%

)

Figure 4: Distribution of the AAD error fromfitting Twu parameters of latent heat of vapor-ization and data point availability. The darkerthe hex, the more data points fall within it.

1 10 100 1000Ncσ

10−3

10−2

10−1

100

101

102

103

AA

Dc σ

(%)

Figure 5: Distribution of the AAD error fromfitting Twu parameters of saturation specificheat cσ and data point availability. The darkerthe hex, the more data points fall within it.

An important check on the behavior of thefitted parameters is an assessment of the shapeof the α function, as described in section 4.2and in Le Guennec et al.8,28 Therefore, the αfunctions were plotted for 50 illustrative fluids,selected by their sorted value of α at Tr = 0.2.Figure 6 shows the results for high values of Tr,and Fig. 7 for low values of Tr, demonstratingby visual inspection that the curves all 1) passthrough (Tr, α) = (1, 1), 2) yield α values abovezero, and 3) have negative slopes and positivesecond derivatives. Therefore, each of these αfunctions satisfy the conditions of consistencylaid out by Le Guennec et al.;8,28 this is as ex-pected because the consistency constraints wereimposed in the fitting procedure.

9

2 4 6 8 10Tr = T/Tc (-)

0.00.20.40.60.81.01.21.4

α (-

)

Figure 6: Plots of α as a function of Tr at highTr for 50 illustrative fluids

0.2 0.4 0.6 0.8 1.0Tr = T/Tc (-)

100

101

102

103

α (-

)

Figure 7: Plots of α as a function of Tr at lowTr for 50 illustrative fluids

The Waring number43 (also known as Riedel’sfactor) is a property derivative that can be usedto check that the shape of the vapor pressurecurve is reasonable. For that reason, it can beinstructive to plot the Waring number, given by

Wa = −R(∂(ln p)

∂(1/T )

)σ

=RT 2

p

(∂p

∂T

)σ

, (22)

as a function of reduced temperature Tr. Whilethere are no hard-and-fast rules for the re-quired shape of the curve of the Waring num-ber, Waring43 suggests that this number shouldhave a minimum value at a Tr value of ap-proximately 0.8 or 0.85, have positive sec-ond derivatives everywhere, and negative first

derivatives for Tr less than the Wa minimumand positive first derivative for Tr greater thanthe Wa minimum. The Waring number isfinite at the critical point (see for instanceWagner44). The Waring number is shown inFig. 8 for the same fluids studied by War-ing:43 methane, ethane, propane, n-butane, n-pentane, n-heptane, ethylene, propylene, 1,3butadiene, benzene, chlorodifluoromethane (re-frigerant 13), methanol, carbon dioxide, car-bon disulfide, sulfur dioxide, and ethylene ox-ide. The Waring numbers for all these fluids,when modeled with the α function from thiswork, demonstrate the qualitatively correct be-havior.

0.4 0.5 0.6 0.7 0.8 0.9 1.0Tr = T/Tc (-)

1.001.021.041.061.081.10

Wa/

min

(Wa)

Figure 8: Plot of Waring numbers divided byits value at the minimum as a function of Tr forthe fluids selected by Waring43

Another conclusion from this fitting exercise,as also noticed by Le Guennec et al. 8 , is thatthe Peng-Robinson + Twu formulation is notalways adequate for representing strongly in-teracting fluids (e.g., acids). As a particu-larly striking demonstration of the challengesinherent in representing the phase equilibria ofstrongly associating fluids, we present in Fig. 9the data representation for both vapor pres-sure and latent heat for acetic acid. The AADfor the vapor pressure, when only the vaporpressure data are included in the fit, is 1.1%,whereas when the latent heat data and satura-tion specific heat data are included, the AADin vapor pressure is more than 19%! For moreweakly associating fluids, the challenges of rep-

10

resentation of the experimental data are muchless severe. Even water, a strongly interactingfluid, has a vapor pressure AAD of 1.2% and alatent heat AAD of 1.0%. Thus, while we haveendeavored to yield as accurate a representationof the phase equilibria that we can, users shouldbe aware of the limitations of these parametersin representing the properties of strongly asso-ciating fluids; a healthy dose of caution is ap-propriate.

0.5 0.7 0.9Tr = T/Tc (-)

101214161820222426

(pca

lc/p

exp−

1)×

100 (

%)

0.5 0.6 0.7Tr = T/Tc (-)

5560657075808590

(Lca

lc/L

exp−

1)×

100 (

%)

Figure 9: Deviations of the property predictionsfor acetic acid with the Twu coefficients as fitin this work

Conclusions

In this work we have developed a database ofconsistent α parameters for the Peng-Robinsonequation of state. These parameters yield ac-curate representations of the thermodynamicproperties of nearly 2600 fluids. A number ofquantitative and qualitative assessments of the

α function have been carried out, demonstrat-ing that the α functions yield consistent, accu-rate, and reasonable behavior of the equationof state. This database of Twu α function pa-rameters therefore forms the basis for the nextgeneration of Peng-Robinson implementations.

Acknowledgement This article is dedicatedto the memory of Marco Satyro, who tragicallypassed away while this work was underway. Theauthors thank 1) Romain Privat and Jean-NoelJaubert of the Universite de Lorraine, Francefor answering numerous questions about theconsistency check and providing check values,2) Vladimir Diky (of NIST) for bringing to ourattention the Waring number, and discussionsof fitting procedures, 3) Kenneth Kroenlein andChris Muzny (both of NIST) for their assis-tance with data extraction from TDE, 4) An-dreas Jager of Technische Universitat Dresdenfor assistance with the analytic derivatives.

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Graphical TOC Entry

Peng-Robinson

+consistent

Twu α

=

L,M,N

&accurate

properties

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