Calculation of critical points from Helmholtz-energy-explicit mixture

models∗

Ian H. Bell, Andreas Jager

October 25, 2016

Abstract

The calculation of the critical points for a mixture of fluids is of practical interest as the calculated critical pointscan be used to more reliably and efficiently construct phase envelopes. The number of stable critical points foundcan also provide insight into whether the mixture has an open or closed phase envelope.

In this work we have developed a reliable method for determining all the critical points for a mixture thatis modeled with Helmholtz-energy-explicit equations of state. This method extends the algorithms developed inthe literature for simpler equations of state to these more complex mixture models. These Helmholtz-energy-explicit equations of state could be either multi-fluid models or transformations of simple cubic equations of stateto Helmholtz-energy-explicit forms. This algorithm locks onto the first criticality contour (the spinodal) and tracesit to high density, thereby locating all relevant critical points. The necessary analytic derivatives of the residualHelmholtz energy, numerically validated values of the derivatives for validation, sample code, and additional figuresand information are provided in the supplemental material.

1 Introduction

The study of the calculation of the critical points (andcritical lines) of mixtures forms a small subset of thebroader field of phase equilibria of pure fluids and mix-tures. If to begin we restrict ourselves to pure fluids, sev-eral phase equilibria are possible - vapor-liquid equilib-ria, solid-vapor equilibria, solid-solid-vapor triple points,and the like. Mixtures, on the other hand, involve allthe complexities of pure fluid phase equilibria, and addanother dimension to the problem - the consideration ofthe mixture composition. The addition of the mixturecomposition introduces a number of more complex phaseequilibria, including critical lines, critical end points,liquid-liquid equilibria, and equilibria with solid phases[1].

The primary goal of this paper is to bring togetherthe work related to the calculation of critical pointsof mixtures of fluids, and that of the developmentsin recent years of the high-accuracy Helmholtz-energy-explicit mixture models. Much of the existing literatureon the calculation of mixture critical points focuses onwell-behaved equations of state (cubics, cubics + excessGibbs energy, etc.), which simplified the analysis some-what. In this case we extend the previous models tomore complex mixture models, which introduces somenumerical challenges, as will be shown below.

The critical line, formed of the critical points, formsan important part of the global phase diagram constitut-ing the vapor pressure curves of the pure components,multi-phase equilibria curves, azeotropic curves, and soon. Global phase diagrams are briefly described in thefollowing section.

∗Commercial equipment, instruments, or materials are identified only in order to adequately specify certain procedures. In no casedoes such identification imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it implythat the products identified are necessarily the best available for the purpose. Contribution of the National Institute of Standards andTechnology, not subject to copyright in the US

1

1.1 Critical Points and Curves

The majority of the scientific literature on the calculationof critical points of multi-component mixtures dates backto the 1970s. In fact, the history starts much earlier, withthe work of Gibbs in 1876, and his establishment of thecritical point conditions for a binary mixture[2]:(

∂2Gm

∂x21

)T,p

= 0 &

(∂3Gm

∂x31

)T,p

= 0, (1)

where Gm is the molar Gibbs energy, T is the tempera-ture, p is the pressure, and x1 and x2 are the mole frac-tions of the first and second components, respectively.

Since 1970, a sizable body of literature has built uprelated to the calculation of critical points and criticallines for mixtures with two or more components. As aresult, the literature survey presented here is intended togive the reader a sense of the more important advancesin this field.

Beegle et al. [3] provide a general description of themathematics involved in assessing the stability of a purefluid or a mixture at a given state through the use ofLegendre transformations applied to the internal energy.

Hicks and Young [4] provide a discussion of the calcu-lation of critical points. They also provide a large bodyof experimental critical point data collected from theirwork and curation of other critical point data in litera-ture.

Reid and Beegle[5] subsequently expressed the crit-ical conditions in terms of Legendre transforms, whichallows for a straightforward application of the mixturemodel to calculate the critical point. In this formulation,two matrix determinants (arising from Legendre trans-forms) must be simultaneously equated with zero, andthis is the method further described and implemented inthe analysis below.

Gubbins and Twu [6] studied some mixtures based onthe combination of model inter-molecular interactions,from which they calculate both phase equilibria and crit-ical loci. They are able to demonstrate a wide rangeof phase equilibrium behaviors through the use of thesesimple models.

Heidemann and Khalil [7] applied the criticality con-ditions and gave practical applications of the calculationof critical points through the use of cubic equations ofstate. Their work remains one of the most-cited contri-butions in this field.

Michelsen et al. [8] provided a novel algorithm forcalculating critical points, as well as a discussion of theshape of the saturation curves in the vicinity of the criti-cal point. Furthermore, extensive discussions of stabilityof the critical points and tangent plane distance analysesare provided.

Sadus [9] provided a review of literature on criticalpoint evaluation and a description of some more complexcritical point behavior, as well as a discussion of criticalpoint transitions.

Kolar [10] applied the Predictive Soave-Redlich-Kwong (PSRK) model to several mixtures and calculatedcritical values with a method similar to that of Heide-mann and Khalil [7]. They provide critical loci for arange of mixtures of technical interest.

Michelsen’s book [11] provides an overview of thestate-of-the-art of calculation of critical points, thoughas usual, the analysis is focused on cubic equations ofstate.

In the work of Deiters and Kraska [12], methods arepresented to express the derivatives along the criticalline, which can then be used to trace the critical line ina computationally efficient manner.

1.2 Global phase diagrams

Some of the work described above (e.g., Heidemann andKhalil) emphasized the calculation of the lowest densitycritical point for a given mixture composition, but therecould be additional critical points. Methods that canfind all critical points are useful from the standpoint ofconstructing global phase diagrams.

van Konynenburg and Scott [13] and Scott and vanKonynenburg [14] developed much of the fundamentalliterature on mixture phase type classification. Thisscheme of classification of mixtures from Type I to TypeV describes the critical curves found in phase diagrams.Type I is the simplest case, a continuous critical linebetween the critical points of the pure fluids in a bi-nary mixture. More complex mixture classifications havebeen discovered; see for instance the book of Deiters andKraska [12], and additional works in the field of globalphase diagrams [15, 16].

Hicks and Young [17] propose an algorithm for find-ing the critical points that is similar to the algorithmpresented here, both in form and description. Their anal-ysis is focused on cubic equations of state, but here weextend their analysis to multi-fluid models.

Cismondi and Michelsen [18] consider global phasediagrams for mixtures. They investigate a wide rangeof phase behaviors, and develop an algorithm that canautomatically generate the phase diagrams for mixturesfrom Type I to Type V.

The extension of phase diagrams into three, four, andmulti-phase equilibria (including solid phases) has beencarried out in the work of Patel [19, 20]. This simula-tion work involved the calculation of critical points andcritical lines, as well as complex multi-phase equilibria,through the use of the homotopy continuation method.

2

van Pelt [21] applied the Hicks and Young algorithm[17] to systems modeled through the use of the perturbedhard chain model.

Hoteit et al. [22] propose a new and more reliablealgorithm for the calculation of critical points (from thePeng-Robinson equation of state) that is based on nestedand bounded iterations of Brent’s method. It is com-pared with the method of Stradi et al. [23] and found tobe more than a thousand times faster while they claimsimilar ability to locate critical points.

1.3 Helmholtz-energy-explicit mixturemodels

Cubic equations were the most-common formulationused to model the thermodynamic properties of mix-tures prior to the advent of multi-fluid Helmholtz-energy-explicit mixture models, and even now are stillin wide use. The majority of the literature dealingwith the calculation of critical points described aboveis thus centered around cubic equation of state models,though much of the analysis can be used for multi-fluidHelmholtz-energy-explicit models as well.

Over the last few decades, multi-fluid Helmholtz-energy-explicit mixture models have been developed, thebest known example being the GERG-2004 [24] andGERG-2008 [25] equations of state. These models arebased on a framework that had been previously devel-oped in parallel efforts by researchers in Germany andthe USA [26, 27, 28].

As there is already a large body of scientific literatureon the use of these multi-fluid Helmholtz-energy-explicitmixture models, and the models themselves are quitecomplex, it is incumbent on those who would extendthese models to continue to use the same or function-ally similar nomenclature. As much as possible, we haveendeavored to ensure consistency with the nomenclatureused in the existing literature. Furthermore, it is beyondthe scope of this work to provide a complete descriptionof the decades of work that has been carried out in thisfield. We therefore direct the reader to the relevant liter-ature where possible and provide here a high-level discus-sion of the multi-fluid Helmholtz-energy-explicit mixturemodels.

The specific Helmholtz energy a is a fundamentalthermodynamic potential; other properties of interestcan be obtained through derivatives of the Helmholtz en-ergy. The Helmholtz energy a is expressed as the sum ofthe ideal-gas contribution a0 and the residual contribu-tion ar. In practice the non-dimensionalized Helmholtzenergy is used rather than the specific Helmholtz energy,given by

α =a

RT= α0 + αr. (2)

In the analysis considered here, the ideal-gas contribu-tion α0 is not required, and will not be discussed further.

The non-dimensionalized residual Helmholtz energyαr can be obtained as a combination of the equationsof state of the pure fluids, through the sum of a linearmixing contribution αr

LM and a departure contributionαr

D. Algebraically, this yields

αr(τ, δ,x ) = αrLM(τ, δ,x ) + αr

D(τ, δ,x ). (3)

The term αrLM is a linear mixing contribution and is

given by

αrLM (τ, δ,x ) =

N∑i=1

xiαroi (τ, δ) , (4)

where αroi is the residual part of the non-dimensionalized

Helmholtz energy of pure component i. The contribu-tion αr

D expresses the departure from ideal correspondingstates behavior:

αrD (τ, δ,x ) =

N−1∑i=1

N∑j=i+1

xixjFijαrij (τ, δ) , (5)

with αrij (τ, δ) being the binary specific departure func-

tions and Fij being a binary specific factor.While the departure term of the multi-fluid model is

commonly referred to as an “excess” contribution, thereis a strong case in the literature [29] against referringto it as the excess contribution and instead referring toit as the residual contribution. Unfortunately, here wealready are using the term “residual” to refer to the non-ideal contribution, and, due to its prevalence in the exist-ing literature, we therefore refer to αr

D as the departureterm.

The independent variables of Eq. 3 are the mole frac-tions x , the reciprocal reduced temperature τ , and thereduced density δ. τ and δ are defined according to:

δ =ρ

ρr (x )and τ =

Tr (x )

T. (6)

Tr (x ) and ρr (x ) are reducing functions for the tempera-ture and density, respectively. These functions solely de-pend on the composition and not on temperature or den-sity. A more detailed treatment of multi-fluid Helmholtz-energy-explicit mixture models is given, for example, inthe publications of GERG 2004 [24] and GERG 2008[25].

3

While the multi-fluid Helmholtz-energy-explicit mix-ture model has great flexibility and offers the promiseof high accuracy, equations of state that are also ofhigh-accuracy are only available for a relatively smallnumber of fluids (approximately 125 of the most techni-cally important fluids). Furthermore, the complexity ofthe multi-parameter equations of state often yield non-physical behavior, as is described in Deiters and Kraska[12], and is demonstrated below. Therefore, it is appeal-ing to use simpler, more well-behaved equations of stateto model fluids for which no multi-parameter equation ofstate exists and/or to replace the entire αr contributionfrom Eq. 3.

The work of Bell and Jager [30] can be used to trans-form cubic equations of state explicit in pressure (forinstance the popular Peng and Robinson [31, 32] andSoave-Redlich-Kwong [33] equations of state) into a formthat is explicit in the residual Helmholtz energy αr, withall derivatives with respect to τ , δ, and composition givenby analytic derivatives. This transformation can takeone of two forms:

• A one-fluid transformation: the transformed cubicequation of state replaces entirely the multi-fluidαr from Eq. 3. The binary interaction parameterskij are taken from tabulated values. The high-accuracy equations of state are not employed.

• One (or more) of the fluids in the multi-fluid modelis replaced by a cubic equation of state transforma-tion. The kij interaction parameters for the equa-tion of state are not considered, and the binaryinteraction parameters are those of the multi-fluidmodel.

2 Critical point derivatives

As described by Reid and Beegle[5] and Akasaka[34], thecriticality conditions defining the location of the criti-cal point can be expressed through the use of Legendretransforms. The advantage of the Legendre transform isthat it allows the user to straightforwardly express thecritical conditions as determinants of matrices. Thereare several equivalent sets of independent variables in thetransformation, and as demonstrated by Akasaka[34],the use of the amount of substance (number of moles)of the components (as opposed to other more readilyhandled independent variables) as independent variablesyields a more well-scaled set of matrices. This in turnallows for a more reliable solution method.

This section expresses the criticality conditions in aform that can be readily implemented into libraries basedon a Helmholtz-energy-explicit mixture modeling frame-work, notably, NIST REFPROP [35], CoolProp [36], andTREND [37]. The analysis in this section is based on theanalysis presented in the GERG model [24, 25] and itsrecent extensions to carbon capture and sequestrationapplications [38, 39].

2.1 First and second criticality condi-tions

In the Legendre-transformed form, the first criticalitycondition that must be fulfilled at the critical point isL1 = 0, where L1 can be given by

L1 = detL =

∣∣∣∣∣∣∣∣∣L11 . . . L1N

L21 . . . L2N

.... . .

...LN1 . . . LNN

∣∣∣∣∣∣∣∣∣ = 0 (7)

where the entries in the L matrix are given by

Lij =1

RT

(∂2A

∂ni∂nj

)T,V

=

(∂ ln fi∂nj

)T,V,ni

, (8)

where fi is the fugacity of the i-th component, as de-scribed by Heidemann and Khalil [7]. Here we use themodified matrix as proposed by Michelsen [11]; that is,we divide the entries in the matrix by the product RT ,which is a valid operation because multiplying (or divid-ing) a row by a non-zero constant does not impact theequality of the determinant with zero.

The second criticality conditionM1 = 0 can be givenby

M1 = detM =

∣∣∣∣∣∣∣∣∣∣∣∣

L11 . . . L1N

L21 . . . L2N

.... . .

...L(N−1)1 . . . L(N−1)N∂L1

∂n1. . .

∂L1

∂nN

∣∣∣∣∣∣∣∣∣∣∣∣= 0 (9)

In order to evaluate the determinants in Eqs. 7 and 9,the entries in the matrices must be scaled by the totalnumber of moles, which allows for evaluation of the con-tributions from Helmholtz-energy-explicit models. Theequality of the critical determinants L1 and M1 withzero are not impacted if we multiply a row by a con-stant, in this case powers of the total number of molesn. Thus we can write

L∗1 = detL∗ =

∣∣∣∣∣∣∣∣∣nL11 . . . nL1N

nL21 . . . nL2N

.... . .

...nLN1 . . . nLNN

∣∣∣∣∣∣∣∣∣ = 0 (10)

4

and

M∗1 = detM∗ =

∣∣∣∣∣∣∣∣∣∣∣∣

nL11 . . . nL1N

nL21 . . . nL2N

.... . .

...nL(N−1)1 . . . nL(N−1)N

n3 ∂L1

∂n1. . . n3 ∂L1

∂nN

∣∣∣∣∣∣∣∣∣∣∣∣= 0

(11)where the entries in L∗

1 are now given by

nLij = n

(∂ ln fi∂nj

)T,V,ni

, (12)

and where the analytic form of Eq. 12 can be directlyevaluated from the mixture model and is given in thesupplemental material. The L∗ matrix is symmetric be-

cause

(∂ ln fi∂nj

)T,V

=

(∂ ln fj∂ni

)T,V

, which allows for

efficient calculation of the entries in the matrix by calcu-lating the upper triangular part of the matrix and mir-roring the data into the lower part of the matrix.

In order to obtain the derivatives of the determinantof the unscaled matrix L with respect to ni that are re-quired to construct the matrix M, Jacobi’s formula (seefor instance Magnus and Neudecker [40] [section 8.3]) forthe derivative of the determinant of a matrix is applied.Jacobi’s formula for a matrix U whose entries are eachexclusively a function of the variable t is given by

d

dtdetU(t) = tr

(adj(U(t))

dU(t)

dt

). (13)

The function tr() is the trace of its argument, and adj()is the adjugate matrix (the transpose of the cofactor ma-trix) of its argument. Thus the partial derivatives in thematrix M are each of the form like(

∂L1

∂ni

)T,V,nj

= tr

(adj(L)

∂L

∂ni

). (14)

Unfortunately, the L matrix cannot be directly con-structed, and thus we must re-introduce the powers ofthe amount of substance (number of moles) into Eq. 14in order to express the equation in terms of L∗. Dis-tributing n3 into the equation yields

n3

(∂L1

∂ni

)T,V,nj

= tr

([n · adj(L)] · n2 ∂L

∂ni

)(15)

which yields

n3

(∂L1

∂ni

)T,V,nj

= tr

(adj(L∗) · n2 ∂L

∂ni

)(16)

because n · adj(L) = adj(L∗) resulting from L∗ = nL. Ingeneral, k · adj(A) = adj(k · A) where k is a constant.The derivative term arising from the unscaled matrix Lis given by

n2 ∂L

∂ni=

n2L11i . . . n2L1Ni

n2L21i . . . n2L2Ni

.... . .

...n2LN1i . . . n2LNNi

(17)

and a term of the form n2Lijk is given by

n2Lijk = n2

(∂Lij∂nk

)= n2 ∂

∂nk

((∂ ln fi∂nj

)T,V

)T,V

.

(18)The analytic form of Eq. 18 is given in the supplementalmaterial. It can be directly evaluated from the mixturemodel.

2.2 τ and δ derivatives of L∗1

For use in later analysis, the derivatives of L∗ with re-spect to each of τ and δ are needed. For instance, thefirst and second partial derivatives of the L∗ matrix withrespect to τ are given by

(∂L∗

∂τ

)δ,x

=

nL11τ . . . nL1Nτ

nL21τ . . . nL2Nτ

.... . .

...nLN1τ . . . nLNNτ

(19)

and

(∂2L∗

∂τ2

)δ,x

=

nL11ττ . . . nL1Nττ

nL21ττ . . . nL2Nττ

.... . .

...nLN1ττ . . . nLNNττ

(20)

where an entry in the first τ derivative matrix is of theform

nLijτ =∂

∂τ

(n∂ ln fi∂nj

)δ,x

(21)

and an entry in the second τ derivative matrix is of theform

nLijττ =∂2

∂τ2

(n∂ ln fi∂nj

)δ,x

. (22)

The analytic forms of these derivatives are given in thesupplemental material.

Similarly, for the partial derivatives with respect toδ, entries in the δ derivative matrices can be given in theform

nLijδ =∂

∂δ

(n∂ ln fi∂nj

)τ,x

(23)

nLijδδ =∂2

∂δ2

(n∂ ln fi∂nj

)τ,x

, (24)

where, again, the form of these derivatives can be foundin the supplemental material.

5

3 Algorithm for finding criticalpoints

For a given mixture composition, the first and secondcriticality conditions L∗

1 = 0 andM∗1 = 0 can each be ex-

pressed as functions of two state variables. In this case,the most readily understandable set of state variableswould be temperature and volume (or density), thoughthe mixture model is based on the use of τ and δ as in-dependent variables. To limit the amount of additionalwork required, τ and δ are selected as the independentvariables to be determined, and a post-processing stepcan be used to obtain the temperature and density.

The goal of the critical point solver is to iterate onτ and δ to find the pair of values that enforce L∗

1 =M∗

1 = 0. To that end, a classical Newton-Raphson it-eration scheme is employed, leveraging the fact that theJacobian matrix can be constructed analytically.

The challenge of the application of the Newton-Raphson method is that quite good estimates of the criti-cal point are required, as noted by several authors in theliterature [7, 11, 12]. Additionally, the mixture modelcan predict numerous values where the criticality condi-tions L∗

1 = M∗1 = 0 are met, some of which correspond

to stable critical points, others which correspond to un-stable critical points, and yet others that are exclusivelynumerical artifacts and have no physical significance.

In this section, a description of the Newton-Raphsonsolver is presented (when a reasonably good guess valueis already known for the critical point). Additionally, amethod is presented that can find all the critical pointsfor a mixture by first finding a point along the L∗

1 = 0contour (the spinodal) at a low density and then tracingthis contour to very high density, thereby finding all therelevant critical points along the L∗

1 = 0 contour.

3.1 Critical point solver

The Newton-Raphson system for the two-dimensionalsolver in τ and δ for the criticality conditions is givenby

J∆X = −[L∗

1(τ, δ)M∗

1(τ, δ)

](25)

where X is given by X = [ τ δ ]T and the step ∆Xis given by ∆X = [ ∆τ ∆δ ]T . The Jacobian is givenby

J =

(∂L∗

1

∂τ

)δ

(∂L∗

1

∂δ

)τ(

∂M∗1

∂τ

)δ

(∂M∗

1

∂δ

)τ

. (26)

The Newton-Raphson algorithm operates by solving Eq.25 for ∆X , and updating X with

X i+1 = X i + ∆X . (27)

Accurate initial values for X 0 are required to yield agood starting point for the method.

Jacobi’s formula (see Eq. 13) is again applied in orderto evaluate the partial derivative terms in the Jacobianmatrix. The derivative terms in Eq. 26 are expanded interms of the matrices L∗ and M∗ (upon which the de-terminants are based). Thus the Jacobian can be givenby

J =

tr

(adj(L∗)

∂L∗

∂τ

)tr

(adj(L∗)

∂L∗

∂δ

)tr

(adj(M∗)

∂M∗

∂τ

)tr

(adj(M∗)

∂M∗

∂δ

) .(28)

The derivatives of entries in the L∗ matrix are givenby equations 21 and 23; the analytic forms of these equa-tions can be found in the supplemental material.

For M∗, the first N-1 rows of the ∂M∗/∂τ matrixare exactly the same as those from the ∂L∗/∂τ matrix.The last row in the ∂M∗/∂τ matrix involves terms ofthe form

∂

∂τ(M∗

Ni) =∂

∂τ

[tr

(adj(L∗) ·

(n2 ∂L

∂ni

))]. (29)

Trace and derivative operations are commutative, so thisderivative can be expressed (after applying the productrule for matrices) as

∂

∂τ(M∗

Ni) = tr

(n2 ∂L

∂ni

)· ∂∂τ

[adj(L∗)]

+adj(L∗)∂

∂τ

[n2 ∂L

∂ni

] . (30)

The derivative term arising from the n2 ·∂L/∂ni term isgiven by

∂

∂τ

(n2 ∂L

∂ni

)=

n2L11iτ . . . n2L1Niτ

n2L21iτ . . . n2L2Niτ

.... . .

...n2LN1iτ . . . n2LNNiτ

, (31)

which involves terms of the form

n2Lijkτ =∂

∂τ

(n2Lijk

), (32)

with n2Lijk from Eq. 18. The derivative in Eq. 32 canbe found in the supplemental material. A similar set ofderivatives is obtained for the partial derivatives takenwith respect to δ, whose forms are also in the supple-mental material.

The final derivative term ∂ [adj(L∗)] /∂τ requires fur-ther discussion. The adjugate of a matrix A (with en-tries that are exclusively a function of t) is given by theoperator AA = adj(A), and has elements given by

AAij = (−1)i+jMji, (33)

6

where Mji is the ji minor of the matrix A.The minor Mji of a matrix A is obtained by taking

the determinant of the sub-matrix A/ji formed by remov-

ing row j and column i from A. This can be expressedas

Mji = det(A/ji). (34)

Thus, in general, the derivative of an entry in the adju-gate matrix is given by

d

dt

(AAij

)=

d

dt

[(−1)i+jdet(A

/ji)], (35)

which can be expanded through the use of Jacobi’s for-mula one final time to read

d

dt

(AAij

)= (−1)i+jtr

(adj(A

/ji)

dA/ji

dt

). (36)

The reader should be advised that the adjugate of a 1x1matrix is equal to 1.

3.2 Critical point finder

While some authors have proposed methods to robustlyfind all the critical points [23, 17, 12], a subset of thatanalysis will be developed here. In practice, the criticalpoints that are of interest are those that correspond topositive pressures, are stable, and lie along the highest-temperature branch of the L∗

1 = 0 contour.As an introduction, we present the criticality con-

tours from a well-behaved mixture model in order to ex-plain some of the more important features. Figure 1demonstrates the critical contours for an equimolar n-hexane+n-heptane mixture as calculated by the multi-fluid model, with the mixture parameters and pure fluidparameters as given in Table 2. A few other curves arealso plotted in order to provide context for the criticalcontours. The phase envelope is shown in order to con-firm that the calculated critical point is co-incident withthe phase envelope. Furthermore, the “ghost” criticalpoints (numerical artifacts described further in section6.1) are entirely enclosed within the phase envelope (to-wards higher τ , or lower temperature). Critical pointsmust lie along a phase envelope because the phase en-velope is the locus of points that satisfies vapor-liquidequilibrium (equality of component fugacities, materialbalance, etc.).

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

τ= Tred/T

0.5

1.0

1.5

2.0

2.5

3.0

δ=ρ/ρ

red ph

ase

enve

lope

( L∗1

τ

)δ=

0

( L ∗1

τ

)δ > 0

Figure 1: Criticality contours (solid: L∗1 = 0, dashed:

M∗1 = 0) of an equimolar n-hexane+n-heptane mixture

from the multi-fluid model, with the model parametersgiven by Table 2. The critical point is identified with amarker.

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

τ= Tred/T

0.5

1.0

1.5

2.0

2.5

3.0

δ=ρ/ρ

red

Figure 2: Criticality contours (solid: L∗1 = 0, dashed:

M∗1 = 0) of an equimolar nitrogen + methane mixture

(red: multi-fluid model, black: SRK transformations ofpure fluids in multi-fluid model), with the model param-eters from Table 2.

7

As a counter-example of a poorly behaved mixturemodel, we overlay the criticality contours for an equimo-lar mixture of nitrogen and methane in Figure 2 as cal-culated with SRK transformations of the pure fluids in amulti-fluid model, and the multi-fluid model itself. Themixture parameters and pure fluid parameters are asgiven in Table 2. This mixture demonstrates challeng-ing behavior; when the multi-fluid model is employed,the L∗

1 = 0 contour is not smooth, which causes signifi-cant challenges for the critical point finding routine. Onthe other hand, when SRK is used, either as a one-fluidmodel or in pure-fluid transformations in the multi-fluidmodel, the critical contours are much more smooth. Fig-ure 2 demonstrates that criticality contour plots serve asa robust check of the physically-reasonable behavior ofthe pure-fluid equations of state when employed in multi-fluid mixture models.

3.2.1 Maximum temperature contour

To begin the critical point finding routine along the max-imum temperature L∗

1 = 0 contour, the value of τ corre-sponding to L∗

1 = 0 is found at a low density value. Thisvalue of density should be selected such that all criti-cal points (if they exist), are above this density value. Inthis work, we have found that selecting a density value of0.5 · ρred(x ) was a good choice for a low-density startingvalue. A first guess for the temperature value is selectedthat should be well above the temperature correspond-ing to the highest temperature L∗

1 = 0 contour, wherewe found that 1.5 · Tred(x ) was a good choice. In thecase that the derivative ∂(L∗

1)/∂τ is positive, the start-ing value of τ is increased so that the proper solution isfound. Figure 1 shows the contour where the derivative∂(L∗

1)/∂τ is zero, and points towards higher values of τare valid starting points. In practice, the user must becareful to select a temperature value that is not too high(τ value that is not too low), otherwise the solver mighttend to overshoot and find a non-physical critical pointinside the spinodal. One means of more reliably find-ing the lowest τ value (highest temperature) satisfyingL∗

1 = 0 for a low density value is to use a higher-orderderivative-based iterative solver. It was found by expe-rience that a first-order derivative-based iterative solver(Newton’s method) required a reasonably accurate guessvalue for the temperature, while the use of a second-order derivative-based iterative solver (Halley’s method)was far less sensitive to the initial guess for τ , allow-ing a higher temperature guess value while also avoidingthe problems of overshoot and the selection of a lower-temperature root.

In Halley’s method, the first and second derivativesof the residual function with respect to the independentvariable are required. The development of the deriva-tives required for the Jacobian above describes the firstpartial derivative of L∗

1 with respect to τ (Eqs. 26 and28), and the second partial derivative of L∗

1 with respectto τ can be given by

(∂2L∗

1

∂τ2

)δ

= tr

(∂L∗

∂τ

)δ,x

∂

∂τ[adj(L∗)]

+adj(L∗)

(∂2L∗

∂τ2

)δ,x

(37)

The next guess for τ can then be found by the itera-tive updating step

τnew = τold −2L∗

1

(∂L∗

1

∂τ

)δ

2

[(∂L∗

1

∂τ

)δ

]2

− L∗1

(∂2L∗

1

∂τ2

)δ

(38)

Eq. 38 is repetitively applied until the value of |L∗1| is

sufficiently close to zero. This therefore defines the firstpoint along the L∗

1 = 0 contour.

3.2.2 L∗1 = 0 contour tracer

After the initial point along the L∗1 = 0 contour has

been found, the solver switches into a new mode whereit traces along the L∗

1 = 0 contour. Along this con-tour, if critical points exist, they are points that satisfyL∗

1 = M∗1 = 0. In order to trace along the contour,

an angle variable θ is introduced that yields a vectorpointing along the contour (in the direction of increas-ing density). At the current value of (τk, δk), the newlypredicted point in τ -δ space is defined by

τk+1 = τk +Rτ cos(θ) (39)

δk+1 = δk +Rδ sin(θ) (40)

The value for θ that yields L∗1 = 0 is iteratively cal-

culated. In the first step, θ is searched in the domain(0, π) through the use of a bounded solver - here Brent’smethod. Once the first value for θ is obtained, the nextstep begins with the guess value that θ equals its pre-vious value. The L∗

1 = 0 contour is in general smooth(see Fig. 1, or as a counter-example, Fig. 2), thereforethe values of θ also change slowly. After the first step,derivative-based methods can be used to obtain θ. Inparticular, Newton’s method is used to iteratively up-date θ; the derivative of L∗

1 with respect to the angle θis required, which can be given by

∂L∗1

∂θ=

(∂L∗

1

∂τ

)δ

dτ

dθ+

(∂L∗

1

∂δ

)τ

dδ

dθ(41)

8

wheredτ

dθ= −Rτ sin θ (42)

anddδ

dθ= Rδ cos θ, (43)

and where θ can be updated by repetitive applicationsof

θnew = θold −L∗

1[∂L∗

1

∂θ

] . (44)

In practice, since the curvature of the L∗1 = 0 contour

is quite gentle for well-behaved mixture models, only oneor two updates of the value of θ are needed, making for acomputationally efficient method of tracing the contour.

In spite of the generally reliable and robust behav-ior of this tracing methodology, there are times whenthe contour tracer would prefer to change direction andtrace back towards lower δ values. In order to avoid this,the obtained value of θ and the previous value of δ arecompared. If their values differ by more than π/2 radi-ans 1, the tracer has made a sharp change in direction.A bounded solver is then used (with a domain centeredaround the previous value of θ) in order to ensure thatthe tracer continues in the same direction.

The values of the radii Rτ and Rδ were selectedby manual iteration to be a compromise between speed(fewer steps are required for larger radii) and reliability(less likely to hit a “ghost” contour for smaller radii).For the multi-fluid model, the values of Rτ = 0.1 andRδ = 0.025 were ultimately selected. Due to the gener-ally more well behaved shapes of the criticality contourswhen using one-fluid SRK model is used (as well as whenSRK transformations in the multi-fluid model are used),the search radii could be much larger. For the SRK mod-els, multiplying the radii for the multi-fluid model by afactor of approximately 5 (after also applying the scalingof section 4.1), proved to be a fair compromise of speedand reliability.

In the process of tracing the L∗1 = 0 contour, the

value ofM∗1 is calculated at each step. If the sign ofM∗

1

changes between the values of (τk, δk) and (τk+1, δk+1),these values for τ and δ must necessarily bound a criti-cal point. A call is then made to the Newton-Raphsoncritical point calculation routine from section 3.1, whichthen efficiently refines the solution for the critical pointbecause the guess value for the critical temperature anddensity is nearly co-incident with the true critical point.

Stopping condition The tracer follows the spinodalfrom lower densities towards higher densities, locatingall the critical points that it can, until it reaches thestopping condition, detailed in this section.

For the one-fluid SRK mixture model, there is a min-imum specific volume that is physically allowed - the co-volume bm. The one-fluid SRK models begin to showsome undesirable behavior in approaching the covolume,and for the one-fluid models, the stopping condition isthat the volume must be greater than 1.1bm. Stradi et al.[23] used the same bound of 1.1bm in their work. If thiscondition is not fulfilled, the tracer terminates. Theremay be some critical points between 1.1bm and bm, butthey are mostly critical points that are of less interest,at negative pressures, unstable, or all of the above.

For the multi-fluid models, the stopping condition issomewhat more difficult to define. While the one-fluidSRK model has a physical bound on the allowed volume,the multi-fluid model is entirely empirical in nature, andless constrained by physical behavior. As such, the stop-ping conditions for the multi-fluid model (any of whichterminate the tracer) are:

• The pressure exceeds 500 MPa.

• The reduced density δ exceeds 5.

• The reciprocal reduced temperature τ exceeds 5.

• The reduced density δ exceeds 1.5 and the tracerhas a sharp change in its direction (other than a U-turn). This can be caused by problems in the pure-fluid equation of state at low temperature (high τ).As an example, see Fig. 2.

3.3 Stability of critical point

It is necessary to implement a full tangent plane distanceanalysis to evaluate whether a critical point is stableor not. The tangent plane distance analysis is a nec-essary and sufficient condition that can guarantee thatthe critical point is stable[11]. Unfortunately, this guar-antee comes at a price. In principle, the tangent planeanalysis requires a global search over the entire test com-position domain, but we simplify the stability search byevaluating two trial compositions for heavy and light testphases.

1The true angle difference function should be used in order to properly handle the fact that θ is 2π periodic. The difference betweenthe angles of 0.01 radians and 2π − 0.01 radians should be 0.02 radians, not 2π − 0.02 radians.

9

In this work we applied the algorithm of Gernert etal. [38] as applied to the multi-fluid Helmholtz-energy-explicit mixture model. The algorithm is based on thework of Michelsen [41, 42] and was developed with anemphasis on multi-fluid Helmholtz models. These mod-els pose significantly more challenging numerical behav-ior than less complex equations of state (like cubic equa-tions of state), which are often used for critical point andphase equilibrium calculations. The applicability and re-liability of this method will be discussed in section 5.

4 Implementation

A C++ implementation of the algorithm for identifyingcritical points and evaluating their stability is providedin the supplemental material; information is availablein the supplemental material about how to compile theC++ sources. Following publication of this work, it isintended to implement this algorithm in the standard ref-erence NIST REFPROP library, where the critical pointcalculation routines will be used in order to more effi-ciently construct phase envelopes for mixtures of fixedcomposition.

4.1 Considerations for one-fluid cubicequations of state

In the generalized transformation of pressure-explicit cu-bic equations of state to Helmholtz energy equations ofstate of Bell and Jager [30], the reducing parameters Tred

and ρred are positive numerical constants (not a func-tion of composition). Unlike the multi-fluid model, inthe one-fluid model there is no link between the criticaldensities and/or temperatures of the pure componentsand the reducing parameters. In the case that a calcu-lation begins with a pair of T, ρ, a round-trip evaluationof T, ρ→ τ, δ → T, ρ poses no numerical problems as the(constant) reducing parameters cancel. On the otherhand, when a calculation begins with a given value of(τ, δ), the differences between the multi-fluid and one-fluid models become evident.

As an example, when determining the first pointalong L∗

1 = 0 at low δ, as described above we begin ourone-dimensional search at a value of τ of approximately2/3 and a value of δ of approximately 1/2. These valueshave a physical significance in the multi-fluid model. Forthe one-fluid model, due to the arbitrary nature of thereducing parameters, it is necessary to rescale τ and δto put them on a more physical footing. In order to dothis, we assume a linear mixing of the critical volumeand critical temperature in the one-fluid model, and usethese linear mixing pseudo-reducing parameters in orderto yield similar values of T and ρ. Mathematically, thisis expressed as:

δOF = δMF1

ρr,OF

(∑i

xivc,i

) (45)

τOF = τMFTr,OF(∑i

xiTc,i

) (46)

where the subscript OF pertains to the one-fluid model,and MF to the multi-fluid model. It is generally the casethat the critical volume vc,i is unknown for cubic equa-tions of state. Therefore, vc,i must be estimated, eitherby applying the volume translation method of Peneloux[43] to critical volumes estimated based on the criticalcompressibility factor of the equation of state (a con-stant), or with the use of an approximate formulation,such as

vc,i ≈ 2.141

(Tc,i

pc,i

)+ 7.773× 10−6 (47)

where vc,i is in m3·mol−1 and Tc,i/pc,i is in K·Pa−1. Thisequation was obtained by fitting the critical parametersfor all the pure fluids with multi-parameter equations ofstate. This equation predicts 94% of the critical volumeswithin 10%, which is adequate for purposes of scaling thevalues for δ.

10

5 Validation of the implementa-tion

In order to demonstrate that our implementations of thecritical point routines as well as the stability calcula-tions in the thermophysical property libraries CoolPropand TREND are correct, a comparison with the workof Stradi et al. [23] is given in Table 1. In this ta-ble, calculated critical points for different compositionsof the binary mixture methane (CH4) + hydrogen sulfide(H2S) are given. The critical points indicated “Stradi etal. - SRK ” are results of Stradi et al. [23], which weretaken from their article and reprinted for comparison.The results indicated “This work - SRK ” are the resultsfor critical points in the binary mixture CH4 + H2S cal-culated with the implementation of the SRK in TREND[37].

In order to get comparable results, values of constantsof the SRK equation were adjusted to match the valuesStradi et al. [23] used. Unfortunately, Stradi et al. didnot provide the value that they used for the gas con-stant R. Hence, we assumed that they used a value ofR = 8.31451 J·mol−1·K−1, which was the standard valueat the time when they did their study. As in Stradiet al. [23], the values for the critical parameters andacentric factors of methane and hydrogen sulfide havebeen taken from Reid et al. [44]; they are: Tc,CH4

=190.4 K, pc,CH4 = 4.6 MPa, ωCH4 = 0.011, Tc,H2S =373.2 K, pc,H2S = 8.94 MPa, andωH2S = 0.097.

Table 1 shows that the calculated critical points andtheir stability are in very good agreement with the re-sults of Stradi et al. If at certain overall compositions novalues for the critical temperature, density, or pressureis given, then according to the equation of state no criti-cal points exist for this overall composition, or they werenot found. With the algorithm presented in this paper,some critical points at negative pressures for methane-rich mixtures have been found, which were not given byStradi et al. The algorithm that was used for testingthe stability [38] is TP -based and thus the stability ofpoints at negative pressures could not be tested, whichis indicated with “NA” in Table 1. The entries “-” inTable 1 indicate critical points that were not able to becalculated with the respective model. Furthermore, thepossible formation of solid phases was not tested either,since no equation of state for the solid phases of CH4

and H2S was employed in this study.

The critical points in Table 1 indicated “This work- Helmholtz” are the predicted critical points fromthe multi-fluid Helmholtz energy equation of state formethane [45] in combination with the multiparameterHelmholtz energy equation of state for hydrogen sul-fide [46] and mixing rules from GERG-2008 [25] (βT =1.011090031, γT = 0.961155729, βv = 1.012599087, andγv = 1.040161207, with no departure function). Cal-culated critical temperatures, molar volumes, and pres-sures of the mixtures differ to some extent from the SRKresults but are in good agreement, especially the tem-peratures and pressures, and less-so the volumes. How-ever, some of the critical points that the one-fluid SRKmodel predicts to be unstable are stable according to themulti-fluid model. Furthermore, the multi-fluid modelpredicts the existence of critical points that are not pre-dicted by the one-fluid model. This behavior, however, isnot unique to the multi-fluid Helmholtz-energy-explicitmodels but it was found that the existence and alsothe stability of critical points is very sensitive to themodel parameters for the binary mixture CH4 + H2S.For example, when changing the critical parameters andacentric factors of methane and hydrogen sulfide in theSRK to the values given by Poling et al. [47] (Tc,CH4

=190.56 K, pc,CH4

= 4.599 MPa, ωCH4= 0.011, Tc,H2S =

373.4 K, pc,H2S = 8.963 andMPa, ωH2S = 0.09) addi-tional critical points at xCH4 = 0.84 can be found. Fur-thermore, the binary mixture of CH4 + H2S was not theprimary focus of the development of the GERG-2008equation of state, because these models were primarilydesigned to describe typical natural gas mixtures. Thismixture was chosen in this work for validation of theproposed algorithms; more detailed results for will bediscussed in the results section of this paper.

11

Tab

le1:

Res

ult

sfo

rth

ecr

itic

alp

oint

calc

ula

tion

sw

ith

the

pro

pose

dalg

ori

thm

for

the

bin

ary

mix

ture

CH

4+

H2S

,ca

lcu

late

dw

ith

the

on

e-fl

uid

SR

Km

od

elw

ithkij

=0.

08,

and

wit

hth

em

ult

i-fl

uid

Hel

mh

olt

z-en

ergy-e

xp

lici

tm

ixtu

rem

od

el.

Th

ere

sult

sare

com

pare

dto

the

resu

lts

of

Str

ad

iet

al.

[23]

Str

adi

etal.

-S

RK

Th

isw

ork

-S

RK

Th

isw

ork

-H

elm

holt

zM

ole

frac

.x

CH

4v c

Tc

pc

Sta

bil

ity

v cT

cp

cS

tab

ilit

yv c

Tc

pc

Sta

bil

ity

cm3·m

ol−

1K

MP

acm

3·m

ol−

1K

MP

acm

3·m

ol−

1K

MP

a0.

998

114.

2619

0.82

4.63

Sta

ble

114.2

6190.8

24.6

3S

tab

le97.7

8191.1

54.6

5S

tab

le0.

998

--

--

--

--

35.0

941.2

1-1

30.4

8N

A0.

9710

7.70

196.

745.

04S

tab

le107.7

0196.7

45.0

4S

tab

le93.3

2197.6

65.1

9S

tab

le0.

97-

--

-38.6

463.7

4-7

2.3

6N

A35.4

646.5

7-1

13.1

8N

A0.

9475

102.

1820

1.37

5.39

Sta

ble

102.1

8201.3

75.3

9S

tab

le89.5

0202.4

85.5

9S

tab

le0.

9475

--

--

42.1

594.4

7-4

2.5

2N

A38.4

275.5

2-6

7.9

1N

A0.

9410

0.30

202.

865.

51U

nst

ab

le100.3

0202.8

65.5

1U

nst

ab

le88.3

8204.0

75.7

3S

tab

le0.

94-

--

-43.2

6103.4

7-3

5.8

1N

A39.3

284.5

8-5

8.4

1N

A0.

9397

.75

204.

785.

67U

nst

able

97.7

5204.7

85.6

7U

nst

able

86.9

6206.1

85.9

2S

tab

le0.

9344

.72

114.

77-2

8.36

NA

44.7

2114.7

7-2

8.3

6N

A40.4

795.9

8-4

7.9

3N

A0.

8677

.95

213.

716.

55U

nst

able

77.9

5213.7

16.5

5U

nst

able

76.2

7219.0

87.2

0U

nst

ab

le0.

8656

.59

181.

31-1

.70

NA

56.5

9181.3

1-0

.17

NA

48.2

5160.8

7-9

.11

NA

0.85

74.0

321

2.99

6.45

Unst

able

74.0

3212.9

96.4

5U

nst

able

74.5

9220.4

77.3

6U

nst

ab

le0.

8559

.41

190.

982.

25U

nst

able

59.4

1190.9

82.2

5U

nst

able

49.4

4168.7

9-6

.12

NA

0.84

--

--

--

--

72.8

9221.7

07.5

0U

nst

ab

le0.

84-

--

--

--

-50.6

9176.5

3-3

.46

NA

0.75

--

--

--

--

--

--

0.53

--

--

--

--

--

--

0.52

59.2

627

0.02

14.6

1S

tab

le59.2

6270.0

214.6

1S

tab

le62.4

2283.0

114.6

7S

tab

le0.

5254

.94

260.

2714

.90

Sta

ble

54.9

4260.2

714.9

0S

tab

le54.9

6266.0

614.5

2S

tab

le0.

5163

.37

279.

2514

.50

Sta

ble

63.3

7279.2

514.5

0S

tab

le64.4

3288.2

614.7

4S

tab

le0.

5150

.31

249.

0116

.01

Sta

ble

50.3

1249.0

116.0

1S

tab

le52.6

3260.9

214.7

6S

tab

le0.

4967

.59

288.

8614

.40

Sta

ble

67.5

9288.8

614.4

0S

tab

le67.1

8295.8

714.7

9S

tab

le0.

4944

.26

231.

6723

.04

Sta

ble

44.2

7231.6

723.0

4S

tab

le49.0

5252.3

416.1

8S

tab

le0.

4934

.89

208.

1518

0.02

Sta

ble

34.8

9208.1

5180.0

2S

tab

le-

--

-0.

3683

.21

323.

0713

.26

Sta

ble

83.2

2323.0

713.2

6S

tab

le78.5

7327.4

614.0

0S

tab

le0.

2494

.58

343.

8611

.84

Sta

ble

94.5

9343.8

611.8

4S

tab

le86.7

6347.2

312.5

8S

tab

le0.

229

95.5

834

5.50

11.7

1S

tab

le95.5

9345.5

011.7

1S

tab

le87.4

3348.7

712.4

3S

tab

le0.

0910

7.91

363.

5810

.00

Sta

ble

107.9

1363.5

810.0

0S

tab

le94.8

0365.1

310.4

4S

tab

le

12

6 Illustrative results

There is a practically infinite set of binary mixtures thatcan be constructed from the fluids for which Tc, pc, andthe acentric factor ω are known. These binary mixturescan result in wildly varying phase equilibrium behav-ior. In this section we focus on a few families of flu-ids that have been studied by other authors in order tovalidate the methodology, to investigate some details ofthe algorithm, and further highlight some of the numeri-cal challenges inherent in calculating critical points withmulti-fluid models.

Table 2 provides the references for the pure fluidsin the multi-fluid model, as well as the pure-componentconstants used in the cubic equations of state.

6.1 Methane + hydrogen sulfide

In this section, the binary mixture of methane and hy-drogen sulfide is studied. This binary mixture is oftenselected for discussions of critical lines, phase diagrams,and related phenomena as it can yield zero, one, two, oreven three critical points depending on the molar com-position and the equation of state used.

Figure 3 shows the critical points calculated withboth the multi-fluid model and the SRK model. Thevalue of kij = 0.08 for the SRK model was selected forconsistency with the work of Stradi et al. [23], who in-vestigated the same system with the same equation ofstate (see section 5). With the selected value of kij forthe SRK model, both models yield qualitatively simi-lar results, demonstrating a single critical point up to amethane mole fraction of approximately 0.4, followed bytwo critical points up to a mole fraction of approximately0.52, no critical points until a methane mole fraction ofapproximately 0.8, two critical points around a methanemole fraction of 0.8, and finally one critical point up tonearly pure methane.

Consideration of the criticality contours can providesome additional insight into the behavior of the criticalpoint solving algorithm and some potential challenges.Figure 4 shows the criticality contours for three differentmolar compositions, demonstrating the range of behav-iors that are possible. In this figure some undesirablebehavior of the mixture model can be seen, caused bynumerical artifacts of the pure-fluid equations of state.Each of the physical critical points along the L∗

1 = 0 con-tour are highlighted with a marker in the figure, whilethere are a number of spurious “ghost” critical pointsthat fulfill the mathematical conditions for the existenceof a critical point. These “ghost” critical points are all

inside the L∗1 = 0 spinodal, and are by definition non-

physical and can be neglected.

In spite of their non-physical nature, the “ghost” crit-ical points cause challenges for the critical point algo-rithm. These problems are manifested in two primaryways:

1. If the initial guess value for the one-dimensionalsolver for L∗

1 = 0 from section 3.2.1 causes thesolver to overshoot and yield a value for τ inside thespinodal, the presence of values fulfilling L∗

1 = 0 in-side the spinodal can cause the solver to get stuck.Beginning the solver fairly close to the spinodal re-moves this potential trap. For instance, Fig. 4bdemonstrates that for δ = 0.5 and at a value ofτ of approximately 1.32, the criticality conditionL∗

1 = 0 is satisfied, though the state is significantlyinside the spinodal.

2. While tracing the L∗1 = 0 contour as described

in section 3.2.2, it is possible that there may be“ghost” critical points that are very near the spin-odal. While the spinodal itself is quite smooth,and can generally be traversed without major chal-lenges, if the search radius contains both the spin-odal curve and a fragment of contour of L∗

1 = 0, theone-dimensional search for the angle θ may resultin the solver selecting the non-physical fragment asopposed to the desired point along the spinodal.

The difficulties caused by the “ghost” critical pointsare primarily caused by the pure-fluid equations ofstate. Figure 1 demonstrates that well-constructed high-accuracy pure-fluid equations of state can still yield rea-sonable behavior inside the spinodal. As evidence thatthe “ghost” critical points are caused by the pure-fluidequations of state, Fig. 5 presents the same contours asFig. 4 except that each pure-fluid equation of state hasbeen replaced with an SRK Helmholtz energy transfor-mation as described in Bell and Jager [30] (with the pure-fluid critical points from Table 2). In comparing Figs. 4and 5, it can be seen that the cubic equations of statehave eliminated the “ghost” critical points, while yield-ing similar qualitative locations for the critical points.Unfortunately, the use of the SRK equations of state inthe multi-fluid model has eliminated the double criticalpoint at a methane mole fraction of 0.49. The supple-mental material includes a number of additional surfaceplots for this mixture (demonstrating that the majorityof the “ghost” critical points are caused by the methaneequation of state) and contour plots for the one-fluidSRK model.

13

Table 2: Pure-fluid and mixture models and critical values used in this study (EOS: reference for equation of state,BIP: reference for binary interaction parameters in the multi-fluid model)

Fluid 1 Fluid 2 EOS 1 EOS 2 BIP Tc,1(K) Tc,2 (K) pc,1 (MPa) pc,2 (MPa) ω1 ω2

n-Hexane n-Heptane [48] [48] [25] 507.820 540.130 3.034 2.736 0.299 0.349Nitrogen Methane [49] [45] [25] 126.192 190.564 3.396 4.599 0.037 0.011Methane H2S [45] [46] [25] 190.564 373.100 4.599 9.000 0.011 0.101Nitrogen Ethane [49] [50] [25] 126.192 305.322 3.396 4.872 0.037 0.099

CO2 n-Propane [51] [52] [25] 304.128 369.890 7.377 4.251 0.224 0.152CO2 n-Pentane [51] [48] [25] 304.128 469.700 7.377 3.370 0.224 0.251CO2 n-Heptane [51] [48] [25] 304.128 540.130 7.377 2.736 0.224 0.349CO2 n-Decane [51] [46] [25] 304.128 617.700 7.377 2.103 0.224 0.488

0.0 0.2 0.4 0.6 0.8 1.0xCH4

5

10

15

20

25

30

ρc (

mol·d

m−

3)

multi-fluid

one-fluid SRK

150 200 250 300 350 400

Tc (K)

10-1

100

101

102

103

pc (

MP

a)

multi-fluid

one-fluid SRK

Figure 3: Critical densities, temperatures, and pressures calculated for methane + hydrogen sulfide binary mixturesthrough the use of the multi-fluid Helmholtz-energy-explicit equation of state and the one-fluid SRK equation ofstate (crossed-out points are determined to be unstable according to the stability evaluation)

14

(a) Methane[0.3] & H2S[0.7] demonstrating one critical point

(b) Methane[0.49] & H2S[0.51] demonstrating two critical points

(c) Methane[0.65] & H2S[0.35] demonstrating zero critical points

Figure 4: Plots of surfaces for L∗1 andM∗

1 for a selection of methane/hydrogen sulfide mixtures (molar composition).The L∗

1 = 0 (solid) and M∗1 = 0 (dashed) contours are plotted on each surface.

15

(a) Methane[0.3] & H2S[0.7] demonstrating one critical point

(b) Methane[0.49] & H2S[0.51] demonstrating zero critical points

(c) Methane[0.65] & H2S[0.35] demonstrating zero critical points

Figure 5: Plots of surfaces for L∗1 andM∗

1 for a methane + H2S mixture when both fluids in the multi-fluid modelare replaced with SRK transformations. See Fig. 4 for a description of the content.

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6.2 Nitrogen + ethane

Figure 6 presents the critical points for the binary mix-ture of nitrogen and ethane. The value of the one-fluidSRK mixture model interaction parameter kij = 0.0407is taken from Ramırez-Jimenez et al. [53]. This mixtureis of interest because even though the constituent fluidsare simple molecules, and form a relatively symmetricmixture, it is still possible to yield interesting criticalpoint behavior. Over a range of nitrogen mole fractionsbetween approximately 0.6 and 0.7, as many as threestable critical points can be found.

Figure 7 shows the criticality contours for a nitrogen+ ethane mixture at a nitrogen mole fraction of 0.65.There are three crossings of the critical contours alongthe main branch of the L∗

1 = 0 contour, demonstratingthat three critical points are predicted. All three of thecritical points are predicted to be stable, but the criticalpoint finder does not find the highest pressure criticalpoint because it is located at a pressure greater than 500MPa, one of the stopping criteria for the spinodal tracer.

6.3 Carbon dioxide + linear alkanes

Another popular family of binary mixtures of technicalrelevance are the mixtures of carbon dioxide and the lin-ear alkanes. These mixtures tend to be of Type I, with acontinuous critical line between the critical points of thepure fluids, at least for the lower alkanes below a carbonnumber of approximately 10 or 11. For higher alkanes(for instance hexadecane), non-Type I phase behavior isseen for mixtures with carbon dioxide.

0.0 0.2 0.4 0.6 0.8 1.0xCO2

300

350

400

450

500

550

600

650

Tc (

K)

n-Propane

n-Pentane

n-Heptane

n-Decane

Figure 8: Critical temperature of binary carbon dioxide+ n-alkane mixtures as in the work of Kolar [10] (solid:multi-fluid model, dashes: SRK as one-fluid model withkij = 0, markers: experimental measurements from theTDE database[54, 55, 56]), with parameters taken fromTable 2

Figure 8 shows the critical lines calculated with themulti-fluid model and the one-fluid SRK model withkij = 0. For the higher alkanes, the predictions of thecritical temperatures are excellent and the SRK modelcaptures the shape of the critical line with high accu-racy. Conversely, for the higher alkanes the multi-fluidmodel provides predictions of the critical temperaturesthat are significantly too high. This deviating behav-ior is a manifestation of the fact that when the binaryinteraction parameters were fit in the work of Kunz &Wagner [25], the critical points were not considered andtherefore the binary interaction parameters were not con-strained to yield the correct critical lines. Furthermore,the GERG model was not primarily designed for carbondioxide rich mixtures. In general, as the mixture be-comes more asymmetric, the deviations from the multi-fluid model increase.

For propane, the conclusions are exactly oppositethose for the higher alkanes. The highest-accuracy pre-dictions of the critical temperatures can be obtainedfrom the multi-fluid model, and the one-fluid SRK modeldeviates more strongly from the experimental data. Thisis likely just a fortuitous behavior, as the critical pointswere not considered in the fitting of Kunz & Wagner [25].

7 Conclusions

In this work we have combined the methods of Heidemanand Khalil [7] and Hicks and Young [17], and applied

17

0.0 0.2 0.4 0.6 0.8 1.0xN2

0

5

10

15

20

25

30

ρc (

mol·d

m−

3)

multi-fluid

one-fluid SRK

50 100 150 200 250 300 350

Tc (K)

100

101

102

103

pc (

MP

a)

multi-fluid

one-fluid SRK

Figure 6: Critical densities, temperatures, and pressures calculated for nitrogen + ethane binary mixtures throughthe use of the multi-fluid Helmholtz-energy-explicit equation of state and the one-fluid SRK equation of state(crossed-out points are determined to be unstable according to the stability evaluation)

Figure 7: Plots of surfaces for L∗1 and M∗

1 for a nitrogen[0.65] + ethane[0.35] mixture demonstrating three criticalpoints when using the multi-fluid mixture model. See Fig. 4 for a description of the content.

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them to multi-fluid Helmholtz-energy-explicit mixturemodels. This new algorithm allows for the robust deter-mination of the critical points of complex mixtures. It isdemonstrated that unphysical behavior of the pure fluidequations of state in the multi-fluid mixture model is ofgreat importance, and poorly behaving pure-fluid equa-tions of state can cause significant challenges in terms oflocating the critical points.

Supplemental Material

Included as supplementary material are:* Additional analytic derivatives required to constructthe determinants.* Numerical values of the newly derived analytic deriva-tives for validation, accompanied by values calculated bynumerical differentiation for additional verification.* Sample code demonstrating how to obtain the criticalpoints and critical contours.

Acknowledgments

The authors thank Eric Lemmon (of NIST), who pro-vided daily inspiration, G. Venkatarathnam (of IITMadras), who helped with the literature survey, BradleyAlpert (of NIST), who provided invaluable assistancewith the linear algebra analysis, Andrei Kazakov (ofNIST), who suggested the form of the fit for vc,i, andArno Laesecke (of NIST), who brought to our attentionthe discussions in literature of the proper use of the term“excess”.

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