vapor-liquid equilibriuiil prediction for iii-defined...
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Vapor-Liquid EquilibriuIIl Studies. Prediction for III-Defined Mixtures
and Modification of a Data Collecting Apparatus
by
Eric Langat Cheluget
A Thesis submitted to the Faculty of Graduate Studies and Research of McGill University in partial fulfillment of
the requirements for the degree of Master of Engineering
Department of Chemical Engineering McGill University Montreal, Canada
@ Eric L. Cheluget, 1988
December 1988
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Abstract
The modeling aspect of this thesis involved the prediction of vapor-liquid equilibria for petroleum fluids using continuous thermodynamic methods. A charaderization scheme for deriving the molar distribution curves of petroleum fractions and gas condensates from True Boiling Point distillations was proposed. The Extended Spline Fit Technique with hoiling point as the distributing variable was found to be an accurate and versatile way of representing the molar distribution curve of these fluids. A continuous Peng-Robinson-Stryjek-Vera equation of state was developed using Generalized Single Carbon Number Properties and by correlating the KI parameter to the hoiling point.
Flash, dew and bubble point calculations were performed for one imaginary and two real semicontinuous systems. For continuously distributed components, following the suggestion of Hendriks, the number of equilibrium and mass balance equations was reduced through integration over the range of the distributing variable using Legendre-Gauss quadrature. The integrated equations were solved using accelerated successive substitution.
For real systems, it was found that binary interaction parameters had varying effects on the calculated vapor-liquid equilibria, although in general these overshadowed that of the KI function. Calculated results are comparable to those obtained by others using different continuous thermodynamic and pseudocomponent methods.
On the experimental side a vapor-liquid equilihria data collecting apparatus was modified. Changes included improvements in the areas of accessibility, operable pressure range and gas phase sampling. The equipment was used to measure vapor-liquid equilibria data for the binary system CO2 -cyclohexane at 313 K and in the pressure range 1300 to 5200 kPa.
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Resumé
l'aspect modellisation de cette thèse comprend la prédiction des équilibres gaz-liquide pour des fluides à base de pétrole à l'aide d'une méthode thermodynamique continue. Un schéma de caractérisation de dérivation des courbes de distribution de fractions molaires de pétrole et de condensats de gaz à partir de distillations du point réel d'ébullition fut proposé. La méthode "Extended Spline Fit Technique" avec le point d'ébullition comme variable de distribution fut trouvée comme étant un moyen à la fois précis et versatile de représenter la courbe de distribution molaire de ces fluides. Une équation d'état Peng- Robinson-Stryjek-Vera à été développée utilisant la méthode "Generalized Single Carbon Number Properties" et en correlant le parametre I\':} au point d'ébullition.
Les calculs du point éclair, de condensation ct d'ébullition, ont été déterminés pour un système imaginaire ainsi que pour deux systèmes semi-continus réels. Pour des composés continuellement distributées, suivant la proposition de I1endricks, le nombre d'équations d'équilibre et de bilans de masses à été l'eduit en intégrant à travers un intervalle de la variable (hstributrice à l'aide de la quadrature Legendre-Gauss. Les équations d'intégration ont été résolues à l'aide d'une substitution successive accélérée.
Pour des systèmes réels, on a trouvé que les paramètres d'interaction du système binaire ont un effet qui varie sur les équilibres vapeur-liquide calculés, bien qu'en général ceux-ci ont tendance à avoir un effet d'écran sur ceux de la fonction I\:}. Les résultats calculés sont comparables à ceux obtenus par d'autres chercheurs, utilisant difr(~rentcs méthodes thermodynamiques continues et pseudo-composantes.
Du côté expérimental un appareil permettant de mesurer des données d'équilibres vapeur-liquide a été modifié. Les changements comprennent des améliorations des parties faciles d'accès, Je l'intervalle de pression opérationnelle, ainsi que de la phase d'échantillonnage du gaz. L'appareil fut utilisé pour mesurer des données des équilibres vapeur- liquide pour le système binaire C02-cyclohexane à 313 K et dans l'intervalle de pression de 1300 à 5200 kPa.
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Acknowledgements
The whole is a sum of parts, the author would like to express thanks and appreciation to various people who contributed to the project:
Prof essor J. H. Vera for invaluable guidance, support and constant encouragement. Messrs A. Krish, H. Alexander, W. Greenwood and A. Gagnon for their assistance in the design and construction of equipment. Messrs J. Dumont, N. Habib, E. Siliauskas and L. Cusmich for their advice and help in running the experiments. Messrs O. Khennache and T. Aguinet for translating the summary. Ms S. Ells for typing assistance The members of the research group; Professor D. Berk, Ms M. Sejnoha. The staff and graduate students in the Department of Chemical Engineering for providing a stimulating and enjoyable working enviroment. The Department of Chemical Engineering of McGill University and the Natural Sciences and Engineering Research Council of Canada for financial assistance.
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Table of Contents .' Abstract ................................................................. .
Resumé ....... .. . ..... . . ...... . . ..... . . . . .. .... . . . . . .... . . ..... . . ..... . . . . 11
Acknowledgernents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III
Table of Contents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV
List of Abbreviations.. . . . ..... . . ..... . . . . .. .... . . . . ..... . . ..... . . ..... . . . . Vll
List of Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIlI
List of Symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XllI
1.0 INTRODUCTION.................................................... 1
1.1 VLE Prediction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 VLE Measurement .................................... ,.. ........... 4
1.3 Summary of Objectives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 VLE Prediction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.2 VLE Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.0 FRAMEWORK FOR PREDICTION OF VLE FOR ILL-DEFINED MIX-
TURES................................................... 9
2.1 Continuous and Semicontinuous Mixtures. . ...... . . ..... . . ..... . . . . . 9
2.1.1 Contim"0us Systems. . ....... . ....... .. . . ..... . . ...... . ..... . .. . 10
2.1.2 Semicontinuous Systems...................... .................. 12
2.2 Characterization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Conversion of TBP Distillation Curve to Molar Distribution Curve 13
2.2.2 Representation of Molar Distribution Curve by Extended Spline
Fit Technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.3 Summary of Characterization Procedure. . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
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2.3.1 The Continuous PRSV Equation of State..... . . ...... . . ....... . 33
2.3.2 Equation of State Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.3 K-factors and Mass Balances. . .. ..... . . . ..... . . . ..... . . ....... . 39
2.3.4 Reduction in Number of Equations... . . ...... . . ........ ....... . 41
2.3.5 Solution of Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3.5.1 Accelerated Successive Substitution. . . . . . . . . . . . . . . . . . . . . . . . 45
2.3.5.2 HandIing of IntegraIs: Legendre-Gauss Quadrature. . . . . . . . . 49
2.3.5.3 Implementation of Algoritbms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.0 PREDICTION OF VLE FOR ILL-DEFINED MIXTURES... . ....... . 53
3.1 VLE for a Model Fluid . . . . .. .. .. .. . .. .. .. .. .. . .. . . . .. .. . . . . .. . .. . . . 53
3.1.1 Flash Calculations ... . . ..... . . ........ . . ..... . . . ...... . . ...... . 57
3.1.2 Saturation Pressure and Temperature Calculations..... . . ..... . . 62
3.2 VLE for Real Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2.1 Radosz et al. Systems.......................................... 67
3.2.2 Hoffmann et al. System........................ . . . . . . . . . . . . . . . . 77
4.0 HIGH PRESSURE VLE DATA COLLECTION: MODIFICATION OF
EQUIPMENT AND MEASUREMENTS FOR THE C02-
CYCLOHEXANE SYSTEM ................. _ . . . . . . . . . . . . . 82
4.1 Description of Apparatus and Experiment _ . . ..... . . . ...... . . ..... . . . 82
4.2 The Gas Sa.mpling Valve. . .. .. .. . . .. . .. .. . . . .. .. . . . . . .. .. . . . .. . .. . .. 85
4.3 Experimental Results and Discussion...... . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.0 CONCLUSIONS AND RECOMMENDATIONS ...... . . ...... . . ...... . 98
5.1 Conclusions......................................................... 98
5.1.1 VLE Prediction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.1.2 VLE Measurement..................... ........ ................ 99
5.2 Recommendations . ...... . . ...... . ........ . . ...... . . ...... . . ...... . . 99
5.2.1 VLE Prediction...... . . ...... . . ....... . . . ..... . . . ...... . ....... 99
5.2.2 VLE Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
REFERENCES. . . . . . . . . . . . .... . . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 101
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Appendix Al Generalized Single Carbon Number Groups...... ............ 104
• Appendix A2 Fugacity Coefficient Expressions for Semioontinuous PRSV EOS 107
Appendix A3 Problem in ICt(Tb) Function............... ...... . . ..... . . . ... 110
Appendix A4 Semicontinuous Rachford-Rice Objective Function. . . . . . . . . . . . 112
Appendix A5 Acceleration of Successive Substitution Method. . . . . . . . . . . . . . . 115
Appendix A6 Computer Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Appendix A7 Calibration of Experiment.al Apparatus.... ...... ............. 121
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API BETAC
CHART EOS ESFC
ESFT EXP GSCNP HPLC KIFIT PRSV EOS SCN SPLNFT TBP TVLET VLE
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List of Abbreviations
American Petroleum Institute. Calculation method where molar distributions are described using beta probability density functions. Fortran program for characterization of oil samples. Equation of State. Calculation method where molar distributions are described using the Extended Spline Fit Technique. The Extended Spline Fit Technique. Experimental results. Generalized Single Carbon N umber Properties. High Performance Liquid Chromatography. Fortran program for evaJuation of KI (n) function. Peng-Robinson-Stryjek-Vera equation of state. Single Carbon Number. Fortran subroutine for Extended Spline Fit Technique. True Boiling Point. Fortran program for multicomponent VLE calculations. Vapor-Liquid Equiibria.
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• List of Figures
Figure Title Page
2.1 Discrete a.nd continuous composition for a multicomponent mixture (not to scale) 11
2.2 Derivatiou of a molar distribution curve from a volume TBP distillation 15
2.3 Derivation of a molar distribution curve from a weight distillation or a simulated TBP analysis. 17
2.4 Molar distribution curve of Jacoby et al. (1959) oil. 23 2.5 Differentiation of unsmoothed distillation curve. 25 2.6 Illustration of Extended Spline Fit Technique. 27 2.7 Aigorithm for correlation of let to Tb. 36 2.8 Accelerated successive substitution algorithm for
isothermal flash calculation. 46 2.9 Accelerated successive subsitution algorithm for bubble
and dew point calculation. 48 2.10 Aigorithm for Newton-Raphson method with numerical
derivatives. 50 3.1 ESFT representation of molar distribution curve of
model fluid. 54 3.2 The ICt(Tb) function for n-alkane family. 56 3.3 Molar distribution in alkane family: flash. 58 3.4 K-factor in alkane family: flash. 58 3.5 Effect of number of ESFT segments on flash calculation results 61 3.6 Molar distribution in alkane family: bubble point. 63 3.7 Molar distribution in alkane family: dew point. 63 3.8 Effect of number of ESFT segments on dew point temperature
calculation results. 65 3.9 Characterization of saturates-rich oil. 68 3.10 Calculated and experimental solubility of propane in liquid
for saturates-rich oil. 72 3.11 Calculated and experimental solubility of propane in liquid
phase for aromatics-rich oil. 74
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( Figure Title Page
4.1 Experimental set up. 83 4.2 Modification of the gas phase sampling valve. 86 4.3 Alternate gas phase sampling valve design. 86 4.4 Cell assembly. 88 4.5 Gas phase sampling operation. 90 4.6 CO2-cyclohexane VLE results using syringe method calibration
constant. 93 4.7 CO2-cyclohexane VLE results using mixture method calibration
constant. 95 4.8 ln(K) versus In(P) plot for CO2-cyclohexane. 97 A3.1 Plot of optimal ICI versus TR • 111 A7.1 Pressure transducer calibration curve. 122 A7.2 Area vs moles for CO2 • 124 A 7.3 Area vs moles for cyclohexane. 125
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an
a.;
a(T)
aJ(I) A bn
b
b(I)
B
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Ci C+ ,
D Fn(I)
FP(I) FOi(v)
FugP
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List of Symbols
Constants in "1 (Tb) polynomial function.
PRSV EOS attractive parameter for phase II.
PRSV EOS discrete pure compound attractive parameter.
PRSV EOS attractive parameter.
Continuous attractive parameter function for ensemble j.
Dimensionless fugacity coefficient variable aPI R2T2•
PRSV EOS excluded volume parameter for phase II.
PRSV EOS excluded volume parameter.
Continuous excluded volume parameter function.
Dimensionless fugacity coefficient variable bPI RT.
Constants in spline fit cubic polynomials.
Total number of discrete components and continuously distributed
ensembles in a semicontinuous system.
Single carbon number group or alkane with i number of carbon atoms.
Single carbon number group or alkane with i or greater number of
carbon atoms.
Number of discrete components in a semicontinuous system.
Molar distribution function for phase II.
Molar distribution function of ensemble j in phase II.
Objective function (i-th) for flash calculations.
Fugacity of component i in phase n. Gradient of Gibbs free energy with respect to vapor composition
for component i.
9 Vector of aIl 9, 's.
i T Transpose of g. 1 Distribution variable.
Ii Value of distribution variable at a quadrature point.
(
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Kco2 Ken KAI) MW n
N
NP P
Pc Pc(l) R sa T
Tb
Tc Tc(l) v
v Va W,
x!1 , Zj
Z~ , z
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Binary interaction parameter between eomponent i and j.
K-factor of ensemble or discrete eomponent j.
Generalized K-Factor for ensemble j: ratio of averaged EOS
a parameter in vapor phase to that in liquid phase.
Generalized K-Factor for ensemble j: ratio of averaged EOS
b parameter in vapor phase to that in liquid phase.
K-factor for carbon dioxide.
K-factor for alkane family.
Continuous K-factor function for ensemble j.
Molecular weight.
Number of quadrature points.
N umber of moles.
Number of points.
Pressure.
Critical pressure.
Continuous critical pressure function.
Gas constant.
Specifie gravity.
Absolute temperature.
Normal boiling point.
Critical temperature.
Continuous critieal temperature function.
Molar volume.
Molar critical volume.
Volume.
Voltage.
Weighting factor at quadrature point i.
Mole fraction of discrete component or ensemble i in phase n. Number of moles.
Mole fraction.
Compressibility factor PV/nRT.
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Greek Letters
oCT) al) , -n !Ji
tJ
tJ(I)
v
4>, (I) 1/1(U) w
Subscripts
J T
Superscripts
F
L V
II
Temperature dependent parameter in PRSV EOS.
Averaged EOS attractive parameter for ensemble j in phase II.
Averaged EOS excluded volume parameter for ensemble j in phase II.
Error term in quadrature rule.
Variable in interaction parameter expression.
Function to be integrated in quadrature rule.
Fraction liquid.
Parameter in PRSV EOS.
Parameter in PRSV EOS.
Parameter in PRSV EOS.
Continuous function relating K} to Tb.
Term in fugacity coeficient expression representing the
differentiation of the mixing rule with respect to composition.
Fraction vapor.
Fugacity coefficient of ensemble j at a given value of J.
Integrated function in quadrature rule.
Acentric factor.
Index representing discrete components or ensembles.
Index representing ensembles.
Iteration index.
Feed Phase
Liquid Phase
Vapor Phase
Phase i.e., feed, vapor, liquid.
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( List of Tables
Table Title Page
3.1 Alkane family II:l(T,,) correlation constants (using GSCNP). 57 3.2 Results of flash calculation at T=500 K and P=3000 kPa. 58 3.3 Effect of 11:1 (T,,) correlation on flash calculations. 60 3.4 Results of bubble point pressure calculation at T=350 K. 62 3.5 Results of dew point temperature calculation at P=350 kPa. 64 3.6 Effect of 1\':1 correlation on calculated saturation pressure
and temperature. 65 3.7 Results of calculations using different EOS constant relations. 65 3.8 Feed mole fraction of Radosz et al. mixtures. 69 3.9 Liquid phase composition of saturates-rich oil system. 71 3.10 Liquid phase composition of aromatics-rich oil system. 73 3.11 Vapor phase composition of oil systems. 76 3.12 Composition at dew point for condensate at 367 K. 78
'1 3.13 Results of flash calculation at 367 K and 13887 kPa. 80 4.1 Experimental VLE results for CO2-cyclohexane 92 4.2 Details of replicate runs. 94 A1.1 Details of GSCNP correlations. 105 A1.2 Generalized single carbon number group properties (GSCNP). 106 A3.1 Values of 11:1 and other parameters in the TR = 0.7 region. 111 A7.1 Calibration data for the pressure transducer. 122 A7.2 Calibration data for COz. 123 A7.3 Calibration data for cyclohexane. 125
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CDPI'IIa 1
Classical separation operations, such as distillation and
absorption constitute a fundamental part of chemical engineering.
While it may be true that newer separation processes such as ion
exchange, reverse osmosis and bioseparations are becoming
increasingly important, due ta their high costs they are likely
to be limited ta the production of specialty products such as
pharmaceuticals. The large scale production of common chemicals,
especially hydrocarbons, and pollution control operations, will
continue ta utilize traditional separation processes for many
years to come.
There are 900d economic reasons for improvinq the efficiency
of classical separation processes. Capi tal costs for separation
equipment are typically in the range 40 - 80 , of total plant
invastment. Another factor is increasing enerqy costs which call
for more enerqy efficient separations.
The quantitative modeling of separation operations requires
accurate and versatile phase equilibria relationships. The design
engineer requires reliable and generalized predictive .odels
that apply over wide ranges of pressure, tempe rature and
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COllpO.ition. OVar the y.ar. che.ical enCJin.er. and acientiat.
have quantitatively solved pha.e equilibria probl ... by r •• ortinCJ
ta cla.aical and statistical theraodynaaica.
The aubject of this thesis i. the .aa.urement and prediction
o~ hiCJh pressure vapor-liquid equilibria (VLE). The direction
chosen for this study is two sided. On one hand an atte.pt was
made to improve the prediction of vapor-liquid equilibriua for so
called ill-defined, or polydi.perae, aixturea. Th •• e are
aul ticomponent fluids, such as heavy fossil fuela and polper
aolutions, that consist of so aany siailar compound. (thousanda)
that it is virtually impossible to determine the type and amount
o~ each of the constituents. The second part of the project
involved the modification of an existinCJ VLE data collectinCJ
apparatus.
1.1 VLB Prediction
The need for a better understandinCJ of VLE for polydisperse
mixtures is immediately evident to those concerned with the
processing of such fluids. In addition to those mentioned above
other examples of polydisperse fluids are natural qas
condensates, coal derivatives, and solutions of fatty acids. An
illustration of a situation requirinq VLE prediction for
polydisperse fluids is the case of a typical oil refinery. The
accurate design and optimal performance of distillation tower. in
thi. case is economically crucial and require. reliable
correlations for equilibrium X-factors. Siailarly, the isothermal
fla.h routine ia probably the most commonly used and perhap. the
moat iaportant routine in a process aimulator (Joulia et al.,
1986) •
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Pipeline engineers also encounter ill-defined .ixtures.
Hydrocarbon condensat. appearance at natural 9a. control stations
or _rket teninals has interrupted continuous 9as servie. on
nuaeroua occasions. Under pipeline conditions this conden.ation
is a VLE phenomenon. Condi tions causing condensation -V be
deterained through dew point calculations. The ob •• rvation of
increased vaporization with higher pressures and condensation
with lowered pressures in natural ga& systems i. known as
retrograde condensation. Prediction of liquid fonaation in the
retrograde region is usually dlfficult (Berpan et al., 1975)
since it requires accurate knowledge of the concentrations of the
constituents, temperature, pressure and equilibriua constant ••
There are two major obstacles to overcoae in atteaptinq to
.odel fluids that have very many components. The first is that of
characterization: how can one establish what compounds make up a
co.plex fluid? In principle it may be possible, usinq analytical
che.istry, to deteraine aIl the components and thair relative
allounts, but in practice this is prohibitively expensive and
tedious. The second is that even if one were able to obtain aIl
the information, subsequent VLE calculations would involve an
unaanaqeably larqe set of equations.
The traditional method of calculatinq VLE for ill-defined
mixtures is to treat the mixture as a finite set (usually less
than fifteen) of representative pure compounds. In such a
procedure, the fluid is characterized by division into fractions,
say by fractional distillation or extraction. Physical prop.rties
such as average boilinq point, .ol.cular weight, density, etc.,
are .ea~ured for each fraction. sasad on the.. phy.ical
prop.rties and the observed VLE of the .ixture as a whole, the
fractions are assigned individual pura co.ponent proparties
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(Peder.en et al., 1983, 1984a, 1984bJ Mebrotra et al., 1985). In
other word. the fractions beco.. p •• udocoaponents vi th a •• igned
critical propertie., acentric factors, and/or equation of state
(EOS) para.eters.
It is important to point out that p •• udoco.ponanta are given
the latter properties based on the .easured values of the
average physical properties And the observed overall VLB. Thus
in a sense the problem i5 one of, usinq standard discrete
component VLE EOS procedures, optimizinq equilibriWl prediction
by .anipulating critical properties and acentric factors vithin
the constraints of the observed physical properties. With so many
different correlations between critical properties/acentric
factors/molecular weight and boilinq point/specifie gravit y
available one is bound to get many different sets of
ps.udocomponents representinq the same fluide This introduces a
measure of arbitrariness into the pseudocomponent procedure.
Another undesirable aspect is the indiscriminate assiqnaent of
physical properties, some of which may be far displaced from the
real values.
These are the reasons, from a chemical engineering point of
view, for interest in polydisperse mixtures. The original
proaptinq for recent work in this area came from studies, along
fundaaental lines, by theoretical physicists (Vrij, 1978 J Blum
and Stell, 1979; Gual tieri et al., 1982) whose main interest
appears to have been an academic one.
On the experimental side, the growth in popularity of
separation processes involvinq a supercritical .olv.nt has
created a demand for hiqh pressure VLE data. Supercritical
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extraction takes advantage of the r •• arkable ability shawn by
.. ny fluids, at temperatures and pressures above cri tical, of
having greatly increased solvent abilities. A compre •• ed fluid,
being in the one phase region, has a liquid-like density and a
ga.-like viscosity. The enhanced solvent qualitie. have been
attributed to a combinat ion of physical effects, the higher
density allowing good extractive qualities and the lower
viscosity ensuring good contact between solvent and solute
molecules (Schneider, 1983).
supercritical extraction is attractive despite the high
pressures involved for several reasons (Bott, 1980). High boiling
components can be solubilized at relatively low temperatures. A
good separation of the sol vent from the extract is obtained by
simple decompression, without the need for addition of another
agent. The low temperatures involved do not affect heat sensitive
compounds and the compressed gases used as solvents are
relatively cheap and non-toxic.
Carbon dioxide is a popular supercri tical sol vent and has
been used to separate organics in several processes. An example
of such a process ls the separation of qlycerides usinq carbon
dioxide (Bott, 1980). It is also a prominent chemical in enhanced
oil recovery, accurate simulatl.on of which requires building
blacks of accurate binary phase equilibria data.
The VLE data collection apparatus was originally designed and
constructed at the Department of Chemical Engineering, McGill
University, Montreal, by Dr. H. Orbey as part of his Ph.D
research. In its original form it was designed ta measure the
co.position of two phases at equilibrium at temperatures in the
range 300-375 K and at moderate pressures (up to 8000 kPa). A
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full description of that apparatu8 ha. been given by Orbey
(1983). In essence, the setup conaiata of a atainlea. .teel VLE
cell equipped with valves which enable aamplinq of the liquid and
vapar phases. Supportinq apparata conaist of qas and liquid
feeding systems, a thermostatic system for cell temperature
control, a vacuum assembly, and a qas chromatograph for
compositional analysis of the phases.
Orbey (1983) encountered some problems in the use of the
apparatus. The major difficulty was with leaks from the cell at
the samplinq valves. The sealinq material used in the valves,
Teflon and eventually Delron, did not perfora weil. There was
deterioration and low compression allowed leaks while hiqher
compression led to deformations into samplinq cavitiea. The high
thermal expansion coefficient permitted leaks from contraction in
runs where the tempe rature was lower than in the prev ious one.
An additional difficulty was with the qas side samplinq valve
which accumulated liquid droplets, a problem then attributed to
the horizontal configuration of the sampling hole.
Signifieant modifications of the equipment were undertaken by
Sejnoha (1986) • Improvements in the areas of samplinq,
temperature control, calibration and operational safety were
made. The qas phase samplinq valve was completely redeaiqned,
with Rulon and Invar beinq used as sealinq materials.
Despite reasonable attempts to eliminate condensation in the
design of the new samplinq valve, this continued to preaent a
problem (Sejnoha, 1986). Various explanations for the cause of
the condensation were put forth, with the two most plausible
being condensation due to rapid adiabatie expansion of
o
n -
7
llydrocarbon vapor into the sa.pline) hol. and condensation due to
pre •• urization with carbon dioxide ot a vapor spac •• aturated
with pure hydrocarbon vapor ( after in situ deqassinq).
Another problem was the presence ot leaks trom the cell into
the gas samplinq loop due to inadequate sealinq. Ironically
enouqh this was the method by which Sejnoha was finally able to
obtain a representative sample of the qas phase. Problems
persisted with the liquid samplinq valve,. with the Rulon seal
havinq to be replaced reqularly due to the formation of a ridqe
caused by the rotation of the valve rode This ridqe resulted in
leaks and material trom it plugged the sample holes.
Sejnoha recommended that a new qas samplinq valve be
designed, one that eliminates the possibility of condensation and
leaks. In addition, the liquid hydrocarbon be degassed
externally, and fed into a cell already containinq some carbon
dioxide. She suqgested that the accessibility of the cell be
improved as the operation of the rxperiment was severely hampered
by its restrictive design. A final point was the installation of
a system to allow measurement at pressures ab ove the saturation
pressure of carbon dioxide at ambient room temperature.
1.3 Su.aary of objectiv ••
1.3.1 VLB pre4ictioD
1) Investigate the possibility of developinq a more
accurate characterization technique for obtaininq the
aolar distribution curve of petroleua fractions and gas
condensates.
11) Find a more versatile and accurate .ethod of
representinq the molar distribution curves ot these
c
c
8
fluid8.
Ill) Deva10p a continuou. Peng-Robin.on-Stryjek-Vara
(PRSV) EOS.
1.) Find or deve10p a .uitable continuou8 theraodynuic.
VLE prediction fOrllu1ation in which to apply the
continuous PRSV EOS to i1l-dafined aixture ••
• , Perfora fIash,dew point and bubb1e point calcu1ation.
i, Moc:lify the VLE apparatua to i.prove the acca •• ibi1ity
of the cel1 and its aampling valves.
il) Desiqn a new gas sampling valve, one that a1i.inates
condensation and Ieakage.
lil) Improve the experi.enta1 technique to .iniaize
chances o~ condensate formation in the qas sampling valve
Iv) Introcluce moc:lifications of the equipaent to a110w
aeasure.ent at pressures abova the saturation pra •• ura of
C02 at ambient room temperature.
v, Measure VLE data for C02-cyc1ohexane at high pressure.
o
o
9
CBaPl'll. 2
....... ou J'O. PUD:tC'l'IOB 0., VLB .oR :tLL-DBPIIIBD Il:tHURBS
This chapter describes the method developed for the
characterization and modeling of VLE for semicontinuous mixtures.
Eacn of the sections presents pertinent background material on
the subj ect and then progresses to the proposed new method.
Continuous and semicontinuous systems are formally defined in the
first section. Next, a method of characterizing polydisperse oi1
fractions beginning with a True Boi1ing Point (TBP) distillation
curve is presented. The final section de scribes the appl~cation
of the PRSV EOS to continuous systems and the solution of
integral equilibrium and mass balance equations.
2.1 cODtinuOU8 and S .. icoDtinuOU8 lIiztur ••
The continuous thermodynamic approach towards phase equilib
ria involves a different way of expressing the relative amounts
of substances in a system. Instead of individual discrete mole
fractions corresponding to an identifiable component, there is a
shift to continuous distribution functions connected with some
observable macroscopic property of the system. There is a
distinction between "continuous systems" , for which the
composition ia entirely deacribed by distribution functions, and
c
c
.. ___ . ____ ._ ... _____ ··_·· __ ·_·~ ___ u_· _________ w _______________________ ·_ • __ ._. ___ • _______ ._ •••••• _______ ._. ___ • __ •• ________ _
10 -
"semicontinuous systems", for which the composi tion is described
by a collbination of discrete mole fractions and distribution
functions.
2.1.1 coatiauous syst ...
The characteristics of a polydisperse mixture are best
illustrated by co:.trast with those of a finite D-component
mixture, a system containinq D discrete components. The
thermodynamic state of a D-component system is described by the
followinq variables, where T is the temperature, P is the
pressure, N is the total number of moles in the system and x j are
the mole fractions.
T,V,N,{X.}
2.1
with
D
[X,-l.O /-1
2.2
In a polydisperse system the identifyinq index i is replaced
by a continuous variable 1 chosen to characterize the mixture.
This variable can be the molecular weiqht, boilinq point,
specific qravity or any other suitable physical macroscopic
quantity. The range of 1 (/.<1<1.) can be finite or infinite,
dependinq on the fluide
An illustration of the difference between the two types of
fluids is given in Figure 2.1 (prausnitz, 1983). In discrete
thermodynamics, the composition of the mixture ia qi ven by mole
fractions, X,. In continuous thermodynamics the composition of
the mixture is given by a distribution function, F(I), of the
o
/
LXI 1
o
- 11-
Coatiauoua Mizture
F(I) l""-
r--- -~ 1--
Coapoaeat .0. 1 Distribution Variable 1
LX,.I.O 1 D(/- b) - i" F(I)dl - 1.0
D(/)
coaponeat .0. Distribution Variable 1
rigure 2.1 Discrete aD4 coatiauous composition for a aul ticoaponeat aizture (Not to scale)
c
c
- 12
continuous variable 1. A continuous systell is one that consists
entirely of polydisperse fluids. For this systell the thermodynam
ic state is deterained by the following variables:
CT, V ,N 1 F(I»
2.3
with
fi,
F(I)dl- 1.0 1.
2.4
Where the functional calculus notation 1 F(J) indicates that
the state depends on the entire function F, not just the value of
F at the point 1.
2.1.2 S .. icontinuous systems
These are mixtures that consist of D discrete and (C- D)
continuously distributed components. Usually the discrete
components have values of the distribution variable 1 that are
outside the ranqe of 1 in the continuously distributed ensembles
(/.</</,). However this is not always the case, and in the
followinq definition this is taken into account by representinq
discrete components by the Dirac delta function, for which the
integral (equation 2.6) is unity. Thus for one phase :
2.5
Where X,.(t- 1 .... D), the discrete mole fraction, is the
weiqhtinq factor of the Dirac delta function.
Assullinq the c less D continuously distributed ensembles, are
described by a normalized distribution function F ,(1), and
weiqhted by overall ensemble mole fraction XI' (i· D + 1 .... C).
Inteqration over aIl values of 1 gives the normalization
o
o
13 -
condi tion for the whole phase since aIl mole fractions sum to
unity.
2.6
leadinq to :
2.7
2.2 Charact.ri.atioD
Characterization is a crucial stage in the prediction of VLE
for a continuous mixture. The characterization of a polydisperse
mixture requires an accurate description of the relationship
between the distributing function, F(I), and the distributing
variable 1 (i.e. the molar distribution curve) •
2.2.1 CODv.raioD of TBP Di.tillation CUrv. to Molar DistributioD
CUrv.
The initial step in the characterization of an oil is the
experimental determination of the relationship between the mole
fraction and the characterizinq variable. This information is not
easily obtained and there exist several approaches to the
problem. What follows is a brief review of the methods that have
been presented.
The most common analysis of a crude oil or a petroleum
fraction is a TBP distillation. In this procedure the mixture is
subjected to a batch distillation wherein, normally, the
cumulative volume percentage distilled is plotted aqainst the
temperature at which it is distilled. Theoretically a TBP
c
_ 14 _
distillation effects complete separation of all compounds in the
.ixture, each at it. own boiling point, a situation rarely
possible in practice. Accompanying measurements of the number
average molecular weight and density of the whole fraction are
often made.
In order to use a TBP distillation, the cumulative volume
percent distilled ordinate ( see Figure 2.2 ) should be converted
to a mole percent axis. This requires a correlation between molar
density and boilinq point. One method is to use a qraphical
correlation such as that of Edmister (1955) or those of available
in the American Petroleum Institute's (API) Data Book (1982).
Such graphs present the relationship between boilinq point and
molecular weiqht and between boilinq point and molar volume for
different oils characterized by their specifie qravity or Watson
characterization factor, a ratio of the cube of the cubic average
boiling point to the specifie qravity. The final result of the
conversion is a cumulative mole percent distilled versus boiling
point curve. The molar distribution curve with respect to normal
boilinq point, is obtained by simple differentiation (numerical
ly) of the cumulative curve , thus passinq from Fiqure 2.2(b) to
Fiqure 2.2 (c) •
TBP distillations are tedious to perform, unstandardized, and
rarely lead to a complete one hundred percent distilled off
analysis due to thermal decomposition of Ct4 plus fractions. This
prompted a search for alternative ways of obtaininq the
information. One solution is the use of simulated true boiling
point analysis by Temperature proqrammed Gas Chromatoqraphy or
High Performance Liquid Chromatoqraphy (HPLC). cependinq on the
specifie columns used the resul ts can be very similar to actual
TBP distillations, and in fact yield much more detail while
o
o
- 15 -
a, Volume TBP Di.tillation ))) Holar HP DistillatioD
cumulative volume percent distille4
•
• •
• • •
•
• • • •
• • Cumulative Hole • percent 4istillad
Bdmistar charts
c, Differentiated CUrve d) Normalized Holar
Distri))ution Curva
Mol .. • • • • • • •
riqure 2.2
• F(l) • • • • z.
Z~ 8--
• L.z! • •
Tb Tb
Deri vatioD of • molar distributioD curv. froa a volume TBP distillation.
c
c
- 16 -
requiring 1 ...... pla (Vogal et. al., 1983). In thi. procedure
th. reaults are uaually preaented in the form of a cumulative
veiqbt percent diatilled versua boilinq point curve. Thus the
tha weight axis has to be convertad to a Ilolar axis in order to
obtain a molar distribution curve. One vay would be ta utilize
the Ecillister graphs, by convertinq the weight at a given boiling
point ta moles using the molecular weight at this boiling point
(knowing the average specifie gravit y or Watson characterization
factor). But the se charts are based on oil stocks from 50 years
ago, and appear to be approximate quidelines rather than accurate
relations (Edmister, 1955).
Another approach is ta use an analytical correlation,
applicable for the qiven oil, between the molecular weight and
boiling point; of the form:
MW - fCT b)
2.8
This approach is illustrated in Figure 2.3. Essentially the
abscissa boiling point values are converted ta molecular weight
values using equation 2.8. Then the resulting curve is
numerically differentiated ta obtain a weight percent versus
molecular weight curve (Figure 2.3(c». Finally the weight
percent value at a qiven molecular weight i8 divided by that
1l01ecular weight, ta obtain the number of moles at that point.
The result ia the unnormalized molar distribution curve of Figure
2.3 Cd). The curve is normalized by dividing discrete molar points
on the curve %, by their sum, ta obtain normalized molar
distribution points %~, as follows:
Il Zi Z ---
1 LZI 2.9
o
o
- 17 -
a) WeiCJht TBP Distillation b) MW CUrve
CUmulative weiqht percent distilled
• •
•
• • •
• • •
Tb
• • • CUmulative weic;iht percent distilled
MW -t(T,,)
MW
c) Dittarentiated OUrve d) Unnormalized Molar
Distribution CUrve
weight • percent • • • • ~
Fiqure 2.3
• Nol ••
• • • • Weight M l
- 0 es • MW
• MW MW
Derivation of a molar distribution curve trom a weiCJht distillation or simulated TBP analysis.
'1
-- ------------- ----------.-- ---_._ .... -----.------------------------------~-_._------,----------
(
(
18 -
Thare ia than a choice of either retaininq the boiling point
as the distributinq variable ; or using the molecular weiqhts
calculated in eguation 2.8, as has been used by Radosz et al.
(1987). In that study the investigators separated oil mixtures
into ensembles consistinq of different homologous groups using
HPLC and used a different molecular weiqht relation for each.
The major difficulty in this approach is findinq an accurate
mo'.ecular weiqht to boiling point correlation for a particu1ar
oi1. This re1ationship is different for each homologous series.
However, over the years there have been attempts to produce
qeneralized correlations applicable for average petro1eum cuts
encountered in industria1 practice. These relations predict not
only the molecular weight but al 50 the critica1 properties and
acentric factors from a know1edge of the boiling point and
specifie gravity. They are usua11y complex polynomial expres
sions, obtained by regressing petroleum fraction and representa
tive pure component data. A review by this author found the two
most reliable for predicting the critical properties of common1y
encountered North American petroleum fractions to be the
relations of Kesler and Lee (1976), and those of Twu (1984).
Thus knowing the specifie qravity and boilinq point of a
qiven oi1 an approximate molecu1ar weiqht is obtained. In reality
every boi1ing point datum in the TBP analysis of the oi1
corresponds to a unique specifie gravit y and molecular weight,
and thus a corresponding specifie qravity curve is required. But
this information is not readily available. Therefore one approach
is to use the averaqe specifie gravit y for the oil, the sinqle
value measured for the whole fluid, with the boilinq points ta
qet the lIolecu1ar weiqht curve. This introc:luces uncertainty but
o - 19
.ay be tolerable for narrow boiling fractions. It is worth noting
that the correlations were developed with sets of average boiling
point, average molecular weight, etc. Another possibility is to
use a specifie gravit y derived from generalized single carbon
number property relations together with the above correlations.
If one is confident that one is dealing with a typical
reservoir fluid, say agas condensate or a petroleum residue of
average paraffinicity, then another option is open. This method
involves the use of the Generalized Single Carbon Number Property
(GSCNP) concept (see Appendix Al). Single carbon number (SCN)
groups are pseudocomponents that "represent" the behavior of all
hydrocarbon compounds with the same number of carbon atoms in an
oil mixture. The number of carbon atoms is expressed by the seN
number. Generalized properties of these groups, obtained as
averages of a selection of natural gas condensa tes, have been
presented (Whitson, 1983). As discussed in Appendix Al the
following expressions reliably represent the relationships
between the normal boiling point and specifie gravit y , acentric
factor (w), and critical properties of SCN groups. Henceforth
these are the GSCNP relations.
MW-98.7-4.773·10- I ·Tb + 1.326' 10-3·T~-1.270·10-7 'T~
2.10a
2.10b
2.10c
2.10d
2.10.
(
j
c
20 -
Thus equation 2.10(a) can be used to convert the bollinq
point axis to molecular weiqht.
2 .2.2 .epr •• eDtation of lIolar Di.tri):)ution CUrv. by Ixtende"
Spline ~it Technique
It ls important to have an accurate molar distribution curve
as errors in the feed phase compositions result in errors in the
equilibrium phase compositions. To date there has been one
major approach to the description of the molar composition of
continuous ensembles. This has involved the use of probability
densi ty functions. The reasons for this are threefold these
functions satisfy the requirement for normalization of the
inteqral to unit y; early studies, in the context of statistical
thermodynamics, (Salacuse and Stell, 1982; Briano and
Glandt, 1983), defined Fel) as the probability that a molecule
ch os en at random from the system will be characterized by the
given value of 1; and finally because these functions have the
same general shape as experimentally determined molar distribu
tion curves of heavy fossil fuels.
The Schultz-Flory distribution function (Schultz, 1935;
Flory, 1936) is one such probability density function that has
been frequently used in continuous thermodynamics. This function,
toqether with the Pearson Type III (Whitson, 1983) and Gamma
probability density functions, aIl closely related, have been
found to describe petroleum residues and fractions reasonably
weIl.
Cotterman and Prausnitz (1985), Gutsche (1986), Shiqaki and
Yoshida (1986) among others have used the GamIna distribution with
molecular weight as the characterizinq variable. Willman and
o
o
- 21 -
Teja (1987a,b) also used this distribution but with the effective
carbon number as the characterizing variable, while Whitson
(1983) and Shibata et al., (1987) used the sinqle carbon number.
For reservoir fI uids,
experi.ental evidence
distribution decreases
especially qas condensates there is some
(V~Jsl et al., 1983) that the molar
exponentially wi th the distributinq
variable, a special case of the Gamma distribution function.
Kehlen and Ratszch (1980, 1987) have analytically solved
sample continuous thermodynaJ'llic VLE problems where molar
distributions in aIl phases were described by the Normal
(Gaussian) distribution function.
Accordinq to correspondinq states analysis a minimum of two
or even three simultaneously distributed parameters are required
to characterize real fluids (Briano and Glandt, 1983). Thus one
characteriz ing variable cannot be expected to y ield a
satisfactory description of real fluids. However there are two
methods of qettinq around this difficulty. The first involves the
use of multivariate distribution functions, to allow additional
deqrees of freedom. The second, perhaps more elegant solution is
to treat the polydisperse fluid as a mixture of several
ensembles, each consistinq of an infinite number of very similar
chemical species. In each family the chemical similarity ensures
that one distributinq variable is sufficient for complete
charaC"'terization. The main shortcoming of this approach is the
need for an experimental method of characterizinq the overall
mixture into different ensembles. The VLE formulation adopted in
this study allows for multiple ensembles.
An example of a multivariate distribution function that has
found application in continuous thermodynamics is the bi variate
J
l, , .
ii . ,
c
" C r •
- 22 -
loq normal distribution. This function was used by Willman and
Teja (1986) to characterize various oils and coa1 liquida. They
found only a small improvement in qoinq from a single variable
function to the double variable one. The likely reason for thia
is the high deqree of correlation between the two variables used,
boiling point and specifie gravity.
The probability functions mentioned so far have distribution
variables that have Infinite ranges. When dealinq with reservoir
fluids there are some problems with usinq values of 1 that extend
to infinity. One reason is that they are unrealistic since it is
rare to find carbon numbers qreater than about 80 (Vogel et al.,
1983, Pedersen et al., 1983). A greater problem is eneountered
when usinq critical property correlations, such as the
Kesler-Lee, or GSCNP, to calculate EOS parameters. These
relations were developed from experimental data of compounds of
relatively low carbon number, and cannot be expeeted to hold to
infinity. Radosz et al., (1987) have used the finite range Beta
probability density function in polydisperse VLE calculations.
Shibata et al., (1987) truncated the SChultz-Flory and Gaussian
distributions in their calculations, as did Willman and Teja
(1986) in their use of the bivariate distribution.
The use of probability density functions to represent molar
distributions has some shortcomings. Apart from the problem with
the range of 1 there are limitations on the shapes of molar
distributions. AlI the previously mentioned probability functions
are unimodal. Although sorne, such as the Gamma and Beta
functions, can be skewed in either of two directions, they
obviously cannot represent aIl possible shapes (see Fiqure 2.4).
One way around this ls to use the multiensemble approach, with a
different funetion for eaeh family (Kehlen and Ratzsch, 1984;
- 23 -
o Gamma FUnction Fit
QOS • g • . -'$ •
,.Q .. . -... • ..a CD O.o.t • .-Q la •
"0 ~
0 200 300 400 500 ·600
Boiling Point (K)
ESFT Fit
0.08
CI 0 • ~ • =' • ,.Q • .- • ... 0.0 A -CD
S • ... .., "0 :s
0 200 300 400 500 600
Boiling Point (K)
o Figure 2.4 Molar Distribution Curve of Jacoby et al. (1959) Oil • .
J
--,
T,
f , , i " r , ,.
(
(
24 -
Cotteraan and Prausnitz, 1985: Radosz et al., 1987). This
approach, although fairly successful, is still limited by the
unimodality of aach function and the difficulty of accurate
characterization into families.
with these limitations in mind an attempt was made to improve
the present method of representing the molar distribution curve.
One of the most flexible and versatile tools for data
representation and smoothing is polynomial interpolation, in
particular spline interpolation. The fa ct that the polynomial
coefficients, the unknowns, are linear functions of the data and
their ease of differentiation make polynomials computationally
advantageous. However, use of high order polynomials often leads
to unwanted oscillations (Ralston, 1965) • To avoid these
oscillations two approaches have been taken.
One method demands only that the interpolating function pass
close to the actual data points. The method of least squares
analytically minimizes the sum of the square of differences
between the observed and interpolated values. Another method
involves the use of different low order polynomials to represent
different parts of the function ta be fit. Spline interpolation
is one version of this technique where different polynomials are
fit between each pair of adjacent data points. However in this
case since the curves pass exactly through each data point, no
smoothing is performed. Thus any errors in the data are
propagated. This is particularly undesirable in situations where
one is interested in the derivatives. Minor bumps and dips in a
curva can lead ta gross errors in the local derivative, as shown
in Figure 2.5, obtained from the TBP data of Radosz et al.
(1987) •
0
o
100
" QI = ... ~ GD ... 'a 50 ~ !
tIO ... ~
0
cc pe4 10 >C c:: o ... ~ 2.0 ..a ... ... ~ GD
:; to ta -
- 25 -
\EIGHT 1. DISTIlLED ~SUS ~ a.RVE
250
IMXBZ IL SHI.!
300 350 400 Molecular weight
RAW ~ZED MOlAR OIS'TRISunON OF' OIL MJOSZ IlL SHI.E
CJ
450
:2 o~------~----~~----~------~--250 300 350 400 450
Molecular weight
.igure 2.5 Diffe~.Ati.tioD of UDsaoothe4 4istillatioD GUZVe.
c
)
c
26 -
One solution to this problem is to combine the Ilethod of
least squares and spline interpolation. This is the Extended
Spline Fit Technique (ESFT), first used by Klaus and Van Ness
(1967) to represent thermodynamic data. It consists of splitting
up the ranqe of the data into intervals (segments). Each interval
contains two or more data points. A sinqle cubic polynomial
interpolates the data in that interval. The cubic goes as close
to the points in the interval as possible in a least squares
sense. In addition the curve passes through the specified
interval boundaries, which may or may not be actual data points,
where the first two derivatives of cubics in adjacent intervals
are matched (See Fiqure 2.6). Additional details of the method
are qiven by Klaus and Van Ness (1967).
This method overcomes the shortcomings of probability density
functions, allowinq multimodality and extreme shape flexibility,
since the user has a choice of the number and location of
intervals.
2.2.3 8uaaary of Characteri.atioD Procedure
SliP 11 Obtain a set of TBP data points, cumulative weight
percent distilled off versus boilinq point. If volume TBP
ia qiven, convert to molar usinq API correlations (or
Edmister chart).
SliP 11. Usinq either GSCNP or the Kesler-Lee relations
with one of the two previously described methods for
obtaininq the specifie qravity, convert the boiling point
axis to Ilolecular weight.
SliP IL Smooth the weight percent distilled versus
Ilolecular weiqht curve usinq the ES FT ; six intervals are
recommended, obtain first derivatives at enouqh points in
o
o
Different cubic describes portion of curve in each interval
Intervall
- 27 -
Interval 2
Interval3
Leut sum of squares minimization
Interval boundary: derivatives matched
piqure 2.1 Illu8tration o~ the azten4e4 spline pit ~ecbDique.
(
c
28
the curve to provide an adequate description or the molar
distribution curve. Divide each derivative datWll (corre
spondinq to the weight of species of the molecular weight
at that point) by the correspondinq molecular weiqht to
obtain a molar value. These values, used as the ordinate
with the abscissa as the original boilinq points describe
an unnormalized molar distribution.
UD J..1. Smooth and normalize the distribution using the
ESFT. Normalization is accomplished by dividinq each of
the spline coefficients by the value of the Integral over
the whole ranqe.
The characterization procedure was implemented in the form of
a Fortran proqram CDRT which incorporates the ESFT in the form
of subroutine 8PLIIPT. The subroutine was obtained from Dr. S.
Sayeqh and required modification in order to run on microcomputer
Fortran and to print out the coefficients of the polynomials.
This program has options for use of several molecular weight
correlations, methods of numerical differentiation, Gamma
distribution function and Beta distribution function fitting and
ESFT fitting. The distributinq variable can either be molecular
weiqht or boiling point. The program is described in Appendix A6
and is included in a diskette in this thesis.
2.3 Probl .. PoraulatioD
Once an ill-defined mixture has been characterized, one can
apply continuous thermodynamic methods to i t. The fundamental
condition for equilibrium in such a system is the equality of
fuqacities in the phases involved. For a component correspondinq
to a qiven value of 1 in the system this is expressed as follows:
Fugv(/) = Fug L (1) 2.11
o
o
29 -
Thera are many .odels for representinq the fuqacity of a
component in a phase, and most of these models are easily
converted from their discreta versions to thair continuous
analoqs (Kehlen et al., 1985: Cotterman et al., 1985: Willman and
Teja, 1987a, b), tharafore there are many ways of calculatinq VLE
in such systems. However these methods fall into two cateqories,
the ~ and the y-~ approaches. The ~ approach utilizes an EOS to
evaluate both the liquid and vapor fuqacities for each species.
Equation 2.11 is then expressed as follows, where ;1'(1) and ,L(I) are
the fuqacity coefficients correspondinq ta a qiven value of 1 in
the vapor and liquid phases, respectively. FV(/) and FL(I) are the
correspondinq distribution functions •
p. ;v(1). FY(I) = p. ;L(/)' FL(/)
2.12
The y-; approach retains the EOS for the vapor phase while
representinq the liquid phase by an excess Gibbs enerqy model
from which an activity coefficient is calculated.
The ; approach has certain advantaqes, especially in qas
processinq where separation is carried out at pressures near the
critical reqion. At these conditions some components are
supercritical and thus reference states for activity coefficients
have ta be hypothetical. Additionally, with the y-; approach it
is difficult to qenerate the complete P-T envelope of a mixture
due to discontinuities in the critical reqion. The popularity of
the; approach has increased dramatically over the last fifteen
years, the major reason beinq the availability of accurate cubic
EOS. CUbi~ equations are advantaqeous over more complex equations
since they versatile, relatively accurata, and allow direct
solution for volume roots.
c
i
c
_ 30
In this study the • approach is followed. The fuqacity
coefficient of a speci .. s represented by 1 in a family J in a phase
is ob'tained from the EOS usinq the followinq exact relation
(Cotterman et al., 1985) :
ln ;,(1)- RIT' f {[ 6Q/:'(I')l,v",., -~ }dV - RTlnZ
2.13
Where P. T • R • v, and Z have their usual thermodynamic meaninq and "J and FJU) are the total moles and distribution function of family }
in the phase. The quantity enclosed in square brackets represents
the functional differentiation of pressure, P, qiven by the EOS,
which is a functional, with respect to the function ('7,'F,(I)}. A
description of functional differentiation has been qiven by
Hansen and Macdonald (1976). However, instead of derivinq the
final expression for the fuqacity coefficient from the above
relation it is equally possible to use correspondinq expressions
valid for discrete mixtures, by interpretinq the index « as the
continuous variable 1 (Hendriks, 1987a). This is the approach
followed in this study, and is summarized in Appendix A2.
Several continuous variable formulations of the VLE problem
usinq the , approach have been presented in the recent past.
Early methods involved the solution of simple, often idealized
problems. In these situations, simplifyinq assumptions allowed
the derivation of analytic expressions for the parameters of
distribution functions in the phase(s) at equilibrium. Gualtieri
et al. (1982) used the continuous Van der Waals EOS to solve the
fractionatj ~n of a polydisperse impuri ty dissol ved in a
solvent,solvinq for the unknown Gamma distribution function
parameters by equatinq moments, as did Cotterman and Prausnitz
(1985). Other, more recent work by Johnson et al. (1985) and
o - 31 -
xincaid et al. (1987) involves the solution of equilibriwa and
critical point conditions for .odel systems using a .athematical
ly rigorous technique (Method of Fredholm) and a perturbation
.ethod. The main interest in such methods appears to be an
acade.ic one, vith the goal being the analytic solution of the
Integral algebraie equilibrium and mass balance equations. These
studies have limited engineering applications since the models
and simplifying assumptions involved reduce the number of real
fluids to which they can be applied. In particular the method of
moments implicitly assumes similar distribution functions in aIl
phases, and yields only an approximate solution.
The second class of methods take a more praqmatic engineering
view of the problem. Attention is paid to the accurate modeling
of real fluids and to computational requirements. One of the most
successful of these techniques ls the quadrature pseudocomponent
method introduced by Cotte man and Prausnitz (1985). The key
point in these methods is the numerical solution of integrals.
The overall formulation of the problem is essentially the same as
in the discrete case. However, in the continuous situation aIl
summations take the fom of integrals covering the range of the
distribution variable 1. These appear in mixing rule and mass
balance summations. Using this approach, Integration is
numerically accomplished using quadrature techniques where the
Integral is replaced by a summation of a finite number of
weighted function evaluations at specified values of the
Integration variable called quadrature points. For n quadrature
points this is expressed as follows, where w, is the weighting
factor at point l,.
2.14
c
r, t ~"
~
.. C , !.
f If
l
- 32 -
This method essentially amounts to solving the equilibrium
and _ass balance equations at fixed values of the distribution
variable 1, given by the quadrature points. As noted by Shibata
et al., in effect this is the pseudocomponent method, only that
pseudocomponents are optimally chosen to accurately represent
integral properties. Cotterman and Prausnitz (1985) have used
Laguerre-Gauss quadrature in conjunction with the Gamma
distribution function. Willman and Teja (1986) used Legen
dre-Gauss quadrature to integrate functions involving the
bivariate log normal distribution function. Radosz et al. (1987)
used finite range Chebyshev-Gauss quadrature for the Beta
distribution function.
VLE
A simplification of this method, in which existing discrete
algorithms are used in conjunction with pseudocomponents
chosen using quadrature points has been presented by Shibata et
al. (1987). The attractiveness of the quadrature approach lies in
the small computation times required, accuracy, as weIl as the
elimination of arbitrariness in the selection of pseudocompo
nents. Another recent method of this kind is the variational
approach to flash calculations of Schiljper (1987). This is a
very qeneral formulation, use fuI for cubic EOS. It is based on a
minimization of the free enerqy in the system and combines a
perturbation expansion with a variational method to solve
(approximately) the integral algebraic equations. Under certain
conditions this method reduces to the quadrature method of
Cotterman and Prausnitz.
Another approach, which could be classified with the above,
is that proposed by Hendriks (1987a,b). In this case the emphasis
o
o
33 -
is ahifted fro. the evaluation of inteqrals to the reduction in
the nWDber of mass balance and equilibriUJD equations requirinq
simultaneous solution. The particular case of VLE for continuous
mixtures is just one of several applications of Hendriks
"Reduction Theorem for Phase Equilibrium Problems" (Hendriks,
1987b).
This approach, with modifications, has been used as a basis
for the solution of flash, dew and bubble point calculations in
this study. The reasons for selecting this method over the
Cotterman quadrature procedure are that it allows a complete
description of distribution functions in the unknown phase(s)
(Le. it provides F(/) over aIl 1) and it allows a multiensemble
approach as weIl as reducing the number of integral algebraic
equations requiring solution. Furthermore it is not tied to a
particular method of numerical integration, thus allowing free
choice of distribution function and quadrature method.
2.3.1 The CODtiDuous PRSV Equation of State
In 1976 peng and Robinson presented a new two constant cubic
EOS. It proved to be remarkably accurate in predictinq phase
equilibria for non polar compounds of industrial interest and as
a resul t i t is now one of the most popular cubic equations of
state in use in the petroleum and qas processing industries. This
study utilizes a variant of the above EOS, the PRSV EOS, which
introduces modifications designed to improve pure component vapor
pressure prediction at low reduced temperatures (Stryjek and
Vera, 1986).
The original PRSV EOS has the form:
p= RT _ a(T) v-b v 2 +2 o b o v-b 2
c
~~ ~ ~- -~-~---~~------
with
b _ 0_,0_7_7_7_9_6_' _R_'_T_c
Pc
- 34 -
J( 0 - 0 .378893 + 1 .4897153, W - O. 17131848 . ru 2 + 0 .0196554 . W 3
2.16
2.17
2.18
2.19
2.20
The distinction between the penq-Robinson and PRSV equations
of state lies in the introdu~tion of the M. parameter, adjustable
for each pure compound and which vanishes if the reduced
tempe rature is above 0.7. The introduction of this parameter led
to reproduction of pure component vapor pressures with deviations
of less than one percent down to 1. 5 kpa for over one hundred
compounds of industrial interest.
In the case of continuous thermodynamics, since the mixture
is seen as consistinq of an infinite number of compounds, and
since the mixture is characterized by one or two variables only,
then the only individual information available on each species is
the value of the distribution variable /. In order to use an EOS
to obtain the fuqacity coefficient of a species represented by a
qiven value of 1, one must calculate the equation of state
parameters a(T) and b that correspond to this compound. This
requires a correlation between a (T), b , and /.
o
.1) -
35 -
For the PRSV EOS, four pure compound pieces of information
are required per compound, P •• T •• w and N, The distributing
variable, 1, chosen in this study is the normal boilinq point.
Therefore a relationship between the boilinq point and the above
parameters is required before the PRSV EOS can be applied to a
polydisperse system.
In the petroleum industry there exist a qreat many predictive
correlations for the critical properties of petroleum fractions.
However, GSCNP relations, described in the previous section, have
been chosen for this study because of their accuracy and
applicability. The Resler-Lee relations could also have been
used, in spite of their requirement for an independent specifie
qravity for each boiling poInt datum. These correlations
therefore provide a method of estimatinq p •• Te. W from the
boiling point. However, N., the adjustable parameter that
minimizes deviations in pure component vapor pressure calcula
tions and has a small value in the discrete PRSV EOS, generally
in the range -0.1 to 0.1 for most compounds of industrial
interest, also requires correlation to the boiling point.
The method of correlating NI to the boilinq point in this
study is analoqous to the approach followed in the case of
discrete compounds. It is illustrated in Figure 2.7 and involves
matching the fuqacities of species represented by a given value
of the boiling point. The TBP distillation data are usually in
the form of a set of weight percent distilled versus boiling
points. Only the boiling points are required for the evaluation
of N. o The lowest boilinq point value is read and used to
estimate the generalized SCN critical properties and acentric
factor. In the first i teration PRSV EOS relations are used to
obtain an initial estimate of a(T) and b. An iteration to correct
c
36
.... D .. T ...... ---...
rDi t:ial •• ~iaa~. y~ fzaa walY zel.~ioD.
a- ffT- r" T •. P •• Jtf.O)
II. f(T •• 1' ,}
T.-r.'-latm
no
... ~Ilod of 1 ... t eqau •• CIO&Te1.tioD of T "". clat.
",(r,) - Clo "a,' T. +aa' r:
1
• 1C)1lZ'. 2.7 AlqoZ'ltbll foZ' .,oZ'Z'.latioD of JI. to T •
o - 37
the value of oCT) usinq the Newton-Raphson (NR) scheme follows,
with the saturation pressure equal to one atmosphare. Once the
value of oCT) is found, the -. value is back calculated.
Successive values of the boiling point and -. are stored and
after the final boiling point datum the -. values are correlated
to the T. values using a polynomial expression. It should be
noted, however, that due to the nature of the functions involved,
a reduced temperature of 0.7 yields an undefined -. value - see
Appendix Al. The solution adopted here is to check the reduced
boilinq points to ensure that none of them is within ~O.02 of the
problematic 0.7 value.
Thus the final continuous PRSV equation has a form similar to
the discrete case only that Te,Pe.w are functions of T. and are
obtained from GSCNP relations Cequations 2.10c to 2 .10e). 1t.(T.) is
a different function for each fluid and is obtained, as has been
described, as part of the characterization procedure. Althouqh
the use of critical properties derived from generalized SCN
properties may be inaccurate for a qiven fluid this is
compensated for by the JC.(T.) value, so that in this procedure
values of this parameter are larger than in the discrete PRSV
case, where the true critical properties of pure compounds are
used. It should be noted that 1t.(TII ) does not vanish even at
reduced temperatures greater than O. 7 • The method has been
implemented in the form of a Fortran program K1'IT, described in
Appendix A6, and is included in a diskette in this thesis.
2.3.2 BquatioD or state par ... ter.
This section presents the relations expressing the
contribut~ ons of different components to the EOS constants in a
semicontinuous system in the context of a flash calculation where
(
c
- 38
thera ara feed, vapor and liquid phases, for which the
superscripts are F.V. and L respectively. The total continuous
fraction of each phase is described by a normalized distribution
function F'(I). r'(/). and FL(I). which is subdivided into family
distributions according to:
c Fn(l)_ L X~'F7(/)
,-D+ •
2.21
Where x7 is the total mole fraction of family J in phase n
Each phase is described by a two constant EOS wi th the
attractive parameter dependinq on discrete mole fractions
X,.CI- 1 •.•• D) and overall family mole fractions x j.CI· D+ 1 .... C) through
a quadratic mixing rule as follows:
c c a - X· X . a . a . l-k n L L n n -n -n ( )
, J '1 '1 , •• J.' 2.22
Where ii~ are family averaged EOS parameters. For the discrete
components, as usual, these are the square roots of the pure
component EOS parameters, calculated as described in the previous
section i.e.,
(i ... l .... D)
2.23
For discrete components the value of iif is independent of the
phase since it is not a function of co:nposition_
However, for the continuously distributed ensemble of overall
mole fraction x, this value is taken as the averaqed square root
value of the a,(1) over the the whole range of the distribution
variable 1. Each value of a~s(r) is weighted by its value of F,(I)
o
il -
· -_ .. _._---_ .... __ . -----------
39 -
and the re.ul t integrated over 1. Thus for phase n:
iif - J Ff(l)· {a J(I)}o.sdl (j- D+ 1 • ••• C)
2.24
In this situation the if is dependent on the phase composition. A
key difference between this and the formulation of Hendriks' is
that QJ(I) differs for each family Cat a given value of 1). The
e:upirical binary interaction parameter, /c'J' is characteristic of
interactions between land } components of the fluide This
parameter accounts for interactions between different families,
different discrete components or between a discrete component and
a family. It is assumed that there are no interaction
coefficients between components of a single family.
The excluded volume parameter depends on composition through
a linear mixing rule as follows:
2.25
Again there is no averaging for the discrete components, and the
value of -,f is simply the pure component parameter b, and is
independent of phase composition. For the continuously
distributed families, this value is equal ta the family averaged
value of b(l) and is dependent on phase composition:
7J~ - J F~ (/). b(l)dl (j-D+I •..• C)
2.26
2.3.3 It-I'aotora aDd .a.. BalaDo ••
From the equality of fuqacities at equilibrium and the
definition of the X-factor K" the following holds for discrete
(
j
c
- 40
components.
xr ;f K ----i X: ;r (i- 1 • ... D)
2.27
The corresponding relationship for a component characterized by 1
in ensemble J is:
Xr'F~(1) ;~(/) K (1)- ---
J X~'F~(I) ;~(/)
From equation 2.27 and 2.28
InK,"ln;~-ln;:'
(j-D+I .... C)
2.28
2.29a
2.29b
The fugacity coefficient expressions for the discrete and
continuous components in a semicontinuous mixture using the PRSV
EOS are derived from the expression valid for a system with
discrete components only, as is shown in Appendix A2.
For flash calculations the second set of equations that must
be satisfied are the mass balances. For discrete components these
take the familiar form:
(i=I .... D)
2.30
Where 1 ls the liquid fraction and u is the fraction vaporized.
The corresponding equation for a component represented by 1 in
continuous ensemble 1 is:
X:' F:(I) = 1· X~' F~(I)+ U· x~· F~(I) (j=D+I .... C)
2.31
8y integrating equation 2.31 over aIl /. making use of the
normalization conditions, and combining equations 2.30 and 2.31,
o
o
- ---- -------------
41
as shown in Appendix A4, one is able to obtain the following
equation, equivalent to the well known Rachford and Rice (1952)
objective function for flash calculations.
f X:'(Kt-l) L.-"';;"-~~~-O t.1 1 + ( Kt - 1 ) . u
2.32
This function is monotonie in u and has a derivative which is
always negative, making it suitable for use with the Newton root
findinq method.
2.3. oC Re4uctioD in tb. lhUlJ)er of BquatioDS
The method used to reduce the number of equations borrows
from that of Hendriks (1987a), with the main difference being the
use of individual family X-factors K /(1). The formulation seeks,
through inteqration of family equations, ta reduce the number of
equations requiring solution while providinq a complete
description of the distributions in the unknown phase(s).
First define a set of generalized K-factors. The first of
which is the ratio, for ensemble J, of the overall mole fraction
in the vapor to that in the 1 iquid:
XV K __ 1
1 XL 1
(j=D+I •..• C)
2.33
Next, define a ratio of the family averaged aJ(/) parameter of
ensemble j
2.34
and a similar X-factor for the family averaged b(l) parameter for
(
c
enaellble J:
- 42 -
-pv K • J
bj -p~
2.35
Introc:lucing Hendriks' notation for the inteqral of the product of
any arbitrary function and the feed distribution function of a
qiven ensemble j:
2.36
Combination of equation 2.28 and 2.31 and elimination of X~'F~(I)
qives
2.37
and similarly elimination of x~· F~(I) qives:
L L X:"F:(l) XJ"FJ(I)-l+u" KJ(l)
2.38
By inteqratinq equations 2.37 and 2.38 over aIl values of 1
and dividinq the integrated equation 2.37 by the integrated
equation 2.38 the following is obtained:
(j=D+l, .. ,C)
2.39
similarly, multiplyinq equation 2.37 and 2.38 by ao.:I(I),
integrating, and then dividing as in above results in:
o - 43 -
(j-D+l, .. ,C)
2.40
Finally, multiplication of the same equations by b(l) and
inteqration yields the following ratio:
( b(l)' K 1(1»)
1 l+u' K 1(1)
K bj = K.· ( b(l) )
J l+u' K 1(1)
(j=D+ 1, .. ,C)
2.41
Overall family mole fractions in the liquid phase are related
to the correspondinq feed mole fractions by the mater ial balance
equation 2.37. Directly integratinq this equation over aIl 1
produces:
(j-D+ 1, .. ,C)
2.42
The family averaged attractive EOS parameters in the liquid phase
can be related ta those in the feed by mu! tiplying equation 2.37
by a U (!) and inteqratinq over aIl 1 givinq:
2.43
A relation, similar ta the above but for the excluded volume
parallt:ter is possible if equation 2.37 is multiplied by b(l) and
inteqra:ted over aIl 1:
2.44
(
/
c
44 -
Tbis completes the fOrJIulation of equations required for
isothermal flash, dew and bubble point calculations in the
semicontinuous system. The next section describes methods used to
solve the equations.
2.3.5 801utioD of BquatioDa
For the typical isothemal flash calculation P. T , and feed
composition are specified and the unknowl1s are the compositions
of the liquid and vapor phases and the fraction vaporized. In a
semicontinuous system the unknown composi tions consist of the
discrete component mole fractions, overall family mole fractions
and distribution functions in the unknown phases. In the solution
schemes adopted here, the independent variables (3e - 2 D + 1) are
considered to be the fraction vaporized and the general ized
K-factors.
{u;K •• (i- 1 •..• C);K QJ.KbJ.(J=D+ 1 .... C)}
2.45
The K. X-factors hold for discrete and continuously distributed
components but the K", and K., are only meaningful for continuously
distributed components since they are equal to one for discrete
components.
The equations derived in the previous sections were presented
in the context of a flash calculation. In the case where the
vapor phase composition and either the pressure or temperature
are specified (i.e. dew point calculation) then the same
equations hold, except that u-J ('-0 ). Similarly, for a bubble
point calculation where the liquid phase composition is known in
advance then 1- 1 ( u - 0 ). In these cases it is no longer possible
o
o
- 45
to prescribe P and T separately, and for a qi ven P or T the other
variables and the composition o~ the incipient phase are
determined by the equations.
In order to solve for the 3C-2D+ 1 variables, a sillilar number
of equations are required. These are the following sets of
equations: the discrete component equilibrium constants froll the
fugacity coefficients, equations 2.29a (D equations): the
Rachford-Rice objective function, equation 2.32 (1 equation): and
the sets of generalized K-factor equations for continuously
distributed ensembles, equations 2.39 (C-D equations), equations
2.40 (C - D equations), and equations 2.41 (C - D equations). Note
that equations 2.39 to 2 • 41 invol ve integrals of functions
involvinq the unknown equilibrium function KJ(I).
2.3.5.1 Accelerate4 Succe •• ive Substitution
Successive substitution is the method that has traditionally
been used to solve isothermal flash problems (Michelsen, 1982:
Mehra et al., 1983: Ammar and Renon, 1987). It is attractive
since it converts a system of non linear equations into a single
equation in one unknown. In its basic form it displays a linear
rate of convergence: but as is described in Appendix A5, there
are methods of accelerating the convergence.
The basic procedure for solving a semicontinuous flash
problem using accelerated successive substitution is illustrated
in Figure 2.8. Ini tially P. T , discrete feed mole fractions,
overall feed family mole fractions and distribution functions are
known. The family averaged parameters for the feed are calculated
using eguation 2.23, 2.17, (discrete components), and 2.24, 2.26,
(ensembles). Initial estillates of equilibrium constants are
obtained from estimated critical properties and acentric factors
c Read P,T x: .FfCI)
J
~
i . riguZ'. 2.8 J ( :
~
t ,. k
" ~ il i-l;
- 46 -
Obtain initial e.tiaate. of K, Ci-l •••• C)
Solve Rachford-Rice objective function for u usinq Newton-Rapbson i teration
U.in; equations 2.27, 2.33-2.35 and 2.30, 2.42-2.44 solve for Xr.X: Ci.I. .•• C}
! {j·D+ I ••.• C}
Calculate EOS parameters for vapor and liquid, and z".~.
Osin; equations 2.27 solve for KI Ci- l, .. ,D)
1 Acceleration 1 ~
Os in; equations 2.39-2.41 and calculate /(J
K (j-D+l, ... C) aJ
K '"
No
Yea
[ STOP l
Accelerat.4 succ.ssiv. substitution alqoZ'itba roZ' isotheraal flash calculation.
o
_ 47 _
using the followinq empirical correlation suitable for
hydrocarbons (Mehra et al., 1983).
ln KI - 5.373' ( 1 + w 1)' (1 - T el IT ) + ln (p ell p) (i-l .... D)
2.46a
1 n K j ( 1) - 5 .373 . ( 1 + w ( 1 ) ) . ( 1 - T e (1) 1 T ) + 1 n ( P e (1) 1 P )
2.46b
Using equation 2.46b in equations 2.39-2.41, initial estimates of
the generalized K-factors are obtained. These estimates are used
in the Rachford-Rice objective function, which is solved for the
fraction vaporized using the Newton-Raphson procedure as shawn in
Fiqure 2.8. After each iteration r a check for convergence is
made; using the following objective function:
2.47
The acceleration of the successive substitution pr~cedure
involves a correction of the calculated K.,(t= 1, .. ,C) before a
return to the Rachford-Rice equation for the next iteration. The
correction is designed to improve the rate of converqence by
seeking a minimum in the maqnitude of the gradient of the overall
system Gibbs free energy along the direction of the search.
Details of this method are available in Appendix AS.
The procedure for dew and bubble point calculations is shown
in Figure 2.9. For these calculations, the inner loop is
essentially the same as that for the flash calculations, except
that the Rachford-Rice equation is not solved since the fraction
vapor ia known apriori. In the iteration of the outer loop new
estimates of P or T based on previous ones are provided by the
c
Raad P or T xr or xt 'HI) or ')(1)
- 48 -
Batiuta P or T K, (i- J .... C)
Ka/ICi - D + 1. ..• C)
K,,/
usinq equations 2.27, 2.33-2.35 and 2.30, 2.42-2.44 solve for xr or xt (1- 1 ••• • C)
;~ ~~ ;; } {} • D + 1. ... C)
Calculate EOS parameters for vapor and liquid, and z" •• ",
Usinq equations 2.27 sol. ve for K, Ci- 1 ... ,D)
1 • .MI:, t 1 ..... : .. B .. ..: ..
New estimate of T or l' usinq Newton-Raphson method wi th numerical derivative
usinq equations 2.39-2.41 and calculate
;:1 1 (j - D + 1 · ... C) KbJ
Acceleration
No
Accalerata4 auac •• aiva au)).ti tutioD .lqor! tu for buJ:tble 04 4 •• poiDt a.lculatioD.
o
o
- 49 -
Newton-Raphson mathod, whera the objective function is the sum of
ovarall family and discrete mole fractions in the incipient phase
and the derivative is evaluated numerically (i.e. 6116P·~I/~P)
The original, and somewhat naive, method used to simultane
ously solve the 3e - 2D+ 1 equations was the Newton-Raphson
procedure with numerical derivatives. These were evaluated as the
change in the objective function, similar to the one in the
previous section, divided by the difference in the variable
causing the change. The procedure is illustrated in Figure 2.10.
This technique proved to be extremely inefficient, in some cases
requiring several hours for the solution of relatively simple
problems.
2.3.5.2 BandliDq of Integrals: LegeDdre-Gauss Quadrature
The integrals in equations 2.39-2.41 are numerically
evaluated using Legendre-Gauss quadrature. The quadrature rule is
expressed as:
fi n
_1 F(U)dU=- ,~w" F(U,)+E
2.48
where w. are weighting factors at u., the quadrature points, and f
is an error term associated with the quadrature rule.
In essence the method is one of f inding a sui table
interpolation for the function F(U) which is known only at
certain discrete pO:Lnts F(U,). By suitably choosing the 2n+2
variables { w., U., (la l, .. ,n)} one can obtain results that are exact
if the function F(U) ls a polynomial of degree 2n+l or less.
A complete discussion of the method has been presented by
Carnahan =t al., (1969). Briefly, in Legendre-Gauss quadrature
(
1
c
Read P, Il
Obtain initial estimatas or KJ Ci-l, .. ,C)
K.I
K ./ Ci - D + l , .. , C)
u
calculate ,Itl to r- as the lert hand side minus the riqht hand side of equations 2.19a and 2.39 to 2.41. These expressions are desiqnated aquations (i) to (v), respactively.
- 50 -
Solve system of linear equations "·-1 usinq qaussian elimination. , is the vector of independent variables and 1 are the variables on the LHS of equations (i) ta CV) •
Calculate j acobian matrix ~ for equations Ci) to Cv) usinq K c. K.v and K'J as independent variables and numerical derivatives based on incremental chanqes (xl.0000001) in the variables
No
Pigur. 2.10 Alqoritba for ••• ton-RaphSOD •• thod vitb nua.ria.l 4.zoiv.tiv •••
o
o
- 51 -
the interpolating polynoaial chosen for F(U) is of the Lagrangian
fora, vith its associated error tera, also a polynomial.
Substitution of this interpolating eguation vith its error term
into the inteqral, and subsequent integration yields equation
2.48 (the weights, 1.'" and error term, ~, are integrals). Using
the properties of the Legendre family of orthogonal polynomials,
it is possible to aliminate the error term F for F(U) that are
pOlynomials of degree 2n+l or less.
The end result is that the n quadrature points Y" lyinq in
the ranqe (-1,1) are given by the roots of the nth deqree
Leqendre polynomial p.(U). These roots, together with their
eorresponding weights, w" have been ealculated by Stroud and
Seerest (1966). They are tabulated as a function of n, the number
of quadrature points.
However, before the tabulated values can be used for VLE
ealeulations, the integration range has to be transformed from
(-1,1) to (/ ... /,,). Equations 2.39-2.41 have integrals of the
product of the feed distribution function with other funetions;
sinee the feed distribution is deseribed by cubie polynomials
then for every ESFT segment equation 2.36 takes the following
form:
2.49
UsinC) a new variable (- 1 < U < 1) to eonvert the range from (/. < / < l,)
the transformation is:
2·[-[ -[ U _ a b
[b-[ a
(
c
- 52 -
for which substitution into equation 2.49 yields:
<fJ(U».I,,-/ajl <CO+cl.«U+ 1)· (J"-/a)+/a)+ 2 -1 2
where fI(U) i8 the function y(l) in terms of the variable U.
2.3.5.3 Iapl ... ntation of Alqorithas
2.51
The semicontinuous VLE prediction scheme, using accelerated
successive substitution, has been implemented in the form of a
Fortran proqram TVLBT. This general purpose proqram is capable of
performing flash, dew point temperature/pressure, and bubble
point pressure/temperature calculations. The proqram allows
choiee of distribution function (Gamma, Beta, ESFT) with
corresponding quadrature method (Laguerre-Gauss, Chebyshev-Gauss,
Legendre-Gauss). The continuous PRSV EOS constants can be
estimated using GSCNP, Kesler-Lee, or n-alkane property
relations, rhe distribution variable can either be the molecular
weight or the normal boil inq point. Addi tional details of the
program are available in Appendix 6 and the program is included
in a diskette in this thesis.
o
o
53 -
CDPDa 3
PUDIC'l'l:OB 01' VLI l'OR ILL-D •• IDD XI ft.,...
In this ehapter the methods described previously are applied
ta sample systems, both model and real fluids.
3.1 VU for a K04.1 .lui4
The aim of th!s section is to illustrate the effect of
various parameters on the predicted VLE using a model fluide
Theae variables are the number of spi ine segments, degree of -.
correlation, number of quadrature points, etc.
The model system consists of a mixture of 40 mole percent
C02, and the balance a combination of n-alkanes, a choiee aimilar
ta that proposed by Cotterman and Prausnitz (1985). Hovever, the
example used here invol ves a more complex molar distribution
curve for the alkane family, as an illustration of the
versatility and flexibility of the ESFT method. It is a truncated
composite of two Gamma functions, with moleeular veight aeans and
variances of 100, 800 and 200, 1600 respectively. The lovest
molecular weight considered is 50 while the maximum is 350. Since
the desired distribution variable is the normal boiling point,
the aolecular weight values vere convertad to boilinq pointa
using the followinq correlation, obtained in this vork a. a laast
~
o ... ~
.15
g .10 .-~ =' .J:2 .-... ~ li) .-Q ... .!! .05' o ~
EXTENDED SPLINEFIT OF MOLAR DISTRIBUTION MOOEL FLUID
o , • 50 ' ASO ' • • 250 3 80iling POint (K)
550
LEGEND • Ddtd a 25eg b 3 Seg C .. 5eg d 5 Seg e 1 Seg
~;t;;;bA • ·&50
~lqur. 3.1 18rr r.pr ••• ntation of aolar 41.trlbutlon curv. of a04.1 P1ui4.
~
Ut ~
~
o - 55 -
.quare. fit of data of nOrJlal albn •• vith 3 to 31 carbon ato_
(CRC, 1986).
T,,-111.1+3.141·MW-6.300·lO-3 ·MW 2 +5.430·lO- 6 ·MW 3
3.1
The representation of the .olar distribution data uaing the
ESFT is shown in Figure 3.1. This figure illustrates the quality
of data representation vith varying nuabers of spline fit
segments. The symbols represent spline fit values vhile the
conneeting lines vere determined by the graphies package (McGill
Plotting Package). As expeeted the representation gets better
vith more secpnents, and for this relatively complicated aolar
distribution curve about 5 secpnents are required for complete
charaeterization. The use of fewer segments resul t. in
unrealistie negative values for the high boiling point end of the
aolar distribution. This is a consequence of the current
iaple.entation of the ESFT and although this ean be prevented, it
requires a significant modification of the ESFT subroutine, and
was not deemed necessary since erroneous values were not obtained
with the 5 or more seCJlllents required for complete charaeteriza
tion.
The boiling point values of normal alkanes vith 3 to 28
carbon atoms (CRC, 1986) were used in program &lFIT to obtain a
II\(T.) function. The second order polynomial fit of the data ia
illustrated in Figure 3.2. The N, shovs a fairly smooth trend with
the higher boiling components displaying the greatest value. of .,
• The choiee of the second order polynomial in the figure ia
arbitrary aince ealculations are done vith thr.e different
correlations for comparison. The coefficients of fir.t, •• cond,
and third order polynomial expressions for N" are pre •• nted in
....... . ~ ';' ';It 1"1~"" .~-;' "'-Il" ... """.. ~ l' ,
~ ~
Correlation of IÇ} with Tb for n-Alkanes
0.00
It}
-0.25
-0.50 1 • •
23 ••
O • i -> • •••
350 470 710 590
80i ling Point (K)
Figure 3.2 ~be III,(T.) fUllctioD for n-alkane r_11y.
Ut 0\
1
1
1
\
o
o
57 -
'l'able 3.1, where the standard d.viation valu •• corr •• pond to
ab.olut. d.viation. of calculated valu.. fra. knovn '" valu.. at
th. boiling points.
Ilbl. ~ AIJtaD. f .. ill NI(T.) aozor.latioD COD.tut. (u8i_CI
G8CJ11t) •
"1(Tb)-aO+aloTb+a2oT!+a3oT!
Degree of "1 a o al a 2 a 3 Polynomial (st. Dev. )
1 (0.0112) 2.553E-l -9.515E-4
2 (0.0073) 1. 990E-2 6.712E-5 -1.005E-6 3 (0.0069) 1.213E-1 -6.325E-4 4.865E-7 -1.0001-9
3.1.1 .1a.b Calculation.
The first calculation is a flash calculation at a temparatura
of 500 K and a pressure of 3000 kPa. 'l'he model fluid is described
using the E8FT, with 7 seC)llents. The integration utilizes 4
quadrature points per segment, the kl(T,) is third order and the
binary interaction parameter between C02 and the alkanes used is
0.12, a typical value for this interaction using the PRSV EOS
based on binary studies by the author. The choice of P-T
conditions is not significant, except perhaps for th. fact that
roughly half of the feed is vaporized. The resulta are .umaarized
in Table 3.2 and Figures 3.3-3.4. The table provide. ov.rall
datail. of the calculation, vith the expected result that .o.t of
the carbon dioxide is in the vapor phase. The two para.eter., K.I
(
c
,.., o )t
c o .... :::J
..c '-4J
Vl
Cl
'd o ~
2.0
:.. e .... ~ 1.0 -~ 1
"" ~
o
58 -
Symbols indicate quadrat.ure pointa. • • o
4.0
2.0
300 400 500 600
801 hng POint (I()
rlqure 3.3 Holar 4istributioD 1ft alkafte f.ailya flash.
\ \ \,
Symbols indicate quadrature points.
300 400 soo 600 Boiling Point (K)
Feed
Vapor Liquid
700
700
o
o
- 59
, and K" are inclu4.4 as illustration of the fact that th.y ar.
ph... composition dependent and redundant for discr.t.
coapon.nts.
Zabl. ~ •• sults of fl.sb aalculation at ~soo K anA ""000
kP ••
Component feed liquid vapor K aJ KbJ mole mole _al. K-factor frac- frac- frac-tion tion tian
C02 0.400 0.092 0.661 7.15 1 1
alkanes 0.600 0.908 0.339 0.374 0.656 0.603
54.1 1Iole perc.nt vaporiz.d
Details of the continuously distributed alkan. fa_ily
compositions are provided in Figures 3.3-3.4. The former is a
plot of feed, liquid and vapor phase molar distributions, scaled
to reflect the liquid/vapor split. The sYJDbols represent the
values at the quadrature points. The liquid retains a pro.inent
bimodal distribution with a predominance of higher boiling
constituents while the vapor phase is essentially uni.odal vith
mainly low boil ing material, as vould be expected. Figur. 3. 4
shows the value of K-factor, KJ{I) vithin the alkane fa.ily as a
function of the boilinq point. At these conditions, vith the
pressure relatively low, the curve is fairly concave and ther. is
a large difference in K J(I) over the range of the distribution
variable illustratinq the relative reluctance of heavi.r boiling
co_pounds to vaporize.
In order ta gain a better understanding of th •• ff.ct of
changes in various parameters on the final VLE pr.diction,
several calculations vere done et these conditions. Th. first
c
1
- 60 -
involv.. the representation of the .olar distribution eurv ••
Calculations were done using ISFT fit. of 2-17 •• pent •• Tbr ••
quadrature points were used par interval and the _.(T.) function
wa. third arder. The fraction vaporized and X-factors for 002 and
th. alkane fa .. ily are presented in Figure 3.5, wh.r. Kc. is the
ov.rall alkane family K-factor. The re.ul ts indicate that for
thi •• odel fluid, six segments are required for conatancy in
re.ulta: i.e., for complete characterization.
Although two Legendre-Gauss quadrature points will exactly
integrate a polynomial of deqree 3 or less, the integrals of
equations 2.24-2.25 and 2.39-2.41 contain products of cubic feed
distribution polynomials and other, different, functions so that
the requisite number of quadrature points is not obvious. However
flash calculations done at these same conditions using 7 ESFT
segments and a third arder k.(T,) correlation yielded results that
were virtually identical for 2, 3 and 4 quadrature point ache •• s.
This was also the case for bubble and dev point calculations.
Additionally, for this flash calculation the k.(T.) function hardly
has an effect on the final results as is evident in Table 3.3
which compares the calculated K-factors and fraction vaporized
using different correlations.
Z.le.l.&.1 8ff.ct of ".(T.) corral.tioD OD fl •• 11 c.lculatioDs.
Degree of k.(T,) N. =0 1 2 3
Fraction Vapor 0.543 0.541 0.543 0.541
KC02 0.375 0.374 0.376 0.374
Kc. 7.15 7.15 7.14 7.15
o
·0
al
='
- 61 -
Flash calculation at T=500K; P=3000kPa 0.75 -,---------.--------------,
0.7
0.65
0.6
iü 0.55
> 0.5
0.45
0.4
2 4 6 8 10 12 14 16
Number of [SfT Segments o fraction Vapor + Kco2x IOE-1 o Kan
ligur. ~ .ff.ct of D~r of .arr •• ga.Dt. OD fl •• h
calcul.tioD r •• ult ••
(
1
c
62 -
3.1.2 .aturatioD pr ••• ur. aa4 ~..,.ratur. CaloulatioDa
Bubble point pressure calculations vere perfonaed at 350 K
for vhich the general resul t ia preaented in Table 3.4 and
Figure 3.6. The calculation involved here utilized 7 ESFT
seCJllents for characterization, 4 quadrature points per segment
and a third arder M.(T,) function. From Table 3.4 it ia evident
that the incipient vapor phase is very rich in C02 , although the
carbon dioxide K-factor is not particularly large. Figure 3.6
shows that the vapor phase molar distribution is uni.odal vi th
virtually no high boiling material. This is corroborated by the
K J{I) curve, not shovn here, vhich is highly concave, indicating
a large difference in KJ(I) over the boiling point range (4 orders
of magnitude) •
TIble ~ ••• ulta of bubbl. point pr ••• ur. calculatioD at ~350 s.
Component liquid vapor Ka} Kbl mole mole K-factor frac- frac-tion tian
C02 0.400 0.967 2.420 1 1
alkanes 0.600 0.033 0.055 0.621 0.560
Bubble point pressure = 6027.5 kPa
Dev point tempe rature calculations at a specified pressure of
350 kpa were perfoned. The resulta of a sample calculation are
presented in Table 3.5 and Figure 3.7. The characterization
paralleters in this example are similar to those of the sample
bubble point calculation. As illustrated in Figure 3.7, the
incipient liquid phase is of high average boiling point and
displays a relatively complicated molar distribution curve. The
K ,(1) curve of the alkanes at this high tellperature has larger
o
)(
c o -::J
.a ï: - 2.0 \1)
éi
-)( c o -::J .a ï: -\1)
c ... «' o ~
2.0.
- 63
Symbola indicate quadrature points. D Liquid
• Vapor
300 400 500 600 700 8011109 POint (K)
'igure 3.~ Kolar 4i.tri~utioa iD alkaa. fa.ilyl bub~l. poiat.
D Vapor
... Liquid
Symbols indicate quadrature points.
300 400 500 600 700
801 hng Poant (K)
c
1
c
Me " IL lW_IIII. Sltlll;_
- 64 -
valu •• althouqh the .hape is .iailar to that of Pigura 3.4, th.
fla.h .ituation. The curve viII flatten out a. one approache. the
critical point, vhere the value of K 1(1)· 1.0 for ail 1.
Zable ~ .e.u1ta of 4 •• point t.aperature oa1culation at 1=350
k.a
Component vapor liquid Kai K "i mole mole K-factor frac- frac-tion tion
C02 0.400 0.006 64.80 1 1
alkanes 0.600 0.994 0.604 0.619 0.571
Dew point tempe rature - 532.8 K
The parameters affecting the calculations vere investigated
in a similar manner ta the flash situation. FiCJUre 3.8 is a plot
of the calculated saturation temperature as a function of the
number of ESFT segments used in characterization. Seven segments
allov for constant resul ts. The calculated saturation pressure
displays the same trend. Since the bubble pressure and dev
temperature are more sensitive than the fraction vapor the curve
does not flatten out entirely due ta small fluctuations in the
molar distribution for large numbers of segments. Table 3.6
presents the results of calculations involving different K.
relations. There is a small effect on the final resulta.
The results of final calculations vith the model fluid are
presented in Table 3.7. In this case three different correlations
vere used to estimate the critical properties and acentric factor
of the continuous ensemble. The first vas si.llar to previous
calculations, involvinq GSCNP relations and a 3rd order
polynomial for the J(.(T,) function. The second set utilized
Kesler-Lee relations for the critical properties, the EdIIliater
o
Il -
- 65 -
equation ror the acentric factor, and a 3rd orcier _,(T.) function.
Tha final calculation involved T.onopoulo. (1987) corralation.
for alkane critical properti •• , vith _.(T.)-O and the follovinC)
8apirical expression for the acentric factor, obtained in thi.
work from n-alkane data (Reid et al., 1977):
,... ~ v
CI ... =' ~ lU ... CI Il. E CI r-I: 0 .. ~ III ... =' ~ lU
(J)
w--O.02785+4.05S4·1O- 3 ·MW-2.978·10-6 ·MW 2
3.2
Model System Dew-T at P=350 kPa 560~-----------------------------------------------,
550
540
530
520
510
500 '--~~--~--~~--~~---r--~~--~--r-~--~~
2 4 6 8 10 12 14 16
Number of Segments
ligur..L.l 8ffect of DUilber of 8.rr .eCJlleDt. OD 4 •• point t .. p.r.tur. calcul.tioD r •• ult ••
(
(
~
-- 66 -
'luI • .L..I .ffect of "1 correlatloD oa a.leul.t.. ..tUZ'atloD pr ••• ure ... t..,.ratllr ••
Dec)ree of NI(T.) J( 1 = 0 1 2 3
Bubb. Press. (kPa) 6167.6 6015.0 6145.2 6027.5
K C02 (Bubble-P) 2.42 2.42 2.42 2.42
Kc. ,Bubble-p) 0.054 0.055 0.054 0.055
Dew Temp. (K) 532.9 532.8 532.7 532.8
K C02 (Dew-T) 64.89 64.81 64.80 64.80
Kc. (Dew-T) 0.604 0.604 0.604 0.604
where the molecular weight is calculated from:
MW - -65.65 + 6.3876' 10 -1 • Tb - 1 .1000, T~ + 1.6024' 10 -6. T~
3.3
As expected, the resul ts of the three calculations are reasonably
close; not only due to similarity in the critical property
relations but aiso because discrepancies are partly compensated
for by the NI(T.) function. However the bubble pressure is
sensitive and shows siqnificant variation.
1II!l. • .L1. R •• ul t. of ca1cul.tioD. a.iDfI dift.r.Dt B08 COD. tut
relation ••
Bubble Point Dew Point Relation for EOS Fraction Pressure Temperature
Constants Vapor (kpa) (K)
GSCNP 0.542 6028 532
Kes1er-Lee 0.553 6237 534
A1kane 0.539 5817 538
o - 67 -
3.2 VLB for a.al syat ...
In this section the model is used to predict properties of
real systems, for which experimental data is available.
3.2.1 aa40 •• et al. sy.t ...
The first example is the supercritical propane-continuous oil
mixture of Radosz et al., (1987). This mixture is suitable for
study because both experimental TBP and phase equilibria data are
available which allows full use of the characterization and VLE
calculation procedure described in the previous chapter. Flash
calculations for two systems were performed over the ranges
374-414K and 3102-5514 kPa. The first system consists of propane
and a saturates-rich oil. The second is made up of propane and an
aromatics-rich oil. Both oils have number average molecular
weights in the region 300-350.
The first calculations involve the saturates-rich mixture.
Using TBP information for the whole oil, obtained from the above
authors, and the characterization procedure of section 2.2.3, a
molar distribution curve was derived. GSCNP relations were used
to convert the boiling points to molecular weights. Figure 3.9
provides a record of the characterization. Eight segments were
chosen for ESFT representation of the molar distribution curve.
The molar distribution curve is skewed slightly to the right and
displays a "bump" on the left hand side. Qualitatively the curve
is similar to tne composite beta function curve of Radosz et al.
(1987). The two extreme points (corresponding to the initial and
final TBP boiling points) for which F(/)-O.O were artificially
generated by the program CHART.
(
j
C
" 100 '" ~ ... .... toi ." .... ~
-... » 50 QI > -... co ~
i '"
0 ... ~
" 0 .... ... ,B .... ~ +' IJ .... Q
; ... a Z
0.3
0.2
0.1
0
- 68 -
DISTlUATIDN CURVE FOR CIL SAMPLE
650 700 750 Boi ung POlot (10
EXTENDED SPUNEFiT OF MOlAR DISTRIBUTION
~ ... ".?' .... .,. - \ ~ \
/ • \
-\
\ \. \. \. ,1 ,
~ ~~ . .... . . " .......... "'" ' . .,- '. , ' .. 1
• 6 0 6 0 1bo 1~0
Boi 1 ill9 Point (X)
PiCJUr. 3.' CharacterizatioD of •• turat •• -rich 011.
o - 69 -
Zabl. aa.A 1' •• 4 .,1. fraot:io.. of .. 40.. .t al. ain ••••
System vith saturat •• -rich oil
Propane/Oil Propane feed Saturates Aroaatic.
veight ratio mole percent .ole per- aole perCel'lt
in feed cent
3.4 95.874 3.423 0.703
3.5 95~ 988 3.329 0.683
3.7 96.196 3.157 0.647
3.8 96.293 3.076 0.631
4.1 96.554 2.869 0.577
system with aromatics~rich oil
Propane/oil Propane feed Saturates Aromatics
Weight Ratio mole percent mole per- mole percent
in Feed cent
3.4 94.780 1.450 3.770
3.5 94.922 1.410 3.668
-3.7 95.183 1.338 3.479
3.8 95.304 1.304 3.392
TBP boiling points vere used in program Kl.I'! to obtain a
IC,(T II ) funetion with Qo, a" a2, and a3 equal to -2.818, 1.370E-02,
-2.269E-05, and 1.142E-08 respectively. Sinee aIl results and
(
(
- 70
.oat input data in the above referance are r.ported in th. fOrll
of veight percentages, .0.. converaion. v.r. n.c •••• ry. UainC)
v.ight fraction infontation in Tabl. 1 and nUllber av.rag •
• olecular veights from Page 735 of th. above reter.nc., the
propane to oil ratios in th. teed vere converted to aole
fractions. The results are displayed in Table 3.8.
Based on these feed mole fractions, flash calculations were
perforaed at values of the temperature and pressure corresponding
to experimental data. In order to compare predicted and
experimental results the mole basis output of program 'l'VLII'1' was
converted to a veight basis. This required average value. of
molecular veight for the oil ensemble. These values were obtained
using another set of calculations at identical values of
te.perature and pressure. This set used beta densi ty functions
vith aolecular weight as the distributing variable to
characterize the oil into tvo families, a
aromatic cut, as described by Radosz et al.
saturated and an
In thi. ca.e the
proqram 'l'VLBT evaluated integrals usillq Chebyshev-Gauss quadra
ture (see Carnahan et al. , 1969). Parameters ot the beta
probability functions vere obtained from Radosz et al. The use of
molecular veight as the distributing variable allow. easy
calculation of the Ilumber average molecular veiqhts of the
equilibrium phases once the equilibrium molar distribution 117\
known. similar calculations vere performed vith the aecond
system.
o
o
- 71 -
Saturates-Rich oil Sy.te. VLE Re.ult.
Liquid Phase Propane Co.position in Percent Weight
Te.p. Pressure EXP ESFC BETAC RADOSZ
(K) (kPa) ( i) ( ii) (iii) (ii) (iv) (ii)
374.4 3102 26.9 26.2 32.8 25.9 32.9 28.5 27.0
392.5 3102 18.0 17.2 21.1 16.9 21.1 20.5 18.5
392.6 4236 28.2 27.9 34.5 27.7 34.7 30.0 27.5
392.2 5514 55.4 56.1 65.7 56.9 67.5 51.0 44.0
413.5 3102 13.1 12.3 14.7 12.1 14.7 15.0 14.0
413.5 4136 19.3 18.5 22.4 18.2 22.4 21.0 19.0
413.5 5514 29.7 29.5 36.0 29.1 36.4 29.0 27.0
EXP - Experimental data. ESFC - oil molar distribution described usinq ESFT. BETAC - Oil molar distributions (2) described by bet. denaity functions. RADOSZ - Resul ts of Radosz et al. calculations, read off graphs. (i) - k 12 - 0.03: l=propane: 2=oil (ii) -all interaction parameters equal to zero (iii) - A: 12 -O.02 and k ,3 -O.04; l=prop.; 2-sat.; 3-aro •• (iv) - interaction parameters given by Radosz et al.
The resul ts of calculations for the two systems are provided
in Tables 3.9-3.11 and Figures 3.10-3.11. For each value of
temperature and pressure the ratio of propane to oil in the feed
ia as indicated in the original reference. Table 3.9 and 3.10
provide the compositions of the liquid phases, .easured and
calculated, using several sche.es. The results of Radosz et al.
were visually estimated from charts in their publication. Figure
3.10 and 3.11 compare calculated propane solubility obtained with
and without the use of interaction paralleters. For both systems
(
~
~ ~
~ '-'
QJ CI) ('0 ,.. -~
"0 .... :::: ~ •• ) ~ c: •• a> c: co ~ 0 ... ~
c
- 72 -
392. 5 , : .. exp. . calc. curve a a
113. 5 , : • exp. cale. curve 1J / /
60 -- (,it : 0 / 112. : 0.03 / a
/ /
/ /
/ 40 /
,/ /
/' b /
,/ ,/
,/
20 ,,/
o~----~----~----~----~----~----~ 30 40 50
Pressure x 10-2 (kPa)
~iCJUJ:. 3.10 Calaulat.4 aJl4 upal.'iaeDt.l .01uJ:tility of pl.'Opu. in liqui4 pb ••• for .atur.t •• -ricb oil.
60
- 73 -
o th. re.ul t. obtainecl using the ESrT .. thod are fairly .ccu~ate
for the liquid phase (excellent wi th th. aid of interaction
par .. etera), co.parable to the those obtain.d by Radoaz et al.
vho u.ed tvo fa.ily characterization and the perturbed-hard-Chain
BOS. It is noteworthy that vi thout interaction par ... ter. the
predictions usinq two beta fa.ilies and those usinq one ESFT
fa.ily are virtually identical. This either indicate. that the
ESFT characterization i8 equivalent to the two family beta or
possibly that the final results are insensitive ta the shape of
the molar distribution curve.
Tible ~ Liquid ph ••• co.poaition of aro .. tica-rich oil
ayat_.
Aromatics-Rich oil System VLE Results
Liquid Phase Propane Composition in Percent
Temp. Pressure EXP ESFC BETAC
(K) (kPa) (i) ( ii) ( iii) (ii)
392.7 3102 14.5 15.7 25.2 14.6 25.2
392.7 4236 22.7 23.7 39.9 22.6 41.0
392.4 5514 42.9 39.5 70.5 37.1 79.2
413.5 3102 10.7 11.9 17.9 10.9 17.9
413.5 4136 15.6 17.2 26.7 15.8 27.0
413.5 5514 23.9 25.5 41.6 23.6 43.5
(i) - all interaction parameters equal to zero (ii) - k. 2 ·O.07; see Table 3.9 (iii) - k. 2 ·-O.16 and k. 3 -O.100; aee Table 3.9 (iv) - interaction parameters qiven by Radosz et al.
Weiqht
RAooSZ
(iv) (ii)
16.5 14.0
23.0 18.0
37.5 28.0
12.0 10.0
15.6 14.0
23.0 18.0
(
,.... ~ ...-~
. ......, QJ fil ~ ..c: ~ ~ .... = 1 cr .... ~
== .... Q)
= Cd ~ 0 5-t ~
- , -:
- 74 -
1 392.5 Je : • exp. cale. curve a / 113.5 Je: • exp. cale. curve b
/ a
/ 60 -- K :; 0 /
" :; 0.065 / /
/ /
/ /
40 / a
/ / /'
/' /'
/' ,/
/' ",
,/ "" ,." ,."
" 20
O~----r---~----~----~----~----30 40 50 60
Pressure x 10-2 (kPa)
l'iqure 3.11 Calcul.tecS and experi.ental solUbiU ty of propane in liqui4 pha.e for aroaatics-rich oil.
o
o
75 -
The vapor phase co.position. are available in Tabl. 3.11. In
thi. caae, for the ESFT calculation, the be.t re.ult i. in error
by 5 percent while the vorst has a 150 percent error, however th.
ab.oluta valuea of the oil co.position are very aull (all 1 •••
than one percent). Results obtained using ESFTC are high. U •• of
interaction parameters leads to opposing resul ts in the tvo
phase. - parametera that give i.proved liquid co.position. reault
in vorse vapor phase predictions, although in a les. aensitive
manner. Qualitatively similar results vere observed when using
beta functions, although in this case the vapor phase predictions
are considerably vorse. The reasons for the poor representation
of the vapor phase are not entirely clear, although the
abnormalities in the interaction parameters, including the
negative values, can be explained by the T-P specifications'
proximity to the critical region and the inherent limitations of
cubic EOS for heavy-hydrocarbons, especially in the critical
ragion, as suqqested by Radosz et al. (1987). Hovever, it is
apparent that the ESFTC method proposed in this work is able to
predict VLE for this system. Resul ts are good for the
propane-rich (liquid) phase and fair for the vapor phase.
- 76 -
labl. ~ Vapor ph ••• ao~.itio. of .7.t ....
vapor 'ha •• oil Coapo.itioD iD '.ra.Dt •• i9ht
Saturat.a-Rich oil sy.t.. VLB •• ault.
'l'_p. Il' BSI'TC BITAC (E) (a) (1)>) (a) (b)
374.4 0.012 0.023 0.027 0.112 0.14'
3'2.' 0.022 0.051 0.054 0.22' 0.285
3'2.' 0.05' 0.115 0.12' 0.34' 0.42'
3'2.2 0.840 1.032 1.181 1."7 2.013
413.5 0.051 0.112 0.114 0.44' 0.53'
413.5 0.0'0 0.1'7 0.201 0.5'2 0.6'3
413.5 0.305 0.580 0.59' 1.183 1.342
Aro.atica-Rich oil Syat_ VLII •• ault.
'l'_p. BI' BSrtC BITAC (1:) Ca, (b) (a) (b'
3'2.7 0.022 0.04' 0.0.' 0.721 0.5"
3'2.7 0.041 0.101 0.11. 0."4 0.'43
3'2 •• 0.780 0.833 0."7 3.587 4.14
413.5 0.052 0.0" 0.100 1.3'7 1.052
413.5 0.083 0.171 0.173 1.675 1.322
413.5 0.2'2 0.488 0.4.7 2.74' 2.332
(a) aIl interaction parameters equal zero (b) binary parameters as qiven in Table 3.9-3.10
c
o - 77 -
3.2.2 Boff.anD et al. syat ..
The second example of a real system invol ves a natural
gas-condensate (Hoffmann et al., 1953) for which experimental
dew-point pressure and flash resul ts in the retrograde region,
at 367 K, are available. These authors give compositions of the
light hydrocarbon components, up to c6 , and for distillation cuts
corresponding to normal alkanes up to c22 • Average molecular
weights and densities of the fractions are provided.
Calculations were performed using three schemes. The first
treated the system as consisting of eight discrete components,
the seven lightest components (up to normal pentane) and one
pseudocomponent representing c!. The second approximated the
system as a discrete mixture of isobutane, isopentane and 22
n-alkanes. The third system consisted of the 5 lightest
components and one continuous ensemble representing c~.
In the case of the discrete components, alkane critical
properties, acentric factors and HI values were obtained from
Stryjek and Vera (1986) except for heavier alkanes, cte, which
came from Reid et al. (1977): for these components the HI value
was fixed at 0.04. For the continuous ensemble, average molecular
weight values of fractions given by Hoffmann et al. corresponded
with alkane values and thus the corresponding alkane boiling
points were used to generate a molar distribution in terms of the
boiling point. These were also used to evaluate the HI(T b) function
with parameters no, al, a2, and aJ eqllal to 3.376E-Ol, -2.229E-05,
4.218E-06, and -J.778E-09 respectively.
o
o
- 78 -
Ilbl. ~ co.,oaitloa at « .. pol.t fo~ co.C ••• at. at S" K.
Co.ponant EXP Sellicontinuoua 24 ps.udocoaponent
Ca) Cb) Ca) (b)
CI 52.00 53.96 53.70 61.42 57.92
C2 3.81 01.35 4.27 4.48 4.07
.-C,
2.37 2.51 2.39 2.42 2.19
tc. 0.76 0.85 0.80 0.78 0.72
nC. 0.96 1.07 0.78 0.96 0.88
c; 40.10 37.26 38.06 29.94 34.22
Ca) - no interaction parameters (b) - interaction parameters equi valent to 9 - 1 .1703
For each of the above cases calculations vere performed vith
and without interaction parameters. Interaction para •• tara for
each of the components in the system (discrete and ensamblas )
were calculated, as a function of V e , the critical volUlles, and
an ampirical parameter 9, usinq a function provided by Nghie. et
al. (1985). For ct5 alkanes values of Vc vere not available in Reid
et al. (1977) and the followinq linear extrapolati~9 function was
usad, based on lighter alkane data:
ve - 24.6+ 4.06' MW
3.4
whar. v c ia in cm3/llol.
(
, v
------ - ------- - -------- ------------------ -----------------------
- 79 -
For the 8 and 24 pseudoco.ponent ca.. th. parlaeter' wa.
adju.ted br trial and error in order to procluce a ri t of th.
experi.ental dew point pressure of 2 .... 70 kPa. For th. 8 coaponent
case, usinq a critical volUlle of 900 ca3/aol for the
pseudocomponent, a value of 6 - 1.034 reproduced the dew point
exactly. The corresponding value for the 24 component cas. was
1.1703. Attempts to fit the dew point pressure for the
semi(lontinuous mixture using an average value of v c .et vith
li.ited success. The bubble point was insensitive to interaction
parameters; using 9 - 1 .1703 produces an interaction parameter of
0.075 between methane and the ensemble but resulta in a dew point
of only 19816 kPa. This is in contra st to the previous problem
where the binary parameters had a larqe effect. However, even at
this incorrect dew point, the semicontinuous system provide. the
most accurate predictions of composition as shown on Table 3.12.
The 24 component system, although predicting the exact bubble
point predicts a hiqh methane concentration.
Flash calculations were performed at 367 K and several
pressures: results of those at 13887 kpa are displayed in Table
3 • 13. For the continuous ensemble the distributions in both
phases vere qualitatively similar to experimental. Fro. the table
it is evident that the 24 pseudocomponent systea with interaction
parueters qives the best predictions. However with no parueters
the se.icontinuous and 24 component case display si.Uar resul ts.
Thu. the weak performance of the se.icontinuou. .y.te. is
attributed to the lack of sensitivity of the interaction
parameters. The 8 component system producea fairly good resul ta
at low computational cast.
o
o
- 80-
Tabl. L.lI R •• ult. of fla.h calculatioD at 317 K and 13887 k'a.
Liquid composition in mole percent
8 24 pseudocomponent Semicont. Comp. EXP pseudo
comp. (a) (h) (c) (a) (b)
Cl 34.19 38.06 42.86 37.89 37.41 41.60 41.33
C 2 3.62 3.67 4.02 3.58 4.05 3.97 3.88
C 3 2.87 2.45 2.60 2.39 2.66 2.59 2.44
IC 4 1.02 0.94 0.96 0.92 0.99 0.96 0.90
nC 4 1.55 1.12 1.23 1.17 1.27 1.24 0.86
c~ 56.75 53.76 51.67 54.05 53.62 49.64 50.59
Vapor composition in mole percent
8 24 pseudocomponent Semicont. Comp. EXP pseudo
comp. (a) (h) (c) (a) (b)
Cl 92.18 92.87 92.05 92.07 92.12 91.82 91.81
C 2 4.03 4.04 4.03 4.04 4.03 4.03 4.03
c 3 1.57 1.50 1.51 1.52 1.51 1.53 1.53
IC 4 0.34 0.37 0.38 0.38 0.38 0.39 0.39
nC 4 0.44 0.41 0.42 0.42 0.42 0.43 0.43
c~ 1.44 0.81 1.61 1.57 1.54 1.80 1.81
no interaction parameters (a) -(h) -(c) -
interaction parameters equivalent to B - 1 .1703
C l7 te t/I = 0.065 for methane wi th c6 to c16 ; k'l = 0.070 for C24 •
methane wi th
(
(
81-
As a way of conclusion of the resul ts in this chapter i t is
possible to formulate a few general observations: The normal
boiling point is a satisfactory distribution variable, however tn
comparison with the molecular weight it does not allow the
automatic calculation of phase densities - an important parameter
in process design; an associated problem is encountered when
derivinq the molar distribution curve usinq the Resler-Lee, Twu
or similar relations, one must calculate the molecular weight in
order to differentiate the curve, however in order to retain the
boilinq point as the distributinq variable it is necessary to
recalculate the boilil1g points from the molecular weiqht, a
cumbersome procedure requirinq i teration - see documentation for
proqram CHAR'!' for additional details. The acceleration of the
successive substitution method improved converqence in most
cases. But in some instances i t worsened converqence and in sorne
bubble and dew point calculations i t promoted convergence ta the
trivial ~olution (aIl R-factors equal to one).
o
o
_ 82
CDft •• 4
.:tGB PRESSURE VLB DA'l'A COLLBC'l'IOIf: 1I0DlI'ICA'l'ION OP
IQUIPIIBRT ABD IIBASURBIIBHTS l'OR C02-CYCLOBBXABB SYSTBM
This chapter presents the experimental aspects of the
project. A brief description of the apparatus as it now exists
is followed by a report of the modifications implemented,
includinq a description of a new qas phase samplinq valve. The
final section discusses the performance of the modified equipment
includinq experimental results.
4.1 D.scription of Apparatu8 and Experiment
A detailed description of most components of the experimental
setup has been presented by Orbey (1983) and by Sejnoha (1986).
Figure 4.1 presents a simplified diaqram of the setup as it now
exists.
The major changes involved the installation of a new gas
sampling valve (3) and the addition of a liquid C02 cylinder
equipped with a siphon to allow liquid C02 injection (10). The
accessibility of the cell assembly was improved by the provision
of a door in the side of the cylindrical aluminum constant
temperature air bath container(8).
~
10
C02
---, ----.3 1
1
Cold Trap
Cell
1 1 1
1 1 1 1 1 11 L __ _
1,2 5-way valves 7 3,4 6-way valves 8 5 3 way valve 9 6 Band operated pump 10
Il
8 1 1
[[ 1
Liquid hydrocarbon flask Temperature controlled air bath Regulating valve C02 siphon cylinder liquid charge valve
~igur ••• 1 Bzperi •• nta1 •• tup.
~
Vacuum Pump
Gas Chromato9raph
He
00 IN
o - 84
pre.sure .a.sure.ents vere made vith a Dynisco PT422A-3. OM
prassura transducer useful in the ranqa 0 to 20 700 kPa.
Temperature readinqa were taken using a Hewlett-Packard 2801
A quartz thermometer with • 2850 C probe. The reported accuracy
of measurements ia 0.05 K. Calibrations was accomplished usinq an
ice bath of distilled water.
The composition of each phase was determined usinq a
Hewlett-Packard 5730 A Gas Chromatograph with a thermal
conductivity detector, as described by Sejnoha. The column type,
operatinq temperatures and qas flow rate were similar to
Sejnoha's.
The experimental procedure employed differs from that
followed by Sejnoha in several respects. Before charginq the cell
the l iquid hydrocarbon was degassed externally, as descr ibed by
Orbey. In charginq the cell with hydrocarbon, throuqh the the
liquid charging valve, a vacuum line was applied to the gas
charginq valve (at the top of the cell) to hasten the process.
The C02 cylinder wi th a siphon was used in cases where the
pressure in the cell was higher th an the saturation pressure of
C02 at ambient room temperature, around 6 300 kPa. In this
instance the hand pump , described by Orbey, was used to wi thdraw
liquid C02 from the cylinder and to compress it into the cell. In
discharqing the cell, the gas feed line was disconnected at the
regulating valve (9) and a line to the fume hood attached. The
cell was decompressed by slowly opening the requlatinq valve (9)
until the internaI cell pressure approached one atmosphere gauqe.
This valve was then closed, the vacuum pump shut off and the
liquid charqe valve (11) opened, with the residual pressure
discharqing the liquid which collected in the cold trap. Shortly
(
(
- 85
after the vacuum pump was turned on to complete the purginq of
the cell and lines. This proved to be a safe and efficient method
of decompressinq the celle
4.2 Tbe Gas s .. p1inq Valve
As described earlier, Sejnoha encountered difficulties in the
use of her design of the gas phase sampling valve, finally
recommending a new design.
The first attempted solution involved a modification of the
design of Sejnoha for the gas sampling valve. This valve is shown
in Figure 4.2. A different valve cap (C) was installed. This cap
has a jacket which is designed to physically prevent any
condensation on the sampling rod (R) frum entering the inclined
sampling hole (H). It is desiqned ta sr,uqly fit around the rod so
as to brush off any condensate drops during the insertion and
withdrawal of the sampling rod. Ad1itionally, two Rulon plugs
were placed in vertical grooves Indchined in the side of the invar
pluq (1). These were designed to prevent leakage from one side of
the sampling loop to the other, via the shell between the Invar
pluq and the stainless steel casing. This was a result of the
realization that the difference in thermal expansion between
Invar and stainless steel would lead to an increased gap as the
temperature was raised from ambient room (295 K) to operatinq
(313 K). Al though not evident in Figure 4.2 (b), the Rulon plugs
were made slightly larger than their housings, so that they fit
tight against the steel casing. The valve was assembled at
ambient room tempe rature and the difference in thermal expansion
between Rulon and stainless steel ensured that the Rulon pluq
was tiqhtly jammed between the Invar and stainless steel.
o
1 - Invar sealant plug C - cell cap R - Valve rOd P - Rulon plue) s - stainl... .te.l cell body (b)
Fiaure 4.2 ModiRcation or the a" ph .. e lamplina valve
p
:t
s
o
A - Central needl. valv. I,e • Sida n-.dl. valv •• D • e.u top ! • Sa~linq cba.ber P - C.1I cap
rivure t.. Alt.zaat. va. pba ..... pliaV valY. d •• iva.
CID 0\
c
c
- 87 -
However the performance of thi. valve was disappointing.
Leaka peraiat.d, both from the oeIl to the sampling loop and from
one side ot the sampling loop ta the other (vith the rod inserted
into the ce11). Full insertion of the samplinq rad into the cell
yielded compositions more representative of a liquid than agas
phase.
A decision vas made to completely redesign the valve. The
intention this time vas ta keep the design simple and eliminate
possible sources of leaks and condensation. The final design
arrived at sacrificed nominal sample size and phase equilibrium
disturbance (through vithdrawal of material) durinq sampling for
simplicity and ease of trouble shooting. The design is presented
in Figure 4. 3 • The previous gas charge valve and vacuum access
valve vere incorporated into the new gas sampling valve, together
with a new needle valve constructed in the center of the cell
top. AlI the valves were operated by threaded knob mechanisms, as
described by Orbey (1983), which allowed vertical but not lateral
movement. Components of the valve were constructed from stainless
steel 316 except the tips of the needle valves. The two side
valves had Teflon tips while that of central valve was made out
of Delrin 150.
In order to sample a phase, the central valve was closed; the
two side valves opened, and the sampling chamber evacuated by
connection to the vacuum line. The two side valves were then
olosed and the central one opened, allowing sampling. Finally,
the central valve was closed and the two side ones opened
fOllowing which the sampling loop was swept with helium carrier
gas ta the chromatograph for analysis.
o
o
- S'8 -
Pressure transducer Gas sampling valve Gas injection valve Equi.li&rium cell M&gnetic stirrer assembly Liqui~ sampling valve
- Temperature probe Saftey rupture disk
~iqa~. 4.4 Cel1 ••• .ably.
c
(
- 89
Unfortunately this desiqn was also plagued by problems.
Initially, there were difficulties in the construction of the the
valve system itself. The welding of the tubes connectinq the
valves proved to be very difficult and required several attempts
to eliminate leaks. Once installed, analysis of the gas phase
yielded unexpected and puzzlinq results. Instead of the presence
of two distinct peaks there were two fuzzy peaks followed by a
steady oscillatinq siqnal which qenerated countless peaks.
Subsequent tests of the valve, at hiqh pressure in the workshop
using helium identified a leak throuqh the central needle valve.
Attempts to solve this problem met with limited success: due to
low tolerances in the machininq of the screw mechanism
controlling the central needle valve, it was difficult to
determine how much downward pressure was beinq applied to the
valve tip (from the torque on the knob); low compression allowed
leaks while high compression led to severe deformation of the
valve tip, and in many cases there was separation of the tip from
the stem. In tho instances when the valve worked, the perforIl\ance
was inconsistent and short lived.
The third and final design of the qas sampling valve is
pre,;;ented in Figure 4.1 and Figure 4.4. In this case the
intention was to avoid havinq any locally constructed high
pressure seals or joints, based on previous experience. The valve
system design centered on a commercially availablp. sampling
valve. The valve chosen for this purpose is the Valco C6PX
six-way sampling valve, rated to 21 000 kPa. It is attached to
the top lid of the VLE cell via standard 0.3175 cm Swaqelock
connectors as shown in Figure 4.4. The samplinq operation is
illustrated in Figure 4.5. Initially the valve is configured as
in Figure 4.5 (a) and the five way switchinq valves connected to
o - 90 -
To Chromatoqraph
CL
A SI 6-way swi tchin9 valve B,C - 5-way switch1n9 valves (a) GV :a Gas phase sampll.ng valve LV - Liquid phase samplinq valve CL :a VLE cell
(b)
He
ligure ".5 Ga. ph ...... pliDg' operation.
(
c
- 91 -
the vacuua lines for evacuation. Then the valve handl. ia turned
to allov sample withdraval as in Figure 4.5 (b). Finally the
valve is aqain switched to th. outer loop and the sample loop
(Valco, SL-250, 2.51-4 Itr volUJIe) svept vith carrier qas for
analysis. This new setup is successful and has not exhibited Any
problems vith leaks or condensation.
4.3 .zperi •• Dtal ••• ult. an4 Diaau •• ioD
Experimental data for the system C02-cyclohexane were
measured at 313.15 K. and in a pressure ranqe of 1300 to 5200
kPa. A full description of the calibration of the pressure
transducer and qas chromatoqraph is available in Appendix 7. This
same system has been studied by sejnoha (1986).
The resul ts are displayed on Table 4. 1. There are 15 vapor
phase and 11 liquid phase data points. The reason for this is the
inconsistent behavior of the liquid samplinq valve vhich failed
severa 1 times durinq experimentation. When rullninq the
experiment, the cell vas charqed to a low pressure and, once
equilibrium was achieved in 16-24 hours, the first datum point
recorded. After this the pressure vas progressively increased by
further charqinq with C02 and nev data measured. For each data
point the liquid sampling valve vas operated several times, and
in some instances the sealing material had to be further
compressed as the hiqher cell pressure led to leaks. This aIl
resulted in deqradation of the sealinq pluq with the Rulon
sealinq material eventually plugqinq the liquid sampling chamber.
By this point the experiment had ta be terminated and the cell
decompressed in order ta change the valve sealant pluq. This
procedure toqether vith the subsequent liquid deqassinq, cell
charqinq and equilibration vas time consuminq, requiring several
92 -
~ day.. Therefore in soae cases the experiment was continued after
failura of the1 liquid valve, yieldinq only vapor phase
co.positions. This problea vith the liquid valve eventually
prevented measurement of data at high pressure, in the reqion
abova the saturation pressure of C02 at ambient room temperature.
The results of Table 4.1 represent several complete experiments.
o
'abla!Ll Zzpari.eDtal VLB ra.ult. for C02-cyclobe.aDe.
'1'= 313.15 Kt
P CkPa) Ya02 Zco2 .21kPa .0.005 .0.009
1311 Ca) (b) (a) (b)
1311 0.'74 0.17' 0.087 0.10'
1"" 0.'74 0.171 0.011 0.111
21 .. 5 0.'82 0.18' 0.13' 0.1'"
2272 0.'82 0.18' 0.1"3 0.172
2 .... 2 0.'83 0.187 0.1'" 0.175
2801 0.'85 0.188 0.172 0.20'
2823 0.'8' 0.188 0.185 0.221
3113 0.'8' 0.'8' 0.208 0.24'
3573 0.187 0."0 0.235 0.277
375' 0.'88 0.110 0.280 0.327
3813 0.'87 0.110 - -.. 2 .... 0.'88 0.1'0 - -.. 555 0.'81 0.110 - -50 .. 3 0.188 0.110 - -5120 0.188 0.111 0.3t7 0.3"
a S rin e () y q calibration. b ( ) Mixture calibration.
The ~omposition values in (a) columns of Table 4.1 were
obtained usinq a calibration constant determined from a syringe
calibration of the chromatoqraph, as described in Appendix 7.
C
-... a.. .:Il. -C'1 1
L&I lS1
• -)(
u L. ~ fit fit u L.
a..
c
- 93 -
7.BB 1 1 • 1 1
V This '-Iork
• 6.BB -. Sejnoha (1986) .
• • S.BB - V ,. •
~ V • ~ 4.BB ~ •• t-V
V •
3.BB .. V V Wy •
2.BB ~ ~. • t-
~y i 1.BB - .
t-
B.BB 1 1 1 1 1
B.BB B.2B B.4B B.6B B.BB 1.BB Hole Fractton CO 2
.iqure 4.S C02-crclohezaDe VLB re.ult. usiDg syriDge .etho4 calibration constant.
o 94 -
Bach of the data points here is ealeulated as an average value
fro. three replieate runs. the reprodueibility differed slightly
depending on data point, but in general is much better for the
vapor than for the liquid phase. Table 4.2 shows the standard
deviations of the compositions (by area) for replieate runs. The
average standard deviation is fifteen times greater for the
liquid than for the vapor phase. This difference is attributed ta
the diffieulties encountered with liquid phase samplinq valve.
Al thouqh Orbey and Sej noha used the same valve, they did not
provide details of any replicate runs and thus it is not possible
ta determine if this phenomenon has been observed before.
Table ~ Detail. or replieate run ••
Liqui4 Pba.e co.p. vapor Phase Co.p.
(pereeDt area, (pereeDt area,
Kaz. st. Dey. 0.88 0.03
KiD. st. Dey. 0.02 0.00
AYeraqe st. Dey. 0.15 0.01
Based on the larqest difference in the replicate samples, it
is estimated that the vapor phase composi tian i8 accurate ta
wlthin 0.0005. Added to the error introduced by calibration
(0.004) the total maximum error in molar composition for the
vapcr phase is • 0.0045. For the 1 iquid phase the correspondinq
total is .0.0090. According ta Orbey, error limits for
composition measurements in the literature are usually between •
0.0010 and .0.0100 by mole, a range encompassinq values in this
work.
(
-• a.. ~ -("rI 1
I.&J tsJ
• -)(
Il &. ~
» ., ., Il &.
a..
c
- 9) -
7.B8 • • 1 1 . 1
"" Thts lIIork y
6.B8 .. y SeJnoha (1986) -• y
S.BB l- V y. y
.. y
4.BB ~ ~ .. t-.. 3.BB .. V V _
~ • t ,-2. BB 1- ..
Q- ~ i 1.BB ~ -
B.BBI"'------··------�------·~--~'--~--~'------'------...... '------~l--B.BB B.2B B.4B B.6B B.8B 1.BB
Mole rraction CO 2
rigur. 4.1 C02-oyclobexan. VLB r.sults using .ixtur ••• tho4 calibration Gonstant.
o
o
96 -
Figura 4.6 compara. tha rasults obtained using syringe
calibration with those of Sejnoha. There is an apparent
syst.matic difference in the liquid phase compositions, with
lower calculated values in this work. Sejnoha was able to obtain
results closer to literature values for the system C02-benzene by
using calibration constants obtained using the mixture method. In
the course of this study a syringe calibration of a C02-benzene
system was carried out using the same column and operating
conditions used by Sejnoha. The calibration constant obtained was
0.6911 compared to Sejnoha's 0.6692. This small (3 percent)
difference shows that the columns performance and retention
selectivity has hardly changed. Based on this compositions were
recalculated using Sejnoha's C02-cyclohexane mixture method
calibration constant, equal to 0.5284. The results are presented
in (b) columns of Table 4.1 and Figure 4.7. As is evident the
systematic displacement has been eliminated and the results agree
weIl except for two data points, corresponding to the lowest and
highest pressure.
Figure 4.8 presents another way
equilibrium data (from Table 4.3), in the
of interpreting the
form of a In(K) versus
ln(p) plot. The results again corroborate those of Sejnoha, and
identical conclusions can be drawn from the plot.
c
J
c
3i PA lU C j .... tr
100 ~ 1-l-• • •
l-
10 :-~ ~ 1-~
~
•
· K 1 ~
~ • • ~
le
•
~ ~
0.1 ~ le
• •
~
0.01 0.01
97 -
• 1 1 •• "' • • 1 IT--.--rT
 Thfs lIIork
• Sejnaha (1986)
1 vapor pressure of cyc1ohexane.
---
. . . . . .. '-' • • • • . 0.1 1
JI
-T • • -. 'TTo;;
: I
1
1
· I
~ ': · 1
~~ · · · ~ ·
1 .. • : · · · · · · ~ · · · · •
• · ~.
~ · • ••• . 1. ••••
10 Pressure x 10-3 (kPa)
,igure 4.8 lD(a) versus ln(P) plot for C02-cyclohezane.
o - 98
CD..,.. 5
CO.CLUSIO.S ABD aBCOKNBRDATIORS
5.1 Conolu.ion.
5.1.1 VLB prediotion
i) A characterization scheme for the derivation
distribution curves of petroleum fractions
of molar
and gas
condensates from TBP distillations or simulated TBP analysis
using HPLC was proposed - one allowing the use of several
molecular weight to boilinq point relations.
ii) The Extended Spline Fit Technique with boiling point as the
distributing variable was found to be a more versatile and
accurate method of representing the molar distribution curve.
iii) A continuous PRSV EOS was developed by usinq GSCNP critical
property and acentrie factor relations and correlating the NI
parameter to the boiling point.
iv) The continuous PRSV EOS was applied to systems using a
modified version of Hendriks (1987a) continuous thermodynam
ies VLE equation formulation.
v) The method was successfully applied to multieomponent model
o and real systems. For the real systems it was found that
binary interaction parameters had varying effects, al thouqh
(
';
c
- 99
in general these overshadowed any effect of the IC.(T.) on the
predicted VLE. Resul ts obtained were comparable to others
obtained using continuous thermodynamics and pseudocomponent
.ethods.
5.1.2 VLI •••• ur ... Dt
i) The accessibility of the VLE cell and its sampling valves
was improved by the provision of a side door in the air bath
container together with a new water circuit line and a
special wrench was installed to allow easier compression of
the liquid sampling valve sealant.
ii) Three new gas sampling valve designs were tried. Two were
unsuccessful due to leaks. The third worked satisfactorily,
giving reproducible results.
iii) A C02 cylinder with a siphon was attached ta the apparatus,
to allow measurement at pressures above saturation at ambient
room temperature. However for the liquid phase this did not
solve the problem as there was was persistent failure of the
liquid sampling valve at high pressure.
iv) Data for the C02-cyclohexane system at 313.15 K and in the
pressure range 1300 to 5200 kPa was measured. Calibration of
the chromatograph was by syringe inj ection, however
compositions derived using Sejnoha's (1986) mixture method
calibration yielded results more coincident with herse
5.2 •• co ... ad.tioa.
5.2.1 VLI Pr.dictioD
It would be beneficial if the prediction method were tested
on further real .ulticomponent systems containing ill-defined
o
o
- 100 -
flu!d. in arder ta gain experience on the effect of binary
interaction parametera on the predicted VLE- eapecially aa
compare~ ta the effect of the IC.(T.) function. It may be the
general caae that the role of the function is superseded by
interaction parametera, as has been observed here.
In order for the equipment to become readily and conveniently
usable a new design of the liquid sampling valve is required.
This author has examined various al ternate solutions, most of
which have been looked at by either Orbey or Sejnoha, and
therefore, taking into consideration the history of the
equipment, it is recommended that the new design do away with
moving parts and sealing materials requiring compression. An
attractive design on which to base the new valve 18 described by
Melpolder (1986). In this case the a new bottom plate would be
constructed and a 0.025 cm ID x 0.16 cm OD sampling tube inserted
(welded or soldered). This tube would be connected to a
commercial switching valve as is the case for the vapor.
It is a180 recommended that in the future the calibration of
the gas chromatograph be accompl ished by the mixture method of
Sejnoha.
c
c
- 101 -
UO"'088
~erican Petroleum Institute (API), (1982), "Technical Data Book - Petroleua Refining" 3rd Ed., American petroleum Institute, Washington, De.
Ammar, M. N. and Renon, H., (1987), AIChE J., 11, 926.
BergJIan, D.F., Tek, M. R. and Katz, D. L., (1975), "Retrograde Condensation in Natural Gas Pipelines", American Gas Association, Arlington, VA.
Blum, L. and Stell, G., (1979), J. Chem. Phys., 11, 42.
Bott, T. R., (1980), Chem. Ind., 228.
Briano, J. G. and Glandt, E. D., (1983), Fluid Phase Equil., 1i, 91.
Carnahan, B., Luther, H. A. and Wilkes, J. O., (1969), "Applied Numerical Methods", John Wiley and Sons, Inc.,NY
Cotterman, R. L. and Prausnitz, J. M., (1985), Ind. Enq. Chem. Process Des. Dev., ~, 434.
Cotterman, R. L., Bender, R. and Prausnitz, J. M., (1985), Ind. Eng. Chem. Process Des. Dev., ~, 194.
CRC, (1986), "CRC Handbook of Chemistry and Physics", CRC Press, Inc., Boca Raton, Florida.
Edmister, W. C., (1955), Ind. Enq. Chem., 47, 1685.
Edmister, W. C., (1958) Petroleum Refiner, 12, 173.
Flory, P.J., (1936), J. Am. Chem. Soc., 58, 1877.
Gualtieri, J. A., Kincaid, J. M. and Morrison, G., (1982), J. Chem. Phys., 11, 521.
Gutsche, B., (1986), Fluid Phase Equil., 30, 65.
Hansen, J. P. and MacDonald, 1. R., (1976), "Theory of Simple Liquids", Academie Press, NY.
Hendriks, E. M., (1987a), Fluid Phase Equil., 33, 207.
Hendriks, E. M., (1987b), Personal Communication
HOffmann, A. E., Crump, J. S. and Hocott, C. R., (1953) Pet. Trans., AIME, liI, 1.
Jacoby R. H., Koeller, R. C. and Berry, V. J., Jr., (1959) J. Pet. Techn., AIME, 58.
Johnson, K. A., Jonah, D. A., Kincaid, J. M. and Morrison, G., (1985), J. Chem. Phys., la, 5178.
Joulia, X., Maggiochi, P., Koehret, B., paradowski, H. and Bartuel, J. J., (1986), Fluid Phase Equil., 11, 15.
o - 102-
Katz, D. L. and Firoozabadi, A., (1978), J. Pet. Tech.,: Trans., AIME, 12.L. 1649.
Kehlen, H. and Ratzseh, M. T., (1980), Proe. 6th Int. Conf. Thermodyn., Merseburg, 41.
Kehlen, H. and Ratzsch, M. T., (1984) Z. Phys. Chemie, Liepzig, ll.2, 1049.
Kehlen, H. and Ratzsch, M. T., (1987), Chem. Enq. Sei., JZ, 221.
Kehlen, H., Ratzsch, M. T. and Berqmann, J., (1985) AIChE J., 11, 1136.
Kesler, M. G. and Lee, B. 1., (1976), Hydrocarbon Proeessinq, 22, [3], 153.
Kincaid, J. M., MacDonald, R. A. and Morrison, G., (1987), J. Chem. Phys., ~, 5425.
Klaus, R. L. and Van Ness, H. C., (1967), AIChE J.,13, 1132.
Mehra, R. K., Heidemann, R. A. and Aziz, K., (1983), Cano J. Chem. Eng., ~, 590.
Mehrotra, A. K., Sarkar, M. and Svrcek, W. Y., (1985), AOSTRA J. Res., 1, 215.
Melpolder, F. W., (1986), Fluid Phase Equil., 26, 279.
Michelsen, M. L., (1982), Fluid Phase Equil., ~, 21.
Nghiem, L. X., Li, Y. and Heidemann, R. A., (1985), Fluid Phase Equil., .a.l, 39.
Orbey, H., (1983) Ph.D Thesis, McGill University, Montreal, Quebec
Pedersen K. S., Thomassen, P. and Fredenslund, A., (1983), Fluid Phase Equil., ~, 209.
Pedersen K. S., Thomassen, P. and Fredenslund, A., (1984a), Ind. En~. Chem. Proeess Des. Dev., li, 163.
Pedersen K. S., Thomassen, P. and Fredenslund, A., (1984b), Ind. Eng. Chem. Process Des. Dev., 23, 566.
Prausnitz, J. M., (1983), Fluid Phase Equil., 14, 1.
Radosz, M.,Cotterman, R. L., and Prausnitz, J. M., (1987), Ind. Eng. Chem. Res., 4, 1§, 731.
Rachford, H. H., Jr, and Rice, J.O., (1952), J. Petrol. Technol., 4(10): sect. 1, 19. and sect. 2, 3.
Ralston, A., (1965) .. A First Course in Numerical Analysis", MeGraw-Hill Book Co., NY.
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Riazi, M. R. and Daubert, T. E., (1980), Hydrocarbon Processing, 2L [3], 115.
c
c
-~-- -- --------
- 103 -
Salacuse, J. J. and Stell, G., (1982), J. Chem. Phys., 11, 3714.
Schlijper, A. G., (1987), Fluid Phase Equil., ~, 149.
Schneider, G. M., (1983), Fluid Phase Equil., ~, 141.
Schultz, G. V., (1935), Z. Phys. Chem., B1Q, 379.
Sejnoha, Mo, (1986), M.Eng Thesis, McGill University, Montreal, Quebec
Shibata, S. K., Sandler, S. 1. and Behrens, R. A., (1987), Chem. Eng. Sei., Al, 19770
Shiqaki, Y. and Yoshida, K., (1986) World Congress III Chem. Eng., Tokyo. 7a-117.
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Stryjek, R. and Vera, J. H., (1986), Cano J. Chem. Eng., 64, 323.
Tsonopoulos, Co, (1987), AIChE J., 33, 2080
Twu, C. Ho, (1984), Fluid Phase Equil., 1§, 137.
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Whitson, C. H., (1983), Soc. Petrol. Eng. J., 23, 683.
Willman B. and Teja, Ao S., (1986), AIChE J., 32,2067.
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Willman B. and Teja, A. S., (1987b), Ind. Eng. Chem. Res., 26, 953.
o
o
-104-
appen4lz Al Generali •• 4 Sinqle Carbon Buaber Group.
In petroleum mixtures, it is readily possible to identify
both isomer and normal paraffinic hydrocarbons constituents for
methane through hexane. But for compounds with higher boiling
points than hexane a large number of paraffinic isomers occur. In
addition aromatic and naphthenic compounds, such as benzene,
toluene and cyclohexane are usually present. The task of
identifying and separating aIl these components is time consuminq
and difficult.
In the past it has been advantageous to treat the mixture as
a set of pseudocomponents, each derived from a distillation eut.
The seN groups are pseudocomponents that "represent" the behavior
of aIl hydrocarbon compounds with the same number of carbon
atoms, equal to the SeN, in an oil mixture. Generalized seN cuts
were obtained as narrow boiling point range fractions from
distillations of a selection of natural qas condensates. The
boiling points, molecular weights, and liquid density of each
fraction were measured. Bergman et al. (1975) used natural gas
condensate data to prepare correlations for the above properties
up to seN group C15 • Katz and Firoozabadi (1978) extendecl the
tabulation of these properties up to C 45 • They extrapolated values
based on paraffin hydrocarbon and literature data to c~. Whitson
(1983) revised the properties, eliminating an inconsistency in
the molecular weiqht and providinq critical properties of seN
groups using a modification of the Riazi and Daubert (1980)
correlations for the critical properties of petroleum fluids. He
also calculated acentric factorR using the Edmister (1958)
relation. These values are presented in Table A1.2. six values of
(
c
-105-
the acentric factor were switched around in order to aliainata an
inconai.tency in Whitson's value. which aro.e tro. u.inq two
ditterent correlations for the critical pre •• ure.
In order to use these critical propel~ie. with the PRSV EOS,
they had to be expressed in analytical for.. Thus the nor.al
boiling point was correlated to the five other variable.,
separately, by a method of least squares fit to cubic
polynomials, chosen for their flexibility and relative
si.plicity. The fit provided is satisfactory and these relations
are presented in Chapter 2, as equations 2.10a to 2.10e. Details
of the fit are provided in Table A1.1, where the average absolute
deviation (AAD) for NP points is calculated ~s:
A1.1
Table ~ Detail. of G8CRP correlation ••
SG MW Tc(K) Pc (kPa) Acen. Fact.
Averaqe Absolute 0.27 0.32 0.06 1.59 1.13 Error(')
Min. Neq. -1.06 -0.92 -0.39 -3.50 -2.48 Deviation('>
Max. Pos. 0.74 1.60 0.24 4.52 3.13 Deviation('>
o
o
-106-
Ilble ~ General1.ea alD91e a.~bo. auaber 9~UP
propertle. CG8a..,.
SCN Tb (1() SG MW Tc(K) Pc (KPa)
6 337 0.690 84 512 3340 7 366 0.727 96 548 3110 8 390 0.749 107 575 2880 9 416 0.768 121 603 2630
10 439 0.782 134 626 2420 11 461 0.793 147 648 2230 12 482 0.804 161 668 2080 13 501 0.815 175 687 1960 14 520 0.826 190 706 1860 15 539 0.836 206 724 1760 16 557 0.843 222 740 1660 17 573 0.851 237 755 1590 18 586 0.856 251 767 1530 19 598 0.861 263 778 1480 20 612 0.866 275 790 1420 21 624 0.871 291 801 1380 22 637 0.876 300 812 1330 23 648 0.881 312 822 1300 24 659 0.885 324 832 1260 25 671 0.888 337 842 1220 26 681 0.892 349 850 1190 27 691 0.896 360 859 1160 28 701 0.899 372 867 1130 29 709 0.902 382 874 1110 30 719 0.905 394 882 1090 31 728 0.909 404 890 984 32 737 0.912 415 898 952 33 745 0.915 426 905 926 34 753 0.917 437 911 896 35 760 0.920 445 917 877 36 768 0.922 456 924 850 37 774 0.925 464 929 836 38 782 0.927 475 935 811 39 788 0.929 484 940 795 40 796 0.931 495 947 771 41 801 0.933 502 951 760 42 807 0.934 512 955 741 43 813 0.936 521 960 727 44 821 0.938 531 967 706 45 826 0.940 539 971 696
*values of acentric factor shifted around.
Acen. ract.
0.250 0.280 0.312 0.348 0.385 0.419 0.454 0.484 0.516 0.550 0.582 0.613 0.638 0.662 0.690 0.717 0.743 0.768 0.793 0.819 0.844 0.868 0.894 0.897 0.909* 0.915* 0.921* 0.932* 0.941· 0.942· 0.954 0.964 0.975 0.985 0.997 1.006 1.016 1.026 1.038 1.048
c
c
-107-
Ipp.a4iz A2 puqacity CoefficieDt .zpre •• ioa. for '-.dcoatiauou •
.... 10.
In this derivation, the fuqacity coefficient uf the
saaicontinuous PRSV EOS is obtained from that of the di.crete
syste. equation by simple analoqy. The continuous variable 1 i.
used to replace the discrete index i. This approach, if applied
judiciously, provides a quick solution as well as avoiding
functional differentiation. There are two expressions: one for
the fuqacity coefficient, .,, of a discrete component in the
semicontinuous mixture and the other for the fuqacity coefficient
correspondinq to a value of variable 1, -1(1), in ensemble i in a
semicontinuous mixture.
The fugacity coefficient of component , in a multicomponent
mixture (discrete) described by the PRSV EOS is:
(n-L,V)
A2.1
The conversion of this equation to that applicable in a
seaicontinuous mixture is straiqht forward except for the term
representinq the derivative of the quadratic mixinq rule with
respect to composition. To help illustrate the procedure used in
obtaininq the results in section A2.1 and A2.2 the followinq
notation is adopted:
A2.2
Whera the subscript di-d refers to discrete-discrete interactions
experienced by discrete component i in a mul ticomponent .ixture of
o -108-
discrete c~~90nents, d. Now define two other subscripts:
di-sc
cj-sc
discrete-semicontinuou. interaction
continuous-semicontinuous interaction
A2.1 Di.oret. co.ponent Bzpre •• ioD
From equation A2.1, the fuqacity coefficient for a discrete
component in a semicontinuous system, in phase n is:
where D C
t: n '" Xn -n '" Xn -n !idl-sc=L k'aA:,+ L ,'a"
A:-I '-0+1
The first term accounts for discrete-discrete component
interactions for which
(ïn_ao.s'ao.s'(l-k) ki i A: iA: (k- 1".,D)
(i= 1,II,D)
A2.3
A2.4
A2.5
The second term accounts for discrete camponent (i) -continuous
onsemble(i) interactions and has the followinq cross term, «7
beinq the family averaqed parameter for family i, as defined in
equation 2. 24 •
(ïn_ao.s,(in'(l-k) Ji i J IJ (j-D+ 1 ,,,,C)
A2.6
A2.2 contiDuous Index Expression
For a cantinuous ensemble i, the fuqacity coefficient of
phase n i!t a value of the index 1 is given by:
(
c
-109-
for which the derivative term ia:
D C t" "Xn -n ~ Xn -n Sc/-sc· L. 1; °al;/+ L. ,oa'i
t- l ,. D+ 1
A2.7
A2.8
The first term on the riqht in this case represents continuous
familY(j)-discrete component(k) interaction for which:
(k- l, oo,D)
A2.9
The second term represents continuous ensemble(j)-continuous
ensemble (t) interaction for which:
(l,=D+l,oo,C)
A2.10
o -110-
App.1l4ix A3 Probl .. in 1t.(T.) Punotion
As described in section 2.3.1 there i. a discontinuity in the
N.(T.) function at a value of the reduced teaperature equal to 0.7.
Thia ia illustrated by the following exa.ple, which invastigates
the magnitude of certain parameters involved in the calculation
of ". in this reqion of the reduced tempe rature •
Boilinq points have been chosen so aa to yield, using GSCNP
critical properties and acentric factors, reduced temperatures
(actually reduced boilinq points) around 0.7. Usinq proqram Kl.IT
optimal ". values were calculated by matchinq the fuqacities as
described in section 2.3.1. The boilinq points, critical
constants and acentric factors are presented toqether with the
optimal EOS parameter a (T) and Je. values are presented in Table
A3.1. In this table it should be noted that the critical
temperature and pressure values have been truncated after the
first decimal point and thus appear to be the same for several
different boilinq points. This has no bearinq on the results for
". since the se were calculated with Fortran double precision
accuracy (15 diqits). The ". values are plotted in Fiqure Al.I, as
a function of the reduced temperature.
It is evident that the value of ". asymptotically approaches
a negatively infinite value as a reduced temperature of 0.7 is
approached from below. Similarly the value becomes infinitely
positive for an approach from above. As expected similar results
are obtained usinq different critical property correlations such
as the Kesler-Lee. An examination of the "explosive" ". raqion
shows tbat it is limited to a very narrow "band" around T.· 0.7 •
c
-111-
By .electively eliminating any boiling points for which
(0.698<T,<0.702) in the input boiling point data curve, on. i. able
ta avoid any indeterminacy in the -.eT) correlation.
libl. Ahl Valu.. of -. an4 oth.r par ... t.r. iD th. T, = 0.7 r.gion.
Tb (R)
425.000 430.000 432.000 434.000 434.800 434.900 434.945 434.950 434.960 434.965 434.970 434.980 435.000 440.000 447.300
Tc Pc (X) (kpa)
611.0 2526.5 616.2 2485.5 618.3 2469.4 620.4 2453.3 621.2 2446.9 621.3 2446.1 621.4 2445.8 621.4 2445.7 621.4 2445.7 621.4 2445.6 621.4 2445.6 621.4 2445.5 621.4 2445.3 626.6 2405.9 634.1 2349.8
70
60
50
40
30
20
10
0
-10
-20
-30
-40
-50
-60
-70 0695 0697
ru
0.368 0.375 0.378 0.381 0.383 0.383 0.383 0.383 0.383 0.383 0.383 0.383 0.383 0.390 0.401
0.699
D
D
D D
o
o
0.701
TI a(T)
IkPalLtr\ (amo~)
0.695581 6176.2 0.697825 6392.0 0.698690 6479.9 0.699859 6568.7 0.699936 6604.5 0.699948 6609.0 0.699949 6611.0 0.699952 6611.2 0.699968 6611.7 0.699980 6611.9 0.699984 6612.1 0.700000 6612.5 0.700032 6613.4 0.702202 6840.5 0.705409 7182.3
0.703 0.7011
ligure AL...l Plot of optimal III versus T,
",
-0.169 -0.207 -0.255 -0.495 -2.313 -6.245 -33.226 -65.180 69.650 34.127 22.568 13.419 7.368
-0.080 -0.126
-112-
~ app.a41a &4 ... iaontlauou8 Raohfor4-aloe Objeo~l~. ~otlo.
For discrete components (i-l .... D) in a flash situation the
follovinq equations hold:
l+u· 1
A4.1
A4.2
xr K .-
1 xf
A4.3
If one eliminates 1 and xr from equations A4.1-3 one obtains:
A4.4
Elimination of 1 and x~ from equations A4.1-3 yields:
A4.5
For the continuously distributed ensembles in the mixture a
similar equation in terms of generalized X-factors is possible.
Examininq the situation of a component represented by the index 1
in ensemble j,(j-D+ 1, .. ,C), the followinq equations hold:
X: . F: (1) = 1· X :(1). F:( 1) + u' X~ . Fr (1)
A4.6
XV. FV(I) K (/) - j j
J X~. FHI)
A4.7
Followin~ the same procedure as for the discrete case, ve
-----,
(
c
-113-
ganerate two new equations from the above thra.:
L L x: . Ff (/) XjoFj(l)-l+(Kj(I)_I)'U
A4.8
and
A4.9
By usinq the qeneralized X-factor, Kil defined in equation
2.33, inteqratinq equation A4. 8 over the ranqe of 1, usinq the
normalization condition of the distribution functions and of
equation A4.1, and substitutinq for KJ in equation A4.10, one
obtains:
A4.10
Similarly, for equation A4.9 the result is:
A4.11
since the mole fractions of the discrete components and the
overall mole fractions of the ensembles are normalized as in
equation 2.7, then the followinq objective functions are
possible. First, summinq of liquid phase mole fractions qiven as
equation A4.4 and A4.10 qives:
A4.12
and addition of equation A4.5 and A4.11 leada to:
o
o
-114 -
A4.13
Subtracting A4.12 from A4.13 produces the Rachford-Rice objective
function.
The advantages of this objective function in the flash
calculation are that it is a monotonie function in u
and the derivative, given below, is always negative.
6FOCu)=_ t X:'(KI;-I)2 6u t-.[l +(Kt- 1)·u]2
A4.14
A4.15
c
-115-
ap,en4iz A5 Acceleration of Succ ••• iv. 8Ub.titution •• tho4
In order to improve the computation effort required, the
successive substitution step was modified, based on a method
applied to discrete systems by Mehra et al., (1983). The
acceleration involves a correction of the mole fraction
generalized K-factor Kj.(r- I .... C) calculated from the fugacity
coefficients of the previous iteration. The change ia _ade ao as
to minimize the qradient of the total Gibbs enerqy of the system
with respect to composition as fully elucidated in the above
reference. In this analysis continuously distributed families are
treated as discrete components whose composition is equivalent to
the overall family mole fraction XJ.,-I , ... C) in that phase.
At equilibrium fuqacities are equal so that:
(FU gr ) gl-ln -- -0 Fugf
AS.1
where gj is the difference in the loqarithms of the fuqacities,
and the gradient of the Gibbs free enerqy with respect to vapor
composition. The fugacity is expanded as
InFug~ = InX~ +ln~~ +lnP en-V,L)
AS.2
substitutinq equation AS.2 into A5.1 yields:
g -ln(X~)-ln(~~) 1 XL ... V
1 .,. ~
AS.3
An examination of equation AS.3 in the context of the successive
substitution method shows that the first term on the right hand
side, the ratio of mole fractions is the value of the equilibrium
ratio in the previous iteration ~-l while the second term, the
o
o
-116-
ratio of fugacity coefficients is the newly generated equilibrium
ratio in iteration f', as defined in equation 2.27 for di.cret.
co.ponents. For the ensembles, there ia no faaily fugacity
coefficient, but equation 2.39 is analoqous to 2.27. Therefore
equation AS.3 can be expressed as:
(i- 1 .... C)
AS.4
Equation AS.4 expresses the the basic successive substitution
method, where the solution is found when equation AS.l ia
satisfied. This can be expressed in vector notation as:
LITIïK =-g AS.S
The successive substitution step of equation AS.S is modified by
the introduction of a step length A., so that:
LITi1K = -Ag AS.6
The step lenqth A. is obtained using Mehra et al.'s Alqorithm
II. This is chosen bec au se of its good performance with
relatively low computational cost. As shown by those authors, one
can expand the JfilK and g in a first order Taylor series expansion
with the number of vapor phase moles as the independent
variables. A minimum in the magnitude of the gradient, g, along
the direction of the search can be found by demandinq that
(-<1 ..... ) 6 g • g ôA - 0
AS.7
subsequ~nt simplifications in the above expression to eliainate
matrices results in the followinq recursive relationahip for the
c
J
- 117-
.tep length in iteratian r:
A5.8
Initiation is with A. - 1. Once). is determined tram the above
equation, the new (K')r are calculated usinq equation AS.6, i.e.
(i-l,,,,C)
A5.9
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app.a4lz a4 computer progr ...
This program derives a molar distribution curve ba.ad on a on
a cumulative weight percent distilled off TBP distillation. It
will also fit one of three distribution functions to the .olar
distribution curve data, providinq spline coefficients or
parameters of the functions. The discussion here is limited; for
full details see the documentation for proqram CHART.
For the derivation of a molar distribution curve, as
described in section 2.2.3 there is a choice of the followinq
boiling point to molecular weight correlations:
1) The Twu (1984) relation. 2) The Kesler and Lee (1976) relation. 3) Radosz et al. (1987) style relations
(allows for different constants). 4) Generalized SeN relations. The cumulative weight percent distilled versus molecular
weiqht curve is differentiated using either a simple two point
scheme or an ESFT derivative. There is a choice of retaining the
molecular weight as the distribution variable or recalculatinq
boiling points as described in the documentation.
The molar distribution can be fit to the followinq
distribution functions:
1) The ESFT function - there is a choice in the number of
equally spaced intervals desired.
2) The Beta density function - in this case the function
parameters are estimated usinq the method of moments.
3) The Gamma density function - with parameters estimated by
c
)
(
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a choice of either:
1) Method of moments.
11) option 3 of Willman and Teja'. (1987a) .ethods.
program K1PIT is the implementation of the continuou8 PRSV
EOS. The details of the method used to obtain the ".(T.) function,
including the flow chart, are provided in section 2.3.1. In
addition to GSCNP critical properties and acentric factors, the
progra. also allows for use of the Kesler and Lee (1976)
relations for critical properties, with acentric factors being
estimated using the Edmister (1958) expression. The progra.
provides a method of least squares fit for ".(T,) • has the ability
to produce a linear, quadratic, cubic and quartic fit of the
data.
This program performs flash, dew-P, dew-T, bubble-P and
bubble-T calculations for discrete, semicontinuoua and continuous
syatems. It can also trace the P-T envelope for a fluide Fugacity
coefficients are evaluated using the discrete and continuous PRSV
EOS.
The program accepts up to a maximum of 25 components, either
as discrete compounds or continuously distributed ensembles.
Equilibrium and mass balance equations are solved using
accelerated successive substitution. For dew and bubble point
problems the correct temperature/pressure is found using
Newton-Raphson iteration.
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Th. feed distribution can be de.eribe4 by .ither diacr.t.
ao1. fractions or continuous ga .. a, bata, or B8FT funetion.. Por
th. eontinuous1y distributed ensembl.. the di.tributinq variable
can be either the .olecular weight or the nora.l boiling point.
Nuaerieal Integration is performed using different type. of
quadrature depending on the the feed distribution funetion:
Laquerre-Gauss for gamma, Chebyshev-Gauss for beta and
Legendre-Gauss for the ESFT.
There is a choice of use of either the GSCNP, Kesler-Lee or
normal alkane correlations for the estimation of critical
properties and acentric factors. For normal alkanes the critical
tempe rature and pressure are estimated using Tsonopoulos (1987)
correlations usinq a function for the carbon number (se.
documentation for TVLBT) while the acentric factor is estimated
from eqn 3.2. In the Kesler-tee case acentric factors are
estimated from critical properties usinq the Edmister (1958)
relation. Four coefficients for a IC.(T.) function are required (can
be equal to zero).
Initial estimates of equilibrium X-factors are obtained from
the Mehra et al. (1983) correlation using critical propertie. and
acentric factors. For dew and bubble point calculations, it is
possible to estimate the temperature or pressure using Raoults
lav, as described by Van Ness and Abbott (1982). For thi.
situation Antoine vapor pressure equation constants are required
for aIl the components(use pseudocomponents for ensemble.).
For additional details consult documentation for proqram
nLB'I'.
;
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appe.4l. &7 Calibration of Bsperi.e.tal apparat ••
Temperature measurements were taken with a Hewlett-Packard
quartz thermometer model 2801A usinq a model 2850C probe. The
_axi_ua deviation from linearity (accordinq to the _anufacturer)
i. 0.05K (see orbey, 1983). Calibration was accomplished usinq an
ice bath of distilled water.
Pressure measurements were performed using a Dynisco
PT422A-3.0M pressure transducer. It is similar in all respects to
that used by Orbey (1983) except for its range of use, thi. being
o to 20700 kPa. The instrument was calibrated several ti.es using
a Chandler high-pressure dead weight tester. A calibration curve
obtained from the test is presented in Figure A7.1. This curve
corresponds to the values used in experiments reported her ••
Additional information is available in Table A7.1 which reports
detaila of the linear least squares fit of the calibration curve.
The relation between pressure (P, kPa) and voltaqe output (vo,
mV) is the followinq:
p= 357 + 706.7· Va
A7.1
When pressure is back-calcul~ted from equation A7.1 and compared
to the oriqinal data, the average nonlinearity ia 12.2 kpa with
the maximum deviation being 27 kPa. Thus the accuracy ia as goo4
as claimed by the manufacturer ( a repeatability of .21 kpa with
a zero shift of 1.9 kpa per deqree K for the diaphraga and 4.0
kpa per degree K for the strain gauge housing).
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• ~abl. AZLl Calibration 4ata for tb. pre •• ure tr ... 4uaer. Transducer Actual Pres- Predicted Deviation
output (mV) sure (kPa) Pre •• ure (kPa) (kPa)
0.45 689 675 15 1.42 1379 1360 18 2.41 2068 2060 8 3.38 2758 2746 12 4.37 3447 3445 2 5.35 4137 4138 1 6.32 4826 4823 3 7.31 5516 5523 7 8.29 6205 6216 11 9.28 6895 6915 21 10.25 7584 7601 17 11.22 8274 8287 13 13.18 9653 9672 19 15.13 11032 11050 18 17.06 12411 12414 3 19.00 13789 13785 5 20.93 15168 15149 19 22.87 16547 16520 27
17
16
15
14
13 ,.... (") 12 1 lai
Il q - 10 ><
" 9
II. ,)/. 8 "" Il 7 .. =' 6 1/1 1/1 Il 5 .. II.
4
3
2
0
0
Output (mV) C Data - Regressed
r,;1gur. ~ pr ••• ur. traDs4ucer oali1»ratioD OUrY ••
o
(
c
--- ~- ~- -------------
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Ga. Cb~o.ato9r.ph
The compositions of the phases were deterained usinq a
Hevlett-Packard 5730A gas chromatograph vith a .ode'. 3380A
integrator. The column and operating conditions were si.ilar to
Sejnoha's (1986). The chromatograph was calibrated usinq the
syringe method, as described in orbey(1983). Results obtained in
the determination of the carbon dioxide response factor are
available in Table A7.2 and Figure A7.2. Each of the data points
is an average value from three different samples. The values are
not tabulated here but the average absolute deviation from the
mean is 6 x 1.0E3 area units (AU) while the largest is 18 x 1.0E3
AU's. The following expresses the relationship between number of
moles (n) and Area in AU x 1.0E-3:
Arga=-162+7.4381·10 7 ·n
A7.2
Calculated areas are compared with those measured in Table A7.2,
and the difference is indicated. A~ shown, the larqest deviation
is 62 while the average nonlinearity is 28. This corresponds to
an maximum absolute nonlinearity error of 0.4 mole percent.
Table A7 2 Calibration data for C02 • • Volume Moles AU x AU x Deviation (cm3) x 1.0E5 1.0E-3 1.0E-3 AU x
(actual) (cale. ) 1.0E-3 0.10 0.409 197 142 55 0.20 0.818 454 446 8 0.30 1.226 731 750 20 0.40 1.635 1059 1054 5 0.50 2.044 1324 1358 35 0.60 2.453 1622 1662 40 0.70 2.861 1941 1966 25 0.80 3.270 2253 2270 17 0.90 3.679 2589 2574 14 1.00 4.088 2941 2878 62
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3.0
2.8
2.6
2.4
" 2.2
(1
~ 2.0 r:: :J 1.8 10 1 1.6 101 q
1.4 ... >C 1.2 ~
CI III 1.0 .. < 0.8
0.6
0.4
0.2
0.0
0.0 1.0 2.0 3.0 4.0
Moles (x 1.00:-S) c Data - Regressed
piqure A7.2 Ar •••• ao1 •• for C02.
A similar calibration was done for cyclohexane. The resulta
of this experiment are displayed on Table A7.3 and Figure A7.3.
For this curve the linear regression equation for area ia:
Area=50+1.1287·10 8 ·n
A7.3
The maximum nonlinearity error is estimated at 0.5 mole percent.
In order to qet a calibration =onstant (Ke)for the syatem,
the slopes of the two curves (area response factors) must be
divided by each other. Thus dividing the cyclohexane response
factor by the carbon dioxide response factor one obtains a value
of Ke equal ta 0.6588. It is estimated that by using thi. factor
the errer introduced in composition calculations is about 0.5
percent.
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~ab1. lILl Calibration data for cyo1obezane. Volume Moles AU x AU x Deviation
uL x 1.0E5 1.0E-3 1.0E-3 AU x (actual) (cale. ) 1.0E-3
0.50 0.460 555 569 14 1.00 0.920 1012 11,)88 76 1.50 1.380 1666 1607 60 2.00 1.839 2192 2126 66 2.50 2.299 2561 2645 84 3.00 2.759 3210 3164 46 3.50 3.219 3717 3683 34 4.00 3.679 4227 4202 25 4.50 4.139 4719 4721 2 5.00 4.598 5184 5240 56
5.5
5.0
4.5
"" 4.0 1/1 !:: c :J 3.5 1() 1
lai q 3.0 ->< 2.5 ..., CI el .. 2.0 <
1.5
1.0
0.5
0.0 1.0 2.0 3.0 4.0 5.0
Moles (x I.OE-5) [] Data - Regressed
ligure lZLl Area v. .01.. ~or oyclob.zan ••
'C