PE2000 Manual

Preface, Contents, Survey, and Introduction

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Preface PE 2000: A Powerful Tool to Correlate Phase Equilibria The first preliminary Windows release of PE had been shown at the 8th International Conference on Properties and Phase Equilibria for Product and Process Design at Noordwijkerhout, The Netherlands, on April 30th, 1998 (Pfohl et al., 1998a). The first official Windows release of PE was finished in July 1998 and shown at the 17th European Seminar on Applied Thermodynamics at Vilamoura, Portugal (Petkov et al., 1999) and on the International Workshop on Scientific Computing in Chemical Engineering II at Hamburg, Germany, (Petkov et al., 1999a / Pfohl et al., 1999a). From 1998 on, the actual releases of PE could be downloaded via internet from http://vt2pc8.vt2.tu-harburg.de for free. Meanwhile more than 50 different research groups use PE for the calculation of fluid-phase equilibria. The first edition of the present book, which had been entitled Usage of PE – A Program to Calculate Phase Equilibria, had been published in December 1998 (Pfohl et al., 1998b). In order to meet the needs of the users, continuous efforts led to significantly improved versions of PE. The latest PE version, released February 2000, additionally enables the users to include new equations of state by their own very easily (see Chapter 4). The newest version of PE will be shown at the ACHEMA at Frankfurt, Germany, in May 2000 and at the 18th European Seminar on Applied Thermodynamics at Kutná Hora, Czech Republic, in June 2000 (Petkov et al., 2000).

February 2000 Oliver Pfohl Stanimir Petkov Gerd Brunner


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Contents Page

Survey 4 Introduction 5 1 Input Files Containing Experimental Data ...............................….....….............. 6 1.1 Pure-Component Data 7 1.1.1 PCFILES 7 1.1.2 Generating PCFILES 12 1.1.3 Generating PCFILES using the data compilation by Daubert&Danner 18 1.1.4 Saving EOS parameters to a PCFILE 20 1.2 Mixture data 21 1.2.1 EXFILES 21 1.2.2 Generating EXFILES 25 1.2.3 Editing EXFILES 32 1.2.4 Estimating Pure-Component Parameters 33 2 Pure-Component Calculations ..........................….............................…............. 41 2.1 Determination of Pure-Component EOS Parameters 41 2.1.1 Visualisation of Grids/Optimisations 46 2.2 Calculation of Vapour Pressure Curves 48 2.2.1 Graphical Output of Vapour Pressure Curves 49 2.3 Calculation of Pρ-Diagrams 51 2.3.1 Graphical Output of Pρ-Diagrams 52 2.4 Calculation of Deviations 53 3 Mixtures .........................................................................................……............... 54 3.1 Mixture Calculations 56 3.1.1 Binary Phase Equilibrium Calculations 59 3.1.2 Ternary Phase Equilibrium Calculations 60 3.1.3 Calculating Phase Equilibria 62 3.1.4 Density Calculations 63 3,2 Optimisation of Binary Interaction Parameters 64 3.2.1 Optimising Parameters (Simplex-Nelder-Mead Method) 65 3.2.2 Scanning Parameters (Grid) 66 3.3 Graphical Output of Mixture Equilibria 68 3.3.1 Pxy- / yP-Diagrams (Binary) 68 3.3.2 Gibbs’ Triangles (Ternary and Quaternary) 71 3.4 Solid-Gas Equilibria 72


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4 Implementation of New EOS by Users ..................................……..................... 73 4.1 Introduction 73 4.2 Files Used for Implementing New EOS in PE 74 4.3 The File pes_EOS_?.for 75 4.3.1 Initialization (calc_what=0) 81 4.3.2 Calculate Pressure (calc_what=1) 81 4.3.3 Calculate Fugacity Coefficients (calc_what=2) 82 4.4 pes_EOS_by_User.FOR 83 4.4.1 Subroutine Parameters 83 4.4.2 Parameters for Simple Peng-Robinson EOS (FORTRAN: save) 85 4.4.3 Initialization (calc_what=0) 85 4.4.4 Calculate Pressure (calc_what=1) 86 4.4.5 Calculate Fugacity Coefficients (calc_what=2) 86 4.4.6 Subroutine si_mix_abc 86 4.5 EOS_P_TEST.FOR, EOS_PHI_TEST.FOR – Programs 87 4.6 FORTRAN Language 87 4.7 EOS_on_DISK.FOR 88 4.7.1 Running EOS_on_DISK.EXE 88 4.8 Some Details 90 4.8.1 Enforcing Tc and Pc during Optimization 90 4.8.2 Using Sites and Interaction Parameters 90 4.8.3 The File EOS_on_DISK.sav 91

5 Export of Data ...............................................................................……................ 92 6 Future Developments ........................................................................……........... 93 Appendix ...............................................................................................….……...... 94 A1: EOS Implemented in PE 94 A2: Mixing Rules for Simple EOS 137 Literature..............................................................................................……........... 143


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Survey The program PE ( = Phase Equilibria ) has been developed in Professor Brunner’s research group at the Technical University of Hamburg-Harburg, starting in 1985. PE was built in order to obtain an efficient tool to correlate phase equilibria - especially those at high pressure that are related to gas-extraction processes. PE 2000 released 2/2000 runs on IBM compatible PC with Windows NT 4.0, Windows 95/98 or Windows 2000. It uses all features supplied by Windows: mouse control, menu bars, dialog boxes as well as text and graphical output to commercial programs like Word, Powerpoint, and Corel Draw. PE offers about 40 different equations of state (EOS) with up to 7 different mixing rules for correlating phase equilibria.

Pure-Component Properties

PE offers subroutines to determine pure-component EOS parameters based on pure-component vapour pressures and densities at saturation conditions. Graphical output routines allow a visual comparison of experimental and calculated data of the vapour pressure curve and PvT data. A useful feature when correlating phase equilibria related to gas-extraction processes is the possibility to enforce pure-component EOS parameters that reproduce the critical temperature and pressure of the supercritical solvent during the pure-component EOS parameter optimisation.

Mixture Data

PE offers subroutines to determine optimum binary interaction parameters for all EOS in order to reproduce the phase behaviour of binary and ternary systems with a given set of EOS and mixing rule, best. Further, PE offers the possibility to calculate phase equilibria and densities with given interaction parameters at any conditions. Graphical output routines provide plots of the miscibility gaps in binary and ternary systems in a quality suitable for publications.

Warranty

Although many efforts have been made in order to supply a program without bugs, no warranty is given for any of the results obtained with PE due to the size of the program. Further, the users are encouraged to check results carefully. Most errors will occur due to a wrong input of data by the user, usage of EOS and mixing rules that are not appropriate for the investigated system, and convergence problems when using inadequate initial guesses.


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Introduction The calculation of phase equilibria may serve as a cheap alternative compared to the measurement of phase equilibria. This is especially true for high pressure where the necessary equipment to measure phase equilibria is more expensive than at low pressure. The calculation may often serve as a fast alternative compared to measurements, also. However, the calculation of phase equilibria requires suitable EOS or gE-models for the system under investigation. When modelling high-pressure fluid-phase equilibria with supercritical components, the use of EOS is advantageous compared to the use of gE-models, because the latter require arbitrary reference points for the pure supercritical compounds. Unfortunately, the successful use of all EOS is limited to certain groups of compounds and certain process conditions, making necessary the use of different EOS for different problems. So-called predictive methods are advanta-geous, if hardly any information is known about the system under investigation. One example of such predictive EOS is the PSRK model (Holderbaum, 1991, Fischer, 1993, Fischer and Gmehling, 1995/1996) which is based on the UNIFAC group contributions. On the other hand such predictive models are inferior to other models with individual adjustable parameters if sufficient is known about the system under investigation. In such case the adjustment of these parameters can be used to make the EOS accurately represent what is already known in order to use the EOS with these parameters afterwards for calculating what is not known from experiments, yet. The program PE was developed for modelling phase equilibria with non-predictive EOS, excluding the necessity to build, handle and maintain large data bases - but enabling best possible correlations, if sufficient is known about the system. PE offers more than 40 different EOS, allowing the user to choose the one which best fits his needs. PE also offers powerful routines to determine the above mentioned adjustable parameters for pure components and mixtures by correlating experimental data and subroutines to use these parameters for predictions of equilibria not necessarily investigated experimentally yet. The use of the EOS that best fits the users needs is the most important step for achieving good results. The choice of the most suitable EOS for a given problem is up to experience. The choice of the most elaborated, complicated and physically sound EOS does not necessarily guarantee best results. One further key to success is the input in order to determine the adjustable EOS parameters. „Garbage in, garbage out“ is also true, here. This manual is intended to give users all necessary information to handle the program PE, which might be seen as the third and final necessity before receiving the desired correlation results. The manual does not intend to solve the first two challenges.

Chapter 1: Input Files Containing Experimental Data

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1 Input Files Containing Experimental Data The use of EOS in order to model phase equilibria requires pure-component and mixture parameters. The van-der-Waals EOS (Eq. 1.1) with one-fluid mixing and combining rules, (Eq. 1.2 and 1.3), serves as a simple example to demonstrate this.

PRT

v bav

=−

− 2 (1.1)

a a a ki j i jj

N

i

N

ij= ∑∑ −==

ξ ξ11

1( ) (1.2)

bb b

li ji j

ijj

N

i

N=

+−∑∑

==ξ ξ

21

11( ) (1.3)

Pure Components The pure-component parameters ai and bi describe the PvT behaviour and phase equilibria of the pure component of type i. These parameters ought to be derived from pure-component properties of component i, only. Chapter 1.1 shows how experimen-tally determined pure-component properties are stored by PE and how these properties have to be entered in PE for each compound of interest. Chapter 2 shows how these data can be used to obtain pure-component parameters for each EOS and how these experimental results can be compared to results calculated with an EOS. Chapter 2 also shows how properties of pure components can be predicted based on the pure-component EOS parameters.

Mixtures The calculation of mixture parameters (a and b) is based on the mixture composition (mole fractions ξi), the pure-component properties (ai and bi ∀ i) and interactions between unlike molecules (i≠j). Mostly, the interactions between unlike molecules are expressed through deviations of the real mixture behaviour from an artificial mixture behaviour with the same composition and a kind of standard interactions. It is common practice to use so-called binary interaction parameters (like kij and lij in Eq. 1.2 and 1.3) in order to express the deviation from standard interactions (kij=lij=0) in each binary system. These binary interaction parameters ought to be derived from binary properties, only. The binary parameters determined this way can be used for calculations in multicomponent systems assuming that binary interactions add up to the multicomponent behaviour. Chapter 1.2 shows how experimentally determined mixture data are stored by PE and how these data have to be entered in PE. Chapter 3 shows how binary (and ternary) data can be used to obtain the binary interaction parameters,


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how these data can be compared to the calculated results and how predictions can be obtained that are based on these interaction parameters.

1.1 Pure-Component Data

1.1.1 PCFILES Pure-component data are stored in so-called PCFILES (= pure-component files) in the subdirectory /PE/PCDATA. Each PCFILE contains experimental data for one compound. A typical PCFILE is shown below. This PCFILE contains data for carbon dioxide. --------------------------------------------------------------------------------------------------------- component a30 : carbon dioxidedata source a30 : Span and Wagner, 1996data input a30 : created on 1998-07-02 at 16:30Mw, g/mol e13.6e2 : 0.440100E+02Tc, K e13.6e2 : 0.304130E+03Pc, bar e13.6e2 : 0.737700E+02omega, - e13.6e2 : 0.225000E+00

Vapour Pressure Curve=====================T = 0 ! 0=K, 1=C, 2=FP = 0 ! 0=bar, 1=MPa, 2=Torr, 3=atm? = 0 ! 0=density, 1=volumem = 3 ! 0=mol, 1=kmol, 2=g, 3=kgv = 2 ! 0=cm3, 1=dm3, 2=m3

ovP T____________ P____________ RhoL_________ RhoG_________111 0.216590E+03 0.517960E+01 0.117846E+04 0.137610E+02111 0.224000E+03 0.706210E+01 0.115140E+04 0.185300E+02111 0.234000E+03 0.103630E+02 0.111308E+04 0.269360E+02111 0.244000E+03 0.146900E+02 0.107213E+04 0.381840E+02111 0.254000E+03 0.202170E+02 0.102772E+04 0.531440E+02111 0.264000E+03 0.271340E+02 0.978510E+03 0.731240E+02111 0.274000E+03 0.356420E+02 0.922300E+03 0.100320E+03111 0.284000E+03 0.459780E+02 0.854740E+03 0.139090E+03111 0.294000E+03 0.584430E+02 0.764090E+03 0.201060E+03001 0.300000E+03 0.671310E+02 0.679240E+03 0.268580E+03001 0.301000E+03 0.686830E+02 0.658690E+03 0.286150E+03001 0.302000E+03 0.702680E+02 0.633690E+03 0.308150E+03001 0.303000E+03 0.718900E+02 0.599860E+03 0.339000E+03001 0.304000E+03 0.735550E+02 0.530300E+03 0.406420E+03

Isotherms (n= 2)================

T= 250.000 K; (n=20); (code=K0032)P____________ Rho__________0.500000E+00 0.106340E+010.100000E+01 0.213630E+010.200000E+01 0.431200E+010.300000E+01 0.652920E+010.500000E+01 0.110970E+020.100000E+02 0.234350E+020.200000E+02 0.104688E+040.300000E+02 0.105102E+040.400000E+02 0.105501E+040.500000E+02 0.105886E+040.600000E+02 0.106259E+040.750000E+02 0.106796E+040.100000E+03 0.107642E+040.150000E+03 0.109177E+040.200000E+03 0.110547E+040.300000E+03 0.112930E+04


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0.400000E+03 0.114968E+040.500000E+03 0.116759E+040.750000E+03 0.120504E+040.100000E+04 0.123557E+04

T= 300.000 K; (n=20); (code=K0032)P____________ Rho__________0.500000E+00 0.884340E+000.100000E+01 0.177300E+010.200000E+01 0.356360E+010.300000E+01 0.537230E+010.500000E+01 0.904560E+010.100000E+02 0.185790E+020.200000E+02 0.394200E+020.300000E+02 0.633760E+020.400000E+02 0.919650E+020.500000E+02 0.128400E+030.600000E+02 0.182310E+030.750000E+02 0.733900E+030.100000E+03 0.801620E+030.150000E+03 0.865820E+030.200000E+03 0.905570E+030.300000E+03 0.959700E+030.400000E+03 0.998350E+030.500000E+03 0.102894E+040.750000E+03 0.108656E+040.100000E+04 0.112945E+04

Pure-Component EOS Parameters=============================

3 0 1 Peng-RobinsonTc 0.304130E+03Pc 0.737700E+02omega 0.225000E+00info >Parameters describe CO2 well and (obviously) match Tc and Pc

7 1 24 SAFT-Conventionalu0/k 0.311490E+03v00 0.260360E-01m 0.842600E+00eps/k 0.000000E+00kap 0.000000E+00eta/k 0.400000E+02sites 0info >Parameters match Tc and Pc, but m<1 (not reasonable) –> do not use !

7 1 24 SAFT-Conventionalu0/k 0.216080E+03v00 0.135780E-01m 0.141700E+01eps/k 0.000000E+00kap 0.000000E+00eta/k 0.400000E+02Sites 0 0 0info >Parameters do not match Tc and Pc but describe CO2 well -> recommended

3 0 25 PC-SAFTu0/k 0.152478E+03 [K]d00 0.256674E+01 [A]m 0.255961E+01 [-]info >Parameters do not match Tc and Pc but describe CO2 well

--------------------------------------------------------------------------------------------------------- Figure 1.1a: Contents of the PCFILE CARBONDIOXIDE.PC. Each PCFILE consists of a header, data for the vapour pressure curve, optional PvT data, and optional pure-component properties for different equations of state.


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The header The header contains the name of the component, the source of the data (typically „Author, Year“), the date when the user created the PCFILE, the molecular weight, the critical temperature, and pressure of the compound, the acentric factor (Pitzer et al., 1957). If the latter three properties are not known from experiment, they have to be estimated, somehow.

The data for the vapour pressure curve The data block for the vapour pressure curve begins with the definition of the units of the data. In the above example the units for temperature and pressure are Kelvin and bar, respectively. The densities/volumes of the coexisting liquid and vapour phase are given as densities in kg/m3 (compare legend in the unit definition block). The below lines contain the data: temperature, saturation pressure, liquid density and vapour density. The data points have to be listed by ascending temperature. „1“ or „0“ in the first column indicates if the data of this line should be used for pure-component parameter optimisation (section 2.1) or not. „1“ or „0“ in the second and third column indicate, if the data of this line should be displayed in plots of the vapour pressure curve and Pρ-diagrams (section 2.2 and 2.3, respectively). It is recommended that this list includes at least three vapour pressures, that are selected to be used for optimisations, and that these data cover a large representative range of the vapour pressure curve. In order to prevent confusion, later, the „1“/„0“ in the first two columns of each line should be identical. For commonly used compounds, it is not recommended to select data for optimisations which are closer than about 5 K to the critical temperature, because convergence there is slow and not always successful. In the above example the data, that are closer than 10 K to Tc are disabled for optimisations, therefore.

Optional PvT data The third block of the above file contains PvT data in blocks of isothermal data. These data are optional, because they are not used for any parameter optimisations. These data can only be displayed together with calculation results in Pρ-diagrams (section 2.3.1). The header of the PvT-data block contains the number of isotherms. The header of each isotherm contains the temperature of the isotherm, the number of points on the isotherm, and a code for the definition of the units. The code (from left to right) uses the same digits as the code from the vapour pressure curve units block (from top to bottom). The input of temperature has to be in Kelvin.

Pure-component EOS parameters Pure-component EOS parameters can be saved in PCFILES using PE. The EOS parameters saved here, will be used as default parameters for filling blank edit boxes


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for the compound “carbondioxide” by PE. Note that the EOS parameters would not be used for the compound “carbon dioxide” in this example, because the filename without the extension “.pc” is used herefore by PE. This EOS parameter database saves a lot of typing efforts, because once a set of parameters is stored here, it has never to be keyed in, again. It is possible to store more than one EOS parameter set for each EOS in a PCFILE, but only the last one for each EOS will be used by PE. This enables storing old parameters which should not be used any more without the possibility that PE ever selects the wrong set. In this example, there are two parameter sets for SAFT-Conventional and only the latter one would ever be used by PE. The information stored together with the parameters tells all users why the first EOS parameter set should not be used. In general, users are urged to specify additional information together with the saved parameters in the “info” line, so that all users can see what the saved parameters are good for. The EOS parameter data are stored in the format shown in Figure 1.1b and explained, below. --------------------------------------------------------------------------------------------------------- 7 1 24 SAFT-Conventional

u0/k 0.216080E+03v00 0.135780E-01m 0.141700E+01eps/k 0.000000E+00kap 0.000000E+00eta/k 0.400000E+02Sites 0 0 0info >Parameters do not match Tc and Pc but describe CO2 well -> recommended

--------------------------------------------------------------------------------------------------------- Figure 1.1b: EOS parameter block of the PCFILE CARBONDIOXIDE.PC. The first line of the data block contains the number of lines containing any parameters for this equation of state (here “7” lines) as well as the number of these lines which are used for saving the number of association sites (here: “1” of 7 lines), and the internal number of the equation of state in PE (here: “24” for SAFT-Conventional, compare Appendix 1), and the name of the EOS (“SAFT-Conventional”) The 6 lines with the EOS parameters are self explanatory because saved with the parameter names and units. The 7th line “Sites 0 0 0” means that there are neither (+)-sites, (-)-sites nor (COOH)-sites on this molecule, i.e. that the total number of sites is 0, here. In previous versions of PE, only the total number of sites was saved, leading to sites combinations which could not be chosen. The table below “translates” the interpretation of lines in old


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format (starting with “sites” instead of “Sites”, also compare both SAFT data sets in Figure 1.1a) by the new version of PE. Old format New format sites 0 Sites 0 0 0 sites 1 Sites 0 0 1 = one (COOH) site sites 2 Sites 1 1 0 = one (+) and one (-) site = alcohol sites 3 Sites 2 1 0 = two (+) and one (-) site = like water sites 4 Sites 2 2 0 = two (+) and two (-) sites = also like water Figure 1.1c: Format of the sites line. The new version of PE reads old and new format.


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1.1.2 Generating PCFILES Selecting FILE/NEW/PURE-COMPONENT DATA from the main menu will cause creating a new file containing pure-component properties (Figure 1.1d).

Figure 1.1d: Selecting FILE/NEW/PURE-COMPONENT DATA to generate a new PCFILE.

Figure 1.1e: Input of some data for the header of a new PCFILE.

Header First, the user will be asked to enter the data necessary for the header of the PCFILE (compare section 1.1.1, see Figure 1.1e). In order to enable the use of other data bases, later, it is recommended to select a component from the data base /PE/NSTOFF.DAT or add a new compound to the data base by pressing the [...]-button. After pressing the [...]-button, the dialog box shown in Figure 1.1f will appear. Pressing the button SELECT there, will select the highlighted component from the data bank and pressing ADD NEW allows to add new components to this data base.


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In order to enable finding the desired compound in the data base quickly, PE provides the possibility to search for compounds in the data base by pressing FIND. The search can either be by COMPONENT NAME or BY CHEMICAL FORMULA (Figure 1.1g). A search by NAME will not be „sharp“ or case sensitive, i.e. searching for „CARBON“ will find „carbon dioxide“, „carbon monoxide“ and similar. Searching by CHEMICAL

FORMULA will be case sensitive and „sharp“, i.e. searching for „H2“ will not find „H2O“. The chemical formula has to be entered like in the list of the appendix by Reid, Prausnitz and Poling (1987), i.e. „C2H6O“ instead of „C2H5OH“ for ethanol. The button FIND AGAIN repeats the search for the next data base entry matching the search criteria.

Figure 1.1f: Selecting a compound from the pure-component data base. PE allows for new components to be added to the data base together with their chemical formula, molecular weight, and pure-component properties such critical pressure, critical temperature, normal boiling temperature, acentric factor when pressing the button ADD NEW. In the case that the latter properties are not known but the user knows the chemical structure of the new substance, they can be estimated using group contribution methods (see chapter 1.2.4).


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Figure 1.1g: Searching for a compound in the pure-component data base NSTOFF.DAT.

Figure 1.1h: Adding further data for the header of the PCFILE and setting the units for the input of the vapour pressure curve data.

Vapour Pressure Curve After choosing the component, the user has to select the units for the input of the data of the vapour pressure curve and the data (Figure 1.1h). Afterwards, the vapour pressure curve data have to be entered by ascending temperature. Each data point on the vapour pressure curve consists of a temperature, the saturation pressure, the densities (or molar/specific volumes) of both coexisting phases and three flags indicating what the data should be used for (Figure 1.1i, compare section 1.1.1).


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Note: In case saturated vapour densities/volumes are not known, the vapour volumes (densities) should be set to arbitrary high (low) values like 1d+99 (1d-99). Arbitrary low (high) vapour volumes (densities) would lead to undesired program crashes during optimisations. „Good guesses“ for vapour densities could be interpreted as experimental data and used for pure-component parameter optimisations by other users, later. Therefore such guesses should be avoided.

The flag PARAMETER OPTIMISATION indicates, if this data point should be used for pure-component parameter optimisations later, VAPOUR PRESSURE DIAGRAM / PV-DIAGRAM indicate, if the point should be plotted in diagrams showing the vapour pressure curve / PvT data, later. Please, compare sections 2.1, 2.2, and 2.3. It is recommended to use the default, i.e. set all flags for all points. However, for usual compounds, for points closer than approximately 5 K to the critical temperature, the flag PARAMETER OPTIMIZATION should not be set in order to improve convergence in the calculations. By pressing ADD, the point will be added to the PCFILE and PE will prompt for the next point. By pressing ADD+EXIT after having entered the last point, the point will be added to the PCFILE and PE will prompt for no further points on the vapour pressure curve. Afterwards PE will ask if the user wants to enter PvT-data (see next section).

Figure 1.1i: Adding a new point to the vapour pressure curve.

PvT data The last step in creating a new PCFILE is the (optional) input of PvT data. These PvT data are not used for pure-component parameter determinations / optimisations. They are only used for graphical output purposes (section 2.3).


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The PvT data have to be entered using isothermal blocks, i.e. isotherms (Figure 1.1j). Selecting NO there, will close the PCFILE and finish the creation process. At the beginning of the input of each new isotherm, the temperature of the isotherm and the units for the input of data for the isotherm have to be specified - similar to the input of the vapour pressure curve (Figure 1.1k). Different input units can be used for different isotherms.

Figure 1.1j: Starting the input of a new isotherm or finishing the creation of the PCFILE.

Figure 1.1k: Setting the temperature and the units for the input of the next isotherm. Each point on an isotherm at a given temperature consists of a pair [pressure, density] or [pressure, volume] (Figure 1.1l). Pressing the button ADD adds the point to the isotherm and asks for the next point on the same isotherm. Pressing ADD+EXIT adds the point to the isotherm and asks if other isotherms should be entered (dialog box 1.1j, again). The isotherms should be sorted by ascending temperature. The points on the isotherms have to be sorted by ascending pressure.


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Figure 1.1l: Input of a point on an isotherm.


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1.1.3 Generating PCFILES using the data compilation by Daubert and Danner Alternatively to the way described in the above section 1.1.2, a PCFILE can also be created very quickly using the pure-component data compilation by Daubert and Danner (1989). This data compilation contains coefficients for functions describing different pure-component properties like vapour pressure and liquid density. Selecting “FILE/NEW/PURE-COMPONENT DATA, D+D” from the main menu will allow for creating a new PCFILE containing pure-component properties (Figure 1.1m).

Figure 1.1m: Selecting FILE/NEW/PURE-COMPONENT DATA, D+D in order to generate a new PCFILE using the data compilation by Daubert and Danner. Using the dialog box shown in Figure 1.1n, data from the data compilation by Daubert and Danner have to be entered in the sections MAIN DESCRIPTION, LIQUID DENSITY, and VAPOUR PRESSURE. The units for this dialog box are the same as used by Daubert and Danner in order to simplify the typing procedure. So far, Daubert and Danner use their “equation 105” for liquid densities, while they use “equation 101” for vapour pressure correlation:

Equation 101 by Daubert and Danner:

+++= EDTTC

TBAY lnexp

Equation 105 by Daubert and Danner:

−+

= D

CT

B

AY11

Further equations will be build in PE upon request.


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Figure 1.1n: Dialog box for the input of coefficients from the data compilation by Daubert and Danner in order to generate a PCFILE using PE. In order to specify the temperature range of the data points to be written to the PCFILE, the minimum and maximum temperature as well as the step size have to be entered in the section TEMPERATURE RANGE. All point written to the PCFILE will automatically have all flags set for PARAMETER OPTIMISATION, VAPOUR PRESSURE CURVE DIAGRAM, and PV-DIAGRAM (compare Figure 1.1i). Like explained in the previous sections, the maximum temperature should be sufficiently below the critical temperature (normally about 5K) in order to be able to achieve convergence, later. The temperature range chosen here, has to be within the ranges where the coefficients / equations by Daubert and Danner are valid. The optional subsequent input of PvT data is identical to the input described in section 1.1.2.


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1.1.4 Saving EOS parameters to a PCFILE In order to append pure-component EOS parameters to an existing PCFILE using the dialog box shown in Figure 1.1p, the menu entry PURE COMPONENT/SAVE

PARAMETERS TO PC-FILE has to be selected: Figure 1.1o.

Figure 1.1o: Selecting PURE COMPONENT/SAVE PARAMETERS TO PC-FILE in order to append pure-component EOS parameters to an (existing) PCFILE.

Figure 1.1p: Dialog box for appending pure-component EOS parameters to a PCFILE. One line with information has to be saved together with the para-meters.


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1.2 Mixture Data

1.2.1 EXFILES Mixture data are stored in so-called EXFILES (= files with experimental mixture data) in the subdirectory /PE/EXDATA. Each EXFILE contains experimental data for one system with at least two components. A typical EXFILE is shown in Figure 1.2a. This EXFILE contains some data for the binary system p-cresol + carbon dioxide. Each EXFILE consists of a header, pure-component data, pure-component EOS parameters, phase equilibrium data of mixtures, and optional binary EOS interaction parameters.

The Header The header contains the name of the system, the source of the data (typically: author, year), the date when the file was created, and complete information about the literature source of the data (typically: authors’ names, manuscript title, journal, vol., no., year ).

Pure-Component Data The pure-component data section includes the number of components (02, here), the names of the components, and pure-component properties (like molecular weight, normal boiling temperature, critical temperature, critical pressure, acentric factor, and solid density for solid-gas-equilibrium calculations). They latter are stored in such a way that one line contains a given property of all components in the form of an alpha numerical identifier (five letters) and the values for each component.


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--------------------------------------------------------------------------------------------------------- >CO2 + p-Cresol <>Pfohl, 1998 <>created on 1998-06-01 at 14:36<

Literature---------->Dissertation, <>TU Hamburg-Harburg, <>Thermische Verfahrenstechnik <> <

02 components: pure component data---------------------------------->carbon dioxide <>p-cresol <

6 propertiesgmg 0.440100E+02 0.108140E+03Tb 0.022222E+02 0.011111E+03Tc 0.304130E+03 0.704650E+03Pc 0.737700E+02 0.515000E+02omega 0.225000E+00 0.513000E+00s.dns 0.000000E+00 0.100000E+04

03 0 01 Peng-RobinsonTc 0.304130E+03 0.704650E+03Pc 0.737700E+02 0.515000E+02omega 0.225000E+00 0.513000E+00

system data, 1 (mol)--------------------01 point(s), T= 290.00----------------------0 EQ=L1L2, P= 100.0000

carbon dioxide 0.9969000000 0.3160000000p-cresol 0.0031000000 0.6840000000

06 point(s), T= 323.15----------------------1 EQ=L1L2, P= 100.0000

carbon dioxide 0.9969000000 0.3160000000p-cresol 0.0031000000 0.6840000000

1 EQ=L1L2, P= 150.0000carbon dioxide 0.9822000000 0.4200000000p-cresol 0.0178000000 0.5800000000





interaction parameter---------------------EOS= 1 MR=42 BL= 1 saved on 1998-06-01 at 14:360.102050.000000.053050.00000 0.000000.00000 0.000000.000000.00000

--------------------------------------------------------------------------------------------------------- Figure 1.2a: Contents of the EXFILE CO2_PCRESOL.EX


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Pure-Component EOS Parameters The pure-component EOS parameter section includes all pure-component EOS parameters (like v00/dm3mol-1 from SAFT) for calculations with this EXFILE. They are stored in such a way that one line contains a given property of all components in the form of an alpha numerical identifier (maximum: five letters) and the values for each component. All EOS parameters for one given EOS are stored in one block with a one-line header. „03_0_01_Peng-Robinson“ stands for a block containing the pure-component parameters for the Peng-Robinson EOS (number 01 in PE, appendix A1) with 03 lines containing the parameters and similar whereof 0 lines contain the number of association sites (compare EOS 23 - 24, 50 - 54, and 90 - 91 in appendix A1). A list of the properties, numerical codes and properties is given in Appendix A1 together with the different EOS available in PE. When generating an EXFILE, these lines are generated for all properties entered by the user automatically. Appendix A1 only becomes important, in case that additions should be made to the EXFILE, later - or that numerical inputs should be double checked.

Phase Equilibrium Data The phase equilibrium data (SYSTEM DATA) are stored in blocks. The usage of weight fractions or mole fractions for phase compositions in the EXFILE is indicated in the line SYSTEM DATA. Each block has its own header, containing the total number of equilibria in the block and common properties of all equilibria in the block (like the temperature for a block containing isothermal data, here). Each entry for an equilibrium consists of a one-line header and one line for each compound which contains the mole or weight fractions of the compound in all equilibrium phases. The header contains the type of equilibrium (either V_L1, V_L2, L1L2, or VL12), properties that might be different from the properties of the next equilibrium (like the pressure in an isothermal block of a binary system, here), and flags indicating what the equilibrium will be used for, later (Table 1.1). Table 1.1: Flags characterising the use of single equilibria in an EXFILE. Flag Usage of equilibrium in PE 0 This equilibrium will not be used for the calculation of the Objective

Function. This equilibrium will not be plotted in diagrams. 1 This equilibrium will be used to compare calculation results with

experimental data, i.e. for calculating the Objective Function. The equilibrium will also be plotted in diagrams.

2 This equilibrium will not be used for the calculation of the Objective Function, but this equilibrium will be plotted in diagrams.


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Table 1.2: Usage of flags for single equilibria in an EXFILE Flag Usage in case: 0 a) If equilibrium calculations should be performed at conditions where no

experimental data are available using ADDITIONAL POINT CALCULA-TIONS (section 3.1), the creation of a block with an INITIAL GUESS for the calculations is necessary. Block 1 (T=290K) in Figure 1.11 contains an INITIAL GUESS, only. The INITIAL GUESS was obtained from T=323.15K (compare block 2).

b) If thermodynamic models are poor, not all experimental data can be correlated, because the flash algorithm does not converge any more. The model might predict a one phase region where two or three phase regions have been measured experimentally. In this case it is advisable to mark the point where the model does not yield calculated data as „point not to be used“. Running optimisations with such EXFILES are much faster, because not all points are used to optimise the parameters any more. Especially the calculations where no convergence can be achieved (and that need a long time therefore) are avoided.

c) In case binary interaction parameters are varied using a large GRID in order to find a suitable pair of binary parameters to start an OPTIMISATION (section 3.2), it is advisable to mark all except one point with „0“ (or „2“) in order to speed up the grid calculations. The point excluded from the calculations this way should be included again before doing the optimisations, afterwards - if possible.

1 This is the normal selection for any experimental equilibrium where all system components are present in all phases (at least traces).

2 a) In binary systems, the vapour pressure points of the pure compounds are often added in order to see them in the binary diagrams, later. Because binary calculations can not be done at pure-component tie lines and optimisations of binary interaction parameters can not be done based on pure-component data, these data must be marked as PLOT, ONLY.

b) In ternary systems the tie lines of the binary subsystems are often included in the EXFILE in order to display them in the same graph as the ternary data (see section 3.4). On the other hand, it might be desired to calculate errors only in the calculated mole fractions of the true ternary tie lines in order to judge about the capabilities of EOS to predict ternary data based on interaction parameters of regressed from binary data (compare section 3.1). Relative errors can not be calculated based on zero mole fractions anyway (division by zero!).


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Binary Interaction Parameters This section contains binary interaction parameters for the EXFILE system and different EOS and mixing rules. When creating a new EXFILE, no binary interaction parameters are stored. After regressing binary interaction parameters (section 3.2) these parameters can be appended to the EXFILE. Each interaction parameter block contains an one-line header and the binary parameters. The header includes the number of the EOS, mixing rule, and system data block for which the parameters have been saved. A list of EOS and mixing rules is given in appendices A1 and A2. This way different interaction parameters can be used for calculating different system data blocks. The parameters are stored in loops like do i = 1, N-1

write k(i,j) (j= i+1, N)end do. Where: N = number of components in the system.

The order of appearance is: k(i,j), l(i,j), λ(i,j), s(1,j), s(j,1), kε(i,j), kκ(i,j). The s(i,j) parameters are only used for the Dohrn-Prausnitz-polar EOS (number 6 in appendix A1). They are saved in a format like: write s(1,j) (j= 1, N)write s(j,1) (j= 1, N).

1.2.2 Generating EXFILES Selecting FILE/NEW/MIXTURE DATA from the main menu initiates the creation of a new EXFILE containing mixture data (Figure 1.2b).

Figure 1.2b: Selecting the menu entry to generate a new EXFILE with mixture data.


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Header First, the user will be asked to enter the data necessary for the header of the EXFILE (see section 1.2.1). The dialog box for this is shown in Figure 1.2c. The user also has to decide if the input of composition will be in mole fractions or weight fractions. PE will save the original numerical input to the EXFILE and do the conversions at the time of phase equilibrium calculation.

Figure 1.2c: Input of data for the header of a new EXFILE.

Pure-Component Data Second, the user will be prompted for an input of all component names and molecular weights (Figure 1.2d). The [...]-BUTTONS allow access to the same pure-component data base /PE/NSTOFF.DAT as already described in section 1.1.2 GENERATING

PCFILES.


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Figure 1.2d: Input of names and molecular weights of all components in the EXFILE. Finally, the user is prompted for the input of pure-component properties and pure-component EOS parameters. Pure-component information from the data base will be used as default values for pure-component properties for the header of the EXFILE (Figure 1.2e). If no experimental pure-component data are available for Tc, Pc, Tb, and ω, these can be estimated by pressing ESTIMATE (details in section 1.2.4). Mixture calculations with a given EOS can only be performed if all necessary parameters for this EOS and all components are entered. Some simple EOS like van-der-Waals, Peng-Robinson and Soave-Redlich-Kwong, which only depend on the parameters Tc, Pc, and ω, will use this information in the pure-component property block for calculation routines as long as no block for the EOS itself is present. EOS parameters for each EOS can be entered after double-clicking the appropriate EOS in the dialog box shown in Figure 1.2f. Boxes like the ones shown in Figure 1.2g will appear for each compound. If a PCFILE with the name of one of the components in the EXFILE exists, PE will try to retrieve EOS parameters from this file as default parameters for the dialog box in Figure 1.2g. Selecting SAVE+EXIT in Figure 1.2f will lead to the next step - the input of the phase equilibrium data.


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Figure 1.2e: Input of pure-component data. Figure 1.2f: Selecting an EOS for the input of pure-component parameters.

Figure 1.2g: Input of pure-component parameters for different EOS.


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Phase Equilibrium Data For binary systems, the phase equilibrium data have to be supplied in blocks containing isothermal data. For ternary systems, the phase equilibrium data have to be supplied in blocks containing experimental data at identical temperature and identical pressure - only varying in the composition of the coexisting phases. PE will assist in generating such blocks in the EXFILE, if the input of the data is in the correct order. Selecting the check box NEW BLOCK will create a new block (Figure 1.2h). The correct order for a binary system is: (compare Figure 1.2k): 1) T1 / P1 \ 2) T1 / P2 > block 1 3) T1 / P3 ... / 4) T2 / P4 \ 5) T2 / P5 > block 2 6) T2 / P6 / 7) T3 ... > block 3 ... The correct order for a ternary system is: 1) T1 / P1 / equilibrium A \ block 1 2) T1 / P1 / equilibrium B ... / 3) T1 / P2 / equilibrium C \ block 2 4) T1 / P2 / equilibrium D ... / 5) T2 / P3 / equilibrium E \ block 3 6) T2 / P3 / equilibrium F ... / 7) T3 ... > block 4 ... Those data in a block which should not be used for the calculation of the OBJECTIVE

FUNCTION, have to be marked as PLOT, ONLY. The first equilibrium in each block will be used as INITIAL GUESS for the CALCULATION

OF ADDITIONAL POINTS (section 3.1). If this first point is not an experimentally determined point, or if it should not be used for the calculation of the OBJECTIVE

FUNCTION for a different reason, the first point can be marked as INITIAL GUESS with the appropriate radio button. See section 1.2.1 (Tables 1.1 and 1.2) for details. It is not possible to add any data with flag=0 to a block except the INITIAL GUESS. The flags of all equilibria can be changed by loading the EXFILE with an ASCII editor of the PE-Viewer, later.


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Figure 1.2h: Dialog box for the input of one phase equilibrium in an EXFILE block. For each phase equilibrium, the type of equilibrium has to be specified. There are four possibilities: a) L1-V, b) L2-V, c) L1-L2, and d) L1-L2-V. Figure 1.2i and Figure 1.2j provide guidance to choose the correct types of equilibrium for each phase equilibrium. The difference between L1-V and L2-V equilibrium becomes important, if different miscibility gaps are encountered in one isothermal phase equilibrium block. If the types of miscibility gaps are not chosen correctly, the plot programs will connect the wrong tie lines in order to receive binodals in diagrams, later. If a three-phase region was observed experimentally at a given temperature and pressure, this phase equilibrium should be entered as the first in the block. This speeds up the calculation of ADDITIONAL POINTS enormously (section 3.1) and prevents the (wrong) calculation of tie lines through the three phase region. If a three phase region is likely to occur / to be calculated - but was not found by experiment - the first equilibrium in the block should be an INITIAL GUESS containing a three phase region. Pressing the button ADD will add the phase equilibrium to the EXFILE and ask for further equilibria. Pressing the button ADD+EXIT will add the phase equilibrium to the EXFILE and finish the generation of the EXFILE.


- 31 -

New: If one phase composition edit box per phase is left empty in the dialog box shown in Figure 1.2h, PE will automatically set the composition for this edit box so, that the sum of fractions in this phase becomes unity and ask the user to confirm this setting.

Figure 1.2i: miscibility gaps in a binary system. Figure 1.2j: miscibility gaps in a ternary system.

Phase: L1 Phase: L2

Phase: V

Phase: V

Phase: L2

Phase: L1

three-phase region


- 32 -

1.2.3 Editing EXFILES Selecting FILE/ADD/EQUILIBRIUM DATA and FILE/ADD/PURE-COMPONENT EOS PARAMETERS allows to add equilibrium data and pure-component parameters to an existing EXFILE, respectively (Figure 1.2k). Note: in order to use an edited EXFILE, that was opened for mixture calculations by PE during editing, for calculations afterwards, the EXFILE has to be reopened (compare 3.1) – PE will ask for reopening such a fíle, automatically.

Figure 1.2k: Selecting a menu entry in order to add equilibrium data to an existing EXFILE. Adding pure-component EOS parameters Selecting FILE/ADD/PURE-COMPONENT EOS PARAMETERS from the main menu allows to add further pure-component EOS parameters to an existing EXFILE and allows to change existing pure-component EOS parameters in an EXFILE. A dialog box (Figure 1.2f) will ask for the EOS for which the parameters should be added/changed. Double clicking an EOS will ask for the new parameters of all compounds for this EOS (Figure 1.2g). Existing EOS parameters will be displayed as defaults. Selecting SAVE+EXIT will save the data to the EXFILE, selecting ABORT will terminate the input without changes in the EXFILE. Adding mixture data Selecting FILE/ADD/EQUILIBRIUM DATA from the main menu allows to add further mixture data to an existing EXFILE. The input of data is arranged in blocks like described in section 1.2.2 GENERATING EXFILES.


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1.2.4 Estimating Pure-Component Parameters When generating the pure-component data block for a new EXFILE (Figure 1.2e), it might occur, that the properties Tc, Pc, Tb, and ω are not known from experimental sources, because of thermal decomposition or experimental difficulties. In order to enable calculations with some simple equations of state, that only need the critical pressure, the critical temperature, and the acentric factor as pure-component parameters (like the Peng-Robinson EOS), PE offers some methods for estimating these properties using group contribution methods or empirical relations.

1.2.4a Group Contribution Methods If the chemical structure of a compound is known, this compound can be split into defined chemical groups whereof each is thought to contribute to a certain pure-component property to a certain extent. The authors of the methods built in PE, have investigated many substances and have determined the average contribution to a certain property for each group. Based on this, the pure-component properties can now be estimated as a sum of the determined contributions of each group in the molecule. The two group contribution methods implemented in PE are the well known ones by Joback and by Fedors for estimating Tc, Pc and Tb (Reid et al., 1987).

1.2.4b The Method by Fedors This method is used for estimating the critical temperature, Tc, only. The method does not require additional data except the tabulated ∆T-values listed below and stored by PE. The Fedors’ relation can be written as:

∑∆= TKTc lg535/

Table 1.3: group contributions according to Fedors Group Implementation in PE T∆

3CH− -CH3 1.79

−− 2CH -CH2 1.34

−> CH >CH- 0.45

<> C >C< -0.22

2CH= =CH2 1.59

−= CH =CH-- 1.40

=> C >C= 0.89

CH≡ =3=CH 1.79

−≡ C =3=C- 2.46

== C =C= 1.03


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COOH− -COOH 10.72

)(anhydrideOCOCO −−−− -CO-O-OC- (anh.) 7.95

−−− OCO -CO-O- 5.32

)(oxalateOCOOCO −−−−− -O-OC-CO-O- (oxalate) 6.25

−− CO -CO- 5.36

−− O -O- 1.56

)(aromaticO −− -O- (aromatic) 2.68

OH− -OH 5.63

)(aromaticOH− -OH (aromatic) 9.65

CHO− -CHO 5.49

NC ≡− -CN 8.49

)(aromaticNC ≡− -CN (aromatic) 9.38

2NH− -NH2 4.56

)(2 aromaticNH− -NH2 (aromatic) 9.20

−− NH -NH- 3.04

)(aromaticNH −− -NH- (aromatic) 7.64

)(aromaticNH −> >N- 0.89

)(aromaticN −> >N (aromatic)- 0.89

−> N >N- 4.74

=− N -N= 4.51

−−− SS -S-S- 9.83

−− S -S- 4.91

SH− -SH 5.36

F− -F 2.10

)(aromaticF− -F (aromatic) 0.45

)( perfluoroF− -F (perfluoro) 0.54

Cl− -Cl 4.20

)( teddisubstituCl− -Cl (disubsituted) 3.71

)( utedtrisubstitCl− -Cl (trisubsituted) 3.17

Br− -Br 5.58

I− -I 8.04

)(aromaticI− -I (aromatic) 10.77

ringmemberedThree − 3-membered ring 0.45

ringmemberedFive − 5-membered ring 2.23

ringmemberedSix − 6-membered ring 2.68

ringinHeteroatom Heteroatom in ring 0.45

)(

cnonaromatibounddoubleaincarbonononSubstituti

Subst. on C in a DB nonar. 0.58

ringbenzeneainitutionOrthosubst o-Subst. in a benzene ring 1.16

bounddoublepernConjugatio , Conjugat. per double bond 0.13

−>− CHofpairsadjacentCH ** CH ** adjacent pairs of >CH- 0.76


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1.2.4c The Method by Joback The method by Joback has been developed for estimating the critical temperature Tc, the critical pressure Pc, the critical volume, and the normal boiling temperature Tb. The method for estimating the critical volume is not implemented in PE. The formula used in PE are:

∑∆+= bb TKT 198/

( )2965,0584,0/

/cc

bc TT

KTKT

∆−∆+=

∑

( )20032,0113,01/

∑∆−+=

cAc

PnbarP ,

where nA is the total number of atoms in the molecule. Table 1.4: group contributions according to Joback Group Implementation in

PE bT∆ cT∆ cP∆

3CH− -CH3 23.58 0.0141 -0.0012

−− 2CH -CH2- 22.88 0.0189 0

−> CH >CH- 21.74 0.0164 0.0020

<> C >C< 18.25 0.0067 0.0043

2CH= =CH2 18.18 0.0113 -0.0028

−= CH =CH- 24.96 0.0129 -0.0006

=> C >C= 24.14 0.0117 0.0011

== C =C= 26.15 0.0026 0.0028

CH≡ =3=CH 9.20 0.0027 -0.0008

−≡ C =3=C- 27.38 0.0020 0.0016

)(2 ringCH −− -CH2-(ring) 27.15 0.0100 0.0025

)(ringCH −> >CH-(ring) 21.78 0.0122 0.0004

)(ringC <> >C<(ring) 21.32 0.0042 0.0061

)(ringCH −= =CH-(ring) 26.73 0.0082 0.0011

)(ringC <= =C<(ring) 31.01 0.0143 0.0008

F− -F -0.03 0.0111 -0.0057

Cl− -Cl 38.13 0.0105 -0.0049

Br− -Br 66.86 0.0133 0.0057

I− -I 93.84 0.0068 -0.0034

)(alcoholOH− -OH (alcohol) 92.88 0.0741 0.0112

)( phenolOH− -OH (phenol) 76.34 0.0240 0.0184

)(nonringO −− -O- (nonring) 22.42 0.0168 0.0015


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)(ringO −− -O- (ring) 31.22 0.0098 0.0048

)(nonringOC => >C=O (nonring) 76.75 0.0380 0.0031

)(ringOC => >C=O (ring) 94.97 0.0284 0.0028

)(aldehydeCHO −= O=CH- (aldehyde) 72.24 0.0379 0.0030

)(acidCOOH− -COOH (acid) 169.09 0.0791 0.0077

)(esterCOO −− -COO- (ester) 81.10 0.0481 0.0005

) ( aboveasexceptO= =O (exept as above) -10.50 0.0143 0.0101

2NH− -NH2 73.23 0.0243 0.0109

)(nonringNH> >NH (nonring) 50.17 0.0295 0.0077

)(ringNH> >NH (ring) 52.82 0.0130 0.0114

)(nonringN −> >N- (nonring) 11.74 0.0169 0.0074

)(nonringN =− -N= (nonring) 74.60 0.0255 -0.0099

)(ringN =− -N= (ring) 57.55 0.0085 0.0076

CN− -CN 125.66 0.0496 -0.0101

2NO− -NO2 152.54 0.0437 0.0064

SH− -SH 63.56 0.0031 0.0084

)(nonringS −− -S- (nonring) 68.78 0.0119 0.0049

)(ringS −− -S- (ring) 52.10 0.0019 0.0051

1.2.4d The Method by Klincewicz A simple relation derived by Klincewicz can also be used for estimating the critical temperature. This relation requires the normal boiling temperature and the molecular weight only:

KTmolgMKT bwc /41,1)//(16,02,50/ +−= If the user did not already enter the required parameters he will be asked to do this: Figure 1.2l.

Figure 1.2l: Input of parameters for estimating the critical temperature according to the equation by Klincewicz.


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1.2.4e Calculation of the Acentric Factor

The definition for the acentric factor is:

1)7,0(lg −=−= rsat

r TPω , where

rT is the reduced temperature cTT / . sat

rP is the reduced vapour pressure csat PP / .

The calculation method for the acentric factor implemented in PE uses the Lee-Kesler vapour pressure relation (Reid et al., 1987):

6

6

43577,0ln4721,136875,152518,15

169347,0ln28862,109648,692714,5/ln

θθθ

β

θθθ

α

+−−=

−++−−= atmPc

,

where c

b

TT

=θ .


- 38 -

1.2.4f Implementation in PE The group contribution methods in PE can be reached by selecting the menu entry FILE/DATABASE in order to edit the pure-component data base nstoff.dat (Figure 1.2m).

Figure 1.2m: Selecting FILE/DATA BASE. If the user chooses the ADD NEW button in the occuring dialog box (Figure 1.1f), another dialog box is displayed where the parameters for the new component can be entered (Figure 1.2n). By pressing the ESTIMATE button in this dialog box, the required property can alternatively be estimated according to the procedures described above.

Figure 1.2n: Dialog box with estimate buttons for group contribution methods.


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If the normal boiling temperature is known from experiment, it is strongly recommended that this value is entered in the appropriate edit box, because the subsequent estimate of the critical temperature will be more precise when using the methods according to Joback or Klincewicz, hereafter. If the normal boiling temperature is not known, it can be estimated with the method of Joback. Depending on the property which should be estimated, the standard dialog box with different radio buttons for the different estimation methods will show those methods disabled, that can not be used (Figure 1.2o).

Figure 1.2o: Choosing a method in order to estimate Tc.

Using the Fedors or Joback method, the dialog box shown in Figure 1.2p is displayed. This dialog box is the same for all group contribution methods – only, the list of groups in the list box on the right side differs depending on the method chosen. There is an example defining the structure of 2,2,3-trimethylpentan in the dialog box, below (Figure 1.1p). 2,2,3-Trimethylpentan contains five methyl groups ( 3CH− ), one 2CH− group, one CH− , and one C− group.

Using the GROUP SELECTOR BOX 1) Selecting a group in the right list box, the user is asked to enter the amount of

groups that should be used for the estimation. Afterwards, this group is shown in the left list box together with the amount chosen for this group.

2) If a group should be removed from the left list of groups to be used for the estimation, the user has to click on the selected group in the right list box again and enter “0” for the amount of groups. Hereafter, the group is automatically deselected in the right list box and removed from the left list box.


- 40 -

Figure 1.2p: Group selector box for group contribution methods.

1.2.4g Calculation of the Acentric Factor The Lee-Kesler method for estimating the acentric factor requires knowledge of the normal boiling temperature, the critical temperature, and the critical pressure. The calculation of the acentric factor is performed by pressing the CALCULATE button (Figure 1.1n). PE will display an additional dialog box (Figure 1.1q) allowing to change the input parameters before they are used for calculation. In this dialog box, the user can change these parameters in order to calculate with other parameters than those in the database dialog box (Figure 1.1n). When pressing OK, the calculated value for the acentric factor will be placed in the database dialog box (Figure 1.1n). If the check box UPDATE PARAMETERS was checked (Figure 1.1q), the parameters Tb, Tc, and Pc in the database dialog box (Figure 1.1n) are additionally replaced by the ones used to calculate ω.

Figure 1.2q: Dialog box for parameter input when estimating the acentric factor.

Chapter 2: Pure-Component Calculations

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2 Pure-Component Calculations PE offers two calculations modes: the pure-component mode and the mixture mode. All calculations in the pure-component mode are based on the experimental data in the PCFILES (section 1.1) and can be accessed by the options from the menu entry PURE

COMPONENT (Figure 2a).

Figure 2a: Menu entries for PURE-COMPONENT subroutines. The entries allow: 1) to DETERMINE EOS PARAMETERS, that reproduce a vapour pressure curve best and

to PLOT diagrams giving evidence about the suitabilities of EOS and parameters to model certain compounds (Section 2.1).

2) to calculate complete VAPOUR PRESSURE CURVES and to PLOT calculated and experimentally determined vapour pressure curves (Section 2.2),

3) to CALCULATE PVT DATA and to PLOT the calculated and experimentally determined PvT data in Pρ-diagrams (Section 2.3),

4) to CALCULATE DEVIATIONS between calculated and experimentally determined properties of the vapour pressure curve (Section 2.4).

5) to append EOS parameters to the file containing the experimental data (Section 1.14)

2.1 Determination of Pure-Component EOS Parameters The calculation of pure-component properties and mixture properties with EOS requires pure-component EOS parameters to describe the pure compounds. There are several methods to obtain these parameters: 1) EOS parameters can be determined from the critical data of the pure component. For cubic

EOS it is common practice to choose two parameters in such a way that the calculated


- 42 -

isotherm with the experimentally determined critical temperature has a point of inflection at the experimentally determined critical pressure. This way the calculated critical temperature and pressure exactly match the experimentally obtained critical temperature and pressure. Such parameters are said to “enforce (the correct reproduction of experimentally determined) Tc and Pc”, below. Further EOS parameters might be chosen in such a way that the calculated critical volume matches the experimentally determined critical volume (Deiters, 1981) or that the temperature dependence of the vapour pressure curve is reproduced well (Soave, 1972).

2) EOS parameters can be chosen in such a way that properties which are calculated with the EOS reproduce certain experimentally determined properties best. The properties to be matched are chosen to be similar to the properties that will be calculated with the EOS, later. For calculating phase equilibria it is common practice to choose the liquid molar volume and the pressure at saturation conditions (Huang and Radosz, 1990). Working with physically sound EOS, the reproduction of the molar liquid volume is also a good measure for liquid densities at higher pressures. However, it is no good measure, if empirical temperature dependent volume translations are used and if the EOS parameters have been adjusted to match the saturated liquid volumes, only (Chou and Prausnitz, 1989, Ji and Lempe, 1997). In general, vapour volumes are not included in the optimisations, because they depend on the ideal gas law to a high extent and because they are not strongly correlated to molecular parameters, therefore.

3) EOS parameters can be obtained by using correlations that are based on EOS parameters of similar compounds that are determined by one of the first two methods (Dohrn and Brunner, 1994).

The menu entry PURE COMPONENT/DETERMINE PARAMETERS allows to determine EOS parameters according to method 1 and 2. Selecting this menu entry, the user will be asked, if calculations with the EOS parameters to be determined should match the critical temperature and critical pressure determined experimentally (Figure 2.1a).

Figure 2.1a: Dialog box to choose if EOS parameters should match Tc and Pc. The EOS parameters will be determined to allow the best possible reproduction of the pure-component vapour pressure curve in the PCFILE of interest. Only points where the o-flag in the PCFILE is set to „1“ are considered (compare section 1.1.1).


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Enforcing EOS parameters that exactly reproduce the experimental values of Tc and Pc from the PCFILE during the optimisations, restricts the choice of parameters significantly. In such case, the first two parameters of the EOS will be set to values so that Tc and Pc from experiments are reproduced exactly, automatically. For certain EOS this procedure can significantly affect the quality of the reproduction of the vapour pressure curve and densities (compare Figures 2.1f - 2.1j). The dialog box shown in Figure 2.1b is the main input form for the optimisation of pure-component parameters. The entries are explained below. If the reproduction of Tc and Pc is enforced and the first two pure-component EOS parameters of the selected EOS are Tc and Pc, the input of both is disabled (Figure 2.1c). If the reproduction of Tc and Pc is enforced and the first two pure-component EOS parameters of the selected EOS are not Tc and Pc, the input of both is enabled for the input of initial guesses - the procedure for this case is described by Pfohl et al., 1998.

Figure 2.1b: Dialog box for an optimisation of pure-component EOS parameters.


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Figure 2.1c: Dialog box for an optimisation of pure-component EOS parameters with an EOS where Tc and Pc are the first two parameters and where the correct reproduction of Tc and Pc should be enforced.

Substance With the button SUBSTANCE, the PCFILE with the pure-component data is selected. By default the directory PE\PCFILE is selected but the user can open any PCFILE on every disk and directory.

EOS With the button EOS, the EOS to be used is selected (compare section 1.2.2).

Parameters The section PARAMETERS displays the EOS parameter names and units. The values for the parameters will be used as initial guesses for the optimisations. It is advisable to give good estimates based on method 3 from above in order to ensure rapid convergence. It is also advisable to use initial guesses which do not underestimate the critical temperature, otherwise the optimisation procedure might not start at all.


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Sites The use of EOS which explicitly account for association like SAFT and the CPA-EOS enforces the input of the number of interacting sites. For one molecule, either COOH-sites can be selected or any combination of (+)- and (-)-sites. (+)-sites will only affect calculation results, if any (-)-sites are present within the system - and vice versa. One COOH-site is used for simple carboxylic acids (dimer formation, only); one (+)-site and one (-)-site are used for simple alcohols (chain formation); two (+)-sites and one (-)-site might be used for water (clusters formation). Two (+)-sites and two (-)-sites may also be used for water (compare with Huang and Radosz (1991) and Fu and Sandler (1995)). Also compare section 1.1.1.

Optimize Parameters where the OPTIMIZE boxes are checked will be optimised using a Simplex-Nelder-Mead algorithm (compare Nelder and Mead, 1965).

Grid One parameter can be selected to be varied on a grid, that is specified by the lower and upper border as well as the grid increments. The one-dimensional output lists can be visualised, later (section 2.1.1). It might be useful to generate such plot in order to judge about the magnitude of the influence of a certain EOS parameter on the calculation results. A parameter can be either optimised or put a grid.

Running an Optimisation/Grid An optimisation can be interrupted by selecting the STOP button from the toolbar. The STOP button and STOP menu entry are only visible during optimisations/grid. After each successful optimisation the achieved results will be displayed in a dialog box (Figure 2.1d).

Figure 2.1d: Optimisation results: Reproduction of vapour pressure curve in the PCFILE.


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The averaged relative errors in vapour pressure, liquid density, and vapour density are shown as well as the value of the OBJECTIVE FUNCTION which was minimised during the optimisation. The OBJECTIVE FUNCTION is the mean of the deviation in vapour pressure and liquid density. The differences between the calculated and experimentally determined critical temperature and pressure are also shown. Such calculation is also available at PURE COMPONENT / CALCULATE DEVIATION.

2.1.1 Visualisation of Grids/Optimisations The results of pure-component parameter determinations can be visualised, if there was a GRID on one parameter - or if Tc and Pc have been enforced during the determination by selecting PURE COMPONENT/DETERMINE PARAMETERS/PLOT from the menu. These deviation diagrams allow to judge about the influence of a certain parameter on the quality of the reproduction of the properties on the vapour pressure curve (compare Pfohl et al., 1998). The dialog box shown in Figure 2.1e allows to set the limits of the axis in the diagram and allows to select the range of points to be shown in the diagram. Figures 2.1f-2.1j show examples of such deviation diagrams.

Figure 2.1e: Setting the limits of the plot axis and determining the points to be shown. Figure 2.1f and 2.1g show diagrams for carbon dioxide and the Peng-Robinson EOS, enforcing and not enforcing experimentally determined Tc and Pc. The diagrams are similar because the constraint to enforce Tc and Pc is not severe, here. It can be seen that the acentric factor has a pronounced influence on the calculation of the vapour


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pressures but not on liquid volumes. Figure 2.1h shows that the volume translation added to the Peng-Robinson EOS (EOS 10 in appendix A1) has no influence on the calculation of vapour pressures, because it was designed to have no influence on it (Peneloux et al., 1982). Figures 2.1i and 2.1j show diagrams for carbon dioxide and SAFT, also enforcing and not enforcing experimentally determined Tc and Pc. The diagrams look very different, because the constraint to enforce Tc and Pc is very severe, here. The choice of the chain length m has a pronounced influence on calculated vapour pressures and liquid volumes.

Figure 2.1f & 2.1g: Deviation analysis for CO2 described by the Peng-Robinson EOS, when the correct reproduction of experimentally determined Tc and Pc is enforced (left side) or not (right side). Vapour pressure deviation: - - - , Density deviation: -----.

Figure 2.1h: Deviation analysis for CO2 described by the PR EOS with volume translation by Peneloux, when the correct reproduction of experimentally determined Tc and Pc is enforced. Vapour pressure deviation: - - - , Density deviation: .


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Figure 2.1i and 2.1j: Deviation analysis for CO2 described by SAFT, when the correct reproduction of experimentally determined Tc and Pc is enforced (right side) or not (left side). Vapour pressure deviation: - - - , Density deviation: .

2.2 Calculation of Vapour Pressure Curves A complete vapour pressure curve, starting from a MINIMUM TEMPERATURE (to be specified) and ending at the critical point, can be calculated by selecting the menu bar entry PURE COMPONENT/VAPOUR PRESSURE CURVE (Figure 2.2a). The settings for the MINIMUM TEMPERATURE must not be too low, otherwise calculations might not converge. The dialog box items (Figure 2.2b) are practically identical to the ones from the dialog boxes described above and therefore not described here.

Figure 2.2a: Selecting to CALCULATE A VAPOUR PRESSURE CURVE.


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Figure 2.2b: Dialog box for the calculation of a vapour pressure curve.

2.2.1 Graphical Output of Vapour Pressure Curves The last vapour pressure curve which has been calculated according to the description in the above section can be visualised by selecting the menu item PURE COMPONENT / VAPOUR PRESSURE CURVE / PLOT. The vapour pressure curve can be either displayed with linear axis or as log(PSAT) against 1/T (Figure 2.2c - 2.2d).


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Figure 2.2c: Vapour pressure curve of carbon dioxide with liner axis.

Figure 2.2d: Vapour pressure curve of carbon dioxide with logarithmic axis


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2.3 Calculation of Pρρρρ-Diagrams A complete Pρ-diagram can be calculated by selecting the menu bar entry PURE

COMPONENT / PVT DIAGRAM (Figure 2.3a). The limits for the ρ-axis have to be specified (Figure 2.3b). First, the extend of the vapour-liquid two-phase region is calculated, then isotherms will be calculated. The minimum accessible temperature is the minimum temperature specified in the box similar to Figure 2.2b. For each temperature of an isotherm specified in the PCFILE, one isotherm will be calculated. Additional isotherms can be calculated by specifying them in the dialog box shown in Figure 2.3b.

Figure 2.3a: Selecting to calculate a Pρ-diagram.

Figure 2.3b: Specifying limits for the ρ-axis in the Pρ-diagram and selecting additional isotherms to be calculated.


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2.3.1 Graphical Output of Pρρρρ-Diagrams The last Pρ-diagram which has been calculated according to the description in the above section can be displayed by selecting the menu item PURE COMPONENT / PVT DIAGRAM / PLOT. The limits of the ρ-axis specified at the time of calculation will be used. The limits of the P-axis have to be specified at the time of the generation of the diagram (Figure 2.3c). A Pρ-diagram for carbon dioxide is shown in Figure 2.3d.

Figure 2.3c: Input of limits for the P-axis of the Pρ-diagram.

Figure 2.3d: Pρ-diagram for carbon dioxide.


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2.4 Calculation of Deviations In order to judge about the quality of given pure-component EOS parameter sets or find good initial guesses for pure-component parameter optimisations, it is possible to calculate the deviation between the experimental vapour pressure curve data in a PCFILE and calculated properties using an EOS by selecting PURE COMPONENT / CALCULATE DEVIATIONS from the main menu (Figure 2.4a).

Figure 2.4a: Selecting CALCULATE DEVIATIONS from the PURE-COMPONENT menu. After selecting PURE COMPONENT / CALCULATE DEVIATIONS from the menu, the VAPOUR PRESSURE CALCULATION DIALOG BOX will appear. The VAPOUR PRESSURE

CALCULATION DIALOG BOX asking for the EOS, PCFILE and EOS parameters is the same as already shown in Figure 2.2b. Because it is handled the same way as already described in Sections 2.1/2.2, this description is not repeated here. In this dialog box, the edit box for the minimum temperature is always disabled when calculating deviations, because all data in the PCFILE with O-FLAG=1 (Section 1.1.1 and 2.1) are used independently from this setting. The calculation results, i.e. deviations between experimental and calculated data are shown in the well known deviation dialog box which has already been shown in Figure 2.1d.

Chapter 3: Mixtures

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3 Mixtures PE offers two calculations modes: the pure-component mode and the mixture mode. All calculations in the mixture mode are based on the experimental data and EOS parameters in the EXFILES (section 1.2). After having chosen an EXFILE to work with by selecting FILE/OPEN, the menu entry MIXTURE will appear (Figure 3a) and the most important information about the next calculations will be displayed in the tool bar of the main window of PE (Figure 3b).

Figure 3a: The menu MIXTURES appears after an EXFILE has been opened.

Figure 3b: Information about the mixture mode in the tool bar of the main window. The first line in the tool bar of the main window displays the number of components in the system (2 = binary system), the EOS (simple Peng-Robinson, appendix A1), mixing rule (Mathias-Klotz-Prausnitz with 3 parameters, appendix A2), and the data block in the EXFILE which should be the basis for the next calculations. The second line in the tool bar of the main window displays the name of the EXFILE and the system name in the EXFILE.

Chapter 3: Mixtures

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It is important to note that PE can exclusively calculate mixture properties with the EOS parameters in the EXFILE and that PE can only perform phase equilibrium calculations at the conditions of a single block in an EXFILE. For binary systems such a block contains isothermal data (section 1.2) in order to enable the calculation of Pxy-diagrams. For ternary systems, such a block only contains data at identical temperature and pressure (section 1.2), so a Gibbs triangle can be calculated for T and P of the block. Future routines will enable additional calculations, also. In case phase equilibrium calculations should be performed at conditions not identical to one of the blocks in the EXFILE, one further block has to be added to the EXFILE with the conditions where the calculations should be performed. The block should have one entry, which will be used as INITIAL GUESS, i.e. starting value, for calculations with ADDITIONAL POINTS (see section 1.2). After selecting an EXFILE by FILE/OPEN, the EOS, mixing rule, and the data block in the EXFILE for the next calculations can be chosen (Figure 3c – 3e). The EOS drop down combo box exclusively contains EOS where all necessary pure-component parameters are given in the EXFILE. If calculations should be performed with an EOS not displayed in the combo box, the pure-component EOS parameters have to be determined according to section 2.1 and keyed in the EXFILE by using FILE / ADD / PURE-COMPONENT EOS PARAMETERS (compare 1.2.3). The mixing rule drop down combo box only contains mixing rules which can be used for the selected EOS. For some EOS only one mixing rule exists, this one is named „Standard“, then. If the user either changes EOS, mixing rule or data block, PE will prompt if the binary interaction parameters should be kept constant or should be set to zero. In case interaction parameters for the new combination EOS + mixing rule + data block are stored in the EXFILE, there is also the possibility to load those. Instead of using the combo boxes for changes, the [...] button on the tool bar can also be used.

Figure 3c: Selecting an EOS for mixture calculations.

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Figure 3d: Selecting a mixing rule for mixture calculations.

Figure 3e: Selecting an data block of the EXFILE for the next calculations

3.1 Mixture Calculations

Mixture calculations using PE are phase equilibrium or density calculations. The calculation of densities is described in section 3.1.4. The phase equilibrium calculations are described in sections 3.1.1 - 3.1.3. The phase equilibrium calculations are carried out for either of the following reasons: 1) compare the calculation results with experimental data from the EXFILE, 2) calculate at conditions not investigated experimentally yet, using the phase equilibria supplied in EXFILE as starting values / initial guesses for the calculations. Binary Interaction Parameters First, the binary interaction parameters have to be keyed in for the next calculations by selecting MIXTURE / SET PARAMETERS (Figures 3a, 3.1a, and 3.1b). If interaction parameters are stored in the EXFILE for the selected EOS, mixing rule, and block (compare section 1.2.1), these parameters can be used as defaults, otherwise only the values from before.

Chapter 3: Mixtures

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Figure 3.1a: Input of interaction parameters for binary and multi component systems.

Figure 3.1b: Input of interaction parameters for a ternary system. Type of deviation calculated After the interaction parameters have been specified and in case of calculations for the above type 1, the type of deviation between calculated and experimentally determined data has to be specified by selecting MIXTURE/OBJECTIVE FUNCTION (Figure 3a). The deviation calculated by PE is based on the mole fractions of the coexisting phases at the temperature and pressure of the data points in the EXFILE. The dialog box PHASES (Figure 3.1c) enables to select the coexisting phases where the deviations should be calculated and averaged. In general, all phases are of interest, but sometimes only the composition of one of the phases has been determined experimen-

Chapter 3: Mixtures

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tally. In the latter case the phase not determined experimentally should be unchecked so that PE uses the values for this phase in the EXFILE as starting values for the iterations, only. The dialog box PHASES (Figure 3.1c) also allows to choose the formula according to which the average deviation is calculated. The buttons are equivalents for the following, where ξi = mole fraction of component i:

( )absoluten i i

calc

i

n

∧ = −=∑2

1 2

1ξ ξexp relative

ni i

calc

ii

n∧ =

−

∑

=2

12

1

ξ ξξ

exp

exp

absoluten i i

calc

i

n= −∑

=

11ξ ξexp relative

ni i

calc

ii

n=

−∑=

11

ξ ξξ

exp

exp

At the end of the calculations the dialog box shown in Figure 3.1d will display the difference between experimentally determined and calculated mole fractions.

Figure 3.1c: Selecting the type of deviation to be calculated.

Figure 3.1d: Calculated difference between experimental and calculated results.

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Start Calculations In order to start calculations with the settings from above, the menu entry MIXTURE/ CALCULATE has to be selected (Figure 3a). PE will display different dialog boxes, depending on the number of components, hereafter. This is described in the sections 3.1.1 and 3.1.2, below.

3.1.1 Binary Phase Equilibrium Calculations Starting binary calculations, the dialog box shown in Figure 3.1e will prompt the user for input. The entries are explained below.

Figure 3.1e: Starting binary calculations. OUPUT FILE The calculation results will be stored in the selected output file PE_1...9.OUT in the directory /PE/. DEBUG INFO PE can be forced to create additional output files that enable the programmer and experienced users to determine the reasons for convergence problems, change the input appropriately, and improve the program due to this information. Creating this debug information can significantly slow down the program. Setting the debug information to MINIMUM is recommended. Warning: Excessive output of MAXIMUM DEBUG INFORMATION may lead to a reduced lifetime of the harddisk.

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ADDITIONAL POINTS PE will - either calculate at the conditions of all equilibria of the current block in the EXFILE, that are marked with flag=1 (compare section 1.2) and determine a deviation between them and the calculated results - or calculate at the first equilibrium of the current block in the EXFILE (independent of the flag) and try to use this equilibrium as starting point for subsequent ADDITIONAL

CALCULATIONS in order to generate a complete Pxy-diagram or similar. By checking ADDITIONAL POINTS, PE can be forced to do the latter calculations. After the calculation of the first equilibrium, the pressure will be increased by the value of FIRST STEP. The pressure will be increased further, until convergence of the iterations fails - then the step size is reduced. After the step size has been reduced several times, the calculations will stop and start at the starting point again, but continuously reducing the pressure by FIRST STEP, now - and decreasing the step size if iterations do not converge any more. Calculations stop after the step size has been reduced several times, again. Using ADDITIONAL CALCULATIONS, PE will not calculate at pressures higher than the product of MAXIMUM PRESSURE FACTOR times the highest pressure of the points with flag=1 in the EXFILE equilibrium block. This enables to obtain diagrams in a certain pressure range, if the miscibility gaps remain open at high pressure, so that phase equilibrium calculations always converge.

3.1.2 Ternary Phase Equilibrium Calculations Starting ternary calculations, the dialog box shown in Figure 3.1f will prompt the user for input. The entries are explained below.

Figure 3.1f: Starting ternary calculations.

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OUTPUT FILE, DEBUG INFO, ADDITIONAL POINTS These three entries have the same meaning as explained for binary systems, above. ADD COMPONENT ADDITIONAL CALCULATIONS start at the first equilibrium in the current EXFILE block, like for binary calculations. The tie line which will be calculated afterwards, will be close to this equilibrium. By using ADD COMPONENT, the user tells PE which component has to be added most significantly in order to reach the next tie lines. For example, starting with a tie line extremely rich in components 2 and 3, ADD

COMPONENT should be set to 1. Choosing the right value here can speed up calculations, choosing the wrong value, might stop calculations at all. If any ternary ADDITIONAL CALCULATIONS do not work well, it is advisable to change ADD

COMPONENT to another value and check if calculation performance is improved. FIRST STEP / LENGTH Similar to the binary system, the user has to tell PE the size of the step from the first to the second tie line. The value is the ratio of the STEP from the middle of the first tie line to the middle of the second tie line divided by the LENGTH of the first tie line itself. In general, values between 1% and 5% work best. If convergence problems arise, this value should be decreased. Three Phase Calculations In order to obtain best calculation results for ADDITIONAL POINTS, the first equilibrium in an EXFILE block should be a three phase region if any was determined experimen-tally and calculation converge there. If no three phase region was detected experimen-tally, but the user knows that there is a three phase region, an INITIAL GUESS for the three phase region should be added to the block in the EXFILE as first point, in order to obtain nice looking diagrams (see section 1.2).

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3.1.3 Calculating Phase Equilibria The status of the last phase equilibrium calculation is shown in the dialog box CALCULATION STATUS shown in Figure 3.1g. The entries are explained below. EXP. Left box: The position of the phase equilibrium in the EXFILE BLOCK that is calculated, is shown here. In case ADDITIONAL POINTS has been checked for additional calculations, the counter will proceed to increase by a step of one for each additional calculation. Right box: The number of experimental points in the block of the EXFILE is indicated here during the calculation without ADDITIONAL POINTS. T, P Temperature and pressure of current calculations. Z Mole fraction of component 1 in the feed for the flash (Bünz et al., 1992). X, X’, Y, TYPE In the upper row the initial guesses for the isothermal isobaric flash calculations are displayed. x, x’, and y are the mole fractions of component 1, TYPE indicates the type of equilibrium expected: 1: V-L1, 2: V-L2, 3: L1-L2, 4: V-L1-L2. In the lower row, the last successfully calculated equilibrium is displayed, using the same output variables. TRY TRY indicates the number of the actual attempt to calculate the current phase equilibrium. If calculations are not successful with the initial guesses, PE itself changes the initial guesses or selects a different kind of miscibility gap, and increases TRY by one each time a new initial guess is tried. Depending on the version of PE, PE tries a different number of times to calculate one equilibrium.

Figure 3.1g: Status of Calculations

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3.1.4 Density Calculations Selecting MIXTURE / CALCULATE DENSITY from the main menu (Figure 3.1h) allows for the calculation of mixture densities for a given temperature, pressure, and composition, i.e. mole fractions (Figure 3.1i). Some calculation results are displayed in Figure 3.1j.

Figure 3.1h: choosing to calculate mixture densities.

Figure 3.1i (left side): input of mixture properties for the calculation of densities. Figure 3.1j (right side): output of the calculated density.

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3.2 Optimisation of Binary Interaction Parameters The optimum representation of phase equilibria in mixtures with an appropriate set of EOS, mixing rule, and pure-component parameters also depends on the correct choice of mixture parameters. PE uses so-called binary interaction parameters as a measure of mixture parameters, as it is common practice in chemical engineering thermodynamics (compare appendix A1 and A2). Theory enforces the adjustment of the pressure-independent binary interaction parameters for an optimum representation of the binary phase equilibrium data, only. Calculations in mixtures with more components should be performed with the so-derived parameters, guaranteeing self-consistent data, and making these calculations predictive. However, it is common practice to adjust the binary parameters to mixtures with more than two components for several reasons. One reason might be that the predictions for the ternary system are not good enough, if they are based on the parameters derived from the binary systems. Another reason might be that no (or not enough) binary information is available to derive the parameters based on these. Therefore, PE allows the optimisation of binary interaction parameters based on binary and ternary phase equilibrium data, although it is recommended to use binary information, only. However, in cases where no binary experimental phase equilibrium data can be obtained for one binary system at the conditions where the ternary system exhibits miscibility gaps, adjusting the parameters for this system to ternary information might be the only way to obtain a good representation of the ternary phase behaviour at all. Adding carbon dioxide to the homogeneous system 2-propanol + water leads to miscibility gaps at 300 K and 10 MPa and serves as an example, here. The ternary miscibility gaps can not be predicted accurately enough with interaction parameters from the binary systems with many cubic EOS, if the system 2-propanol + water at the same temperature at low pressure is used to regress the parameters of this system. Using parameters for this binary system that are optimised to represent the ternary system, calculations can come close to the experimentally determined equilibria (compare Adrian, 1997). Using PE, optimisations of interaction parameters can be done according to one of two different ways. Using a Simplex-Nelder-Mead algorithm in order to obtain the optimum interaction parameters is the standard method. However, the Simplex-Nelder-Mead algorithm requires initial guesses for the interaction parameters and will not find the way from one local optimum to another optimum. Therefore a second method - the so-called grid - enables to find initial guesses and different minima on the expense of increased time consumption. The application of both methods is explained in section 3.2.1 and 3.2.2.

Chapter 3: Mixtures

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3.2.1 Optimising Parameters (Simplex-Nelder-Mead Method) Selecting MIXTURE / OPTIMIZE from the main menu enables the optimisation of binary interaction parameters for the selected system, EXFILE block, EOS, and mixing rule, using a Simplex-Nelder-Mead algorithm (Nelder and Mead, 1965). The user is prompted for the input of the initial guesses of the parameters and for any binary parameters to be held constant during optimisations (Figure 3.2a). By checking OPTIMIZE in the dialog box, the user chooses to optimise the appropriate parameter with the given value as an initial guess. By unchecking OPTIMIZE in the dialog box, the user chooses to hold the value of this parameter constant during the whole optimisation. The results of the last calculation are stored in the specified OUTPUT FILE. The optimisation is stopped when the given TOLERANCE is reached. If VISUALIZATION

DURING OPTIMIZATION is checked, a graphic window is shown during the optimisation. A trace of the optimisation is stored in the file PE/PAROPT.OUT. If the optimisation is successful the optimised parameters can be stored to the EXFILE by selecting MIXTURE/SAVE PARAMETERS (Figure 3a and 3.2b and section 1.2).

Figure 3.2a: Input of interaction parameters for optimisation.

Figure 3.2b: Saving interaction parameters to the EXFILE.

Chapter 3: Mixtures

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3.2.2 Scanning Parameters (Grid) Selecting MIXTURE/GRID from the main menu enables finding good sets of binary interaction parameters for the selected system, EXFILE BLOCK, EOS, and mixing rule by calculating deviations between experimental points and calculated points with different binary interaction parameters on a rectangular grid, automatically. For a ternary system the binary interaction parameters from one binary subsystem can be varied on a GRID (Figure 3.2c) while holding all other parameters constant. The dialog box shown in Figure 3.2d prompts for three inputs per binary parameter: the lower and upper bound of the grid interval and the step between two different calculations. In general, this method requires a lot of computing time. Using a step width of 0,01 for kij and λ ij and the limits shown in Figure 3.2d enforces ( |-0,2 - 0,5| / 0,01 + 1 ) × ( |-0,3 - 0,4| / 0,01 + 1) = 5041 calculations:

kij λ ij -0,20 -0,30; -0,29; ... +0,39; +0,40 -0,19 -0,30; -0,29; ... +0,39; +0,40 ... ... ... ... +0,49 -0,30; -0,29; ... +0,39; +0,40 +0,50 -0,30; -0,29; ... +0,39; +0,40

Figure 3.2c: Choosing the binary subsystem of a ternary system for GRID parameters.

Chapter 3: Mixtures

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Figure 3.2d: Input of binary interaction parameters for calculations on a GRID ENABLE TIME LIMIT Because many parameter sets will be tested that will not lead to convergence of the flash calculations, it is possible to enable a TIME LIMIT for each calculation of a block in the EXFILE. If convergence is not achieved within this time for all equilibria of the block, the calculations with the current set of parameters is interrupted and the calculated deviation between experimental and calculated points is set to arbitrary high values around 1000%. The required time is depending on the number of experimental points in the EXFILE BLOCK, the complexity of the EOS (especially weather explicitly accounting for association or not), the number of components, and the performance of the PC. If the purpose of the GRID is to find starting values that allow convergence at all points for an optimisation according to section 3.2.1, it can be advisable to mark only the point at which convergence is most difficult with flag=1 in the EXFILE (section 1.2) in order to speed up the GRID significantly. The parameter sets and results with these parameters are stored in the file GRID.OUT.

Chapter 3: Mixtures

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3.3 Graphical Output of Mixture Equilibria The calculation results for mixtures are stored in the files PE_1..9.OUT (compare section 3.1). The calculation results can be displayed in form of different diagrams together with the data from the EXFILE, which are not marked with flag=0 (section 1.2). Depending on the number of components in the system of interest, which is displayed in the menu bar (section 3.1), different graphs are possible. The different kinds of graphical routines are described in sections 3.3.1 - 3.3.3.

3.3.1 Pxy- / yP-Diagrams (Binary) Selecting MIXTURE / PLOT from the main menu will display the dialog box shown in Figure 3.3a if the two-component mode is active and prompt for input of the plot parameters that are described below.

Figure 3.3a: Input of plot parameters for binary systems.

Chapter 3: Mixtures

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PLOT FILES It is possible to check a box corresponding to a PE_1..9.OUT file, if this file contains binary phase equilibrium data (otherwise the check box will be disabled). Checking a box will cause graphical output of the calculated data of this file and the corresponding experimental data from the EXFILE. It is possible to check more than one file for graphical output in order to receive a diagram with more than one set of experimental + calculated data. DISPLAY The compositions of the coexisting phases can be either plotted in mole fractions or weight fractions. ENLARGEMENT By checking ENLARGEMENT, the regions with the fractions of component 1 ranging a) from 0 to an upper bound and b) from a lower bound to 1 are enlarged, if a PXY-DIAGRAM is plotted. This option is useful, if the miscibility of the components is low: compare Figure 3.3b with Figure 3.3c. Further Options - It is possible to show the experimental points corresponding to the first (left) selected

PE_?.OUT file, only. This option is used if different EOS or parameters are used in order to correlate the same experimental data to avoid multiple different symbols for the same data.

- It is possible, to suppress the output of the calculated data in order to present the experimental results from the EXFILE, alone.

- Using the yP-DIAGRAM (see below), it is possible to choose a logarithmic axis for the composition. This option is useful for supercritical extractions of solids for example, where the solubility of the solid in the gas phase can cover many orders of magnitude.

Types of diagrams It is either possible to create a PXY-DIAGRAM or a yP-DIAGRAM. Figure 3.3b, 3.3c and 3.3d demonstrate the different graphical outputs.

Chapter 3: Mixtures

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Figure 3.3b: Pxy-diagram without enlargement

Figure 3.3c: Pxy-diagram with enlargement at 0-0.7 and 0.9-1

Figure 3.3d: yP-diagram (here: with logarithmic axis)

Chapter 3: Mixtures

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3.3.2 Gibbs’ Triangles (Ternary and Quaternary) Selecting MIXTURE / PLOT from the main menu will display the dialog box shown in Figure 3.3e, if a ternary or quaternary system is selected in the menu bar. This box prompts for input of the plot parameters, which are identical to the ones described in section 3.3.1 for the binary systems. Quaternary System Trying to display equilibria of quaternary systems will result in ternary projections of the equilibria onto the Gibbs’ Triangle without the fourth component. The composition fractions of the first three components are normalised so that the sum is unity again. This type of projection is useful, if only traces of the fourth component are present in all phases (compare Pfohl et al, 1996, 1996a, 1997). ENLARGEMENT Checking ENLARGEMENT will cause the enlarged display of the triangle corner with more than the specified amount of the specified COMPONENT.

Figure 3.3e: Input of plot parameters for a ternary or quaternary system.

Chapter 3: Mixtures

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3.4 Solid-Gas Equilibria Using PE, it is possible to calculate solid-gas equilibria in binary systems. For these calculations it is necessary that the density of the solid to be dissolved in the gas is given in the pure-component data block of the EXFILE (compare Figure 1.2a / Section 1.2.1). The solid component has to be component 2 in the EXFILE. If a suitable value for “s.dns” of component 2 is stored in the pure-component data block of the EXFILE, the check box SOLID in the dialog box for binary calculations (Figure 3.4a) will be enabled.

Figure 3.4a: Binary calculations for solid-gas equilibria If this check box is checked by the user, an algorithm for calculating solid-gas equilibria is been used instead of the vapour-liquid flash:

SV ff 22 = , i.e. ( )

−=RT

MwPPPPy S

satsatsatS

2

222222 exp

ρϕϕ

Where: f = fugacity ϕ = fugacity coefficient y = mole fraction in gas phase P = system pressure and Psat = saturation pressure T = system temperature R = universal gas constant Mw = molecular weight ρS = solid density (S.DNS from EXFILE) 2 = component 2 = solid compound V = vapour phase S = solid phase (pure compound 2) sat = state of saturation

Chapter 4: Implementation of New EOS by Users

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4 Implementation of New EOS by Users

4.1 Introduction PE 2000 has been designed in such a way that for each equation of state (EOS) only one subroutine in one single FORTRAN file is necessary. This modular structure allows to implement new EOS very easily – without the necessity to know anything about flash, isofugacity equations, vapor pressure calculations and so on. Because PE is designed in such a way that PE.EXE starts density and phase equilibrium calculations using one standard interface for all EOS, these EOS subroutines have a standard format. With such an EOS standard subroutine, PE can do all calculations with the given EOS, which are implemented in PE: Pure component calculations:

- calculate and plot Pv-diagram - calculate and plot vapor pressure curve - test and optimize pure-component parameters

Multicomponent mixture calculations: - density calculations - calculate and plot Px-diagrams (LLE/VLE/SGE) - calculate and plot Gibbs triangles (LLE/VLE/VLLE) - optimize interaction parameters

Such EOS standard subroutine can be written by anyone using a standard F77 compiler and the guidelines listed in this chapter. Researchers who are familiar with FORTRAN will be able to set up an EOS standard subroutine for simple EOS like Soave-Redlich-Kwong within one day. These standard files can be tested and used by the users themselves using the normal dialog boxes provided by PE.EXE the same day by following the description of the program EOS_on_DISK.EXE in Section 4.7. Upon request, such an EOS standard subroutine will be implemented in PE.EXE for free, if a MS-Word file with an appropriate description for appendix1 is attached. A fixed implementation in PE will speed up all subroutine calculations by a factor of ~100 compared to the use of EOS_on_DISK.EXE. Such implementation will enable all users world wide to work with the new EOS, test it and speed up the publication of its beneficial capabilities. This is seen as the major benefit of the PE 2000 project.


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4.2 Files Used for Implementing New EOS in PE Users willing to implement new EOS in PE are encouraged to do so and will receive help by the authors of this book in addition to what is written here. When installing the program PE, a directory PE/FORTRAN will be created in addition to the program PE. This directory contains all FORTRAN files necessary for implementing a new EOS in PE. But, only two of these files (pes_EOS_by_User.FOR and pes_RW_EOS_save.FOR) have to be modified when programming a new EOS. 1) File containing the EOS “pes_EOS_by_User.FOR” 2) Files necessary for testing the EOS without connection to PE.EXE “pes_EOS_by_User.FOR” – see above “EOS_TEST.FOR” – main program “EOS_P_TEST.FOR” – subroutine for testing P(T,rho) calculation “EOS_PHI_TEST.FOR” – subroutine for testing ϕ(T,P) calculation “EOS_TEST_epk” – interface for subroutine pedens – do not change ! “pes_dens_1.FOR” – interface for “pedens2.FOR” – ignore all “historic” lines for the

Peng-Robinson-EOS, here. These lines are not used ! Do not change ! “pes_dens_2.FOR” – subroutine iteratively solving ρ(T,P) – do not change ! “pes_RW_EOS_save.FOR” – subroutines handling the save-parameters. These

subroutines could be dummy subroutines for testing, here. 3) Files necessary for building EOS_on_DISK.EXE (testing with connection to PE.EXE) “pes_EOS_by_User.FOR” – see above “pes_dens_1.FOR” – see above and do not change ! “pes_dens_2.FOR” – see above and do not change ! “EOS_on_DISK.FOR” – main program called by PE.EXE – do not change ! “pes_RW_EOS_save.FOR” – subroutines handling the save-parameters. These

subroutines have to be adjusted by the user depending on the EOS, here. “pes_RW_EOS_para.FOR” – subroutines handling the parameter list of the

User_EOS, i.e. the file transfer between “PE.EXE” and “EOS_on_DISK.EXE” - Do not change !

“EOS_DISK_pepres.FOR” – interface for pedens – do not change ! 4) files necessary for running EOS_on_DISK.EXE “EOS_on_DISK.EXE” – program with User EOS and I/O subroutines “EOS_on_DISK.in” – optional file with names and units of parameters for User EOS “EOS_on_DISK.sav” – will be created by EOS_on_DISK.EXE, contains save-

variables “EOS_on_DISK.par” – will be created by PEINT.EXE and EOS_on_DISK.EXE,

contains all parameters of the subroutine User_EOS with the User EOS.


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4.3 The File pes_EOS_?.FOR The EOS standard subroutines used by PE are in files named pes_EOS_?.FOR where “?” stands for the name of the EOS being in this file. The subroutine in this file contains 3 different parts, which are used, depending on the value of the variable calc_what in the parameter list (see section 4.3.1 – 4.3.3). A demo file pes_EOS_by_User.FOR which is in the PE/FORTRAN directory contains the simple Peng-Robinson EOS. The file is shown in Figure 4.3a. !==================================================================! Standard EOS subroutine for PE!! 2000-01-30 by Oliver Pfohl!! This subroutine does all calculations with the Peng-Robinson EOS:! a) P(rho,T,x)! b) rho(P,T,x)! c) phi(P,T,x,phase)!==================================================================

subroutine User_EOS ( calc_what, nsteu, mixrule, nco,& nmax, NAv, Tau, Pi, R, T, rho, PuCoPa, sts,& k, l, la, kep, kkap,& pp, dp, iphase, iphaseerr, mode,& xi, rhol, phil, errliq,& yi, rhog, phig, errvap, fk, called_by_EOS_on_DISK)

implicit none

! parameters of subroutine EOS_from_DISK - DO NOT CHANGE !!!----------------------------------------------------------logical called_by_EOS_on_DISKinteger calc_what, nsteu, mixrule, nco, nmax, mode,

& iphase, iphaseerr, errliq, errvap,& sts(nmax,0:3)double precision r, t, Nav, Tau, Pi, rho, PuCoPa(7,*),

& k(nmax,*),l(nmax,*),la(nmax,*),kep(nmax,*),kkap(nmax,*),& pp, dp, xi(*), yi(*), rhol, rhog, phil(*), phig(*), fk(*)

! parameters of Subroutine PengRobinson! (CHANGE ACCORDING TO YOUR EOS)!-----------------------------------------------------------double precision ai(100,100), bi(100,100), ci(100,100),

& ax, ay, bx, by, cx, cy, tc(100), pc(100), omega(100)

! keep in mind the "global" variables for Peng-Robinson! after exiting the subroutine. Remember for next call!------------------------------------------------------save tc, pc, omega, ai,bi,ci, ax,ay,bx,by,cx,cy


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! temporary "local" variables!----------------------------integer i, ierr

double precision ac(nmax),tr(nmax),xo,xt,& thet, dthet, am, bm, cm, rbm, rt,& pideal, zett, xa(nmax),& p, rhomx, pold, prho, dprho, prhol, prhog,& sq2, rgp, rgm, rlp, rlm, rll, rlg, rt2sq,& arax, aray, arx, ary, arbx, arby, thetx, thety,& suphiax, suphiay, suphibx, suphiby,& dadx(nmax), dady(nmax), dbdx(nmax), dbdy(nmax),& dcdx(nmax), dcdy(nmax), philogl(nmax), philogg(nmax)

! read all save parameters from the file EOS_SAVE.DAT if! EOS_on_DISK.EXE calls the subroutine (then ´save´ does not work)!----------------------------------------------------------if (called_by_EOS_on_DISK)

& call read_eos_save(nco,tc,pc,omega,ai,bi,ci,ax,ay,bx,by,cx,cy)

!==========================================================! INITIALISATION (calc_what = 0)! -------------------------------!! Here the "save" variables for Peng-Robinson are initialized.! I.e. constants and parameters that are only depending on! temperature (not on composition, density, pressure) are calculated! from pure-component parameters and temperature.!! In order to obtain a "global" character of the variables, the! values of the parameters are stored via "save"!! input: pure-component EOS parameters: ´PuCoPa´ (and T)! intermediate output: Tc, Pc, omega! final output: ai(i,i), bi(i,i)!==========================================================

if (calc_what.eq.0) then

do i = 1, nco

tc(i) = PuCoPa(1,i)pc(i) = PuCoPa(2,i)omega(i) = PuCoPa(3,i)

tr(i)=t/tc(i)

ac(i)=0.457235529d0*r*r*tc(i)*tc(i)/pc(i)bi(i,i)=0.077796074d0*r*tc(i)/pc(i)

xo = 0.37464d0+1.54226d0*omega(i)-.26992d0*omega(i)**2xt = 1.d0+xo*(1.d0-dsqrt(tr(i)))

ai(i,i)=ac(i)*xt**2

enddoendif ! INIT (calc_what=0)


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!==========================================================! CALCULATE PRESSURE P(x,rho,T) (calc_what = 1)! ----------------------------------------------!! input: T, rho, iphase/iphaseerr AND (ax,ay,bx,by,cx,cy)!! output: pp(rho,T), dp(rho,T)!!==========================================================if (calc_what.eq.1) then

rt = r * t

if (iphaseerr.eq.1) thenif (iphase.eq.1) then

iphase = 0else

iphase = 1endif

endif

! GAS PHASE!----------if (iphase.eq.0) then

do i = 1, nco! mole fractions: not used here but maybe for other EOSxa(i) = yi(i)

enddoam = aybm = by! c: not used herecm = cy

! LIQUID PHASE!-------------else

do i = 1, ncoxa(i) = xi(i)

enddoam = axbm = bx! c: not used herecm = cx

endif

if (iphaseerr.eq.1) thenif (iphase.eq.1) then

iphase = 0else

iphase = 1endif

endif

! just an abbreviationrbm = rho * bm

! thet = (Z-1)/rhothet = bm/(1.d0-rbm)-am/rt/(1.d0+2.d0*rbm-rbm*rbm)! dthet = d_thet / d_rhodthet = bm*(bm/ dexp(dlog(1.d0-rbm)*2.d0) +

& 2.d0*am/rt*(1.d0-rbm)/& dexp(dlog(1.d0+2.d0*rbm-rbm*rbm)*2.d0))

! pressure from ideal gas lawpideal=rho*rt

! pressure with Peng-Robinsonpp=pideal*(1.0d0+rho*thet)

! dp = d_P / d_rho (used for Newton method)dp=rt*(1.0d0+rho*(2.0d0*thet+rho*dthet))

endif ! CALC PRESSURE (calc_what=1)


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!==========================================================! CALCULATE FUGACITIES phi(x,P,T) (calc_what = 2)! ------------------------------------------------!! input: T, P, xi, yi!! output: rhol, prhol, phil(), errliq! rhog, prhog, phig(), errvap,! for pedens: rhomx!!==========================================================if (calc_what.eq.2) then

p = pprt = r * t

! determine mixture parameters!-----------------------------call si_mix_abc ( nsteu, mixrule, nco,nmax, R, T,

& ai, bi, ci, k, l, la, xi, yi,& ax, ay, dadx, dady, bx, by, dbdx, dbdy, cx, cy, dcdx, dcdy )

! write all save parameters to the file EOS_SAVE.DAT if! EOS_on_DISK.EXE calls the subroutine (then ´save´ does not work)!-----------------------------------------------------------------if (called_by_EOS_on_DISK)

& call write_eos_save(nco,tc,pc,omega,ai,bi,ci,ax,ay,bx,by,cx,cy)

if(p.lt.0.0) return

! determine liquid phase density!-------------------------------iphase=1rhomx=1.d0/bx ! max. density (1/covolume) so that p -> infinitypold=pcall pedens(rhomx,pold,rho,prho,dprho,ierr,

& pi,r,t,nsteu,iphase,mode,ax,ay,bx,by,& iphaseerr, mixrule, nco,nmax, nav, tau,& PuCoPa, sts, k, l, la, kep, kkap,& xi, yi, rhol, rhog, phil, phig, errliq, errvap)

rhol=rhoprhol=prhoerrliq=ierr! determine vapour phase density!-------------------------------iphase=0rhomx=1.d0/by ! max. density (1/covolume) so that p -> infinitypold=pcall pedens(rhomx,pold,rho,prho,dprho,ierr,

& pi,r,t,nsteu,iphase,mode,ax,ay,bx,by,& iphaseerr, mixrule, nco,nmax, nav, tau,& PuCoPa, sts, k, l, la, kep, kkap,& xi, yi, rhol, rhog, phil, phig, errliq, errvap)

rhog=rhoprhog=prhoerrvap=ierr

! abbreviations!--------------sq2=dsqrt(2.d0)rgp=by*rhog*(1.d0+sq2)+1.d0rgm=by*rhog*(1.d0-sq2)+1.d0rlp=bx*rhol*(1.d0+sq2)+1.d0rlm=bx*rhol*(1.d0-sq2)+1.d0rll=dlog(rlp/rlm)rlg=dlog(rgp/rgm)rt2sq=rt*2.d0*sq2

! d(ar)/d(a)!-----------arax=-1.d0/(bx*rt2sq)*rllaray=-1.d0/(by*rt2sq)*rlg


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! ar!---arx=-dlog(1.d0-bx*rhol)+ax*araxary=-dlog(1.d0-by*rhog)+ay*aray

! d(ar)/d(b)!-----------arbx=rhol/(1.d0-bx*rhol)+ax/bx/bx/rt2sq *rll-ax/bx*

& rhol/r/t/(1.d0+2.d0*bx*rhol-(bx*rhol)**2)arby=rhog/(1.d0-by*rhog)+ay/by/by/rt2sq *rlg-ay/by*

& rhog/r/t/(1.d0+2.d0*by*rhog-(by*rhog)**2)

! (Z-1)/rho!----------thetx=bx/(1.d0-bx*rhol)-ax/rt/(1.d0+2.d0*bx*rhol-(bx*rhol)**2)thety=by/(1.d0-by*rhog)-ay/rt/(1.d0+2.d0*by*rhog-(by*rhog)**2)

suphiax=0.d0suphiay=0.d0suphibx=0.d0suphiby=0.d0do i=1,nco

suphiax=suphiax+xi(i)*arax*dadx(i)suphiay=suphiay+yi(i)*aray*dady(i)suphibx=suphibx+xi(i)*arbx*dbdx(i)suphiby=suphiby+yi(i)*arby*dbdy(i)

enddo

do i=1,ncozett=p/(rhol*r*t)philogl(i)=arx+arax*dadx(i)+arbx*dbdx(i)-suphiax-suphibx+

& zett-1.d0-dlog(zett)zett=p/(rhog*r*t)philogg(i)=ary+aray*dady(i)+arby*dbdy(i)-suphiay-suphiby+

& zett-1.d0-dlog(zett)phil(i)=dexp(philogl(i))phig(i)=dexp(philogg(i))

enddo

endif ! CALC FUGACITIES (calc_what=2)

! write all save parameters to the file EOS_SAVE.DAT if! EOS_on_DISK.EXE calls the subroutine (then ´save´ does not work)!-----------------------------------------------------------------if (called_by_EOS_on_DISK)

& call write_eos_save(nco,tc,pc,omega,ai,bi,ci,ax,ay,bx,by,cx,cy)

end

subroutine si_mix_abc ( nsteu, mixrule, nco,nmax, R, T,& ai, bi, ci, k, l, la, xi, yi,& ax, ay, dadx, dady, bx, by, dbdx, dbdy, cx, cy, dcdx, dcdy )

!============================================================! subroutine of program pe (calculates phase equilibria)! calculates the mixture parameters a,b,c for most cubic EOS!! nco = number of components! R = universal gas constant! T = temperatue [K]! ai, bi, ci = pure-component parameters! k, l, la = binary interaction parameters! ax, ay, bx, by (c) = a and b for liquid and gas phase! dadx, dady, dbdx, dbdy (c) = derivatives of a and b for mole fr!!============================================================implicit none


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! parameters!-----------integer nsteu, mixrule, nco, nmaxdouble precision r, t, ai(100,100), bi(100,100), ci(100,100),

& k(nmax,*), l(nmax,*), la(nmax,*),& xi(*), yi(*),& ax, ay, dadx(nmax), dady(nmax),& bx, by, dbdx(nmax), dbdy(nmax),& cx, cy, dcdx(nmax), dcdy(nmax),& asx(nmax,nmax),asy(nmax,nmax)! counter!--------integer ii,jj

! avoid compiler warnings!(these variables are not used in this mini-version)!---------------------------------------------------if ((nsteu.eq.-1).or.(mixrule.eq.-1).or.(R .eq.-1) .or.

& (T .eq.-1).or.(ci(1,1).eq.-1).or.(la(1,1).eq.-1)) continue

!-----------------------------------------------------! Set all mixture parameters (and derivatives) to ZERO!-----------------------------------------------------ax=0.0d0ay=0.0d0bx=0.0d0by=0.0d0cx=0.0d0cy=0.0d0do ii=1,nco

dadx(ii)=0.0d0dady(ii)=0.0d0dbdx(ii)=0.0d0dbdy(ii)=0.0d0dcdx(ii)=0.0d0dcdy(ii)=0.0d0

enddo

!-----------------------------------------------------! calculation of parameter "a" (quadratic mixing rule)!-----------------------------------------------------do ii=1,nco

do jj=1,ncoai(ii,jj) = (1.d0-k(ii,jj))*dsqrt(ai(ii,ii) * ai(jj,jj))asx(ii,jj) = ai(ii,jj)asy(ii,jj) = ai(ii,jj)ax = ax+xi(ii)*xi(jj)*asx(ii,jj)ay = ay+yi(ii)*yi(jj)*asy(ii,jj)dadx(ii)=dadx(ii)+2.d0*xi(jj)*ai(ii,jj)dady(ii)=dady(ii)+2.d0*yi(jj)*ai(ii,jj)

enddoenddo!----------------------------------------! calculation of parameter "b" (standard)!----------------------------------------do ii=1,nco

do jj=1,ncobi(ii,jj) = (1.0d0-l(ii,jj))*(bi(ii,ii)+bi(jj,jj))/2.d0bx=bx+xi(ii)*xi(jj)*bi(ii,jj)by=by+yi(ii)*yi(jj)*bi(ii,jj)dbdx(ii)=dbdx(ii)+2.d0*xi(jj)*bi(ii,jj)dbdy(ii)=dbdy(ii)+2.d0*yi(jj)*bi(ii,jj)

enddoenddo

end

--------------------------------------------------------------------------------------------------------- Figure 4.3a: demo file pes_EOS_by_User.FOR


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4.3.1 Initialization (calc_what=0) PE calls the EOS subroutine using the pure-component parameters PuCoPa (1:7,1:100) containing up to 7 different pure-component parameters (1:7) for up to 100 compounds in a mixture (1:100). Doing the initialization, these parameters are copied to local variables (named Tc , Pc , omega or a, b, c for example) which are stored within the subroutine using the FORTRAN save-command. Further, all EOS parameters (or parts of the EOS) which are not depending on composition, density and pressure should be saved in save-variables. These save-variables (which may only depend on pure-component properties and temperature according to the above definition) will be kept constant during (isothermal) flash and density calculations, speeding up the calculations enormously for most EOS, because there is no necessity to recalculate the variables again and again in each iteration. 4.3.2 Calculate Pressure (calc_what=1) Most EOS are written in a pressure-explicit form P=P(T,v,z), where P is pressure, T is temperature, v is the molar volume and z the molar composition of a phase. For calc_what=1, this calculation is performed as well as the calculation of (∂P/∂ρ)T,z using the parameters from the save-paragraph and the information which phase is looked at (iphase=0 means gas phase, i.e. P=P(T,vG,y). iphase=1 means liquid phase, i.e. P=P(T,vL,x)). Because the formula for the residual Helmholtz ar energy is needed for the calculation of fugacity coefficients later anyway, the calculation of P(T,v,z) and (∂P/∂ρ)T,z in PE is generally done using the abbreviation θ (Topliss, 1985). But: this way of calculation is not mandatory.

Definition of θθθθ

ρρ 1],,[ −=Θ ZzT

with RTPZ

ρ=

i.e. ∫=

=

Θ==vr

r dzTRT

zvTazvTa/1

0

],,[],,[],,[~ρ

ρ

ρρ

Calculation of P, using ΘΘΘΘ

RTzTPideal ρρ =],,[

]),,[*1(*],,[],,[ zTzTPzTP ideal ρρρρ Θ+=


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Calculation of (dP/dρρρρ)T , using ΘΘΘΘ and (dΘΘΘΘ/dρρρρ)T

Θ+Θ+=

zTzT ddRT

ddP

,,

**2*1*ρ

ρρρ

4.3.3 Calculate Fugacity Coefficients (calc_what=2) The isofugacity equation for the determination of equilibria enforces the knowledge of fugacity coefficients ϕi given by

PzTzf PhPhPhi

Phi

Phi *],,[* ρϕ=

Here: fi

Ph = fugacity of component “i” in phase “Ph” zi

Ph = mole fraction of component “i” in phase “Ph” ϕi

Ph = fugacity coefficient of component “i” in phase “Ph” ρPh = density of phase “Ph” According to the parameterization used by Topliss (1985), the fugacity coefficient is derived very easily by the following formula.

ZZdzd

dadz

dzd

dada

n

j bak j

k

k

r

jbak i

k

k

rr

i ln1~~~ln

1 ...,...,

−−+

Ψ

Ψ

−

Ψ

Ψ

+= ∑ ∑∑= =Ψ=Ψ

ϕ

ψk are different parts of the EOS (not necessarily parameters) which depend on composition. Their individual choice is up to the programmer. All ψk together must cover all parts of the EOS which depend on composition, when ρ and T are given and held constant. For the PR-EOS (see below) it is sufficient to use ψ1=a and ψ2=b. For SAFT, Huang and Radosz (1990-1993) used 8 different parts in order to minimize the size of the resulting single equations. A good parameterization saves a lot of programming work: In the limit when the programmer sets ψ= ra~ , no simplification is achieved at all. Calling the subroutine with calc_what=2, the fugacity coefficients for both phases are calculated. Because for phase equilibrium calculations, there is mechanical and thermal stability between the phases, T and P are given in order to calculate the fugacity coefficients for both phases. This enforces the calculation of ρL=ρ(T,P,x,iphase=1) and ρG=ρ(T,P,y,iphase=0) prior to the fugacity coefficients. The information of iphase has to be included here, because of the van-der-Waals loop in the calculated isotherms. Because it is hard work to derive analytical expressions for ρ(T,P,z) for most (complicate) EOS, PE offers an algorithm for finding the solution numerically. This


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algorithm pedens was written by Topliss (1985) and slightly adjusted in order to meet the needs of PE. This algorithm repeatedly calls the above described routine for the calculation of P(T,ρ,z) with calc_what=1 (section 4.3.2) similar to routines for finding roots of normal equations. Because T and z do not change during these iterations, those parts of the EOS that only depend on T and z should be calculated once before doing these iterations and stored in local variables using the save-paragraph (compare example with PR-EOS, below): 1) Doing the initialization (calc_what=0), all variables that only depend on

temperature T are set. 2) The variables that depend on composition z are set just before running the iterations

by calling pedens when calling the subroutine with calc_what=2. All these variables have to be saved – see PR-example, below.

Finally, P, T, x {FORTRAN: xi}, y {yi}, ρL {rhol}, ρG {rhog} are known and the fugacity coefficients ϕi,L {phil(i)} and ϕi,G {phig(i)} can be calculated. 4.4 pes_EOS_by_User.FOR The FORTRAN file pes_EOS_by_User.FOR (Figure 4.3a, see above) serves as an example for an EOS standard subroutine. The file contains the simple Peng-Robinson EOS with the simple quadratic mixing rule. The below sections 4.4.1 – 4.4.6 describe the specific parts of the subroutine, so that the implementation of the Peng-Robinson EOS can be understood by anyone trying to implement another EOS.

4.4.1 Subroutine Parameters The subroutine parameters are the same for all different EOS and the list of parameters must not be changed: - numerical constants: - nmax is the maximum number of components and is set outside of the subroutine - NAv = 6.022045 D+23 (Avogadro number) - Tau = 1/6 * π * 2 = 0.74048... (closest packing fraction) - Pi = π = 3.1415926535897932385... - R = 0.0831439 (universal gas constant. Unit in PE: bar * dm³ / mol / K) - nsteu = number of EOS to be used (vdW=0, PR=1, SRK=3, EOS_on_DISK=99...) - mixrule = number of mixrule to be used. The chosen mixrule must have no influence

on calculated pure-component properties. Calculations with mixrule=0 have to be possible.


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- nco = number of compounds in mixture (1=pure, 2=binary, 3=ternary ...), nco≤nmax! - T = Kelvin temperature [Unit = K] - PuCoPa = up to 7 pure-component parameters for each compound - sts(i,0:3) number of association sites selected for the pure components “i”:

(i,1)= number of (+)-sites, (i,2)= number of (-)-sites, (i,3) = number of (COOH)-sites, (i,0) = sum of entry 1,2,3 = total number of sites on molecule “i”.

- Calc_what (CW) determines what should be calculated in the subroutine - CW=0: initialization, input: PuCoPa, sts, T as well as numerical constants output: temperature-dependent parameters which are saved. - CW=1: calculate P[rho,T,iphase,(xi or yi), (k,l,la,kep,kkap) and save- parameters]. The save-parameters are set before by calling CW=0 and CW=2 - CW=2: calculate ϕL(T,P=pp,x) and ϕG(T,P=pp,y) by:

a) set save-parameters that depend on composition b) set rhomx – that is the maximum density for the given EOS, T and z c) calculate ρL(T,P=pp,x) and ρG(T,P=pp,y) with pedens, iteratively d) calculate ϕL(ρL,T,P=pp,x) and ϕG(ρG,T,P=pp,y)

- binary interaction parameters with kii = l ii = λ ii = kε ii = kκ ii = 0 - k: kij=kji - l: lij = lji - la: λ ij = -λ ji (i.e. asymmetric!) - kep: kεij = kεji - kkap: kκij = kκji - pp = P(T,rho) =

1) output of calculation with calc_what=1 [unit = bar] 2) input for calculation with calc_what=2 [unit = bar]

- dp = ∂P/∂ρ(T,ρ) = output of calculation with calc_what=1 [unit = bar /(mol/dm³)] - rho = input of density for P(T,rho) calculation (calc_what=1) [mol/dm³ = kmol/m³] - iphase determines which phase should be used for calculations with calc_what=1 iphase=0: gas phase, iphase=1: liquid phase - iphaseerr = error flag set outside the subroutine. It inverts the meaning of iphase. - mode = variable for pedens, set outside to a value of “3” (see pedens header) - xi, yi = mole fractions of the coexisting phases (x: liquid, y: gas) - rhol, rhog = density results of calc_what=2 calculations, i.e. ρL(T,P=pp,x) and ρG(T,P=pp,y) [mol/dm³ = kmol/m³] - errliq, errvap = error flags which are set if pedens was not successful (see pedens header) - phil, phig = fugacity coefficients: results of calc_what=2 calculations, i.e. ϕL(ρL,T,P=pp,x) and ϕG(ρG,T,P=pp,y) - called_by_EOS_on_DISK = logical flag which is true if the subroutine was called by EOS_on_DISK.EXE. Then the parameters from the save-command have to be

read from a file and saved to the file EOS_on_DISK.sav (see below and example). Otherwise the flag is false.


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4.4.2 Parameters for Simple Peng-Robinson EOS (FORTRAN: save)

In order to speedup calculations, the equations within the subroutine blocks (calc_what=1,2,3) are arranged in such a way, that multiple calculations of variables that do not change within iterations are avoided: The equations for calculating the pressure P(T,ρ,z) (calc_what=1) are constructed in such a way that single parts of the equations that only depend on temperature and composition - but not on density - are calculated and stored in save-variables before the iterations in pedens are started. There are three types of possible save-variables 1) pure-component parameters like Tc , Pc , ω (they need not be saved, here, but

demonstrate how it could look like for other EOS like SAFT). 2) Pure-component parameters like ai , bi , ci that do not depend on composition but

may depend on temperature (ci is not used for the PR-EOS, here) 3) Mixture-parameters that also depend on composition but not on density (ax, ay, bx,

by ...) Variables of type 1 and 2 which do only depend on temperature are set at “initialization” (calc_what=0). The variables of type 3 are set at “calculate fugacities” (calc_what=2) directly before the FORTRAN statement “if (p.lt.0) return” which is directly placed before the calls of pedens. The FORTRAN save-statement ensures that the variables which are set once, can be used in the next calls of the subroutine, again, i.e. that they are not forgotten. The save-statement can only be used, if the subroutine is directly included in the PE.EXE / PE_INT.EXE files or in EOS_P_TEST.FOR / EOS_PHI_TEST.FOR. Save can not be used by EOS_on_DISK.EXE (see Section 4.7). Important: In case, only the pressure P(ρ,T,z) should be calculated, PE calls the subroutine with calc_what=0, first. Then the subroutine is called with calc_what=2 and pp=-1 (negative!). This ensures that the parameters ax, ay... are set and present at the subsequent call with “calculate pressure” (calc_what=1, see EOS_P_TEST.FOR as example). On the other hand, pp=-1 prevents the superfluous calculation of ρ(P,T,z) and ϕ(P,T,z) in the call with calc_what=2 because of the statement “if (p.lt.0) return”.

4.4.3 Initialization (calc_what=0) In case of calc_what=0, there have to be done two things: a) store the pure-component parameters PuCoPa(), which come from PE.EXE in local

variables, which have to be saved if they should be used for calc_what≠0 calls, again.

b) In order to speedup calculations, all variables (or parts of the EOS) which only depend on temperature but not on composition, density or pressure should be set here. For the Peng-Robinson EOS, all ai(T) and bi are set here. These variables have to be saved, because they are used for calc_what≠0 calls later, again.


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4.4.4 Calculate Pressure (calc_what=1) Here, either P(T,ρL,x) or P(T,ρG,y) is calculated. Iphase (and iphaseerr) are used by PE to decide which of either has to be done. Depending on iphase (and iphaseerr) the variables am/bm (a and b for mixture) are set equal to ax/bx or ay/by. (Note: ax, ay, bx, by are set in the section calc_what=2 before the statement “if(p.lt.0)return”). Afterwards, Θ {thet}, ∂Θ/∂ρ {dthet}, P {pp}and ∂P/∂ρ {dP} are calculated. {pp} and {dp} are returned to PE via the parameter list.

4.4.5 Calculate Fugacity Coefficients (calc_what=2) First, mixture parameters that only depend on temperature and composition but not on density (ax, ay, bx, by...) are calculated. Afterwards, there are two possibilities: 1) For P>0, ρL(P,T,x), ρG(P,T,y), ϕi,L(P,T,x), and ϕi,G(P,T,y) should be calculated:

first, rhomx is determined. Rhomx is the maximum achievable density for the given EOS, i.e. the density for which the EOS calculates P→∞. Rhomx is required by pedens. Afterwards, pedens is called. Pedens will call the subroutine again multiple times with calc_what=1. Therefore the saved parameters ax, bx, ay, by have to be present. Within PE.EXE and EOS_P/PHI_TEST.EXE this is achieved using the save-statement. Two calls of pedens determine ρL(P,T,x), ρG(P,T,y). Afterwards ϕi,L(P,T,x), and ϕi,G(P,T,y), i.e. {phil(i)} and {phig(i)} are calculated and returned to PE together with {rhol, rhog, errliq, and errvap}.

2) For P<0, the subroutine has only been called in order to set the mixture parameters {ax, ay, bx, by} that are needed for a call of “calculate pressure” (calc_what=1), later. Afterwards the subroutine is finished, by the statement “if(p.lt.0)return”. P<0 is obviously (mis)used as a flag here: see EOS_P_TEST.FOR.

4.4.6 Subroutine si_mix_abc Si_mix_abc is a simplified subroutine of the subroutine Mix_abc used in original PE. The parameter list is the same, thus the subroutines are interchangeable. The subroutine Mix_abc is used for practically all cubic EOS in PE and should be used for all new (cubic) EOS, also. This saves enormous amounts of listing, because Mix_abc contains many different mixing rules which can be chosen by selecting the appropriate value for the variable {mixrule}: Mixrule=0 → Quadratic Mixrule=4 → Panagiotopoulos-Reid Mixrule=1 → Adachi-Sugie Mixrule=42 → Mathias-Klotz-Prausnitz Mixrule=2 → Melhem Mixrule=43 → Mathias-Klotz-Prausnitz, Wong-Sandler Mixrule=3 → Stryjek-Vera The documentation of the mixing rules and their parameters is in appendix A2. Mix_abc calculates ax, ay, bx, by, cx, cy and their derivatives with respect to the mole fractions which are needed for the calculation of the fugacity coefficients according to the parameterization by Topliss (see above).


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4.5 EOS_P_TEST.FOR, EOS_PHI_TEST.FOR - Programs The (sub)programs EOS_P_TEST.FOR and EOS_PHI_TEST.FOR called by EOS_TEST.FOR demonstrate the use of the subroutine User_EOS in the file pes_EOS_by_User.FOR. These two little programs should serve as first tests for new written subroutines in pes_EOS_by_User.FOR files. The files are self explanatory. The results can easily be compared with calculation results obtained with other programs. - EOS_P_TEST.FOR demonstrates the calculation of P(T,ρ,x). - EOS_PHI_TEST.FOR demonstrates the calculation of ρL(T,P,x) and ρG(T,P,y) as

well as ϕL(ρL,T,P,x) and ϕG(ρG,T,P,y). For these two programs it is necessary to use the save-statement like in PE.EXE and NOT the subroutines read_eos_save and write_eos_save like in EOS_on_DISK.FOR – thus the variable called_by_EOS_on_DISK is set to false. Thus, for these tests, the subroutines read_eos_save and write_eos_save can be dummy subroutines.

4.6 FORTRAN Language In case one likes to implement the EOS routines in PE.EXE, later, please follow the below instructions for the FORTRAN language, that avoid typical errors: 1) Because the new FORTRAN files should work independently of the compiler used,

please, do only use standard FORTRAN 77 with two exceptions: a) in order to avoid the use of labels, try to write

do i=1,10enddo

instead of do 123 i=1,10123 continue

b) Try to use “implicit none” instead of “implicit logical a-z” in order to guarantee that all variables have been defined correctly.

2) Please do start the program lines in column 7 according to F77 standard although the compiler might allow to start with the FORTRAN code in column 1. Please, do not use “Tabs” – please use “Spaces”.

3) Please, do use double precision variables and the appropriate functions “dsqrt”, “dlog”, “dexp”, “double”. Example: if “r” and “s” are double precision and “i” is an integer, do not write “r=i*s“, but “r=double(i)*s“.

4) Please do not write “a=b**3.d0“ but “a=b**3“, because for b<0 the latter one is interpreted as a=b*b*b by some compilers while the former leads to runtime errors for b<0.

5) Please do use the predefined constants R, Pi, NAv, and TAU and do not redefine them.


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4.7 EOS_on_DISK.FOR After having performed (and passed) the above test programs EOS_P_TEST.FOR and EOS_PHI_TEST.FOR, the FORTRAN file EOS_on_DISK.FOR allows the user to connect the new EOS in the pes_EOS_by_User.FOR file with PE.EXE within a few minutes and without the necessity of having any other FORTRAN code of the PE project. Important: Because EOS_on_DISK.EXE is a program (and not a subroutine) which repeatedly calls itself, the FORTRAN save-command does not work here in the same way as it is used in the above sections. In order to compensate for the loss of this save-buffer, EOS_on_DISK.EXE uses the subroutines read_eos_save and write_eos_save in pes_RW_EOS_Save.FOR which have to be adjusted for each EOS so that all save-parameters are written to (and read from) the file EOS_on_DISK.sav (compare example with PR-EOS). Setting the variable called_by_EOS_on_DISK to true in EOS_on_DISK.FOR, will automatically lead to calls of read_eos_save and write_eos_save in pes_EOS_by_User.FOR. 4.7.1 Running EOS_on_DISK.EXE Choosing the equation “EOS_on_DISK” in PE.EXE, PE.EXE will call EOS_on_DISK.EXE written by the user and placed in the same directory as PE.EXE instead of calling the internal EOS subroutines: Figure 4.7a and 4.7b.

Figure 4.7a Choosing EOS_on_DISK for mixture calculations


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Figure 4.7b: Choosing EOS_on_DISK for pure-component calculations The pure-component parameter names and units can be supplied in the file EOS_on_DISK.in placed in the directory of PE.EXE. The included file for the “Simple Peng-Robinson” EOS in the subroutine User_EOS in the file pes_EOS_by_User.FOR is shown in Figure 4.7c. Line 1 and column 1 and 2 are used for documentation. Column 3-7 contain the parameter names, column 8 is blank, column 9-17 contain the units of the parameters. Please fill all lines for all parameters with blanks up to column 17 in order to prevent I/O-errors for your EOS. These parameter names and units are displayed in the appropriate PE dialog boxes, then (Figure 4.7d).

name- unit-----1 Tc [K]2 Pc [bar]3 omega [-]4 - -5 - -6 - -7 - -

Figure 4.7c: file EOS_on_DISK.in Figure 4.7d:

Parameter names and units from file EOS_on_DISK.in


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4.8 Some Details 4.8.1 Enforcing Tc and Pc During Optimization The PE feature of optimizing pure-component parameters while simultaneously enforcing the correct reproduction of Tc and Pc, requires that the first two parameters are the major parameters of the EOS. 1) Parameter 1 must be a parameter closely connected to Tc or an energy (Tc, Tc’, a,

u0/k ...). 2) Parameter 2 must be a parameter closely connected to Pc or a volume (Pc, Pc’, b, v00

...). 3) Parameters connected to non-spherical shape (ω, ω’, c from Prigogine, chain length

m) or purely empirical fit parameters (p1 in the PR-Boston-Mathias EOS) must not be parameter 1 or 2.

Please, compare the choice of parameters with the choice of EOS parameters for EOS already implemented in PE. Further, the name of parameter 1 and 2 must not be Tc and Pc if the calculated critical properties are not equal to these parameters: compare the use of Tc’ and Pc’ for the Anderko-EOS or the PR-CPA-EOS. Generally, parameters which are closely related to experimental properties and parameters with simple units should and be preferred, so that the user can easily find initial guesses for parameter optimisations. For the Peng-Robinson EOS, Tc and Pc are used instead of a(Tc) and b, for example. For the volume-translated PR- and SRK-EOS c/b is used instead of c, because this simplifies finding initial guesses, because c/b can believed to be between – 20% and + 20%. 4.8.2 Using Sites and Interaction Parameters Using sites, parameter 4 and 5 should be used to describe the property of the sites, because these parameters are enabled if sites are used and disabled if the number of sites is set to zero in PE.EXE. The binary interaction parameters kε and kκ should be used for interaction between sites for the same reasons. kij (λ ij) should be the first (second) parameter to fit aij and uij, i.e. energy parameters. lij should be used to fit bij, i.e. volume parameters. Please note, that kij=kji, lij=lji, kεij = kεji, kκij=kκji but λ ij = - λ ji.


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4.8.3 The File EOS_on_DISK.sav Because EOS_on_DISK.EXE will run as a program and call itself, it is not possible to benefit of the FORTRAN save-command. Instead, the subroutines read_eos_save and write_eos_save have to be used. All save-parameters have to be saved in the file EOS_on_DISK.sav which serves as (slow) save-buffer. The subroutines read_eos_save and write_eos_save in the file pes_RW_EOS_Save.FOR are used to handle this buffer. These two subroutines have to be modified according to the needs of a new EOS in order to read/write all save-variables. In the beginning of the EOS standard subroutine in pes_EOS_by_User.FOR, read_eos_save has to be called. At the end of the subroutine write_eos_save has to be called in order to save the parameters. One additional call of write_eos_save has to be placed just in front of the calls of pedens / the statement “if (p.lt.0.0) return”. It is important to note, that for one single flash calculation, PE will call the user subroutine with the EOS (placed in EOS_on_DISK.EXE) once with calc_what=0 and several times with calc_what=2. Each call with calc_what=2 (and P>0) will lead to 2 calls of pedens, each leading to about 10 calls of EOS_on_DISK.EXE with calc_what=1. Thus for 20 points on a vapor pressure curve, EOS_on_DISK.EXE will be called about 20 x (1 + 10 x 2 x 10) = ~ 4000 times – so be patient. Parameter optimizations will lead to multiple calls in the range 50 x 4000 = 200.000 and can not be recommended, therefore. Submission of the source code for fixed implementation is recommended, then. Using the included EOS_on_DISK.EXE file, each call of EOS_on_DISK.EXE will lead to one DOS window opening and closing automatically in order to show that PE.EXE and EOS_on_DISK.EXE are working.

Chapter 5: Export of Data

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5 Export of Data PE allows the export of all graphics in Windows metafile format and any printer language like Postscript or HPGL. Each graphic window has its own menu bar offering EXPORT / WINDOWS METAFILE, EXPORT / PRINT and EXPORT / PRINT TO FILE (Figure 5a). Selecting EXPORT / WINDOWS METAFILE will create a Windows metafile file (*.wmf). Selecting EXPORT / PRINT TO FILE will use any available printer driver to create a file. Many Postscript and HPGL printer drivers come along with Windows. Thus, using a Postscript printer driver will create Postscript files (*.ps / *.eps), and using an HPGL / HPGL II printer driver will create HPGL / HPGL II files (*.hgl). These formats allow the export of all graphics to commercial programs like Microsoft Word, Microsoft Powerpoint, Corel Draw, Picture Publisher, and similar. Selecting EXPORT / PRINT will print the graph directly to the printer. Note: In order to keep the efforts to generate different files low, standard printer driver interfaces are used. When generating Postscript files, these files will have a header and footer not appropriate for file processing other than direct printing. It is therefore necessary to delete header and footer. In PE the user can give the file to be generated the extension *.PS and then select the brandnew file with the next file selector box, again. The second selection will start a program, to erase header and footer so that the file can be loaded by CorelDraw, for example.

Figure 5a: Export of graphics to files and printers.

Chapter 6: Future Developments

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6 Future Developments The first preliminary Windows release of PE was shown at the 8th International Conference on Properties and Phase Equilibria for Product and Process Design held at Noordwijkerhout, The Netherlands, on April 30th, 1998 (Pfohl et al., 1998a). The first official Windows release of PE was finished in July 1998 and shown at the 17th European Seminar on Applied Thermodynamics at Vilamoura, Portugal, (Petkov et al., 1999) and on the International Workshop on Scientific Computing in Chemical Engineering II at Hamburg, Germany, (Petkov et al., 1999a, Pfohl et al., 1999a). The present manual describes the PE version released in February 2000. The Windows versions of PE are based on the latest (and last) hp-UNIX version of PE finished by Pfohl (1998) in December 1997. Although enormous efforts have been made, the Windows version of PE does not contain all routines from the UNIX version, yet. The first future development will be the implementation of the remaining routines, especially:

- pure-component EOS parameter optimisations for more than one parameter when enforcing critical temperature and pressure.

- advanced output of data and warnings / error messages to screen and files. Further future developments might also include:

- improving the performance of the flash, including further stability tests - improving three-phase calculations for binary systems - improving azeotropic calculations for binary systems - improving the quality of the graphical output - implementing further possibilities to calculate other phase diagrams

(Txy diagrams, composition tetraeders for quaternary systems) - implementing four-phase algorithms for ternary systems - implementing new EOS, like SAFT-Dimer and EOS with near-critical corrections - implementing export of data to Microsoft Excel.

If users desire special routines or special EOS, these can also be built in.

Appendix A1: EOS in PE

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Appendix A1: EOS Implemented in PE Appendix A1 lists all EOS implemented in the program package PE. The numbers of the sections are identical to the numbers that are used by PE to address the different EOS for internal data processing. This appendix also contains the names of the different pure-component and mixture parameters for each EOS used in PE. This appendix also contains the syntax used in the pure-component parameter blocks in the EXFILES. For many EOS the first two PE parameters have been chosen to be the critical temperature and critical pressure calculated by the EOS** . PE calculates two real EOS parameters (like a and b) from this information. Setting these PE parameters equal to the experimentally determined values of Tc and Pc might allow sufficient calculations with many of these EOS. However, it is common practice to fine tune the real EOS parameters, in order to reproduce special properties with the EOS, best. PE allows this by optimising Tc and Pc to be calculated with the EOS - i.e. changing the real parameters a and b. 0. Van-der-Waals EOS The van-der-Waals EOS is the simplest EOS implemented in PE. It should only be used in order to demonstrate the abilities and inabilities of simple EOS. Use of the van-der-Waals EOS only requires the input of Tc and Pc - corresponding to a and b - for each component. EOS:

P RTv b

av

=−

− 2

PURE-COMPONENT PARAMETERS:

a R TP

c

c

= 0 4218752 2

,

b RT

Pc

c

= 0 125,

** It is important to note, that these are not necessarily identical to the experimentally determined properties of a pure compound.


- 95 -

EXFILE EOS PARAMETER INPUT: 2 0 00 van der Waals

Tc 0.000000 0.000000

Pc 0.000000 0.000000

MIXTURE PARAMETERS: The mixture parameters are calculated according to the mixing rules shown in Appendix A2. LITERATURE: van der Waals (1899) 1. Peng-Robinson EOS The Peng-Robinson EOS is the EOS most widely used in chemical engineering thermodynamics. It is known to give slightly better predications of liquid densities than the Soave-Redlich-Kwong EOS by Soave (1972). The EOS requires three inputs per compound: Tc, Pc and the acentric factor ω. Different modified Peng-Robinson EOS with volume-translations and modified alpha functions are listed in sections 10-19. EOS:

( )P RTv b

a Tv bv b

=−

−+ −2 22


a T a T Tc r( ) ( ) ( , )= ⋅α ω with a T R TPc

c

c

( ) ,= 0 457242 2

( )α ω κ( ) ( ),T Tr r= + −1 1

2 with κ ω ω= + −0 37464 1 54226 0 26992 2, , ,

b RT

Pc

c

= 0 0778,

EXFILE EOS PARAMETER INPUT:

3 0 01 Peng-Robinson

Tc 0.000000 0.000000

Pc 0.000000 0.000000

omega 0.000000 0.000000


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MIXTURE PARAMETERS: The mixture parameters are calculated according to the mixing rules shown in Appendix A2. LITERATURE: Peng and Robinson (1976) 2. Hederer-Peter-Wenzel EOS The Hederer-Peter-Wenzel EOS was presented in the same year as the Peng-Robinson EOS. The EOS needs three inputs per pure compound: Tc, Pc and α. α is a measure for the slope of the vapour pressure curve. In case α=-0.5, the simple Redlich-Kwong EOS (Redlich and Kwong, 1949) is obtained. EOS:

P RTv b

a Tv v b

=−

−+

( )( )


a T a Tc r( ) = ⋅ α with a R TPc

c

c

= 0 427472 2

,

b RT

Pc

c

= 0 08664,


3 0 02 Hederer-Peter-Wenzel

Tc 0.000000 0.000000

Pc 0.000000 0.000000

alpha 0.000000 0.000000

MIXTURE PARAMETERS: The mixture parameters are calculated according to the mixing rules shown in Appendix A2. LITERATURE: Hederer, Peter, and Wenzel (1976)


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3. Soave-Redlich-Kwong EOS The Soave-Redlich-Kwong EOS was the first modification of the simple Redlich-Kwong EOS where the parameter a was made temperature dependent in such a way that the vapour pressure curve could be reproduced well. The EOS requires three input parameters per pure compound: Tc, Pc and ω. Different modified Soave-Redlich-Kwong EOS with volume-translations and modified alpha functions are listed in subsections 30-39. EOS: P RT

v ba T

v v b=

−−

+( )

( )


a T a Tc( ) ( )= ⋅α with a R TPc

c

c

= 0 427472 2

,

( )( )α ( )T m Tr= + −1 1

2 with m = + −0 48 1 574 0 176 2, , ,ω ω

b RT

Pc

c

= 0 08664,


3 0 03 Soave-Redlich-Kwong

Tc 0.000000 0.000000

Pc 0.000000 0.000000

omega 0.000000 0.000000

MIXTURE PARAMETERS: The mixture parameters are calculated according to the mixing rules shown in Appendix A2. LITERATURE: Soave (1972)


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4. Redlich-Kwong EOS The Redlich-Kwong EOS by Redlich and Kwong (1949) is a modification of the van-der-Waals EOS. Similar to the van-der-Waals EOS it should only be used in order to demonstrate the abilities and inabilities of simple EOS, because better EOS exist. Use of this EOS only requires the input of Tc and Pc - corresponding to the parameters a and b - for each component. EOS:

PRT

v ba Tv v b

=−

−+

/( )


aR T

Pc

c= 0 42747

2 2 5

..

b RT

Pc

c

= 0 08664,


2 0 04 Redlich-Kwong

Tc 0.000000 0.000000

Pc 0.000000 0.000000

MIXTURE PARAMETERS: The mixture parameters are calculated according to the mixing rules shown in Appendix A2. LITERATURE: Redlich and Kwong (1949) 5. Dohrn-Prausnitz, nonpolar The Dohrn-Prausnitz EOS is based on the repulsive term of hard spheres by Carnahan and Starling (1969) for pure compounds and the reference term by Boublik (1970) and Mansoori et al. (1971) for mixtures instead of on the van-der-Waals repulsive term. The EOS in not cubic in the volume, therefore. The attractive term of the Carnahan-Starling-vdW EOS (Carnahan and Starling, 1972) was modified in such a way that critical isotherms of many compounds can be reproduced well. The EOS needs three inputs for each pure compound: Tc, Pc and ω.


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EOS: Z Z Zref pert= +

( ) ( )

Zref

DEF

EF

DEF

EF=

+ − + − + −

−

1 2 11

3 3 3 2 3

3

3

2

3

2η η ηη( )

with D i ii

N

==∑ξ σ

1

E i ii

N

==∑ξ σ 2

1

F i ii

N

==∑ξ σ 3

1

σπi

i

A

bN

=⋅

32

13

η π ρ ρ ξ= ⋅ ==∑6 4 1

N F bA i ii

N

Z aRT

b bpert = − − +

ρ ρ ρ1 1 414

5 074

2

, ,

PURE-COMPONENT PARAMETERS: ( )a a a a Tc r= − +( ) ( ) .

tanh( )1 2 0 71 1

with a (1) , ,= +0 367845 0 055966ω a m( ) ( ) ( , , )2 1 0 604709 0 008477= − ⋅ + ω

a R TPc

c

c

= 0 5504082 2

,

( )b b b b Tc r= +( ) ( ) .tanh( ln )1 2 0 8

1

with b(1) , ,= −0 356983 0 190003ω b m( ) ( ) ( , , )2 1 1 37 1 898981= − ⋅ − ω

b RTPc

c

c

= 0 187276,

and m=1 for Tr≥1 m=0 for Tr<1


3 0 05 Dohrn-Prausnitz, nonpolar

Tc 0.000000 0.000000

Pc 0.000000 0.000000

omega 0.000000 0.000000


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MIXTURE PARAMETERS: The mixture parameters are calculated according to the mixing rules shown in Appendix A2. LITERATURE: Dohrn and Prausnitz (1990) 6. Dohrn-Prausnitz EOS, polar The implementation of this EOS is not released for official use, because no tests have been carried out, yet. LITERATURE: Dohrn and Prausnitz (1990, 1990b) 7. Elliott-Suresh-Donohue EOS For non-associating fluids the ESD EOS is a cubic in volume. The non-spherity of molecules is taken into account by the theory of Prigogine (1957). The proposed extension to associating fluids has not been implemented in PE. EOS:

PvRT

c z q Yk Y

m= +−

−+

14

1 19 1 1

ηη

ηη.


ak

Tc c

c cc= =+ − + −

+ − + −ε 1 0 945 1 0134 1

1023 2 225 1 0 478 1

2

2. ( ) . ( )

. . ( ) . ( ) **

b vRTP

c cc c

c

c= =

+ − + −+ − + −

* . . ( ) . ( ). ( ) . ( )

0 0312 0 087 1 0 008 11 2 455 1 0 732 1

2

2

c = + +1 3535 0 533 2. .ω ω

** It is not clearly stated in the original paper, if the formulas for a and b are analytical solutions in order to obtain the point of inflection at Tc and Pc or if the formulas are correlations. There is some evidence that these formulas are correlations.


- 101 -

k1 = 1.7745 k2 = 1.0617 k3 = 1.90476

zm = 9.49 q = 1 + k3(c-1)


3 0 07 Elliott-Suresh-Donohue

Tc 0.000000 0.000000

Pc 0.000000 0.000000

omega 0.000000 0.000000

MIXTURE PARAMETERS:

ηξ

=∑ i iv

v

*

44

ccv

vi j ijη

ξ ξ=

∑∑ ( )*

q Yqv Y

vi j ij ijη

ξ ξ=

∑∑ ( )*

( )** *

cvc v c v

iji j j i=

+2

( )** *

qvq v q v

iji j j i=

+2

ε ε εij i j ijk= −( )1

YkT

kijij= −exp( )

ε2

ηηξ

Yq Y

qi i=∑

LITERATURE: Elliott, Suresh, and Donohue (1990)


- 102 -

8. Sako-Wu-Prausnitz The Sako-Wu-Prausnitz EOS is a modification of the Soave-Redlich-Kwong EOS, where a fourth parameter c is used to explicitly account for non-spherity of molecules. EOS: P RT v b bc

v v ba

v v b= − +

−−

+( )

( ) ( )

PURE-COMPONENT EOS PARAMETERS: a a Tc= α ( )

with α α( ) ( )T T TT

R R

R

= − ++

02 2

21 2

1

α 0 11920 0 11060 1000 0 30734= + +, , ln( ) ,V Vw w

aD cD D cD D

D DR T

Pcc

c=

− + + − +− +

⋅( )( )

( ) ( )1 2 2 1

3 1 20 0 0

202

02

02

0

2 2

D c D D0

302

06 3 3 1 0+ − + − =( )

bD RT

Pc

c= ⋅0

3

Vw is the van der Waals volume and can either be calculated by the group-contribution method of Bondi (1968) or treated as adjustable parameter.


- 103 -

EXFILE EOS PARAMETER INPUT: 4 0 08 Sako-Wu-Prausnitz

Tc 0.000000 0.000000

Pc 0.000000 0.000000

c 0.000000 0.000000

v00 0.000000 0.000000 [dm3/mol]

MIXTURE PARAMETERS The parameters a and b are calculated according to appendix A2. The parameter c is the arithmetic mean of the ci

LITERATURE: Sako, Wu, and Prausnitz (1989) 10-19. Modified Peng-Robinson EOS The Peng-Robinson EOS (Peng and Robinson, 1976) was modified by several researchers. Peneloux et al. (1982) suggested a volume translation which does not affect the calculation of vapour pressures. Many researchers proposed new alpha functions, introducing a new temperature dependence of the parameter a(T). All modified PR EOS implemented in PE include the possibility of a volume translation according to Peneloux et al. (1982). Temperature dependent volume translations like those by Chou and Prausnitz (1989) or Ji and Lempe (1997) are not included in PE, because they lead to unphysical behaviour in the high density region (Pfohl, 1999). The only differences between the different modifications are the different alpha functions with a different number of adjustable parameters. EOS:

PRT

v ba T

v b v b=

−−

+ −~ ~( )

~ ~~ ~2 22 with ~v v c= + and ~b b c= +


a T R TPc

c

c

( ) ,= 0 457242 2

with a T a T Tc r( ) ( ) ( , )= ⋅α ω

~ .bRTP

c

c= 0 0778


- 104 -

10 (PR-VT, Peneloux): κ ω ω= + −0 37464 1 54226 0 26992 2, , ,

( )α ω κ( ) ( ),T Tr r= + −1 12

EXFILE EOS PARAMETER INPUT: 4 0 10 PR-VT, Peneloux

Tc 0.000000 0.000000

Pc 0.000000 0.000000

omega 0.000000 0.000000

c/b 0.000000 0.000000 [ratio of c b/ ~ ] LITERATURE:

Peng and Robinson (1976) Peneloux, Rauzy, and Freze (1982)

11 (PR-VT, Mathias):

κ ω ω= + −0 37464 1 54226 0 26992 2, , , if Tr<1: ( )α κ( ) ( ) ( )( . )T T p T Tr r r r= + − − − −1 1 1 0 71

2

if Tr>1: ( )( )α ( ) exp ( )T c Tr rd= −1

2

c p= + +12

0 3 1κ

.

dc

c=

−1

EXFILE EOS PARAMETER INPUT: 5 0 11 PR-VT, Mathias

Tc 0.000000 0.000000

Pc 0.000000 0.000000

omega 0.000000 0.000000

p1 0.000000 0.000000


Peng and Robinson (1976) Peneloux, Rauzy, and Freze (1982) Mathias (1983)


- 105 -

12 (PR-VT, Stryjek-Vera): κ ω ω ω= + − +0 378893 14897153 017131848 0 01965542 3. . . .

( )( )( )α κ κ( ) ( )( . )T T T Tr r r r= + + + − −1 1 0 7 112

EXFILE EOS PARAMETER INPUT: 5 0 12 PR-VT, Stryjek-Vera

Tc 0.000000 0.000000

Pc 0.000000 0.000000

omega 0.000000 0.000000

kap_1 0.000000 0.000000


Peng and Robinson (1976) Peneloux, Rauzy, and Freze (1982) Stryjek and Vera (1986)

13 (PR-VT, Melhem):

( )α ( ) exp ( ) ( )T m T n Tr r r= − + −1 1 2

EXFILE EOS PARAMETER INPUT: 5 0 13 PR-VT, Melhem

Tc 0.000000 0.000000

Pc 0.000000 0.000000

m 0.000000 0.000000

n 0.000000 0.000000

c/b 0.000000 0.000000 [ratio of c b/ ~ ]LITERATURE:

Peng and Robinson (1976) Peneloux, Rauzy, and Freze (1982) Melhem, Saini, and Goodwin (1989)


- 106 -

14 (PR-VT, Thorwart-Daubert): κ ω ω= + −0 37464 1 54226 0 26992 2, , ,

α κ( ) ( )( )

T T nT

Tr rr

r= + − +

−

1 1

12

EXFILE EOS PARAMETER INPUT: 5 0 14 PR-VT, Thorwart-Daubert

Tc 0.000000 0.000000

Pc 0.000000 0.000000

omega 0.000000 0.000000

n 0.000000 0.000000


Peng and Robinson (1976) Peneloux, Rauzy, and Freze (1982) Thorwart and Daubert (1993)

15 (PR-VT, Boston-Mathias): κ ω ω= + −0 37464 1 54226 0 26992 2, , , if Tr<1: ( )α κ( ) ( ) ( )( . )T T p T Tr r r r= + − − − −1 1 1 0 71

2

if Tr>1: α ( ) exp ( )( )Td

Tr rd= − −

11

12

d p= + +12

0 3 1κ

.

EXFILE EOS PARAMETER INPUT: 5 0 15 PR-VT, Boston-Mathias

Tc 0.000000 0.000000

Pc 0.000000 0.000000

omega 0.000000 0.000000

p1 0.000000 0.000000


Peng and Robinson (1976) Peneloux, Rauzy, and Freze (1982) Boston and Mathias (1980)

MIXTURE PARAMETERS: The mixture parameters are calculated according to the mixing rules shown in Appendix A2. In order to avoid inconsistencies the parameters lij are set to 0. The parameter c is the arithmetic mean of the ci


- 107 -

20. Pfennig EOS The EOS by Pfennig (1988) is a modification of PHCT by Beret an Prausnitz. The EOS is based on the hard sphere repulsion term by Carnahan and Starling (1969), Boublik (1970), and Mansoori et al. (1971). Instead of using the polynomial series by Alder et al. (1972) for attractive forces like in PHCT, Pfennig developed a polynomial

series in λη

η=−1

in order to obtain a pole at closest packing (η=1), automatically.

The EOS has three input parameters for pure compounds: the core volume v0, the potential depth characterised by T*, and the parameter c after Prigogine (1957) characterising deviations from spherical shape. EOS:

aRT

c aT

r

ij

i

jj

n

i

m= ∑∑

==

λ~

01

~*T

TT

= , ητ

=vv

0 , τπ

= =2

60 74048. ... (closest packing fraction)

PURE-COMPONENT EOS PARAMETERS: For the Pfennig EOS no conversion formulas are needed. EXFILE EOS PARAMETER INPUT:

3 0 20 Pfennig

T* 0.000000 0.000000

v0 0.000000 0.000000

c 0.000000 0.000000

MIXTURE PARAMETERS

T T T ki j i j ijj

N

i

N* * * ( )= −∑∑

==ξ ξ 1

11 with kij = kji

vv v

i ji j

j

N

i

N

001 3

01 3 3

11 2=

+

∑∑

==ξ ξ ,

/,/

c ci ii

N= ∑

=ξ

1

LITERATURE:

Pfennig (1988)


- 108 -

21. BACK The BACK EOS for pure components was proposed by Chen and Kreglewski (1977). Chen and Kreglewski used the reference term for hard convex bodies by Boublik (1975) and the dispersion term by Alder et al. (1972) with refitted constants in order to obtain better representations of real fluids (Alder et al. fitted the constants to molecular dynamics data). For each component four parameters are needed: the core volume V00, the potential depth u0/k, α for characterising non-spherity and η/k influencing the temperature dependence of the potential depth. Later, Simnick et al. (1979) as well as Machat and Boublik (1985b) extended the BACK EOS to mixtures by proposing different mixing rules. Pfohl and Brunner (1998) proposed a new mixing rule and claimed it to be more efficient as the best mixing rule „Set B“ by Machat and Boublik (1985b). The „Set B“ mixing rule by Machat and Boublik (1985b) and the mixing rule by Pfohl and Brunner (1998) are implemented, here. It should be noted, that using the EOS number 23 in PE, i.e. „SAFT-ConveyBody“ and setting all parameters m=1 the BACK EOS is also obtained. The pure component results are the same as obtained here. Using SAFT-ConvexBody, setting all m=1, offers two further mixing rules, whereof the „van-der-Waals mixing rule“ is even better than the two supplied for EOS 21, here (compare: Pfohl and Brunner, 1998). EOS: Z Z Zh a= +

Zy

yy

y yy

h

s

s s

s

s s s

s

=−

+−

+ −−

11

31

312

2 2

3( ) ( )( )

( )α α with y V

Vss

s

=

π 26

0

Z mD ukT

VV

anm

sn

s

smn

m

=

==∑∑

0

1

9

1

4

with the universal constants Dij:

D11 = -8,8043 D21 = 2,9396 D31 = -2,8225 D12 = 4,164627 D22 = -6,0865383 D32 = 4,7600148 D13 = -48,203555 D23 = 40,137956 D33 = 11,257177

D14 = 140,4362 D24 = -76,230797 D34 = -66,382743 D15 = -195,23339 D25 = -133,70055 D35 = 69,248785

D16 = 113,515 D26 = 860,25349 D27 = -1535,3224 D41 = 0,34 D28 = 1221,4261 D42 = -3,1875014 D29 = -409,10539 D43 = 12,231796

D44 = -12,110681


- 109 -

PURE-COMPONENT EOS PARAMETERS:

uk

uk kT

= +

0

1 η

V V ukT

0 000 3

1 0 12 3= − −

, exp


4 0 21 BACK

u0/k 0.000000 0.000000

v00 0.000000 0.000000

alpha 0.000000 0.000000

eta/k 0.000000 0.000000

MIXTURE PARAMETERS V x Vs i i

0 0=∑ α α αs

si i i i i iV

x V x V= ⋅ ∑∑10

0 013

23( ) ( )

„Set B“ by Machat and Boublik, 1985b:

uk

x x uk

uk

ksi j

i jij= −∑∑ ( )1

„Set B’“ by Pfohl and Brunner, 1998:

uk

x x k Vx x V

s i juk

uk ij ij

i j ij

i j

=− ⋅∑∑

∑∑( )1 0

0 with: VV V

iji j00 0 31

31

3

2=

+

( ) ( )

LITERATURE: Alder, Young, and Mark (1972) Boublik (1975) Chen and Kreglewski (1977) Simnick, Lin, and Chao (1979) Machat and Boublik (1985a,b) Pfohl and Brunner (1998)


- 110 -

22. SAFT-Pfennig/Pfohl This EOS is a modification of SAFT and was derived by Pfohl. It is not published, yet. It uses the original SAFT equations by Huang and Radosz (1991, compare section 23) but uses the term for the attractive interactions between spheres from the Pfennig EOS (Pfennig, 1988, compare section 20) instead of the term from the BACK EOS. The SAFT-Pfennig/Pfohl EOS therefore needs one pure-component parameter per pure component less than the original SAFT EOS: - T* = depth of the potential

- v0 = volume of one sphere - m = the number of spheres in one molecule - same parameters for association as SAFT

The implementation of this EOS in PE has not been tested thoroughly, yet. It was not shown yet that the use of this EOS is advantageous compared to the use of original SAFT, yet. Therefore the use of this EOS is not recommended at this state.


- 111 -

23. SAFT-ConvexBody The SAFT-ConvexBody EOS is a modification of original SAFT by Huang and Radosz (1990-1993) presented by Pfohl and Brunner (1998). This EOS allows to mix the sphere chains from original SAFT with convex bodies from original BACK. Thus, each molecule can either be modelled as a sphere chain with the possibility of association or as a convex body. In case no convex body is present, the EOS exactly reduces to original SAFT by Huang and Radosz (1991). In case no sphere chain is present, the EOS exactly reduces to original BACK (see section 21). A molecule in SAFT-ConvexBody is described by either of the following pure-component parameter sets in Table A1.1. Table A1.1: parameters for pure components, either sphere chain or convex body. Parameter sphere-chain (SAFT) convex body (BACK) potential depth u0/k u0/k core size v00 v00 chain length m m≡1 T-dependence of u/k η/k η/k convex parameter α≡1 α number of asso. sites sites sites≡0 asso. Potential depth ε A no association possible asso. Volume κ A no association possible Note: it is clearly shown in Table A1.1, that either m or α has to equal unity in order to chose the SAFT or BACK model for each molecule. Further, it is not possible to create convex bodies with α≠1 and association sites. EOS: a a a a ar hs disp chain asso= + + +


- 112 -

aRT N

hs

A

=⋅

+ −−

− −

−

6 3 31

123

1 2 3 1 2 32

3 32 0

23

32 3π ρ

ξ ξ ξ ξ ξ ξ ξξ ξ

ξ ξξ

ξ( )

ln( )

with ξ π ρ0 6

= ⋅N AA A X m mi i= =∑

ξ π ρ1 6= ⋅N BA B X m di i i i=∑ ( )α 1

3

ξ π ρ2 6= ⋅N CA C X m di i i i=∑ ( )α 1

3 2

ξ π ρ η3 6= ⋅ =N DA D X m di i i=∑

aRT

m D ukT

disp

ij

i

ji

j

=

==∑∑ η

τ1

9

1

4

aRT

X m gchain

i i iji

N

= −=∑ ( ) ln1

1

with gd d

d dd d

d diji j

i j

i j

i j

=−

+− +

+− +

11

31

213

2

32

22

33

2

ξξξ

ξξ( ) ( )

aRT

X XXasso

iAi

Ai

Ai

sites i

i

N= +

−

∑∑

==ln

( ) 1211

with XN X X

Ai

A jB j Ai B j

B j

sites j

j

N=+ ∑∑

==

1

111

ρ ∆( )

∆Ai B jij

Ai B j

ijAi B jg

kT=

−

expε

σ κ1 3

σσ σ

iji j=

+2

ε ε ε εAi B j Ai B j

ijk= −( )1

κ κ κ κAi B j Ai B j

ijk= −( )1

The constants Dij are listed in section 21 („BACK EOS“)


- 113 -


V V ukT

0 000 3

1 0 12 3= − −

, exp

d VN A

=

0 21

3

uk

uk kT

= +

0

1 η


8 1 23 SAFT-ConvexBody

u0/k 0.000000 0.000000

v00 0.000000 0.000000

m 0.000000 0.000000

eps/k 0.000000 0.000000 [ε A ]kap 0.000000 0.000000 [κ A ] eta/k 0.000000 0.000000 [η/k] alpha 0.000000 0.000000 [α]Sites 0 0 0 0 0 0


- 114 -

MIXTURE PARAMETERS SAFT-ConvexBody offers two mixing rules for the parameter u/k. Both mixing rules require two binary interaction parameters per binary system and exactly reduce to the original SAFT mixing rules by Huang and Radosz (1991) if all λ=0. The advantages of these two-parameter mixing rules are explained in detail by Pfohl and Brunner (1998). „2-parameter van-der-Waals mixing rule“

ukT

ukT

ukT

= +( ) (1)0

with ukT

X X m m k VX X m m V

i j i ju ukT ij ij

i j i j ij

i j( ) ( )0 0

0

1=

−∑∑∑∑

( )

ukT

X m X m V

X X m m V X m

i i j j iju ukT ij

i j i j ij i i

i j

(1)

=

∑∑

∑∑∑

λ 03

0 2

13

„2-parameter volume-fraction mixing rule“

ukT

f fu ukT

k f fu ukTi j

i jij i j

i jij= − +

∑∑ ∑∑( )1

13

3

λ

with the volume fraction f X m VX mVkk k k

i i i

=∑

0

0

LITERATURE: Chapman et al. (1986-1990) Huang and Radosz (1990-1993) Pfohl and Brunner (1998) Boublik (1975, 1981) Chen and Kreglewski (1977) Alder, Young, and Mark (1972) Wertheim (1984-1987)


- 115 -

24. SAFT-Conventional This EOS is identical to SAFT-ConvexBody (section 23) but it does allow no use of convex bodies (i.e. all α=1). This implementation allows to save different sets of pure component parameter sets (and kij) in the same EXFILE - one with convex bodies using EOS 23 and one without convex bodies using this EOS. This way it is easy to compare calculation results, later. This implementation of SAFT should be used, if calculations are intended with original SAFT by Huang and Radosz (1990-1993) in order to keep calculations as easy as possible. EXFILE EOS PARAMETER INPUT:

7 1 24 SAFT-Conventional

u0/k 0.000000 0.000000

v00 0.000000 0.000000

m 0.000000 0.000000

eps/k 0.000000 0.000000 [ε A]

kap 0.000000 0.000000 [κ A ]

eta/k 0.000000 0.000000 [η/k]Sites 0 0 0 0 0 0

25. Perturbed-Chain SAFT The FORTRAN code for the EOS in PE and this description have been supplied by:

J. Groß, F. Tumakaka, G. Sadowski, and W. Arlt TU Berlin Institut für Verfahrenstechnik Forschungsgruppe Polymerthermodynamik Sekr. TK 7 Straße des 17. Juni 135 10623 Berlin Germany

The Perturbed-Chain SAFT EOS was proposed by Groß and Sadowski (1999). Similar to original SAFT, molecules are assumed to be chains composed of spherical segments. In the dispersion term of the original SAFT model, the chain-like structure of polymer compounds is not taken into account, since a hard-sphere reference is used. Whereas, in the PC-SAFT model, the dispersive forces are accounted for applying a perturbation theory of second order (Barker and Henderson), using an expression for the radial pair distribution of a hard-chain reference fluid.


- 116 -

Three pure component parameters are required for each non-associating component: the segment number m, the segment diameter σii , and the attraction parameter εii. The association expression of the SAFT-model is used for associating compounds. The additional association parameters are then: the number of association sites, the association potential depth, and the association volume. EOS: Besides the association part, which is identical to original SAFT, the Helmholtz energy of the EOS is split in a hard-chain contribution and a perturbation contribution:

NkTA

NkTA

NkTA perthc

+=

Hard-Chain Contribution:

( )iii

hsiiii

S

hsidealhc

gmxkTN

AmNkTA

NkTA σ∑ −−⋅=− ln)1(

with ( )

−⋅

−+

−+

−= 302

3

32

333

32

3

21

0

1ln)1()1(

31 ζζζζ

ζζζ

ζζζ

ζkTNA

S

hs

and 3

3

22

2

23

2

3 )1(2

)1(3

)1(1

ζζ

ζζ

ζ −

++

−

++

−=

ji

ji

ji

jihsij dd

dddd

ddg

where ∑=i

miiim dmxρπζ

6

Perturbation Contribution

NkTA

NkTA

NkTA pert

21 +=

with ∑∑

⋅⋅−=

i jij

ijjiji kT

mmxxmINkT

A 31

1 ),(2 σε

ηπρ

and ∑∑

⋅⋅

∂

∂+⋅⋅−=−

i jij

ijjiji

hchc

kTmmxxmIZZm

NkTA 3

2

2

1

2 ),( σε

ηρ

ρπρ

The one-fluid mixing concept is applied to the compressibility term of the second order perturbation term, i.e.

[ ]

−−−+−−+

−−+=

∂

∂+ 2

432

4

2

)2)(1(2122720)1(

)1(281

ηηηηηη

ηηη

ρρ mmZZ

hchc


- 117 -

In these equations, the integrals over the radial pair distribution function of chain-molecules were replaced by power series in density of sixth order

i

iiii a

mm

mma

mmamI η⋅

−−+−+=η ∑

=

6

02101

211),(

i

iiii b

mm

mmb

mmbmI η⋅

−−+−+=η ∑

=

6

02102

211),(

with the universal constants a0i, a1i, a2i and b0i, b1i, b2i :

i a0i a1i a2i b0i b1i b2i 0 0.79198281 -0.62311554 -0.06777556 0.79198281 -0.62311554 -0.067775561 1.07148651 0.48573437 0.02837411 2.14297303 0.97146874 0.056748232 0.91474661 1.12485267 0.09612281 2.74423982 3.37455809 0.288368413 -7.81060651 -2.09485016 0.06815027 -31.2424260 -8.37940062 0.272601104 25.7855977 9.45049823 0.05980187 128.927988 47.2524911 0.299009335 -56.9822877 -17.1027262 0.28660979 -341.893726 -102.616357 1.719658746 41.9308941 7.77610281 -0.74701698 293.516259 54.4327197 -5.22911885

PURE-COMPONENT PARAMETERS: di is the temperature dependent segment diameter of component i:

−⋅−⋅= kTd ii

iiiεσ exp12.01


6 1 25 PC-SAFT

u0/k 0.000000 0.000000 [ εii ]

d00 0.000000 0.000000 [ σii, unit: Å ]

m 0.000000 0.000000

eps/k 0.000000 0.000000 [ε A for association]

kap 0.000000 0.000000 [κ A for association]

Sites 0 0 0 0 0 0

MIXTURE PARAMETERS The PC-SAFT EOS is applied to mixtures using one-fluid mixing rules. Conventional combining rules are employed to determine the parameters between a pair of unlike segments


- 118 -

( )jjiiij σσσ +=21

and

)1( ijjjiiij k−εε=ε

LITERATURE: Groß and Sadowski (1999) Groß and Sadowski (2000) Chapman et al. (1988) Barker and Henderson (1967)

! For further in-depth information, please contact the authors at http://pcitr4.fb10.tu-berlin.de/pc-saft/pc-saft-home.html

27 PHSCT Perturbed Hard-Sphere-Chain Theory was developed by Song, Lambert, and Prausnitz (1994) in order to obtain an EOS for normal fluids and polymers. In PHSCT each molecule is modelled as a chain of chemically bonded spheres similar to SAFT (compare section 23/24). PHSCT was extended to copolymers by Song, Lambert, and Prausnitz (1994b), later. The latter extension is not included in PE. Thus, for each component only v00 and ε, which characterise one sphere as well as the number r of spheres in a molecule have to be known. EOS:

Z r r b g r gkT

r r ai j i j ij ij i i iii

N

i j i j ijj

N

i

N

j

N

i

N= + − − −∑ − ∑∑∑∑

= ====1 1 1

1 1111ρ ξ ξ ξ

ρξ ξ( )( )


- 119 -


a v FkTs ra= 4 00ε ε(( )

)

b v FkTs rb= 4 00 (( )

)ε

πd T b( )3

6 4=

( )F x xa ( ) . . exp .= + −0 717 19003 0 5152

( ) ( )[ ]F x x xb ( ) . exp . . exp . /= − + − − −0 5849 0 4772 0 4151 1 10669 1 4

s(r) is calculated from a FORTRAN function obtained from Professor Prausnitz, University of California at Berkeley.


3 0 27 PHSCT

eps/k 0.000000 0.000000

v00 0.000000 0.000000

r 0.000000 0.000000

MIXTURE PARAMETERS

a v FkTs r

FkTs r

kij ij ai i

aj j

i j ij= −4 100 (( )

) (( )

) ( )ε ε ε ε

vv v

iji j00003 003

3

2=

+

, b

b bij

i j=

+

3 33

2 , d

d dij

i j=+2

gijij ij=

−+

−+

−1

132 1

12 12

2

3ηζ

ηζ

η( ) ( )

ηρ

ξ= ∑=4 1

i i ii

Nr b

ζρ

ξiji j

ijk k k

k

Nb bb

r b=

∑

=42 3

1

/

LITERATURE: Song, Lambert, and Prausnitz (1994)


- 120 -

30-39. Modified Soave-Redlich-Kwong EOS The Soave-Redlich-Kwong EOS (Soave, 1972) was modified by several researchers. Peneloux et al. (1982) suggested a volume translation which does not affect the calculation of vapour pressures. Many researchers proposed new alpha functions, introducing a new temperature dependence of parameter a(T). All modified SRK EOS implemented in PE include the possibility of a volume translation according to Peneloux et al. (1982). Temperature dependent volume translations like those by Chou and Prausnitz (1989) or Ji and Lempe (1997) are not included, because they lead to unphysical behaviour in the high density region (Pfohl, 1999). The only differences between the different modifications are the different alpha functions with a different number of adjustable parameters. EOS:

PRT

v ba T

v v b=

−−

+~ ~( )

~(~ ~) with ~v v c= + and ~b b c= +


a T a Tc( ) ( )= ⋅α with a R TPc

c

c

= 0 427472 2

,

~ .bRTP

c

c= 0 08664


- 121 -

30 (SRK-VT, Peneloux): m = + −0 48 1 574 0 176 2, , ,ω ω

( )α ( ) ( )T m Tr r= + −1 12

EXFILE EOS PARAMETER INPUT: 4 0 30 SRK-VT, Peneloux

Tc 0.000000 0.000000

Pc 0.000000 0.000000

omega 0.000000 0.000000


Soave (1972) Peneloux, Rauzy, and Freze (1982)

31 (SRK-VT, Mathias):

m = + −0 48 1 574 0 176 2, , ,ω ω if Tr<1: ( )α ( ) ( ) ( )( . )T m T p T Tr r r r= + − − − −1 1 1 0 71

2

if Tr>1: ( )( )α ( ) exp ( )T c Tr rd= −1

2

cm

p= + +12

0 3 1.

dc

c=

−1

EXFILE EOS PARAMETER INPUT: 5 0 31 SRK-VT, Mathias

Tc 0.000000 0.000000

Pc 0.000000 0.000000

omega 0.000000 0.000000

p1 0.000000 0.000000


Soave (1972) Peneloux, Rauzy, and Freze (1982) Mathias (1983)


- 122 -

32 (SRK-VT, Schwartzentruber-Renon): m = + −0 48508 1 55171 0 15613 2, , ,ω ω if Tr<1: ( )α ( ) ( ) ( )( )T m T p T p T p Tr r r r r= + − − − + +1 1 1 11 2 3

2 2

if Tr>1: α ( ) exp ( )( )Td

Tr rd= − −

1 1 1

2

d m p p p= + − + +12

11 2 3( )

EXFILE EOS PARAMETER INPUT: 7 0 32 SRK-VT, Schwartzentruber-Renon

Tc 0.000000 0.000000

Pc 0.000000 0.000000

omega 0.000000 0.000000

p1 0.000000 0.000000

p2 0.000000 0.000000

p3 0.000000 0.000000


Soave (1972) Peneloux, Rauzy, and Freze (1982) Watanasiri and Gallier (1988) Schwartzentruber, Renon (1989) Schwartzentruber, Renon, and Watanasiri (1990) Aspen Plus 9.3 Handbook, model: „SR-polar“


- 123 -

33 (SRK-VT, Thorwart-Daubert): m = + −0 48508 1 55171 0 15613 2, , ,ω ω

α ( ) ( )( )

T m T nT

Tr rr

r= + − +

−

1 1

12

EXFILE EOS PARAMETER INPUT: 5 0 33 SRK-VT, Thorwart-Daubert

Tc 0.000000 0.000000

Pc 0.000000 0.000000

omega 0.000000 0.000000

n 0.000000 0.000000


Soave (1972) Peneloux, Rauzy, and Freze (1982) Thorwart and Daubert (1993)

34 (SRK-VT, Peneloux-3P):

m = + −0 48 1 574 0 176 2, , ,ω ω

( )α ( ) ( )T m Tr r= + −1 12

cRTP

c

c= +0 40768 0 00385 0 08775. ( . . )ω

EXFILE EOS PARAMETER INPUT: 3 0 34 SRK-VT, Peneloux-3P

Tc 0.000000 0.000000

Pc 0.000000 0.000000

omega 0.000000 0.000000

LITERATURE: Soave (1972) Peneloux, Rauzy, and Freze (1982), model „RSKc3“


- 124 -

35 (SRK-VT, Boston-Mathias): m = + −0 48 1 574 0 176 2, , ,ω ω if Tr<1: ( )α ( ) ( ) ( )( . )T m T p T Tr r r r= + − − − −1 1 1 0 71

2

if Tr>1: α ( ) exp ( )( )Td

Tr rd= − −

11

12

dm

p= + +12

0 3 1.

EXFILE EOS PARAMETER INPUT: 5 0 35 SRK-VT, Boston-Mathias

Tc 0.000000 0.000000

Pc 0.000000 0.000000

omega 0.000000 0.000000

p1 0.000000 0.000000


Soave (1972) Peneloux, Rauzy, and Freze (1982) Boston and Mathias (1980)

MIXTURE PARAMETERS: The mixture parameters are calculated according to the mixing rules shown in Appendix A2. In order to avoid inconsistencies, the parameters lij are set to 0. The parameter c is the arithmetic mean of the ci.

40 3P1T The 3P1T EOS by Yu and Lu (1987) contains three EOS parameters (=3P) whereof one is temperature dependent (1T). The EOS is the result of the work by Yu et al. (1986) who evaluated the abilities of many different cubic EOS to correlated different properties of the first 10 members of the homologue series of n-alkanes. The use of this EOS requires Tc, Pc and ω as pure-component parameters for each pure compound. A brief comparison of this EOS and the Peng-Robinson EOS by Pfohl et al. (1997a), Pfohl et al. (1998), and Pfohl (1998) showed that this EOS correlated pure-component properties as well or slightly better than the PR EOS but gave slightly weaker mixture correlations. EOS: P RT

v ba

v v c b v c=

−−

+ + +( ) ( )3


- 125 -

PURE-COMPONENT EOS PARAMETERS: a a Tc r= ⋅α ω( , )

a R TPc

c

c

= − +( , , , )0 46863 0 0378304 0 00751969 22 2

ω ω

if Tr<1: log ( ) ( ) ( )( ),10 0 1 2

2 1α ω ωT M A A T A T Tr r r r= ⋅ + + − if Tr≥1: log ( ) ( ) ( )( ),10 0 1 2 1α ω ωT M A A A Tr r= ⋅ + + −

if ω≤0,49 M( ) , , , ,ω ω ω ω= + − +0 406846 1 87907 0 792636 0 7375192 3 A0 = 0,536843 A1 = -0,39244 A2 = 0,26507 if 0,49<ω≤1 M( ) , , , ,ω ω ω ω= − + −0 581981 0 171416 1 84441 1190472 3 A0 = 0,79355 A1 = -0,53409 A2 = 0,37273

b RTP

c

c

= − −( , , , )0 0892828 0 0340903 0 00518289 2ω ω

cb

= − + +1 29917 0 648463 0 895926 2, , ,ω ω


3 0 40 3P1T

Tc 0.000000 0.000000

Pc 0.000000 0.000000

omega 0.000000 0.000000

MIXTURE PARAMETERS The mixture parameters are calculated according to the mixing rules shown in Appendix A2. LITERATURE: Yu and Lu (1987) Yu, Adachi, and Lu (1986)


- 126 -

41 Anderko, linear The Anderko EOS implemented in PE is the EOS proposed by Anderko (1989 and 1989a) applicable to model systems containing any number of associating and inert compounds. This AEOS (association + equation of state) uses the 3P1T EOS (section 40) as a so-called „physical“ part in order to model non-specific interactions between molecules and a linear association model by Kempter and Mecke (1940) according to the chemical theory in order to explicitly account for association (Eq. 32 in Anderko, 1989a). The EOS therefore requires the same parameters as the 3P1T EOS for non-associating („inert“) compounds and as much as six pure-component parameters for each associating compound. If association theory is included to model a pure compound, the critical temperature calculated with the EOS is not longer Tc and Pc used as input for the physical part of the AEOS, i.e. the 3P1T EOS. The inputs required for the 3P1T part of the AEOS have therefore renamed into Tc’, Pc’, and ω’ by Anderko in order to prevent confusion. In addition to these three pure-component parameters, the input of ∆H 0 , ∆S 0 , and ∆Cp

0

which characterise hydrogen bonds is required for each associating component. EOS:

P P T v T T P PRTv RT

vK

P T EOSc c c c

i

ij jj

Ni

N= = = = +

+ + ∑∑−

=

=

3 1

1

1

2

1 1 4( , , , , , )' ' '

r r r r r r rξ ω ω

ξ

ξ

PURE-COMPONENT EOS PARAMETERS: a) compare 3P1T EOS in section 40 with

r rT Tc c= ' ,

r rP Pc c= ' , and r r

ω ω= ' .

b) )ln()(ˆln 000000

TTCSTTTCHKRT ppi +∆−−∆+∆=−

Note (also true for PE V1.0): In order to be compatible with Anderko (1989-1992), the unit of K̂ is [atm], so that KK ˆ01323.1= , because the unit of K is [bar].


- 127 -

EXFILE EOS PARAMETER INPUT: 6 0 41 Anderko, linear

Tc’ 0.000000 0.000000

Pc’ 0.000000 0.000000

omga’ 0.000000 0.000000 **

H 0.000000 0.000000 [kJ/mol]

S 0.000000 0.000000 [J/mol/K]

CP 0.000000 0.000000 [J/mol/K]

MIXTURE PARAMETERS The mixture parameters for the physical part given by the 3P1T EOS are calculated according to the mixing rules shown in Appendix A2. The cross association constants Kij are zero if either i or j is an inert compound, otherwise: K K K kij i j ij= −( )1 ε

cc c

i ji j

j

N

i

N=

+∑∑==

ξ ξ211

LITERATURE: Anderko (1989-1992) Yu and Lu (1987) Kempter and Mecke (1940) Dolezalek (1908) 50 CPA-SRK-Tassios Kontogeorgis, Voutsas, Yakoumis, and Tassios (1996) presented an EOS named Cubic-Plus-Association EOS. Similar to Anderko (1989, section 41) they used a simple cubic EOS as a reference system for physical interactions and added a second part to this cubic EOS in order to account for association. Their proposed CPA EOS is based on the Soave-Redlich-Kwong-EOS (Soave, 1972, section 3) and uses the powerful association part from SAFT (compare section 23). However, Kontogeorgis et al. (1996) use 4 pure-component parameters for each non-associating compound although they use the Soave-Redlich-Kwong EOS, that generally only requires 3 pure-component parameters. They use the parameters a0, b, c1, and Tc

EXP , whereof TcEXP is set equal to the experimentally determined critical

temperature of the pure component.

** „omga’“ is no spelling mistake. No identifier may exceed 5 characters.


- 128 -

Instead of using the radial distribution function for pure compounds for mixtures like Tassios et al. did, the version implemented in PE uses the gij for mixtures by Boublik (1970) and Mansoori et al. (1971). EOS:

PRT

v ba T

v v bRTv

a asso

=−

−+

+

( )( )

~∂∂ρ

ρ

~ ln( )

aaRT

X XXasso

asso

iA

A

A

sites i

i

Ni

i

i

= = +−

==∑∑ 1

211

PURE-COMPONENT PARAMETERS: a T a c T Ti i i c i

EXP( ) ( ( / )),= + −0 1 21 1

b Nd

i Ai= 4

6

3π

σ ii

Av

bN

=

1 3/

Note: σ i id≠


7 1 50 CPA-SRK-Tassios

a0 0.000000 0.000000

b 0.000000 0.000000

c1 0.000000 0.000000

TcEXP 0.000000 0.000000

eps/k 0.000000 0.000000 [K]

beta 0.000000 0.000000

Sites 0 0 0 0 0 0


- 129 -

MIXTURE PARAMETERS: The mixture parameters for the physical part (i.e. the SRK EOS) are calculated according to the mixing rules shown in Appendix A2. The association equations are essentially identical to the ones of SAFT.

XN X X

Ai

A jB j Ai B j

B j

sites j

j

N=+ ∑∑

==

1

111

ρ ∆( )

∆Ai B jij

Ai B j

ijAi B jg

kT=

−

expε

σ κ1 3

with

σσ σ

iji j=+2

ε ε ε εAi B j Ai B j

ijk= −( )1

κ κ κ κAi B j Ai B j

ijk= −( )1

gd d

d dd d

d diji j

i j

i j

i j

=−

+− +

+− +

11

31

213

2

32

22

33

2

ξξξ

ξξ( ) ( )

ξπ

ρk A i ik

i

NN X d= ∑

=6 1

LITERATURE: Kontogeorgis, Voutsas, Yakoumis, Tassios (1996) Yakoumis, Kontogeorgis, Voutsas, Tassios (1997) Voutsas, Kontogeorgis, Yakoumis, Tassios (1997) Soave (1972) Pfohl (1998)


- 130 -

51 CPA-SRK Kontogeorgis et al. (1996, compare section 50) use 4 pure-component parameters for each non-associating compound although they use the Soave-Redlich-Kwong EOS, that generally requires 3 pure-component parameters, only. EOS 51, 52, 53, and 54 are designed in such a way that only three parameters are necessary for the physical part of the EOS, i.e. the cubic SRK, PR, or 3P1T EOS. These three parameters are named Tc’, Pc’ and ω’, here. The prime (‘) indicates that the critical temperature (Tc), critical pressure (Pc), and acentric factor (ω) calculated with the EOS are not longer identical or even similar to the inputs of the physical terms, if association is included (compare section 41). Because the incorporation of association can lead to Tc’<<Tc and thus easier to Tc’<T, the alpha functions of the SRK and PR EOS 51 and 52 have been extended to „supercritical temperatures concerning the physical term“ for T>Tc’ according to the approach by Boston and Mathias (1980, compare section 15 and 35 with p1=0). The equations of EOS 51, 52, 54 concerning the physical part are thus identical to the ones described in section 35, 15, and 40, respectively. The equations for the association part are those listed in section 50. EXFILE EOS PARAMETER INPUT:

6 1 51 CPA-SRK

Tc’ 0.000000 0.000000

Pc’ 0.000000 0.000000

omga’ 0.000000 0.000000

eps/k 0.000000 0.000000

beta 0.000000 0.000000

Sites 0 0 0 0 0 0


- 131 -

52 CPA-PR This EOS is practically identical to the one described in section 51. The only difference is that the cubic EOS describing the physical part is the PR EOS from section 15 instead of the SRK EOS from section 35. EXFILE EOS PARAMETER INPUT:

6 1 52 CPA-PR

Tc’ 0.000000 0.000000

Pc’ 0.000000 0.000000

omga’ 0.000000 0.000000

eps/k 0.000000 0.000000

beta 0.000000 0.000000

Sites 0 0 0 0 0 0

53 CPA-PR (a,b,om) This EOS is essentially identical to the one described in section 52. The only difference is that a0 (=a(Tc’)) and b are used as first two parameters instead of Tc’ and Pc’ and that the alpha function by Mathias (1983) is used (compare EOS 11 with p1=0). It is recommended to use EOS 52 (CPA-PR) because much better convergence can be achieved at pure-component parameter determinations. EXFILE EOS PARAMETER INPUT:

6 1 53 CPA-PR (a,b,om)

a0 0.000000 0.000000

b 0.000000 0.000000

omga’ 0.000000 0.000000

eps/k 0.000000 0.000000

beta 0.000000 0.000000

Sites 0 0 0 0 0 0


- 132 -

54 CPA-3P1T This EOS is practically identical to the one described in the previous sections 51 and 52. The only difference is that the cubic EOS describing the physical part is the 3P1T EOS from section 40. EXFILE EOS PARAMETER INPUT:

6 1 54 CPA-3P1T

Tc’ 0.000000 0.000000

Pc’ 0.000000 0.000000

omga’ 0.000000 0.000000

eps/k 0.000000 0.000000

beta 0.000000 0.000000

Sites 0 0 0 0 0 0

90 SAFT-HR/vdW This is the SAFT EOS by Huang and Radosz (1990-1991). The EOS 90 is a very old implementation of SAFT in PE using the van-der-Waals mixing rule. The new implementation by EOS=23/24 should be preferred, because it is much faster. This implementation can be used to double check the implementation of EOS 23/24. EXFILE EOS PARAMETER INPUT:

7 1 90 SAFT-HR/vdW

u0/k 0.000000 0.000000

v00 0.000000 0.000000

m 0.000000 0.000000

eta/k 0.000000 0.000000 [η/k]kap 0.000000 0.000000 [κ A ]

eps/k 0.000000 0.000000 [ε A]

Sites 0 0 0 0 0 0


- 133 -

MIXTURE PARAMETERS: The mixture parameters are calculated according to the way described by Huang and Radosz (1991) which is nearly identical to the way described for EOS 23. However, the second binary interaction parameter lij used here, is used to modify „m“ and not used to modify „u“:

m mi j ijj

N

i

N= ∑∑

==ξ ξ

11 with m

m mlij

i jij=

+−

21( )

∑∑

∑∑

= =

= == N

i

N

jijjiji

N

i

N

jijijjiji

vmm

vummu

1 1

0

1 1

0

ξξ

ξξ with

33 03 0

0

2

+= ji

ij

vvv and u u u kij i j ij= −( )1 ∇

91 SAFT-HR/vf This is the SAFT EOS by Huang and Radosz (1990-1991). The EOS 91 is also a very old implementation of SAFT in PE. The new implementation by EOS=23/24 should be preferred, because it is much faster. In contrast to EOS 90 it uses the volume fraction mixing rule. EXFILE EOS PARAMETER INPUT:

7 1 91 SAFT-HR/vf

u0/k 0.000000 0.000000

v00 0.000000 0.000000

m 0.000000 0.000000

eta/k 0.000000 0.000000 [η/k]kap 0.000000 0.000000 [κ A ]

eps/k 0.000000 0.000000 [ε A]

Sites 0 0 0 0 0 0

MIXTURE PARAMETERS:

m mi j ijj

N

i

N= ∑∑

==ξ ξ

11 with m

m mlij

i jij=

+−

21( )

∑∑= =

=N

i

N

jijji uffu

1 1

with ∑

=

= N

jjjj

iiii

vm

vmf

1

0

0

ξ

ξ and u u u kij i j ij= −( )1 ∇

That means γ = 1 for volume fraction according to Huang and Radosz (1991).

∇ The formulas for deriving „u“ had been wrong in the manual for PE V1.0. The program had been correct.


- 134 -

94 Deiters Deiters (1981, 1981a, 1982) constructed an EOS based on perturbed hard chain theory with the aim to obtain an EOS which yields correct critical temperatures, pressures, and densities of pure components. Calculations of phase equilibria of mixtures under elevated pressure may be affected by the wrong representation of the pure components’ critical points, seriously (compare: Pfohl et al., 1998). However, the extension of this EOS to mixtures is difficult - leading to an implementation in PE for pure components, only. Deiters (1982) proposed an extension to binary mixtures. Because the EOS originally proposed by Deiters (1981) could not be integrated analytically, Deiters (1982) exchanged a part by a polynomial series. The EOS with the polynomial function Ft is implemented in PE. The three pure-component parameters needed for the Deiters EOS are: a, b, and c. EOS:

( )Z ccabc T

h T Fi= +−−

− + +14 2

10

2

3 2 02

1ξ ξ

ξρ

γ ψ( )

( ~)

with F p c T b tt ijkk j i

ijk= −∑∑∑ −

===( ) ~ ( )1

0

10

0

6

0

3ρ

where pijk = constants obtained from Deiters. ψ ( ~) ~(exp( ~ ) )T T T= −−1 1 γ = − −1 0 697816 1 2. ( )c

h0 7 0794046= . , c0 0 6887= . , ~TckT

=ε

, ξπ

ρ=2

6b


εk

a=

EXFILE EOS PARAMETER INPUT: The EOS implemented in PE is for pure compounds, only. MIXTURE PARAMETERS The EOS implemented in PE is for pure compounds, only. LITERATURE: Deiters (1981, 1981b, 1983) Pfohl et al. (1998)


- 135 -

95 PHCT The Perturbed Hard Chain Theory by Beret and Prausnitz (1975) was the first attempt to construct a hard chain EOS which should cover the whole density region from ideal gas to close packing and all kinds of molecular complexity by incorporation of Prigogines factor c also covering other than translational degrees of freedom (Prigogine, 1957). However, the original paper by Beret and Prausnitz is restricted to pure components - and so is the implementation of PHCT in PE. Many groups did pick up the idea by Beret and Prausnitz afterwards. An early extension of PHCT to mixtures is the one by Donohue and Prausnitz (1978). The modification of PHCT by Pfennig (1988) which is also applicable to mixtures is described in section 20. The EOS 95 needs three inputs for a pure compound: <eqk>, <rv*>, and <c>. EOS:

Z cmAv T

nmm nmn

= +−−

+ ∑∑

==1

4 21

2

31

9

1

4η ηη( ) ~ ~

PURE-COMPONENT EOS PARAMETERS: ~ *v rv− =1 ρ

~/

TcTq k

=ε

EXFILE EOS PARAMETER INPUT: The EOS implemented in PE is for pure compounds, only. MIXTURE PARAMETERS The EOS implemented in PE is for pure compounds, only. LITERATURE: Beret and Prausnitz (1975) Donohue and Prausnitz (1978) Carnahan and Starling (1969) Prigogine (1957)


- 136 -

96 LJ-SAFT, Kraska This EOS is the Lennard-Jones SAFT by Kraska and Gubbins (1996). The implementation in PE only works for pure components, yet. 97 LJ-SAFT, JZG This Lennard-Jones SAFT is similar to that by Kraska and Gubbins (1996, see above section). The only difference between both EOS is that EOS 97 uses the equation of Johnson, Zollweg, and Gubbins (1993) instead of the equation of Kolafa and Nezbeda (1994) to describe the LJ-contribution. This implementation in PE suffers the same deficits like EOS 96, because most equations are the same.

Appendix A2: Mixing Rules for Cubic EOS

- 137 -

Appendix A2: Mixing Rules for Simple EOS PE offers several mixing rules for the following simple EOS, which are mostly cubic EOS or based on a cubic EOS:

- van der Waals (EOS 0) - Peng-Robinson (EOS 1) and modifications (EOS 10-15) - Hederer-Peter-Wenzel (EOS 2) - Soave-Redlich-Kwong (EOS 3) and modifications (EOS 30-35) - Redlich-Kwong (EOS 4) - Dohrn-Prausnitz, nonpolar (EOS 5) - Sako-Wu-Prausnitz (EOS 8) - 3P1T (EOS 40) - Anderko (EOS 41) - CPA EOS (EOS 50-55)

Most of the mixing rules for these EOS calculate the mixture parameters a and b for the EOS according to the one-fluid mixing rules (Eq. A2.1 and A2.2) and the only difference between them is the combining rule that determines how the cross coefficients aij and bij are calculated. However, the Mathias-Klotz-Prausnitz mixing rule and the MKP-WS mixing rule can not be written in terms of equations A2.1 and A2.2. In order to minimise confusion, the term „mixing rule“ will be used instead of „combining rule“, also.

a x x ai j ijj

N

i

N= ∑∑

== 11 (A2.1)

b x x bi j ijj

N

i

N= ∑∑

== 11 (A2.2)

All mixing rules offer three adjustable binary interaction parameters, except the simple quadratic mixing rule which only offers two adjustable parameters per binary system. One binary interaction parameter is used to adjust the mixture parameter b, the other parameters are used to adjust the mixture parameter a. It should be noted that the Adachi-Sugie-, Panagiotopoulos-Reid-, Melhem-, and Stryjeck-Vera mixing rule suffer the so-called Michelsen-Kistenmacher-Syndrome (Michelsen and Kistenmacher, 1990). Using them yields different calculation results if λ≠0 and the mole fraction xi of one component is arbitrarily split into two mole fractions xi’ and xi“ with xi’+xi“=xi. The use of these mixing rules is not recommended, because this theoretical disadvantage is strongly correlated to inabilities to model real systems as shown by Mathias et al. (1991), Pfohl (1998), and Pfohl et al. (1999). Instead, it is recommended to use the Mathias-Klotz-Prausnitz mixing rule which does


- 138 -

not suffer this deficit, although it is identical to all above mixing rules for binary systems. The MKP-WS mixing rule is a modification of the Mathias-Klotz-Prausnitz mixing rule which also reproduces the quadratic dependence of the second virial coefficient enforced by theory, correctly. A2.1 Quadratic mixing rule The quadratic mixing rule is the only mixing rule described here which only offers a maximum of two binary interaction parameters per binary system, whereof only one is used to adjust the parameter a. All other mixing rules described hereafter offer a maximum of three adjustable binary parameters, whereof two are used to adjust the parameter a. These mixing rules exactly reduce to quadratic mixing rule, if the third parameter λ is not used, i.e. set to zero. Often, the quadratic mixing rule is sufficient for the correlation of equilibria in systems that do not contain specific interactions. Mostly, the use of the second adjustable parameter l ij is avoided, by setting all l = 0 .

a a a kij i j ij= −( )1 with k kji ij= (A2.3)

bb b

iji j

ij=+

−2

1( )l with l lji ij= (A2.4)

A2.2 Panagiotopoulos-Reid Mixing Rule (1985) The quadratic mixing rule (A2.1) is often only sufficient for the correlation of equilibria in systems where no specific interactions are involved. Panagiotopoulos and Reid (1985) have been the first to use two binary interaction parameters per binary system in order to fine-tune the parameter a (Eq. A2.5 and A2.6). In most cases, the parameters l ij are set to zero, because the use of kij and λ ij already allows excellent

representations of many highly non-ideal binary phase equilibria. The use of the two parameters kij and λ ij is much more powerful than the use of the two parameters kij and l ij of the quadratic mixing rule (compare: Schwartzentruber and Renon, 1989). If all λ

are set to zero, the quadratic mixing rule is obtained again. a a a k xij i j ij

PRijPR

i= − +( )1 λ with λ λijPR

ijPR

jiPR

jiPRk k= − = − (A2.5)

bb b

iji j

ij=+

−2



- 139 -

The original formula for the calculation of parameter a proposed by Panagiotopoulos and Reid (1985) is shown in Eq. A2.7. a a a k k k xij i j ij ij ji i= − + −( ( ) )1 with k kij ji≠ (A2.7)

A2.3 Adachi-Sugie Mixing Rule (1986) Adachi and Sugie (1986, „A new Mixing Rule“) proposed their mixing rule (Eq. A2.8)1 slightly later than Panagiotopoulos and Reid (1985) and claimed that this mixing rule is identical with the Panagiotopoulos-Reid mixing rule for binary systems but not identical with it for multicomponent mixtures. However, Pfohl (1998) showed that both mixing rules are absolutely identical. The incorporation of both mixing rules in PE can be stated as superfluous, therefore. a a a k x xij i j ij

ASijAS

i j= − − −( ( ))1 λ with k kjiAS

ijAS= and λ λji

ASijAS= − (A2.8)

bb b

iji j

ij=+

−2


The original formula for the calculation of parameter a proposed by Adachi and Sugie (1986) is shown in Eq. (A2.10). a a a l m x xij i j ij ij i j= − − −( ( ))1 with l lij ji= and m mij ji= − (A2.10)

1 Equations (4), (5), (6), (13), and (14) in this publication. It should be noted that these equations are different from the equations (7) and (8) in this publication, although Adachi and Sugie claim that they are identical.

Equations (7) and (8) would lead to λ λijAS

jiAS= + and therefore to a complete elimination of this parameter in

calculations, i.e. calculation results independent of the value of λijAS .


- 140 -

A2.4 Stryjek-Vera Mixing Rule (1986) Stryjek and Vera (1986) claimed that their „Van Laar-Type“ mixing rule (A2.11)2 gives better correlation results than their „Margules-Type“ mixing rule3 which is identical to the Panagiotopoulos-Reid mixing rule programmed in PE.

a a ak k

x k x kij i jijSV

jiSV

i ijSV

j jiSV= −

+( )1 with λ λij

SVijSV

jiSV

jiSVk k:= − = − (A2.11)

bb b

iji j

ij=+

−2


A2.5 Melhem Mixing Rule (1991) Melhem et al. (1991) proposed a new mixing rule (A2.13)4 which does not suffer of what they call a dilution problem, i.e. that the concentration dependency of the mixing rule decreases as the number of components increases. But, this mixing rule still suffers of the Michelsen-Kistenmacher Syndrome detected by Michelsen and Kistenmacher (1990), before (see above).

a a a kx

x xij i j ijM

ijM i

i j= − +

+( )1 λ with λ λij

MijM

jiM

jiMk k= − = − (A2.13)

bb b

iji j

ij=+

−2


A2.6 Mathias-Klotz-Prausnitz Mixing Rule (1991) Mathias et al. (1991) proposed a new mixing rule (A2.15)5 which does not suffer of the Michelsen-Kistenmacher Syndrome (Michelsen and Kistenmacher, 1990). Using this mixing rule yields the same calculation results if the mole fraction xi of a component is arbitrarily split into two mole fractions xi’ and xi“ with xi’+xi“=xi. This theoretical advantage is strongly correlated to the ability to model real systems with similar compounds as shown by Mathias et al. (1991) for benzene + cyclohexane and Pfohl (1998) for two cresol isomers. It is noteworthy that this mixing rule is identical with the above mixing rules by Panagiotopoulos-Reid (= Adachi + Sugie) and Melhem for

2 Equation (13) in this publication. 3 Equation (12) in this publication. 4 Equation (6) in this publication. 5 Equations (4), (5), (6), (12), and (13) in this publication.


- 141 -

binary systems, so that binary interaction parameters regressed from binary systems in literature can be used for this mixing rule without the necessity to optimise the parameters, again (compare A2.8).

( )a x x a a k x x a ai j i j ijMKP

j

N

i

N

i j i j jiMKP

j

N

i

N= −∑∑ + ∑

∑

== ==( )

/

111 1

1 3 3

1λ

with k kjiMKP

ijMKP= and λ λji

MKPijMKP= − (A2.15)

bb b

iji j

ij=+

−2


Mathias et al. (1991) used the abbreviation lij instead of λ ij in Eq. A2.15.

A2.7 MKP-WS Mixing Rule Bünz (1995) modified the above Panagiotopoulos-Reid mixing rule in order to retrieve a mixing rule which correctly reproduces the quadratic dependence of the second virial coefficient enforced by theory (Eq. A2.17), according to the ideas of Wong and Sandler (1992).

ba

RTB x x B x x b

aRTi

j

N

i

N

j ij ij

N

i

N

jij

− = = ∑∑ = ∑∑ −

== ==11 11

(A2.17)

Since the Panagiotopoulos-Reid mixing rule suffers the Michelsen-Kistenmacher Syndrome, the implementation in PE by Pfohl (Eq. A2.18 and A2.19) is based on the Mathias-Klotz-Prausnitz mixing rule.

( )a x x a a k x x a ai j i j ijBP

j

N

i

N

i j i j jiBP

j

N

i

N= −∑∑ + ∑

∑

== ==( )

/

111 1

1 3 3

1λ (A2.18)

( )b x xb b

RTx x a ai j

i j

j

N

i

N

ij i j i j jiBP

j

N

i

N=

+∑∑ − + ∑

∑

== ==21

111 1

1 3 3

1( )

/

l λ (A2.19)

with k kji

BPijBP= , λ λji

BPijBP= − , and l lji ij=

The influence of a volume translation by introducing parameter c≠0 in EOS 10-19/30-39 and using Eq. A2.19 has not been investigated. A usage of a combination of both is not recommended therefore - although the usage is possible.


- 142 -

A2.8 Conversion of Interaction Parameters The Panagiotopoulos-Reid mixing rule ( = Adachi-Sugie mixing rule), the Melhem mixing rule, and the Mathias-Klotz-Prausnitz mixing rule are identical if applied to binary systems but yield different results if applied to systems with more than two components. If optimum binary interaction parameters have been determined by fitting binary data, these parameters are also the optimum parameters for the other above mixing rules, therefore. The following table contains the conversion formulas between the binary parameters of the different mixing rules in order to obtain identical calculation results in binary systems. The Melhem mixing rule is not included, because it obviously gives the same calculation results in binary systems as the Panagiotopoulos-Reid mixing rule, if k kij

MijPR= and λ λij

MijPR= (compare Eq. (A2.5) and (A2.13), if xi + xj = 1).

Table A2.1: Conversion of binary interaction parameters between different mixing rules Panagiotopoulos-

Reid Adachi-Sugie Mathias-Klotz-

Prausnitz Panagiotopoulos-Reid

- k kijPR

ijAS

ijAS

ijPR

ijAS

= −

= −

λ

λ λ2

k kijPR

ijMKP

ijMKP

ijPR

ijMKP

= +

=

0 5, λ

λ λ

Adachi-Sugie k kij

ASijPR

ijPR

ijAS

ijPR

= −

= −

05

0 5

.

.

λ

λ λ

- k kijAS

ijMKP

ijAS

ijMKP

=

= −λ λ0 5,

Mathias-Klotz-Prausnitz

k kijMKP

ijPR

ijPR

ijMKP

ijPR

= −

=

0 5, λ

λ λ

k kijMKP

ijAS

ijMKP

ijAS

=

= −λ λ2 -

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