XSEOSThermodynamic Properties using Excess Gibbs

Free Energy Models and Equations of State

Marcelo Castier

Department of Chemical and Petroleum EngineeringUnited Arab Emirates University

P.O. Box 17555 - Al Ain - United Arab Emiratesmcastier@uaeu.ac.aemcastier@gmail.com

February 21, 2010

Abstract

XSEOS is an Excel R© add-in to calculate physical properties using excess Gibbs free energymodels and equations of state. This manual describes the add-in installation and use, lists themodels and properties implemented in the package, and presents the function name conventionadopted.

IntroductionXSEOS - Excess Gibbs Free Energy Models and Equations of State - is a freely available Excel R©

add-in to compute thermodynamic properties using many traditional and modern thermodynamicmodels. XSEOS also includes correlations to characterize oil fractions. These properties areprogrammed as functions in Visual Basic for Applications (VBA), totaling more than 22000lines of code. Users can call these functions from their Excel R© spreadsheets.

XSEOS is primarily intended for use in undergraduate courses, but may also be useful ingraduate courses and research. The underlying concept in XSEOS is that students should haveeasy access to calculations using modern thermodynamic models, to become aware of their exis-tence and learn how to use them. The calculation experience, however, should not be limited togetting results from ready-to-use programs. Instead, students should engage in developing cal-culation procedures but not in long program development. For example, students can implementa bubble point algorithm at Excel R© spreadsheet level but use the functions available in XSEOSto compute the activity or fugacity coefficients required by their calculations. Therefore, theXSEOS add-in does not contain ready-to-use VBA procedures for chemical equilibrium, flash,dew, and bubble point calculations —XSEOS provides the physical properties required by thesecalculations. Furthermore, the XSEOS add-in does not contain VBA functions for ideal gas andideal solution properties, as they are simple enough to be quickly programmed at spreadsheetlevel, and their implementation is a good exercise for students.

XSEOS has open source code and is distributed under the GNU General Public License,version 3 (available in Appendix B). Therefore, it is an expandable platform and will hopefullycontribute to the exchange of experiences in the teaching of Chemical Engineering Thermody-namics. XSEOS runs in Excel R©, which is part of the Microsoft R© Office suite, available fromMicrosoft R© resellers.

DistributionThe XSEOS distribution contains:

1. this manual in PDF format;

2. file XSEOS-installer.xls;

3. file XSEOS-Add-in.xla with the models implemented in Visual Basic for Applications;

4. file XSEOS-spreadsheet.xls, which is an Excel R© spreadsheet with examples.

InstallationThe installation procedures are different for Excel 2003 and 2007. Please refer to the section thatcorresponds to the version you have.

1

Excel R© 2003 installationInstallation requires two parts. The first part sets Excel R© security levels and installs the Solverpackage.

1. start Excel R©;

2. click on Tools, then on Macro, and then on Security. Set the security level to Medium.Navigate back to the main Excel R© window;

3. click on Tools and then on Add-ins;

4. check the Solver Add-in box;

5. click on OK to close the Add-ins box;

6. close Excel R©;

The second part installs the add-in files. The details differ depending on whether the usercopies the add-in files to a directory called C:\XSEOS (simpler installation) or elsewhere.

Installation in the C:\XSEOS directory - Recommended

1. copy all files to a directory called C:\XSEOS;

2. open Excel R©;

3. open the file XSEOS-installer.xls;

4. at the Security Warning, choose Enable Macros;

5. click on Tools, then on Macro, then on Macros..., and press the Run button to execute themacro Add_an_addin. This may take a few seconds;

6. close Excel R©.

After installation, the functions of the XSEOS add-in will be automatically available everytime you open Excel R©.

Installation in a user-defined directory

Let us refer to the user-defined directory as D:\ userdir.

1. copy all files to directory D:\userdir;

2. open Excel R©;

3. open the file XSEOS-installer.xls;

2

4. at the Security Warning, choose Enable Macros;

5. click on Tools, then on Macro, then on Macros..., and press the Edit button;

6. in the Visual Basic code, replace C:\XSEOS\XSEOS-Add-in.xla with D:\userdir\XSEOS-Add-in.xla and save the Visual Basic code.

7. click on Tools, then on Macro, then on Macros..., and press the Run button to execute themacro Add_an_addin. This may take a few seconds;

8. close Excel R©.

After installation, the functions of the XSEOS add-in will be automatically available everytime you open Excel R©.

Excel R© 2007 installationInstallation requires two parts. The first part sets Excel R© security levels and installs the Solverpackage.

1. start Excel R©;

2. click on button located at the upper left corner of the screen and then on the button ExcelOptions;

3. click on Add-ins and on the Manage pull-down menu, select Excel Add-ins and click onthe Go... button;

4. click on the Solver Add-in box and then on OK. Excel may ask for permission to installthe Solver Add-in: answer Yes to grant permission. The installation procedure may take afew minutes;

5. click again on button located at the upper left corner of the screen and then on the buttonExcel Options;

6. click on Trust Center, then on Trust Center Settings and then on Add-ins. Make sure allboxes are unchecked;

The second part installs the add-in files.

1. copy all the files in the XSEOS distribution to a directory of your choice;

2. click on button located at the upper right corner of the screen and then on the button ExcelOptions;

3. click on Add-ins and on the Manage pull-down menu, select Excel Add-ins and click onthe Go... button;

3

4. click on the Browse button, navigate to the directory that contains the XSEOS files, selectthe XSEOS-Add-in.xla file, and click OK to return to Excel’s main window;

5. close Excel R©.

After installation, the functions of the XSEOS add-in will be automatically available everytime you open Excel R©.

Use

General issuesTo use XSEOS, some basic familiarity with Excel R© is required. The tutorial:

http://getit.rutgers.edu/tutorials/excel/media/excel.pdf

describes the most important Excel R© features needed to use XSEOS. Use of the Excel R© Solveris described at:

http://www.vertex42.com/ExcelArticles/excel-solver-examples.html.

Microsoft R© provides several more detailed tutorials:

http://office.microsoft.com/en-us/training/CR061831141033.aspx

It is important to emphasize two useful features of Excel R© operation:

1. absolute and relative cell addresses: it is simple to pull rows or columns in Excel R© tobuild tables in which the entries refer to similar calculations. In these pulling operations,Excel R© will utilize relative addresses by guessing which cells will be used in the newcalculation. Some of the arguments of the XSEOS functions, however, such as the modelparameters and the universal gas constant, will have the same values for any temperature,pressure, and/or composition. Therefore, it is more convenient to use absolute addresses(i.e., addresses that Excel R© will not change) to the cells that contain these values. Thequickest way to impose that Excel R© will refer to a cell by its absolute address is to selectthe cell reference in a formula and then press F4 only once. By pressing F4 more thanonce, other effects can be obtained as explained in the site:

http://office.microsoft.com/en-us/excel/HP101023451033.aspx

2. array functions: some of the XSEOS functions return arrays and not only a single value,such as the functions that compute activity coefficients and the fugacity coefficients ofthe components of a mixture. In the case of these two functions, the number of valuesreturned is equal to number of components in the mixture under evaluation. To use an

4

array function, the first step is to use the mouse to select the necessary number of cells (forexample, three cells in a ternary mixture). Making sure these cells remain selected, thefunction call should be typed. Typing is completed by pressing CTRL+SHIFT+ENTERsimultaneously. A detailed explanation about array formulas in Excel R© is available in thesite:

http://office.microsoft.com/en-us/excel/HA010872901033.aspx

Before implementing new calculations, it is strongly suggested that you:

1. watch the movies available at http://www.engg.uaeu.ac.ae/mcastier/downloads.htm andpractice with the templates available there;

2. try some of the examples in the file XSEOS-spreadsheet.xls. This file presents applicationsof the models available in XSEOS to different situations, such as vapor-liquid equilibrium,liquid-liquid equilibrium, flashes, chemical equilibrium, parameter fitting, and speed ofsound calculations. The file contains documentation about each example.

Each XSEOS function for excess Gibbs free energy models has a name and a list of inputparameters, which typically include the universal gas constant, temperature, mole fractions, andthe set of parameters used by each thermodynamic model to characterize fluid behavior (suchas critical properties, binary interaction parameters, etc...). Functions for equations of state alsoinclude the pressure (or the molar volume) in their list of input parameters. The function nameconvention used in XSEOS is described in the next subsection.

Models and properties availableTable 1 shows the excess Gibbs free energy models and properties currently available in XSEOS,along with their name convention. In the lower part of Table 1, mname stands for the name con-vention, as defined in the upper part of the Table. For example, the functions to compute thelogarithms of activity coefficients using the regular solution theory and the modified UNIFAC(Dortmund) are called regsol and modunifac, respectively. The values of gE

RT , hE

RT , sE

R , and cEPR are

not returned by individual functions. Instead, all four values are returned by an array function.Again using the regular solution theory and the modified UNIFAC (Dortmund) models as exam-ples, the names of the corresponding functions are regsolxs and modunifacxs, respectively. Thesubscript values 1 to 4 in Table 1, as in mnamexs1 and mnamexs4, are included for illustrativepurposes, only to indicate the array position in which each property is returned. Two of the prop-erties listed in Table 1 are tagged as internal functions to indicate that the corresponding formulasare programmed in the XSEOS add-in but a user-friendly function has not been developed yet.

Table 2 shows the equations of state and properties currently available in XSEOS, along withtheir name convention. mname in the lower part of Table 2 stands for the model name convention,as defined in the upper part of the Table. For example, the functions to compute the logarithmsof fugacity coefficients of the components of a mixture in the liquid and vapor phases, using thePeng-Robinson equation of state, are called prlnphil and prlnphiv, respectively. A third type of

5

Table 1: Excess Gibbs free energy models and properties available in XSEOS

Excess Gibbs free energy Name conventionMargules 2-,3-,4-suffix margulesRegular solution theory regsol

Flory-Huggins fhWilson wilson

Models TK-Wilson tkwilsonNRTL nrtl

UNIQUAC uniquacUNIFAC unifac

Modified UNIFAC (Dortmund) modunifacgE

RT mnamexs1hE

RT mnamexs2sE

R mnamexs3

Properties cEPR mnamexs4

lnγi mname(∂ lnγi

∂T

)x

internal function

n(

∂ lnγi∂n j

)T,nk 6= j

internal function

6

function is available to compute the logarithms of fugacity coefficients of the components of amixture when the user wishes to specify molar volume instead of pressure. For example, such afunction for the Peng-Robinson equation of state is called prlnphi.

The values of lnφ , hR

RT , sR

R , and cRP

R are not returned by individual functions. Instead, all fourvalues are returned by an array function. Again using the Peng-Robinson equation of state asexample, the names of the corresponding functions are prresl and prresv, depending on whetherliquid or vapor phase properties should be calculated. Similarly, the values of ρ ,

(∂P∂ρ

)T,x

, cRP

R ,

and(

∂P∂T

)v,x

are calculated by array functions, whose names for the Peng-Robinson equation of

state are prsoundspeedl and prsoundspeedv.The subscript values 1 to 4 in Table 2 that appear in array functions are included for illus-

trative purposes, only to indicate the array position in which each property is returned. In Table2, the properties tagged as internal functions are programmed in the XSEOS add-in but a user-friendly function has not been developed yet.

Literature references about the implemented models and their parameters are provided in fileXSEOS-spreadsheet.xls.

Modeling petroleum fractionsSeveral correlations to predict the properties of petroleum fractions are available in XSEOS. Theyallow the computation of properties such as critical temperature, critical pressure, acentric factor,molar mass, etc. More information about these models should be included in future versions ofthe manual. Please refer to the Oil Examples sheet of the XSEOS-spreadsheet.xls for the currentdocumentation.

An example of correlation is the the Whitson interpolation tables to estimate the values ofcritical temperature, critical pressure, acentric factor, and molar mass of oil fractions are availablethrough the functions whitsontc, whitsonpc, whitsonomega, and whitsonmw. These functionshave a single argument, which is the number of carbon atoms in the hydrocarbon molecule.This must be a number between 6 and 45. The functions accept non-integer values because thismethod is normally used to model pseudocomponents in phase equilibrium calculations that maybe represented by a fractional number of Carbon atoms.

Surface tensionXSEOS includes the parachor, Miqueu, and Danesh methods to predict surface tension. Moreinformation about these methods should be included in future versions of the manual. Pleaserefer to the Surface tension sheet of the XSEOS-spreadsheet.xls for the current documentation.

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Table 2: Equations of state and properties available in XSEOS

Equations of state Name conventionvan der Waals vdw

Redlich-Kwong rkSoave-Redlich-Kwong (SRK) srk

Models Peng-Robinson (PR) prPR, quadratic mixing rule for b prb2

Stryjek-Vera svPredictive SRK (PSRK) psrk

Mattedi-Tavares-Castier (MTC) mtcP mnamePρ mnamesoundspeedl1, mnamesoundspeedv1(

∂P∂ρ

)T,x

mnamesoundspeedl2, mnamesoundspeedv2(∂ 2P∂ρ2

)T,x

internal function(∂ 3P∂ρ3

)T,x

internal function

lnφ mnameresl1, mnameresv1mnameresidual1

lnφi mnamelnphi, mnamelnphil, mnamelnphiv(∂ lnφi

∂P

)T,x

internal function

Properties n(

∂ lnφi∂n j

)T,P,nk 6= j

internal function

n(

∂V∂n j

)T,P,nk 6= j

internal function(∂ lnφi

∂T

)P,x

internal function(∂v∂T

)P,x

internal functionhR

RT mnameresl2, mnameresv2mnameresidual2

sR

R mnameresl3, mnameresv3mnameresidual3

cRP

R mnameresl4, mnameresv4,mnamesoundspeedl3, mnamesoundspeedv3

mnameresidual4(∂P∂T

)v,x

mnamesoundspeedl4, mnamesoundspeedv4

8

FeedbackSuggestions of corrections and improvements to the add-in, its supporting spreadsheet, and man-ual or proposals of new examples are welcome. The author’s e-mail addresses are available atthe front page of this manual.

AcknowledgmentsThe author acknowledges Dr. Silvana Mattedi (Federal University of Bahia, Brazil) and the stu-dents of the Chemical Engineering Thermodynamics and Fluid Phase Equilibria courses of theUnited Arab Emirates University, 2007-9, for testing the program and suggesting many improve-ments.

List of Symbols

Roman LetterscE

P : molar excess heat capacity at constant pressurecR

P : molar residual heat capacity at constant pressurehE : molar excess enthalpyhR : molar residual enthalpygE : molar excess Gibbs free energyn : number of molesP : pressureR : universal gas constantsE : molar excess entropysR : molar residual entropyT : temperaturev : molar volumeV : volumex : mole fraction

Greek Lettersγi : activity coefficient of component i in a mixtureφ : fugacity coefficient of a pure component or of a mixtureφi : fugacity coefficient of component i in a mixtureρ : molar density

9

Appendix A

Thermodynamic models

This appendix presents the formulas of the thermodynamic models as implemented in XSEOS.To keep the lists of symbols as short as possible, intermediate variables used in the model defi-nition are omitted.

A.1 Excess Gibbs free energy models

A.1.1 Margules modelThe four-suffix Margules formula is implemented. It is possible to make calculations with the2- or 3-suffix formulas by setting proper values for the model parameters, as explained below, inthe list of symbols.

(A.1)ge

RT=

x1

(A + B (x1 − x2) +C (x1 − x2)

2)

x2

RTList of symbols:A: model parameterB: model parameter (should be set to zero when using the Margules 2-suffix formula)C: model parameter (should be set to zero when using the Margules 2- or 3-suffix formula)ge: molar excess Gibbs free energyR: universal gas constantT : temperaturex1: mole fraction of component 1x2: mole fraction of component 2

A.1.2 Regular solution theory

(A.2)ge

RT=

(−(δm (∑nc

i=1 (δ (i) φ (i)))) + ∑nci=1

(δ (i)2

φ (i)))

(∑nck=1 (v(k) x(k)))

RT

10

with:(A.3)δm =

nc

∑j=1

(δ ( j) φ ( j))

and:

(A.4)φ ( j) =v( j) x( j)

∑ncie=1 (v(ie) x(ie))

List of symbols:ge: molar excess Gibbs free energync: number of components in the mixtureR: universal gas constantT : temperaturev: molar volume of pure component (model parameter)x: mole fractionδ : solubility parameter (model parameter)

A.1.3 Flory-Huggins

ge

RT=

nc

∑i=1

(ln(

m(i)∑

nckn=1 (m(kn) x(kn))

)x(i)

)+

∑nci=1

(m(i)

(∑

ncj=1 (χ (i, j) m( j) x( j))

)x(i)

)2 (∑nc

km=1 (m(km) x(km)))

(A.5)

List of symbols:ge: molar excess Gibbs free energym: size parameter (model parameter)nc: number of components in the mixtureR: universal gas constantT : temperaturex: mole fractionχ: Flory-Huggins energy parameter (model parameter)

A.1.4 Wilson

(A.6)ge

RT= −

nc

∑i=1

ln

∑ncj=1

(v( j)x( j)

e∆g(i, j)

RT

)v(i)

x(i)

11

List of symbols:ge: molar excess Gibbs free energync: number of components in the mixtureR: universal gas constantT : temperaturev: molar volume of pure component (model parameter)x: mole fraction∆g: binary interaction parameter (model parameter)

A.1.5 TK-Wilson

(A.7)ge

RT=

nc

∑i=1

ln

∑ncj=1 (v( j) x( j))

∑ncie=1

(v(ie)x(ie)

e∆g(i,ie)

RT

) x(i)

List of symbols:ge: molar excess Gibbs free energync: number of components in the mixtureR: universal gas constantT : temperaturev: molar volume of pure component (model parameter)x: mole fraction∆g: binary interaction parameter (model parameter)

A.1.6 NRTL

(A.8)ge

RT=

∑nci=1

(

∑ncj=1

(a( j,i)x( j)

ea( j,i)α( j,i)

RT

))x(i)

∑ncm=1

(x(m)

ea(m,i)α(m,i)

RT

)

RTList of symbols:a: binary interaction parameter (model parameter)ge: molar excess Gibbs free energync: number of components in the mixtureR: universal gas constantT : temperaturex: mole fractionα: non-randomness parameter (model parameter)

12

A.1.7 UNIQUAC

(A.9)

ge

RT=

nc

∑i=1

(ln(

φ (i)x(i)

)x(i)

)−

nc

∑i=1

(ln

(nc

∑j=1

(θ ( j)

ea( j,i)RT

))q(i) x(i)

)

+z(

∑nci=1

(ln(

θ(i)φ(i)

)q(i) x(i)

))2

with:(A.10)φ (i) =

r (i) x(i)∑

nckn=1 (r (kn) x(kn))

and:

(A.11)θ (i) =q(i) x(i)

∑nckn=1 (q(kn) x(kn))

List of symbols:ge: molar excess Gibbs free energync: number of components in the mixtureR: universal gas constantT : temperaturer: UNIQUAC volume parameterq: UNIQUAC area parameterx: mole fractionz: coordination number (usually equal to 10 in UNIQUAC)

A.1.8 UNIFAC

(A.12)ge

RT=

gec

RT+

ger

RT

with:

(A.13)ge

cRT

=nc

∑i=1

(ln(

φ (i)x(i)

)x(i)

)+

z(

∑nci=1

(ln(

θ(i)φ(i)

)q(i) x(i)

))2

(A.14)ge

rRT

=nc

∑i=1

((nsg

∑ksg=1

( f (i,ksg) (lngr (ksg)− lnigr (i,ksg)))

)x(i)

)

13

(A.15)lngr (isg) = qsg(isg)

(1− ln

(nsg

∑msg3=1

(τ (msg3, isg) thsgm(msg3))

)

−nsg

∑msg4=1

(τ (isg,msg4) thsgm(msg4)

∑nsgmsg5=1 (τ (msg5,msg4) thsgm(msg5))

))

thsgm(isg)

=qsg(isg) (∑nc

i2=1 ( f (i2, isg) x(i2)))(∑

nsgmsg2=1

(qsg(msg2)(∑

nci2=1( f (i2,msg2)x(i2)))

∑nci1=1

((∑

nsgmsg1=1( f (i1,msg1))

)x(i1)

))) (

∑nci1=1

((∑

nsgmsg1=1 ( f (i1,msg1))

)x(i1)

))(A.16)

(A.17)lnigr (i, isg) = qsg(isg)

(1− ln

(nsg

∑lsg3=1

(τ (lsg3, isg) thsgp(i, lsg3))

)

−nsg

∑lsg4=1

(τ (isg, lsg4) thsgp(i, lsg4)

∑nsglsg5=1 (τ (lsg5, lsg4) thsgp(i, lsg5))

))

(A.18)thsgp(i, isg) =f (i, isg) qsg(isg)

∑nsglsg2=1 ( f (i, lsg2) qsg(lsg2))

(A.19)r (i) =nsg

∑ksg=1

( f (i,ksg) rsg(ksg))

(A.20)q(i) =nsg

∑ksg=1

( f (i,ksg) qsg(ksg))

(A.21)φ (i) =r (i) x(i)

∑ncknc=1 (r (knc) x(knc))

(A.22)θ (i) =q(i) x(i)

∑ncknc=1 (q(knc) x(knc))

(A.23)τ (i, j) = e−(

a(i, j)RT

)

14

List of symbols:a: binary interaction parameter (model parameter)f : number of occurrences of a subgroup in a given moleculege: molar excess Gibbs free energync: number of components in the mixtureR: universal gas constantT : temperaturer: molecular volume parameterrsg: subgroup volume parameterq: molecular area parameterqsg: subgroup area parameterx: mole fractionz: coordination number (usually equal to 10 in UNIFAC)

A.1.9 Modified UNIFAC (Dortmund version)The equations are similar to those of the UNIFAC model with the following exceptions:

(A.24)ge

cRT

=nc

∑i=1

(ln

(φ′(i)

x(i)

)x(i)

)+

z(

∑nci=1

(ln(

θ(i)φ(i)

)q(i) x(i)

))2

(A.25)φ′(i) =

r (i)0.75 x(i)

∑ncknc=1

(r (knc)0.75 x(knc)

)

(A.26)a(i, j) = a′(i, j) + T b

′(i, j) + T 2 c

′(i, j)

List of additional symbols compared to the UNIFAC model:a′: binary interaction parameter (model parameter)

b′: binary interaction parameter (model parameter)

c′: binary interaction parameter (model parameter)

A.2 Equations of state

A.2.1 van der Waals

(A.27)P =RT

−bm + v−(am

v2

)

15

with:

(A.28)am =nc

∑i=1

((nc

∑j=1

(a(i, j) x( j))

)x(i)

)

(A.29)bm =nc

∑i=1

(b(i) x(i))

(A.30)a(i, j) =√

apure(i)√

apure( j) (1− k (i, j))

(A.31)apure(i) =27R2 Tc (i)

2

64Pc (i)

(A.32)b(i) =RTc (i)8Pc (i)

List of symbols:k: binary interaction parameter (model parameter)P: pressurePc: critical pressure (model parameter)R: universal gas constantT : temperatureTc: critical temperature (model parameter)v: molar volumex: mole fraction

A.2.2 Redlich-Kwong

(A.33)P =RT

−bm + v− am

v (bm + v)

with:

(A.34)am =nc

∑i=1

((nc

∑j=1

(a(i, j) x( j))

)x(i)

)

(A.35)bm =nc

∑i=1

(b(i) x(i))

16

(A.36)a(i, j) =√

apure(i)√

apure( j) (1− k (i, j))

(A.37)apure(i) =0.42748R2 Tc (i)

2.5√

T Pc (i)

(A.38)b(i) =0.08662RTc (i)

Pc (i)

List of symbols:k: binary interaction parameter (model parameter)P: pressurePc: critical pressure (model parameter)R: universal gas constantT : temperatureTc: critical temperature (model parameter)v: molar volumex: mole fraction

A.2.3 Soave-Redlich-Kwong (SRK)

(A.39)P =RT

−bm + v− am

v (bm + v)

with:

(A.40)am =nc

∑i=1

((nc

∑j=1

(a(i, j) x( j))

)x(i)

)

(A.41)bm =nc

∑i=1

(b(i) x(i))

(A.42)a(i, j) =√

apure(i)√

apure( j) (1− k (i, j))

apure(i)

=0.42747R2

(1 +

(0.48508 + 1.55171ω (i)− 0.15613ω (i)2

) (1−

√T

Tc(i)

))2Tc (i)

2

Pc (i)(A.43)

17

(A.44)b(i) =0.08664RTc (i)

Pc (i)

List of symbols:k: binary interaction parameter (model parameter)P: pressurePc: critical pressure (model parameter)R: universal gas constantT : temperatureTc: critical temperature (model parameter)v: molar volumex: mole fractionω: acentric factor (model parameter)

A.2.4 Peng-Robinson

(A.45)P =RT

−bm + v+

am

bm (bm − 2v)− v2

with:

(A.46)am =nc

∑i=1

((nc

∑j=1

(a(i, j) x( j))

)x(i)

)

(A.47)bm =nc

∑i=1

(b(i) x(i))

(A.48)a(i, j) =√

apure(i)√

apure( j) (1− k (i, j))

apure(i)

=0.45724R2

(1 +

(0.37464 + 1.54226ω (i)− 0.26992ω (i)2

) (1−

√T

Tc(i)

))2Tc (i)

2

Pc (i)(A.49)

(A.50)b(i) =0.0778RTc (i)

Pc (i)

List of symbols:k: binary interaction parameter (model parameter)

18

P: pressurePc: critical pressure (model parameter)R: universal gas constantT : temperatureTc: critical temperature (model parameter)v: molar volumex: mole fractionω: acentric factor (model parameter)

A.2.5 Peng-Robinson with quadratic mixing rule for the b parameterThe model is identical to the Peng-Robinson EOS except for the expression that computes thevalue of b for a mixture, here denoted as bm:

(A.51)bm =nc

∑i=1

nc

∑j=1

xix j

(bi + b j

2

)(1− `i j

)A.2.6 Stryjek-Vera

(A.52)P =RT

−bm + v+

am

bm (bm − 2v)− v2

with:

(A.53)am =nc

∑i=1

((nc

∑j=1

(a(i, j) x( j))

)x(i)

)

(A.54)bm =nc

∑i=1

(b(i) x(i))

(A.55)a(i, j) =√

apure(i)√

apure( j) (1− k (i, j))

(A.56)apure(i) =0.45724R2

(1 + f w(i)

(1−

√T

Tc(i)

))2Tc (i)

2

Pc (i)

(A.57)f w(i) = 0.378893 + 1.48972ω (i)− 0.171318ω (i)2

+ 0.0196554ω (i)3 + k1 (i)

(1 +

√T

Tc (i)

)(0.7− T

Tc (i)

)19

(A.58)b(i) =0.0778RTc (i)

Pc (i)

List of symbols:k: binary interaction parameter (model parameter)k1: pure component parameter (model parameter)P: pressurePc: critical pressure (model parameter)R: universal gas constantT : temperatureTc: critical temperature (model parameter)v: molar volumex: mole fractionω: acentric factor (model parameter)

A.2.7 Predictive Soave-Redlich-Kwong (PSRK)

(A.59)P =RT

−bm + v− am

v (bm + v)

with:

(A.60)am = bm RT

nc

∑i=1

(apure(i) x(i)

RT b(i)

)+

ge

RT + ∑nci=1

(ln(

b(i)bm

)x(i)

)ln( u

1+u

)

(A.61)u = 1.1

(A.62)bm =nc

∑i=1

(b(i) x(i))

(A.63)apure(i)

=

0.42748R2(

1 + c1 (i)(

1−√

TTc(i)

)+ c2 (i)

(1−

√T

Tc(i)

)2+ c3 (i)

(1−

√T

Tc(i)

)3)2

Tc (i)2

Pc (i)

(A.64)b(i) =0.08664RTc (i)

Pc (i)

In Eq. A.60, ge denotes the molar excess Gibss free energy from the UNIFAC model withtemperature-dependent parameters, as follows:

20

(A.65)a(i, j) = a′(i, j) + T b

′(i, j) + T 2 c

′(i, j)

List of symbols:a′: binary interaction parameter (model parameter)

b′: binary interaction parameter (model parameter)

c′: binary interaction parameter (model parameter)

c1: pure component parameter (model parameter)c2: pure component parameter (model parameter)c3: pure component parameter (model parameter)ge: molar excess Gibss free energy from UNIFAC with temperature-dependent parametersP: pressurePc: critical pressure (model parameter)R: universal gas constantT : temperatureTc: critical temperature (model parameter)v: molar volumex: mole fraction

A.2.8 Mattedi-Tavares-Castier (MTC)The compressibility factor is given by:

z = vr ln[

vv− 1

]+

Z2

vr ln[

v− 1 + (q/r)v

]+ l− vΨ(q/r)

v− 1 + (q/r)

nc

∑i=1

ng

∑a=1

xiνai Qa (Γa − 1)

v− 1 + (q/r)Γa

(A.66)

with:

(A.67)l = (Z/2)(r - q) - (r - 1)

(A.68)r =nc

∑i=1

xi

ng

∑a=1

νai Ra

(A.69)Zq =nc

∑i=1

xi

ng

∑a=1

νai Z Qa

(A.70)rV ∗ =nc

∑i=1

xi

ng

∑a=1

νai V a

(A.71)v =V

NrV ∗=

vrv∗

21

(A.72)rv∗ =nc

∑i=1

xi

ng

∑a=1

νai va

(A.73)Γa =

ng

∑m=1

Smγ

ma

(A.74)Sm =

nc∑

i=1νm

i xiQm

q

(A.75)γma = exp(−uma/(RT ))

(A.76)uba

R=

uba0R

(1 +

Bba

T

)List of symbols:Bba: model parameter related to the interaction energyP: pressureQa: surfarce area of a group of type a (model parameter)R: universal gas constantRa: group contribution to the number of segmentsT : temperatureuma: interaction energy between groups m and auba

0 : : model parameter related to the interaction energyv: molar volumev∗: lattice cell volume (assumed equal to 5 cm3)V a: group contribution to the hard core volume (model parameter)x: mole fractionz: compressibility factorZ: lattice coordination number (equal to 10)νa

i : number of occurrences of group a in molecule iΨ: lattice constant (assumed equal to 1)

22

Appendix B

GNU GPL License

Copyright c© 2007 Free Software Foundation, Inc. http://fsf.org/

Everyone is permitted to copy and distribute verbatim copies of thislicense document, but changing it is not allowed.

23

Preamble

The GNU General Public License is a free, copyleft license for software and other kinds ofworks.

The licenses for most software and other practical works are designed to take away yourfreedom to share and change the works. By contrast, the GNU General Public License is intendedto guarantee your freedom to share and change all versions of a program–to make sure it remainsfree software for all its users. We, the Free Software Foundation, use the GNU General PublicLicense for most of our software; it applies also to any other work released this way by itsauthors. You can apply it to your programs, too.

When we speak of free software, we are referring to freedom, not price. Our General PublicLicenses are designed to make sure that you have the freedom to distribute copies of free software(and charge for them if you wish), that you receive source code or can get it if you want it, thatyou can change the software or use pieces of it in new free programs, and that you know you cando these things.

To protect your rights, we need to prevent others from denying you these rights or asking youto surrender the rights. Therefore, you have certain responsibilities if you distribute copies of thesoftware, or if you modify it: responsibilities to respect the freedom of others.

For example, if you distribute copies of such a program, whether gratis or for a fee, you mustpass on to the recipients the same freedoms that you received. You must make sure that they,too, receive or can get the source code. And you must show them these terms so they know theirrights.

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COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM “ASIS” WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED,INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MER-CHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRERISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITHYOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COSTOF ALL NECESSARY SERVICING, REPAIR OR CORRECTION.

16. Limitation of Liability.

IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO INWRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MOD-IFIES AND/OR CONVEYS THE PROGRAM AS PERMITTED ABOVE, BE LIABLETO YOU FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTALOR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TOUSE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF DATA ORDATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU ORTHIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANYOTHER PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEENADVISED OF THE POSSIBILITY OF SUCH DAMAGES.

17. Interpretation of Sections 15 and 16.

If the disclaimer of warranty and limitation of liability provided above cannot be givenlocal legal effect according to their terms, reviewing courts shall apply local law that mostclosely approximates an absolute waiver of all civil liability in connection with the Pro-gram, unless a warranty or assumption of liability accompanies a copy of the Program inreturn for a fee.

END OF TERMS AND CONDITIONSHow to Apply These Terms to Your New Programs

If you develop a new program, and you want it to be of the greatest possible use to the pub-lic, the best way to achieve this is to make it free software which everyone can redistributeand change under these terms.

To do so, attach the following notices to the program. It is safest to attach them to the startof each source file to most effectively state the exclusion of warranty; and each file shouldhave at least the “copyright” line and a pointer to where the full notice is found.

<one line to give the program’s name and a brief idea of what it does.>Copyright (C) <year> <name of author>This program is free software: you can redistribute it and/or modify it under theterms of the GNU General Public License as published by the Free SoftwareFoundation, either version 3 of the License, or (at your option) any later version.

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This program is distributed in the hope that it will be useful, but WITHOUTANY WARRANTY; without even the implied warranty of MERCHANTABIL-ITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU GeneralPublic License for more details.You should have received a copy of the GNU General Public License along withthis program. If not, see http://www.gnu.org/licenses/.

Also add information on how to contact you by electronic and paper mail.

If the program does terminal interaction, make it output a short notice like this when itstarts in an interactive mode:

<program> Copyright (C) <year> <name of author>This program comes with ABSOLUTELY NO WARRANTY; for details typeshow w. This is free software, and you are welcome to redistribute it undercertain conditions; type show c for details.

The hypothetical commands show w and show c should show the appropriate parts ofthe General Public License. Of course, your program’s commands might be different; fora GUI interface, you would use an “about box”.

You should also get your employer (if you work as a programmer) or school, if any, to signa “copyright disclaimer” for the program, if necessary. For more information on this, andhow to apply and follow the GNU GPL, see http://www.gnu.org/licenses/.

The GNU General Public License does not permit incorporating your program into propri-etary programs. If your program is a subroutine library, you may consider it more usefulto permit linking proprietary applications with the library. If this is what you want to do,use the GNU Lesser General Public License instead of this License. But first, please readhttp://www.gnu.org/philosophy/why-not-lgpl.html.

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