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Graduate Studies The Vault: Electronic Theses and Dissertations
2014-01-27
Error Estimation and Reliability in Process
Calculations Subject to Uncertainties on Physical
Properties and Thermodynamic Models
Hajipour, Samaneh
Hajipour, S. (2014). Error Estimation and Reliability in Process Calculations Subject to
Uncertainties on Physical Properties and Thermodynamic Models (Unpublished doctoral thesis).
University of Calgary, Calgary, AB. doi:10.11575/PRISM/25935
http://hdl.handle.net/11023/1287
doctoral thesis
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UNIVERSITY OF CALGARY
Error Estimation and Reliability in Process Calculations Subject to Uncertainties on Physical
Properties and Thermodynamic Models
by
Samaneh Hajipour
A THESIS
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
DEPARTMENT OF CHEMICAL AND PETROLEUM ENGINEERING
CALGARY, ALBERTA
JANUARY, 2014
© Samaneh Hajipour 2014
ii
Abstract
The issues related to error propagation from uncertainties in physical properties and
thermodynamic models involved in process modelling and simulation are examined.
Traditionally, the effect of these basic parameters are ignored in chemical and process
engineering and designers make the final decision on determining equipment parameters such as
sizing and residence time in an ad-hoc manner based on their prior experience with similar
problems. The objective of this dissertation is to develop a self-contained and consistent
mathematical procedure to quantify the effect of uncertainties related to thermodynamic models
on process design calculations for flow sheets of any complexity. The methodology is based on
the Monte Carlo technique along with Latin Hypercube Sampling (LHS) method.
The development of such an error propagation algorithm requires that the uncertainty
information of physical properties of pure compounds and vapour-liquid equilibrium (VLE) data
of binary mixtures be readily available. A pure component database was developed for 176 pure
hydrocarbons in the range of C5 to C36 based on NIST’s ThermoData Engine (TDE) system. Two
generalized correlations for the calculation of critical properties and acentric factors
parameterized by the normal boiling point and specific gravity were re-parameterized. The
Peng–Robinson (PR) equation of state was re-parameterized against the pure component
database using a weighted nonlinear least squares method for the determination of its
dependency on acentric factors and the definition of the uncertainty of its generalized
parameters. The variance-covariance matrices for error propagation calculations were also
determined for each model.
Binary mixture database was also developed containing experimental VLE data and their
uncertainties taken from TDE for 87 binary mixtures present in natural gas processing. The
iii
quality of each isothermal VLE dataset was investigated using a thermodynamic consistency test.
The binary interaction parameters associated with their uncertainties for the re-parameterized PR
equation of state along with the van der Waals quadratic mixing rules were evaluated against the
consistent VLE data using nonlinear optimization coupled with the Monte Carlo method taking
into account the uncertainties of input parameters.
Using the databases developed in this study, a simple and general error propagation algorithm
based on the Monte Carlo technique combined with the LHS sampling method was developed
and coupled with the VMGSim™ process simulator to analyze the effect of uncertainties on
chemical process design and simulation. The method was applied to simplified cases of industrial
interest such as gasoline blending and injection of liquid hydrocarbon to the existing natural gas
pipeline. The results show how the new approach can guide process engineers in revisiting
process design decisions affected by uncertainties related to thermodynamics.
iv
Preface
This paper-based Ph.D. thesis includes the results of studies conducted at the Department of
Chemical and Petroleum Engineering of the University of Calgary and funded by Shell Canada
Ltd. The main chapters of this thesis have been published in reputable peer-reviewed journals in
the field of chemical engineering. All papers have been reused with the permission of copyright
owners and reformatted to conform to the University of Calgary formatting requirements.
A version of Chapter 2, along with Appendices A and B, has been published as S.
Hajipour, M.A. Satyro, Uncertainty Analysis Applied to Thermodynamic Models and Process
Design - 1. Pure Components, Fluid Phase Equilibria 307 (2011) 78-94.
A version of Chapter 3, along with Appendices C and D, has been published as S.
Hajipour, M.A. Satyro, M.W. Foley, Uncertainty Analysis Applied to Thermodynamic Models
and Process Design - 2. Binary Mixtures, Fluid Phase Equilibria 364 (2013) 15-30.
A version of Chapter 4 has been published online as S. Hajipour, M.A. Satyro, M.W.
Foley, Uncertainty Analysis Applied to Thermodynamic Models and Fuel Properties - Natural
Gas Dew Points and Gasoline Reid Vapour Pressures, Energy Fuels (2013), DOI:
10.1021/ef4019838.
For all three papers, I was the lead investigator and intellectually responsible for concept
formation, literature review, data collection and analysis, mathematical modelling, simulation
and optimization, graphical and tabular results preparation, as well as manuscript composition.
The first paper was written under supervision of Dr. Satyro and two others were supervised by
Dr. Foley. Dr. Satyro was involved throughout the research in forming concepts, identifying the
research questions, reviewing the research findings, and editing the manuscripts. Dr. Foley was
involved in this project as a supervisor and contributed in discussions and manuscript edits.
v
Acknowledgements
The accomplishment of a doctoral research is fundamentally a collaborative process and it has
happened because of those who supported and encouraged me on this path, emotionally,
academically, and financially. My expressions and feeling of gratitude to compensate their
efforts are not bounded by these brief remarks in these pages.
Above all, I would like to express my sincerest gratitude and appreciation to my
supervisor, Dr. Michael W. Foley, and my ex-supervisor, Dr. Marco A. Satyro, for their
continuous encouragement and unconditional support during these challenging years with both
ups and downs. This work would not have been possible without Dr. Foley’s kind support and
agreement to take me on as a Ph.D. student in the middle of my research, despite the tenuous
connection between his research and my own. I am grateful to him for always being believing
me and letting me pursue my ideas and being available for guidance whenever required. My
most important coach and advisor, Dr. Satyro, deserves very special thanks for the initiation of
this study and continuous technical and emotional support at all times. I can never express my
gratitude to him for eagerly sharing his valuable knowledge and ideas with me, and being always
ready to take time out from his busy schedule to guide me and keep me on the right track.
Working with him was a “dream come true” and I am very proud of being his student.
I am very grateful to my supervisory committee members, Dr. William Y. Svrcek and Dr.
Harvey W. Yarranton, for their precious time and constructive feedback and suggestions.
Furthermore, I would like to acknowledge Dr. Laurence R. Lines for his time to review this
thesis and being my examiner and Dr. Vladimir V. Diky from National Institute of Standards and
Technology (NIST) for agreeing to act as an external examiner and sharing his insights on
TDE’s uncertainty evaluation.
vi
Thanks go to my previous teachers from the University of Tehran, Dr. Mohsen Edalat for
providing me with a strong background in Thermodynamics and Dr. Rahmat Sotudeh-Garebagh
for introducing me to Dr. Satyro and motivating me to continue my graduate studies at the
University of Calgary. My appreciation goes Dr. José O. Valderrama from the Universidad de la
Serena for valuable discussions on his proposed thermodynamic consistency test method. I also
gratefully acknowledge Virtual Materials Group Inc. for providing access to NIST’s TDE
software and a copy of the VMGSim process simulator.
I would like to acknowledge Shell Canada Ltd. For funding this research and offer many
thanks towards the Ursula and Herbert Zandmer Graduate Scholarship, the Graduate Students’
Association and the Department of Chemical and Petroleum Engineering at the University of
Calgary for their scholarships and financial support. I would also like to thank the administrative
staff of the department specially Dolly Parmar and Arlene Wallwork for their help.
Great appreciation goes to my officemates for providing an extraordinary working
environment in our office and close friends of many years in Iran and Canada for their
friendship, good humor, faith, understanding, respect, and emotional support.
At last but definitely not least, I owe the deepest gratitude to my parents, my brother,
Meisam, and my sister in law, Behnaz, in Iran who have always believed in me and understood
my desire to study abroad and being by my side virtually with full encouragement, compassion
and love. I would like to extend my thanks to my two-year-old niece, Tina, for putting a smile on
my face from miles apart. Their memories and their support from long distance by emails and
phone calls kept me focused on my goals and provided me with hope and confidence.
vii
Dedication
Dedicated to my Mom and Dad
for their endless support and unconditional love.
viii
Table of Contents
Abstract ........................................................................................................................................... ii Preface............................................................................................................................................ iv Acknowledgements ..........................................................................................................................v Table of Contents ......................................................................................................................... viii
List of Tables ...................................................................................................................................x List of Figures and Illustrations .................................................................................................... xii List of Symbols, Abbreviations and Nomenclature .......................................................................xv
CHAPTER ONE: INTRODUCTION ..............................................................................................1
1.1 Overview ................................................................................................................................1 1.2 Research Objectives ...............................................................................................................6
1.3 Thesis Structure .....................................................................................................................9
CHAPTER TWO: UNCERTAINTY ANALYSIS APPLIED TO THERMODYNAMIC
MODELS AND PROCESS DESIGN – 1. PURE COMPONENTS ...................................11 2.1 Abstract ................................................................................................................................11 2.2 Introduction ..........................................................................................................................11
2.3 Pure Component Database Development ............................................................................17 2.3.1 Uncertainty on Standard Specific Gravity ...................................................................18
2.3.2 Uncertainty on Pitzer Acentric Factor .........................................................................19 2.4 Development of A New Correlation for Critical Temperature, Critical Pressure and
Acentric Factor Using Uncertainties in Physical Property Data ........................................21
2.4.1 Computational Approach .............................................................................................23
2.4.1.1 Linear Regression ..............................................................................................28 2.4.1.2 Nonlinear Regression .........................................................................................30
2.4.2 Results and Discussion ................................................................................................32
2.4.2.1 Examples ............................................................................................................41 2.5 Effect of Uncertainties in Thermodynamic Data on Calculated Thermo-physical
Properties ...........................................................................................................................42 2.5.1 Notes on the Uncertainty of Input Variables ...............................................................42
2.5.2 The Monte Carlo Technique and Sampling .................................................................42 2.6 Re-parameterization of the Peng–Robinson Equation of State ...........................................46 2.7 Natural Gas Processing Examples .......................................................................................49 2.8 Conclusions ..........................................................................................................................53
CHAPTER THREE: UNCERTAINTY ANALYSIS APPLIED TO THERMODYNAMIC
MODELS AND PROCESS DESIGN – 2. BINARY MIXTURES .....................................55 3.1 Abstract ................................................................................................................................55
3.2 Introduction ..........................................................................................................................56 3.3 Thermodynamic Consistency Test .......................................................................................66
3.3.1 Computational Approach for Modelling of VLE Data ................................................70 3.4 Binary VLE Database Development ....................................................................................72
3.4.1 Application of the Selected Consistency Test in This Study ......................................74
ix
3.5 Estimation of Binary Interaction Parameters Associated with Uncertainties ......................80
3.5.1 Input Variables and Their Uncertainties ......................................................................80 3.5.2 The Monte Carlo Technique and Sampling Method ...................................................81
3.6 Results and Discussion ........................................................................................................87 3.6.1 Saturation Point Calculation ........................................................................................87 3.6.2 De-ethanizer Example .................................................................................................92
3.6.3 Natural Gas Processing Example ................................................................................94 3.7 Conclusions ..........................................................................................................................97
CHAPTER FOUR: UNCERTAINTY ANALYSIS APPLIED TO THERMODYNAMIC
MODELS AND FUEL PROPERTIES – NATURAL GAS DEW POINTS AND
GASOLINE REID VAPOUR PRESSURES .......................................................................99 4.1 Abstract ................................................................................................................................99
4.2 Introduction ........................................................................................................................100 4.2.1 Liquid Hydrocarbon Injection into an Existing Natural Gas Pipeline ......................100
4.2.2 Gasoline Blending .....................................................................................................102 4.3 Development of the Error Propagation Algorithm ............................................................103 4.4 Case Study Problems .........................................................................................................106
4.4.1 Injection of Liquid n-Butane into an Existing Natural Gas Pipeline .........................106 4.4.2 Gasoline Blending .....................................................................................................111
4.5 Uncertainty Analysis Results and Discussion ...................................................................114 4.6 Conclusions ........................................................................................................................128
CHAPTER FIVE: CONCLUSIONS AND RECOMMENDATIONS ........................................130
5.1 Conclusions ........................................................................................................................130
5.2 Recommendations ..............................................................................................................133
APPENDIX A: DATABASE FOR PURE HYDROCARBONS FROM C5 TO C36 ..................135
APPENDIX B: CALCULATED UNCERTAINTY OF VAPOUR PRESSURE USING NEW
3-PARAMETER PENG-ROBINSON EQUATION OF STATE BY COVARIANCE
APPROACH .......................................................................................................................143
APPENDIX C: DETAILS ON THE DEVELOPED VLE DATABASE ...................................145
APPENDIX D: BINARY INTERACTION PARAMETERS AND THEIR
UNCERTAINTIES ............................................................................................................149
REFERENCES ............................................................................................................................153
x
List of Tables
Table 2.1. General forms of correlations. ..................................................................................... 24
Table 2.2. Fitted parameters and covariance matrices for new Riazi–Daubert correlations
obtained from nonlinear regression....................................................................................... 33
Table 2.3. Fitted parameters and covariance matrix for the re-parameterized Lee–Kesler
correlation for critical temperature. ...................................................................................... 33
Table 2.4. Fitted parameters and covariance matrix for the re-parameterized Lee–Kesler
correlation for critical pressure. ............................................................................................ 34
Table 2.5. Fitted parameters and covariance matrix for the re-parameterized Lee–Kesler
correlation for acentric factor. ............................................................................................... 35
Table 2.6. Comparison of re-evaluated correlations using weighted deviation. ........................... 36
Table 2.7. Comparison of the experimental and calculated critical properties and acentric
factors of n-hexane and n-dodecane...................................................................................... 41
Table 2.8. Monte Carlo sampling for normal boiling point. ......................................................... 45
Table 2.9. Peng-Robinson equation of state refitted parameters and covariance matrix. ............. 47
Table 2.10. Comparison results of the original PR equation and the refitted equations............... 47
Table 2.11. Comparison of vapour pressure and its uncertainty calculated using the
covariance-based approach and the Monte Carlo simulation. .............................................. 49
Table 2.12. Critical point, cricondenbar and cricondentherm coordinates when compressing
lean natural gas prototype mixtures. ..................................................................................... 50
Table 2.13. Basic equipment performance data estimated using uncertainty information. .......... 51
Table 3.1. Sample of developed VLE database for the ethane/propane mixture. ......................... 73
Table 3.2. Range of VLE data used for the consistency test. ....................................................... 74
Table 3.3. Critical properties and acentric factors of pure components. ...................................... 75
Table 3.4. Thermodynamic consistency data for ethane/propane and methane/H2S. ................... 76
Table 3.5. Input variables for estimation of a binary interaction parameter. ................................ 81
Table 3.6. Temperature and pressure ranges of consistent VLE data for ethane/propane and
methane/H2S binary mixtures. .............................................................................................. 83
xi
Table 3.7. Monte Carlo simulation results for binary interaction parameters (k12) with
different sample sizes. ........................................................................................................... 86
Table 3.8. Calculated VLE data and their uncertainties for ethane/propane mixture at P=2758
kPa using the technique developed in this work. .................................................................. 89
Table 3.9. The de-ethanizer product specifications (ethane(1)/propane(2)) at P=2758 kPa. ....... 92
Table 3.10. Comparison of the minimum number of stages using different approaches applied
in this work. .......................................................................................................................... 94
Table 3.11. Basic equipment performance data and their uncertainties revisited in this work. ... 95
Table 3.12. Positions of the cricondenbar, cricondentherm and critical point calculated using
the Monte Carlo simulation. ................................................................................................. 96
Table 4.1. Composition of natural gas used in this study. .......................................................... 108
Table 4.2. Existing natural gas pipeline specifications used in this work. ................................. 108
Table 4.3. Existing pipeline equipment performance data. ........................................................ 110
Table 4.4. Low RVP gasoline blend chemical composition. ...................................................... 112
Table 4.5. Properties of pure components. ................................................................................. 113
Table 4.6. Results of the phase envelopes uncertainty analysis. ................................................ 116
Table 4.7. Physical properties of the gas and the pipeline equipment performance data before
and after the injection. ......................................................................................................... 121
Table 4.8. Vapour pressures and uncertainties calculated using the Monte Carlo simulation
for the gasoline before and after n-butane blending at different temperatures. .................. 124
Table 4.9. Results of uncertainty analysis of RVP calculation depending on the volume ratio
of the blended n-butane to gasoline at standard conditions. ............................................... 126
Table C.1. Detailed information about the developed binary VLE database. ............................ 145
Table D.1. Binary interaction parameters associated uncertainties. ........................................... 149
xii
List of Figures and Illustrations
Figure 2.1. Effect of error in the critical temperature on the predicted vapour pressure using
the Peng–Robinson equation of state for simple paraffins. .................................................. 13
Figure 2.2. Calculated acentric factor associated with uncertainty as a function of vapour
pressure @ Tr=0.7. ................................................................................................................ 20
Figure 2.3. Calculated acentric factor associated with uncertainty as a function of critical
pressure. ................................................................................................................................ 20
Figure 2.4. Calculated acentric factor associated with uncertainty as a function of critical
temperature. .......................................................................................................................... 21
Figure 2.5. Uncertainties of normal boiling point and specific gravity. ....................................... 37
Figure 2.6. Critical temperature versus normal boiling point. ...................................................... 39
Figure 2.7. Critical pressure versus normal boiling point. ............................................................ 39
Figure 2.8. Acentric factor versus normal boiling point. .............................................................. 40
Figure 2.9. Critical temperature normal distributions for (a) n-hexane (b) n-dodecane. .............. 44
Figure 2.10. Comparison of vapour pressure calculated using the original and the improved
Peng–Robinson equations of state. ....................................................................................... 48
Figure 2.11. Pressure–temperature envelope for Composition 1 (methane and n-hexane). ......... 52
Figure 2.12. Pressure–temperature envelope for Composition 2 (methane and n-dodecane). ..... 52
Figure 3.1. Temperature-composition diagram for ethane/propane system at 2758 kPa. Note
that the thickness of the TXY “curves” actually represents the uncertainties associated
with the bubble and dew points curves. ................................................................................ 59
Figure 3.2. Effect of uncertainties in compositions on (a) vapour–liquid equilibrium constant
(Ki), and (b) relative volatility (α) for ethane/propane system at pressure of 2758 kPa. ...... 61
Figure 3.3. Illustration for the calculation of AP between two consecutive points of r and s. ...... 68
Figure 3.4. System ethane/propane, (a-c) Pressure-composition diagrams at 270.00, 310.93,
and 273.20 K, (d-f) Deviations in the individual areas ( iA% ) for liquid phase () and
vapour phase ( ). ................................................................................................................... 77
xiii
Figure 3.5. System methane/H2S, (a-c) Pressure-composition diagrams at 273.20, 277.59, and
310.93 K, (d-f) Deviations in the individual areas ( iA% ) for liquid phase () and
vapour phase ( ). ................................................................................................................... 78
Figure 3.6. Conceptual scheme of the approach used for uncertainty estimation of the fitted
parameter. .............................................................................................................................. 82
Figure 3.7. Histogram of calculated binary interaction parameters by different sample sizes
(a) ethane/propane, (b) methane /H2S. .................................................................................. 84
Figure 3.8. Binary interaction parameter distribution for ethane/propane with sample size
100. ........................................................................................................................................ 85
Figure 3.9. Temperature-composition diagram for (a) ethane/propane at 2758 kPa, and (b)
methane/H2S at 6894.8 kPa. .................................................................................................. 88
Figure 3.10. Pressure-composition diagram for (a) ethane/propane at 310 K, and (b)
methane/H2S at 320 K. .......................................................................................................... 91
Figure 3.11. Schematic diagram of natural gas processing example. ........................................... 95
Figure 3.12. Pressure-temperature envelope for Composition 2 (methane/n-decane). ................. 96
Figure 4.1. Sequence of overall error propagation evaluation process. ...................................... 105
Figure 4.2. Pressure-temperature (PT) envelope for a natural gas and thermodynamic
positions of the pipeline with temperatures of higher (T1) and lower (T2) than dew point
temperature at pressure of P................................................................................................ 107
Figure 4.3. Schematic view of the existing natural gas pipeline used in this work. ................... 109
Figure 4.4. Pressure-temperature envelopes for a natural gas before and after the liquid n-
butane injection. .................................................................................................................. 115
Figure 4.5. (a) The zoomed-in version of Figure 4.4 for pressure-temperature envelope of gas
after the injection of 137.52 m3/hr, and (b) Monte Carlo simulation results for dew point
calculation at 5515.8 kPa. ................................................................................................... 117
Figure 4.6. (a) Calculated dew point and associated uncertainty at 5515.8 kPa against the
injected liquid/gas standard volume ratio, and (b) zoomed-in version of (a) in the
vicinity of maximum dew point. ......................................................................................... 119
Figure 4.7. Monte Carlo simulation results for the dew point calculation at 5515.8 kPa after
the injection of 135.45 m3/hr n-butane................................................................................ 120
Figure 4.8. Pressure-temperature envelopes for the gasoline before and after n-butane
blending. .............................................................................................................................. 123
xiv
Figure 4.9. Monte Carlo simulation results for the RVP calculation of the final gasoline blend
with 7.17 volume percent of blended n-butane. .................................................................. 125
Figure 4.10. Calculated RVP and associated uncertainty against the blended n-
butane/gasoline standard volume ratio. ............................................................................... 126
Figure 4.11. Monte Carlo simulation results for RVP calculation of the final gasoline blend
with 6.86 volume percent of blended n-butane. .................................................................. 128
xv
List of Symbols, Abbreviations and Nomenclature
Abbreviation Definition
AAD Average Absolute Deviation
AD Absolute Deviation
APR Advanced Peng–Robinson
ASTM American Society for Testing and Materials
CI Confidence Interval
DIPPR Design Institute for Physical Properties
EOS Equation of State
EPS Equal Probability Sampling
LHS Latin Hypercube Sampling
LK Lee–Kesler
LNG Liquefied Natural Gas
LPG Liquefied Petroleum Gas
MAOP Maximum Allowable Operating Pressure, kPa
Max. Maximum
MC Monte Carlo
MCS Monte Carlo Sampling
MCSE Monte Carlo Standard Error
Min. Minimum
NFC Not Fully Consistent
NIST National Institute of Standards and Technology
NPS Nominal Pipe Size
PR Peng–Robinson
RD
RK
Riazi–Daubert
Redlich–Kwong
RVP Reid Vapour Pressure, kPa
SHS Shifted Hammersley Sampling
SRK Soave–Redlich–Kwong
xvi
TC Thermodynamically Consistent
TDE ThermoData Engine
TI Thermodynamically Inconsistent
TRC Thermodynamic Research Centre
VLE Vapour–Liquid Equilibria
WS Wong–Sandler
Symbol (Context Dependent)
A
Wagner equation parameters (Equation 2.6) or
molar Helmholtz energy in Chapter 3, kJ/kmol
A area deviation
A calculated area
AP experimental area
a vector of model parameters
a model parameter in Chapter 2 or
attraction parameter in Chapter 3, kPa.(m3/kmol)
2
b van der Waals co-volume, m3/kmol
C variance-covariance matrix
C element of matrix C
Fobj objective function
f vector of independent variables
f independent variable
fω PR acentric factor function
G molar Gibbs energy, kJ/kmol
H molar enthalpy, kJ/kmol
K vapour-liquid equilibrium constant (K-value)
Kw Watson characterization factor
kij van der Waals mixing rule binary interaction parameter
l interval identification
MW molecular weight, kg/kmol
xvii
m
m
number of fitted parameters in Chapter 2 or
re-parameterized PR parameters in Chapter 3
m vector of re-parameterized PR parameters
m' number of independent variables
n number of data points in Chapter 2
n' sample size
NC number of components
Nmin minimum number of column stages (at total reflux)
NP number of experimental VLE data points
NT=cte. number of isothermal datasets
P pressure, kPa (or psia in Chapter 1 for LK model)
Psat
vapour pressure, kPa
Q heat duty, kJ/hr
q weighting factor
R universal gas constant, kJ/kmol.K
S standard deviation
SG specific gravity
T absolute temperature, K (or R in Chapter 1 for LK model)
Tb normal boiling point, K (or R in Chapter 1 for LK model)
U symmetric matrix (Equation 2.31)
U element of matrix U in Chapter 2
U' symmetric matrix (Equation 2.40)
U' element of matrix U' (Equation 2.38)
V molar volume, m3/kmol
W work, HP (or kW in Chapter 4)
x liquid phase composition, mole fraction
y vapour phase composition, mole fraction
Z compressibility factor
xviii
Greek letters
α relative volatility in Chapter 3
αPR PR alpha function
β row matrix (Equation 2.30)
β element of matrix β
β' row matrix (Equation 2.40)
β' element of matrix β' (Equation 2.35)
γ activity coefficient
δij WS mixing rule binary interaction parameter
ε inverse matrix of U (Equation 2.33)
ε element of matrix ε
ξ phase composition, mole fraction (Chapter 3)
dependent variable (Chapter 2)
Λ12 van Laar model parameter
Λ21 van Laar model parameter
damping factor (Equation 2.38)
μ mean value
standard liquid density, kg/m3
uncertainty
fugacity coefficient
χ2 objective function in Chapter 2
Ω WS mixing rule constant (Ω = –0.62322 for PR equation)
acentric factor
Subscript
avg. average
B bottom product
c critical property
xix
D distillate
m mixture
r reduced property
tra. transferred
Superscript
cal. calculated
E excess property
exp. experimental
L liquid phase
R residual property
V vapour phase
1
Chapter One: Introduction
1.1 Overview
Physical and thermo-physical property data for pure components and mixtures are essential in
the field of chemical engineering for the simulation, design, optimization, and debottlenecking of
industrial facilities. Vapour pressure, for example, is required for the design of almost all
equipment and processes in which both liquid and vapour phases are present. Vapour–liquid
equilibrium data are used for the simulation and design of separation equipment and used
throughout the design of a plant or a fluid transportation facility. Critical properties and acentric
factors are essential for vapour–liquid processes simulated using equations of state. The
reliability and accuracy of these properties are essential for the proper understanding and
modelling of processes. Physical properties are invariably derived from experimental
measurements and therefore subject to uncertainties associated with measured values. The errors
associated with physical properties can have costly consequences such as unnecessarily large
overdesign with corresponding high capital or operating costs or, at its worst, designs that cannot
be made to provide products within desired specifications.
Today, commercial process simulation software such as VMGSim™ and Aspen
HYSYS® are routinely used to quickly simulate, design, develop and optimize processes.
Physical and thermo-physical property data are the most important ingredients for the
development of thermodynamic models used in such simulators. These mathematical models and
correlations are used to estimate the physical properties of pure components and oil fractions
and/or to predict phase equilibria and physical properties of mixtures such as activity and
fugacity coefficients. These models contain undefined parameters that are determined from
2
available experimental data using linear or nonlinear regression procedures. Errors in
experimental data used to determine the model parameters propagate and hence simulation
results are also subject to errors inherited from the original data used to develop the
thermodynamic models.
Currently the basic input properties such as critical properties and acentric factors of pure
components are used in simulators without statistical uncertainty information. These properties
are commonly used as input parameters in thermodynamic models, such as cubic equations of
state, and are extensively used in process and reservoir engineering for the prediction of phase
equilibrium and thermo-physical properties of material streams. Although reliable prediction of
thermodynamic properties relies on the regressed model parameters, the uncertainties of
estimated parameters propagated from the uncertainties in experimental data are traditionally
overlooked in simulators.
In addition to pure components and thermodynamic models, the quality of the binary
interaction parameters used in equations of state to improve their ability to predict the fluid phase
behaviour of mixtures is a key component for the development of statistically meaningful VLE
data. Interaction parameters greatly affect the accuracy of VLE calculations and therefore
estimated equipment performance of separators and distillation columns. Binary interaction
parameters are estimated by data regression using the available experimental binary VLE data.
While uncertainties in VLE data propagate through the estimation procedure to the adjusted
parameters, they are defined in simulators as deterministic inputs and their uncertainties are also
not taken into account.
Avoiding uncertainties in simulators due to lack of statistical information can have
serious consequences, since no qualification of the accuracy of input parameters is known and no
3
information on the way errors in parameters propagate through the computations is provided to
users. Consequently, experience-based risk assessment and safety measures have to be used
without the benefit of a tool to assist in the critical analysis of the quality of the results.
The availability of the uncertainty information associated with the basic properties and all
thermodynamic model parameters was the main reason for the development of a computational
procedure for uncertainty analysis. The final result is an increase in the knowledge of how these
uncertainties propagate through common process calculations, how they affect the estimated
performance and, most important, assist designers in defining equipment overdesign consistently
thus optimizing the process design.
Several previous studies on the uncertainty analysis in the field of process engineering
did not provide a general way to quantitatively determine the associated uncertainties of physical
properties and model parameters but rather determined the uncertainties in an ad-hoc manner
based on the estimated average errors frequently considered as a percentage of the reported
values. Using sources for critically evaluated thermophysical property data developed during the
last decades, some studies were done to quantify the effect of physical property uncertainties on
process design using limited uncertainty information available at that time, such as through the
DIPPR® 801 database where pure component uncertainties were roughly estimated based on the
average absolute deviation between reported and calculated physical properties.
Traditionally, critical evaluation of data for a particular chemical system or property
group is a time and resource-consuming process and must be performed far in advance of need.
As a result, a significant part of the existing data has never been evaluated. Moreover, since it is
quite common that significant new data have become available during the data evaluation
project, data analysis and fitting model parameters, such as equations of state models, must be
4
updated in order to provide up-to-date predictions. This type of data evaluation is slow and
inflexible [1].
Recently, the National Institute of Standards and Technology (NIST) developed the first
software in the form of ThermoData Engine (TDE) [2] to: (1) automatically generate
recommended data based on all available, up-to-date experimental data with assigned
uncertainties for pure compounds, binary and ternary mixtures, and reaction systems stored in
the SOURCE electronic database [3, 4]; (2) produce critically evaluated data dynamically or “in
order” using an automated system when information is required. Dynamic data evaluation
contrasts with the traditional evaluation of data which must be initiated in advance of anticipated
need. The unique feature of this software is that all calculated numerical values include estimates
of uncertainties. In this study, TDE is used as the most comprehensive source of experimental
data and their uncertainties in the "world" for both pure components and binary and ternary
mixtures and allows the determination of not only the model parameters but rather statistically
significant model parameters weighted based on the quality of the physical property data as well
as model parameter uncertainties. The most important aspect of this software and fundamental to
this work is the estimation of uncertainty based on the normal (Gaussian) distribution density
function with a level of confidence of approximately 95%.
After recognizing the uncertainty in the various input variables, a practical method to deal
with the uncertainty propagation arising from complex models is required to analyze and
quantify uncertainty induced from errors in input variables. The Monte Carlo simulation
technique coupled with a sampling method is the most popular and useful computer-based
approach and has been commonly applied in the uncertainty propagation analysis of chemical
plants. This kind of analysis corresponds to the probabilistic approach where all the input
5
variables are characterized by probability distributions representing the full range of possible
values and the uncertainties propagate in the model prediction such that the result is also a
probability distribution. The normal distribution is the most popular probability density function
used for characterized the uncertain variables to represent the uncertainty resulting from
unbiased measurement errors. It is recognized that errors in physical properties are not
necessarily distributed according to a normal distribution since other sources of errors such as
equipment systematic deviations may be present in the data. These are usually not available to
the modeller and therefore considered outside the scope of this study.
In this study, the Monte Carlo simulation was used for the uncertainty propagation
calculations for complex flow sheets. A sample set from the probability distribution of all
uncertain input parameters is generated using an appropriate sampling technique and the values
of the desired equipment or process parameters are repeatedly calculated using different sampled
values of each input variable. The number of samples and the sampling technique are the two
main factors in the sampling process in order to get reliable samples for estimation of output
variables. Examples of these techniques are random Monte Carlo Sampling (MCS) [5] and Latin
Hypercube Sampling (LHS) [6] which are used in this research and will be discussed in the
following chapters.
Several studies of uncertainty analysis applied to the chemical industry emphasized its
importance in process design. However, a comprehensive computational procedure has not yet
been presented to systematically analyze the effect of thermodynamic model parameter
uncertainties on the results of process simulation/design. In this study, this procedure is
developed through the linkage between the TDE derived evaluated physical property database
and the VMGSim process simulator [7] for quantification and analysis of process uncertainties
6
via the Monte Carlo method coupled with nonlinear regression algorithms and sampling
techniques.
1.2 Research Objectives
The main objective of this dissertation is the development a self-contained and consistent
computational procedure to quantify the uncertainties in basic physical properties,
thermodynamic model parameters, and binary interaction parameters and how these uncertainties
affect the calculated thermo-physical properties, material and energy balances, equipment
parameters, and product properties. To achieve this goal, the following specific challenges were
overcome:
Develop a comprehensive pure component database capable of storing all experimental
and predicted data with associated uncertainties. The database contains the physical
properties (molecular weight, normal boiling temperature, standard liquid density,
standard liquid specific gravity, and vapour pressure), critical properties (critical
temperature and critical pressure) and acentric factor for pure components commonly
present in natural gas. The experimental values and their relevant uncertainties were
taken from TDE version 5.0 [2] , while the predicted values of acentric factor and
standard specific gravity were calculated as part of this work and their uncertainties were
predicted using the principles of error propagation based on the first order Taylor series
linearization.
Re-parameterize estimation models for the proper characterization of petroleum using
pseudo-components. The majority of naturally occurring hydrocarbon systems contain
some undefined heavy fractions that are lumped together and identified as the “plus”
7
fraction, or are determined using oil assays using distillation curves. In order to predict
the phase behaviour of hydrocarbon systems using a thermodynamic model, the acentric
factor and critical properties of these undefined fractions are required together with
estimated uncertainties. Estimation methods were re-parameterized by taking into
account the uncertainties of both dependent and independent variables, and their
associated variance-covariance matrices of model parameters were provided. These
newly developed estimation methods allow the rigorous estimation of uncertainties in
physical properties of undefined oil fractions.
Re-parameterize equation of state models and provide the uncertainty information related
to model parameters. The availability of uncertainties in the necessary input parameters
required for developing a thermodynamic model makes it possible to re-parameterize the
model and provide variance-covariance matrix of its parameters. While the objective of
this thesis is to provide a method generally applicable to any thermodynamic model,
specific examples will used. A re-parameterized version of the Peng-Robinson equation
of state where its parameters were determined using statistically meaningful vapour
pressures, critical pressures, critical temperatures, and acentric factors will be developed.
The parameters of the corresponding equation of state were determined with their
associated uncertainties.
Develop a database for binary mixtures containing the experimental vapour–liquid
equilibrium (VLE) data and their uncertainties and perform the thermodynamic
consistency test to check the reliability of the experimental data. The first step in
expanding the thermodynamic model for mixtures is providing the database including the
experimental data of pressures, temperatures, and both liquid and vapour phase
8
compositions and their uncertainties as determined by TDE version 5.0. Thermodynamic
consistency of VLE data was checked through the use of the Gibbs-Duhem equation.
Estimate binary interaction parameters and their associated uncertainties. The prediction
of phase and volumetric behaviour of mixtures using equations of state is done through
the use of mixing rules for model parameters. Since the binary interaction parameters are
traditionally determined using the VLE data regression, the uncertainties in the original
data necessarily affect the quality of the estimated parameters. Therefore, the binary
interaction parameters and associated uncertainties have to be determined to meet the
research objective. The re-parameterized Peng-Robinson equation of state was used along
with van der Waals mixing rules with a single adjustable binary interaction parameter and
the interaction parameter associated with its uncertainty was obtained by simultaneously
taking into account the uncertainties in the binary vapour–liquid equilibrium data,
physical properties of pure components, and the Peng-Robinson model parameters.
Develop an efficient error propagation algorithm to perform uncertainty analysis for
generic flow sheets. The algorithm must be able to propagate the uncertainties from pure
component physical properties and thermodynamic model parameters through all
material balances, energy balances, and equilibrium relationships and provide uncertainty
estimations for the quantities resulting from these process calculations. The computer-
based error propagation algorithm must be coupled with chemical plants flow sheets of
any complexity.
9
1.3 Thesis Structure
This is a paper-based dissertation consisting of five chapters structured as follows. The main
findings of this research study are presented in the next three chapters consisting of published
papers in peer-reviewed journals. There is, therefore, some repetition such as introduction,
mathematical models, and numerical methods. Each chapter presents background information
and a literature review of the principal subjects as well as brief reviews of the pertinent methods.
Chapter 2 focuses on the uncertainties in physical property data of pure components and
their effect on thermodynamic models and process simulation/design. Two estimation models for
characterization of undefined oil fractions and a cubic equation of state are re-parameterized
based on the data and associated uncertainties taken from the developed database for pure
components. In addition, the variance-covariance matrices of model parameters are evaluated. A
version of this chapter was published in the Fluid Phase Equilibria journal [8].
Chapter 3 deals with the binary mixtures and quantification of the uncertainties in
estimated binary interaction parameters propagated from uncertainties in pure components
physical properties, VLE data, and thermodynamic model parameters. The thermodynamic
consistency test is performed for each isothermal VLE dataset in order to determine the quality
of the associated data. A version of this chapter was published in the Fluid Phase Equilibria
journal [9].
Chapter 4 presents the methodology used for the development of a consistent and self-
contained error propagation algorithm and the application of the proposed algorithm is illustrated
through two case studies related to hydrocarbon processing. The first case study is related to
liquid hydrocarbon injection into an existing natural gas pipeline and the second case study is a
gasoline blending process. In both cases, the process conditions are revisited and the safety
10
factors are determined in light of the uncertainty analysis. A version of this chapter has been
published online in the Energy & Fuels journal [10].
Finally, general conclusions and recommendations and suggestions for future studies are
presented in Chapter 5.
11
Chapter Two: Uncertainty Analysis Applied to Thermodynamic Models and Process
Design – 1. Pure Components 1
2.1 Abstract
A simple model is proposed to estimate the critical temperature and critical pressure of
hydrocarbons in the range of C5–C36 with parameters determined using weighted linear least
squares and weighted nonlinear least squares taking into consideration the experimental
uncertainty in the data as well as in the correlating parameters. The correlation model was
parameterized using the normal boiling point and specific gravity at 60 ˚F. The uncertainties of
parameters and associated covariance matrix necessary for error propagation calculations are
reported and a comprehensive evaluation of acentric factors uncertainties based on the
experimental vapour pressures was conducted. In addition, a simple sensitivity analysis designed
to determine how the uncertainty of properties used for calculations based on equations of state
propagate thorough the model and affect the final results. The normal boiling points of two pure
components, n-hexane and n-dodecane were calculated using an equation of state and the
estimated error in the calculations is presented together with estimated uncertainties for the
prototype pressure-temperature envelopes for two binary mixtures of methane/n-hexane and
methane/n-dodecane.
2.2 Introduction
Critical properties and acentric factors are important for prediction of thermodynamic and
physical properties of fluids and commonly used as input parameters in cubic equations of state
1 Reprinted from Fluid Phase Equilibria, Vol. 307, S. Hajipour, M.A. Satyro, Uncertainty Analysis Applied to
Thermodynamic Models and Process Design – 1. Pure Components, 78-94, Copyright (2011), with permission from
Elsevier. It should be noted that the content of this chapter includes the extensions beyond the cited journal paper.
12
such as Soave–Redlich–Kwong (SRK) [11] and Peng–Robinson (PR) [12]. These equations of
state are extensively used in process and reservoir engineering and routinely used for the design,
optimization and debottlenecking of industrial facilities via process simulators. Currently
equation of state input values such as critical temperatures and pressures are used in simulators
without statistical uncertainty information. Therefore safety measures such as equipment
overdesign have to be applied in an ad-hoc manner.
The objective of this work is to expand on ideas put forth by Whiting and co-workers
[13-18] in uncertainty analysis of chemical processes through the development of a
comprehensive database of critical parameters, acentric factors, interaction parameters, and
estimation techniques for pure component physical properties and binary interaction parameters,
taking advantage of new uncertainty on fundamental physical property data available now
through NIST’s ThermoData Engine (TDE) [2]. A unique feature of this development is related
to the uncertainty information encoded in this new database as well as in estimation methods
necessary for the modelling of pseudo-components used to model complex hydrocarbon fluids.
The availability of uncertainty information associated with all the model parameters
allows us to estimate in a rigorous way what is the uncertainty in values calculated from the
model and put the analysis of the quality of results from a thermodynamic model on solid footing
thus making this important information available to process engineers and assist them in
determining if critical parts of a process being designed require more information before
significant investment is committed or if process conditions must be revisited due to
uncertainties on the state of the process fluid at certain process conditions such as proximity to
the dew point line at the inlet of a compressor.
13
It is well known that small errors in the critical properties used in equations of state affect
the quality of final results, sometimes in a dramatic fashion. For example, the effect of errors in
the critical temperature of different compounds using the Peng–Robinson equation of state is
shown in Figure 2.1. If the critical temperature is under-estimated by 2% from its accepted value,
errors in vapour pressure between 20 and 60% for several evaluated compounds are obtained.
Note, the deviation curves do not go through the zero-zero point due to the inherent inaccuracy
of the model used for the calculations.
Figure 2.1. Effect of error in the critical temperature on the predicted vapour pressure
using the Peng–Robinson equation of state for simple paraffins.
There are several studies reported in the literature, for example, Zudkevitch [19],
Zudkevitch and Gray [20], Larsen [21], and Zeck [22], that illustrated the effects of uncertain
thermodynamic data and corresponding effects in the accuracy of models in several specific
-60
-20
20
60
100
140
180
-6 -4 -2 0 2 4 6
% D
evia
tio
n i
n V
ap
ou
r P
ress
ure
@ T
= 0
.7 T
c
% Deviation in Critical Temperature
n-Hexane
n-Dodecane
n-Eicosane
n-Tetracosane
14
cases, but these studies did not provide a way to quantitatively determine the uncertainties but
rather associated uncertainties of physical property data in an ad-hoc manner based on the
estimated average errors. Uncertainty analysis in the field of process engineering was studied by
Halemane and Grossmann [23], Diwekar and Rubin [24], Pistikopoulos and Ierapetritou [25] and
Chaudhuri and Diwekar [26].
Notably Whiting and co-workers [13-18] showed the importance of the effect of physical
property inaccuracies on process design. At the time, little quantitative information related to
uncertainty was available and these earlier studies were performed using average uncertainties
estimated for different physical properties such as the evaluations performed by DIPPR (Design
Institute for Physical Properties) [27]. Recent developments in chemical engineering data
collection and correlation by the National Institute of Standards and Technology (NIST) in the
form of the ThermoData Engine (TDE) and the SOURCE database allow now for the
development of databases and correlations that reflect the uncertainty of physical properties and
the determination of not only model parameters but also model parameters weighted based on the
quality of physical property data as well as model parameter uncertainties.
TDE is the first software that implements the concept of dynamic data evaluation to
thermo-physical property data. TDE uses experimental data stored in the TDE–SOURCE
database, predicted data (obtained through application of several predictive methods), and user-
supplied property values for dynamic evaluation process. All experimental properties archived in
the TDE–SOURCE originate from journals, articles, reports, and theses and it is a subset of the
Thermodynamic Research Centre (TRC) SOURCE, an extensive relational data archival system
for thermo-physical and thermo-chemical properties reported in the scientific literature. The
artificial intelligence (expert-system) software built into TDE automatically generates critically
15
evaluated data on demand through assessment of available experimental and predicted data. The
estimation of uncertainties with a confidence level of 95% for all numerical property values used
in TDE is the most important aspect of this software [1] and is fundamental to this thesis work.
In TDE, it is assumed that the uncertainty of each property of a pure component is
characterized by a normal (Gaussian) distribution. For a confidence level of 95% with the
evaluated true value set as the mean value (), the standard deviation (S) is half of the evaluated
uncertainty. The range of values that each property can take in the 95% confidence interval
would lie in the interval S2 . In this work, the estimated uncertainties are used in a weighted
least squares regression procedure as weighting factors of data points and in the error
propagation procedure. This ensures that the best possible models are developed from statistical
and data quality points of view.
The objective is to develop a carefully evaluated database of pure component properties,
interaction parameters, and parameters used to estimate pure component properties and
interaction parameters for mixtures of interest to the natural gas processing industry together
with the necessary statistical uncertainty information for each piece of information present in the
database. With this information, Monte Carlo techniques are used to evaluate the effect of
physical property uncertainties in process simulation, with the final goal of providing a sound
background for the re-evaluation of process equipment design parameters such as heat transfer
correlations thus bringing us closer to the goal of providing process engineers with better tools to
access the feasibility, quality and safety of new industrial processes or processes being modified
or revamped.
There are dozens of correlations available in the literature to estimate critical properties
[28-40]. These properties often depend on some easily measurable physical properties such as
16
molecular weight, normal boiling point, and standard liquid density (or specific gravity). Lee and
Kesler [35, 36] and Riazi and Daubert [37] proposed models dependent on normal boiling
temperature (used as a crude energy parameter) and specific gravity (used as a crude size
parameter). These simple two-parameter correlations can be applied only to hydrocarbon and
non-polar compounds. Wilson et al. [38] and Brule et al. [39] suggested two-parameter
correlations for coal liquids. Another two-parameter correlation was developed by Twu [40] both
for petroleum and coal liquids. All these correlations involve only the boiling point and the
specific gravity as input parameters. The parameterization using normal boiling point and
specific gravity is of particular importance to the oil industry, since usually only these properties
are available, as a result of an oil characterization procedure and if critical properties, acentric
factors and ideal gas heat capacities can be reliably estimated from these basic properties then a
complete simulation model can be constructed.
The parameters used in these models were obtained from regressions using independent
and dependent variables experimental values. Notwithstanding the usefulness of these estimation
methods, they were presented without uncertainty information, such as uncertainty related to the
dependent variables (in this case critical pressure and temperature), independent variables
(normal boiling point and specific gravity) and model parameter uncertainties. Due to the need to
deal with undefined components present in refining and natural gas systems, there is the need to
redevelop the estimation methods taking into account errors in the dependent and independent
variables, and to present the associated covariance matrix of model parameters for error
propagation calculations.
In this work, the Riazi and Daubert’s model [37] and Lee and Kesler’s model [35, 36]
were chosen to re-evaluate a wide variety of hydrocarbons in the range of C5–C36, although the
17
procedure is entirely general and other methods could be used. These models depend on the
normal boiling point and the specific gravity, readily available properties from oil
characterization, for prediction of critical temperature, critical pressure and acentric factor. We
chose the Riazi and Daubert’s model due to its simplicity and accuracy in prediction of critical
properties and Lee and Kesler’s model because of its accuracy in prediction of critical properties
and acentric factors. Both methods are widely used in the hydrocarbon industry. The uncertainty
on normal boiling point and specific gravity were taken into account together with the
uncertainty of critical pressures or temperatures while developing the correlation. To support this
effort a database containing critical temperature (Tc), critical pressure (Pc), normal boiling point
(Tb), and specific gravity (SG) for hydrocarbons associated with their uncertainties was prepared.
2.3 Pure Component Database Development
Re-evaluation of estimation models taking into account the uncertainties required the
development of a database capable of storing all relevant experimental and predicted data
associated with uncertainties. The database contains physical properties (molecular weight,
normal boiling point, standard liquid density, standard liquid specific gravity, and vapour
pressure), critical properties (critical temperature, critical pressure) and acentric factor for 176
pure hydrocarbons in the range of C5–C36. The selection of hydrocarbons is based on compounds
commonly present in natural gas with normal boiling points above 290 K. These are necessary to
redevelop estimation models for characterization of undefined oil fraction such as C7+. The
experimental values of molecular weight (MW), normal boiling point (Tb), standard liquid
density (l), critical temperature (Tc), critical pressure (Pc) and vapour pressure data (Psat
) at
reduced temperature (Tr) of 0.7 and their relevant uncertainties were taken from TDE version 5.0
18
[2]. After selecting a compound, TDE was used to gather the experimental data for each property
from the TDE–SOURCE database and to evaluate these data dynamically using an internal
algorithm [1]. To redevelop the estimation methods, standard specific gravity and acentric factor
data and their uncertainties were required. Since this type of information is not available directly
in TDE, they were specially calculated as part of this work and their predicted values are listed in
the database presented in Appendix A.
2.3.1 Uncertainty on Standard Specific Gravity
The standard specific gravity is defined in Equation 2.1:
OH
iiSG
2
2.1
where i and OH2 are the standard liquid density of the selected compound and water at 60 ˚F.
The uncertainty on specific gravity was determined using the standard error propagation
equations, Equations 2.2 and 2.3 [41]:
2
2
2
2
2
2
2
OHii
OH
i
i
iSG
SGSG
2.2
22
2
2
OHii
SG OHii
SG
2.3
where iSG is the standard specific gravity uncertainty of the selected component, and
i and
OH2 are the standard liquid density uncertainties of the selected component and water,
respectively.
19
2.3.2 Uncertainty on Pitzer Acentric Factor
The Pitzer correlation [42], Equation 2.4, was used for calculation of the acentric factor ():
1)(log 7.010 rT
sat
rP 2.4
where Pr and Tr are the reduced pressure and temperature. The acentric factor uncertainty was
calculated from the propagation of the vapour pressure uncertainty using Equation 2.5:
22
10ln10ln
c
P
sat
P
PP
csat
2.5
Vapour pressures and critical pressures data associated with their uncertainties are taken
from TDE Version 5.0 [2] in the form of a 5-parameter Wagner equation [43], Equation 2.6:
)(lnln 5
4
5.2
3
5.1
21 AAAAT
TPP c
c
sat 2.6
where )(1 cTT .
Figures 2.2 to 2.4 show the uncertainty of acentric factor versus vapour pressure at
reduced temperature of 0.7, critical pressure, and critical temperature. The values of the
parameters and their uncertainties were obtained from TDE Version 5.0. In Figures 2.2 and 2.3,
all of the data follow a common trend; however, Figure 2.4 does show three distinct trends for
the acentric factor as a function of the critical temperature. The lower trend line shows the
uncertainty of compounds with two rings in their structures such as naphthalene and 1,1-
bicyclopentyl, the middle trend line shows the trend for compounds which have one ring in their
structure (aromatic and/or naphthenic) such as benzene and cyclohexene, and the upper trend line
is the trend for the other compounds in the database. This figure indicates that the acentric factor
of a hydrocarbon is a function of its structure. This relationship could be further explored for the
development of better acentric factor correlations.
20
Figure 2.2. Calculated acentric factor associated with uncertainty as a function of vapour
pressure @ Tr=0.7.
Figure 2.3. Calculated acentric factor associated with uncertainty as a function of critical
pressure.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0 40 80 120 160 200 240 280 320
Ace
ntr
ic F
act
or
Vapour Pressure @ Tr=0.7 (kPa)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Ace
ntr
ic F
act
or
Critical Pressure (kPa)
21
Figure 2.4. Calculated acentric factor associated with uncertainty as a function of critical
temperature.
2.4 Development of A New Correlation for Critical Temperature, Critical Pressure and
Acentric Factor Using Uncertainties in Physical Property Data
In this study, the Riazi and Daubert’s model (RD) [37] and Lee and Kesler’s model (LK) [35,
36] were re-evaluated. The Riazi and Daubert method is a simple correlation expressed by the
multiplication of two power functions. Voulgaris et al. [44] did a comparative study of the
accuracy of several calculation methods and recommended the Riazi and Daubert method for the
critical properties and Lee and Kesler method for acentric factor estimation.
Riazi and Daubert [37] proposed a simple two-parameter equations to correlate the critical
temperatures and critical pressures of hydrocarbons in the C5–C20 range, Equations 2.7 and 2.8.
3596.058848.006232.19 SG T T bc 2.7
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
450 500 550 600 650 700 750 800 850 900 950
Ace
ntr
ic F
act
or
Critical Temperature (K)
22
3201.23125.29 1053027.5 SGTP bc
2.8
where Tc is the critical temperature in K, Pc is the critical pressure in kPa, Tb is the normal
boiling point in K, and SG is specific gravity of the liquid at 60 ˚F.
The original Lee and Kesler [35, 36] correlations for critical temperatures, critical
pressures, and acentric factors are as follows:
b
bcT
SGTSGSGT510
)2623.34669.0()1174.04244.0(8117.341 2.9
310
2
27
2
3
2
106977.1
42019.01047227.0648.3
4685.1
1011857.02898.2
24244.0566.0
3634.8ln
bb
bc
TSG
TSGSG
TSGSGSG
P
2.10
For Tbr ≤ 0.8
6
6
4357.0ln4721.136875.152518.15
169347.0ln28862.109648.692714.5ln
brbrbr
brbrbrbr
TTT
TTTP
2.11
For Tbr > 0.8
br
wbrww
T
KTKK
01063.0408.1359.8007465.01352.0904.7 2
2.12
where Tc is critical temperature in Rankine, Pc is critical pressure in psia, is acentric factor, Tb
is the normal boiling point in Rankine, SG is specific gravity of the liquid at 60 ˚F, Tbr and Pbr are
the reduced normal boiling temperature and pressure, respectively, and Kw is the Watson
characterization factor which is a function of the normal boiling point in Rankine and standard
specific gravity, Equation 2.13:
SG
TK b
w
31)8.1( 2.13
23
In this study, new parameters for the Riazi–Daubert (RD) and Lee–Kesler (LK) models
were recalculated using new data to extend them into the C5–C36 range. These correlations can
be written in the general forms shown in Table 2.1. In this table, the dependent variable (a
function of the critical property or acentric factor) is expressed as a function of independent
variables f and a vector of model parameters a . The numerical values of model parameters are
unknown and to be determined using the experimental data of dependent and independent
variables. Linear and nonlinear regressions using the Levenberg–Marquardt [45] method were
used to determine the parameters for the different models. In this procedure the uncertainties in
dependent (Tc, Pc or ) and independent variables (Tb and SG) were taken into account.
2.4.1 Computational Approach
The model parameters were estimated using the weighted least squares method, a special case of
the more general maximum likelihood estimation procedure [46]. The method essentially
involves the minimization of an objective function based on the model, model parameters,
experimental data and associated uncertainties of experimental data, Equation 2.14.
n
i
ii
i1
2
2
2 );(1
)( afa
2.14
where n is the number of selected compounds from the experimental database, i are the
experimental values of dependent variables, );( af i are the model variables calculated using the
true parameter values a . i represents the value of the total uncertainty calculated using both
dependent and independent variable uncertainties.
24
Table 2.1. General forms of correlations.
Model General Form Correlation Structure
RD Nonlinear form: 32
211);(aa
ffaaf SGT
aaa
PT
b
cc
f
a 321
or
RD Linear form:
3
1
);(k
kk faaf SGT
aaa
PT
b
cc
lnln1
lnor ln
321
f
a
LK
6
1
);(k
kk faaf
bb
bb
c
T
SG
TTSGTSG
aaaa
T
55
6521
1010 .1
...
f
a
LK
10
1
);(k
kk faaf
2
310310
2
272727
2
333
10921
1010
101010
101010
11
...
ln
SG
TT
SG
T
SG
TT
SG
T
SG
TT
SG
aaaa
P
bb
bbb
bbb
c
f
a
LK For Tbr ≤ 0.8
4
1
4
4
1);(
k
kk
k
kk
fa
fac
af
6
8721
ln1
1
...
ln
brbr
br
br
TTT
aaaa
Pc
f
a
For Tbr > 0.8
6
1
);(k
kk faaf
br
w
br
brwwT
K
TTKK
aaaa
11
...
2
6521
f
a
25
Data regression problems can be significantly simplified if the uncertainties in the
independent variables are neglected. Since we want to determine model parameters that encode
uncertainty information of dependent and independent variables, this simplification is not
warranted and the procedure proposed by Fornasini [47] is used and briefly described below:
1. As a first step, only the uncertainty of the dependent variable (or the experimental
uncertainty .exp)(i
) is taken into account and the approximate values of parameters are
obtained by the least squares method with weighting factors set equal to 2
.exp)(1i
.
2
.exp
2 )(ii 2.15
2. The uncertainties of independent variables are transferred into contributions to the
uncertainty of the dependent variable of the model by the propagation procedure for each
point. The squared uncertainty of the dependent variable calculated using the error
propagation procedure is shown in Equation 2.16:
2
1
2
tra.
);()(
i
m
j
f
jji f
af 2.16
where m' is the number of independent variables and tra.)(i
is the transferred contribution
to the uncertainty of the dependent variable.
3. The squared experimental uncertainty and the squared transferred contributions to the
uncertainty of the dependent variable are then summed for each point.
2
tra.
2
.exp
2 )()(iii 2.17
4. The least squares method is again used to estimate the vector of model parameters (a), but in
this step the weighting factors are equal to 21 i . Weighting factors are updated using
26
Equations 2.16 and 2.17 with the values of a re-estimated at each iteration. The procedure is
repeated until the sum of the absolute differences between two values of total uncertainties
i( ) in two successive iterations is less than the specified tolerance of 10-8
or some fractional
amount like 10-6
.
5. Once acceptable parameters were found using Fornasini’s iterative procedure, the probable
uncertainties in the fitted parameters must be estimated. The covariance between two
parameters aj and al, or variance for j=l, is the sum of the variances of each of the data points
( 2
i ) multiplied by the effect of each data point has on the determination of each parameter
and by assuming that there are no correlations between uncertainties in the measured
variables i , is given by Equation 2.18.
jl
n
i i
l
i
j
iaa Caa
lj
1
22
2.18
The mm symmetric matrix C is commonly known as the covariance matrix or error matrix
where m is the number of fitted parameters. The diagonal elements of matrix C are the
variances (squared uncertainties) associated with estimates of fitted parameters and its off-
diagonal elements are covariances between aj and al, Equation 2.19:
222
222
222
1
1
111
mlmm
lll
ml
aaaaa
amaaaa
aaaaa
C 2.19
6. The parameters associated with their uncertainties are shown in Equation 2.20:
kkk Ca 2.20
27
Now, knowing the uncertainties of all input properties, available model parameters, and
covariance matrix, the squared uncertainty of the calculated dependent variable can be
calculated, Equation 2.21:
1
1 1
1
2
1
2
2
);();(2
);();()(
m
k
m
kl
kl
ilik
m
k
kk
ik
m
j i
f
j
i
Caa
Caf j
afaf
afaf
2.21
The first term shows the effect of uncertainties of all input properties, the second term
models the influence of the variances of estimated parameters, and the third term indicates
the impact of covariances between two parameters on the uncertainty of the calculated
dependent variable.
An important result of this detailed regression procedure is including input parameter
uncertainties and the ability to estimate the uncertainty of estimated critical properties or acentric
factor as a function of the input variables.
The weighted bias, the percentage weighted average bias, the weighted absolute deviation
(AD), and the percentage weighted average absolute deviation (AAD%) were also calculated in
order to compare the updated models and the original correlations, Equations 2.22 to 2.25.
n
i
ii
in1
));((11
Bias
af 2.22
n
i
ii
in1
);(11
AD
af 2.23
n
i i
ii
in1
);(1100%Bias
af 2.24
28
n
i i
ii
in1
);(1100%AAD
af 2.25
where i are experimental values of critical properties and acentric factors, );( af i are
calculated values of critical properties or acentric factors using the models and the parameter
values developed in this work and i are the corresponding experimental uncertainties.
2.4.1.1 Linear Regression
The general form of the linear functions is shown as Equation 2.26 [41]:
m
k
kk fa
1
);( af 2.26
where the dependent variable is expressed as a function of dependent variables f , composed
of a vector of independent variables as functions of the normal boiling point (Tb) and standard
specific gravity (SG), and a is a vector of unknown model parameters, Table 2.1.
The minimum of the objective function (Equation 2.14) is determined by taking partial
derivatives with respect to each parameter and setting them to zero, Equation 2.27:
n
i
ik
m
j
ijji
ik
ffaa
1 1
2
2
0)()(1
2
2.27
Equation 2.27 can be rewritten as:
m
j
n
i
ikij
i
jiki
n
i i
ffaf
1 1
2
1
2)()(
1)(
1
2.28
A set of simultaneous linear equations for parameters la can then be expressed in matrix
form, Equation 2.29:
29
aUβ 2.29
where the matrix a is a row matrix of parameters to fit and the elements of matrix β and
symmetric matrix U are defined by Equations 2.30 and 2.31:
n
i
iki
i
k f
1
2)(
1
2.30
kj
n
i
ikij
i
jkaa
ffU
22
1
2 2
1)()(
1
2.31
By multiplying both sides of the Equation 2.29 by the inverse of matrix U (1Uε ), the
parameter matrix a is obtained as:
m
k
m
k
n
i
iki
i
jkkjkj fa
1 1 1
2)(
1)(
2.32
The covariance of two parameters ja and la , or variance for lj , using Equation 2.18 is
given by:
n
i
jl
m
p
ip
i
lp
m
k
ik
i
jkiaa fflj
1 1
2
1
2
22 )(1
)(1
2.33
The covariance matrix C in a linear regression is the inverse ε of the symmetric matrixU .
The squared uncertainty of the calculated dependent variable can be calculated using
Equation 2.21 and can be expressed as Equation 2.34:
klil
m
k
m
k
m
kl
ikkkik
m
k
ifki fffak
1
1
1 1
2
1
22 2)( 2.34
30
2.4.1.2 Nonlinear Regression
In this study, the procedure was repeated using a nonlinear regression. The Levenberg–
Marquardt [45], as suggested by Press et al. [48], used a general nonlinear form of each model
for each property, Table 2.1. With a nonlinear model the minimization must proceed iteratively
using a selected minimization algorithm. In this study, the Levenberg–Marquardt [45]
minimization method was used. It is an elegant technique that combines advantages of the
Gauss–Newton [46] method for solving a set of linear system of equations and the steepest
descent method [46]. This method inherits its accelerated convergence near the minimum from
the Gauss–Newton iteration method and derives its stability from the steepest descent method.
Given an initial guess for the parameter vector a , a procedure is developed that improves
the trial solution. The procedure is then repeated until2 , Equation 2.14, stops decreasing. The
total uncertainty for each point is updated in each iteration using Equation 2.17.
The gradient of 2 with respect to the parameters a will be zero at the
2 minimum. In
the Levenberg-Marquardt method, described by Press et al. [48], the following formulae are
used to define the components of matrix β , equal to minus one-half times the Gradient matrix,
and matrix U whose components are equal to one-half times the first-derivative terms of the
components of the Hessian matrix, Equations 2.35 and 2.36:
k
i
n
i
ii
ik
ka
;;
a
)()(
1
2
1
1
2
2 afaf
2.35
n
i kj
iii
k
i
j
i
ikj
jkaa
;;
a
;
a
;
aaU
1
2
2
22 )()(
)()(1
2
1 afaf
afaf
2.36
31
Equation 2.36 can be written as Equation 2.37 by ignoring the second partial derivative.
Firstly, this term is small enough to be negligible when compared to the first derivative term.
Secondly, the factor in the second derivative term representing the error of measured value from
the calculated value, )( af ;ii , can have either sign, so the second term tends to cancel when
summed over i.
n
i k
i
j
i
i
jka
;
a
;U
1
2
)()(1 afaf
2.37
The basis of the method to find the parameters a is that when the current estimated
parameter vector ( currenta ) is far from the nexta , then the steepest descent method is best and when
currenta is close to nexta , then Gauss-Newton method is best. So, a new matrix, U , is defined by
Equation 2.38:
)(
)1(
kjUU
UU
jkjk
jjjj
2.38
where is a non-dimensional positive damping factor, 1 to allow switching between the
two methods, steepest descent and Gauss-Newton. When is very large the diagonal elements of
the matrix U dominate, so the method becomes identical to the steepest descent method. On the
other hand, if is very small, the method becomes more Gauss-Newton like. A value equal to
0.001 was assumed for the first iteration for . To determine the parameters, the following set of
linear equations must be solved for the increments ja that, added to the current approximation,
provide the next approximation ) ( currentnext aaa , Equations 2.39 and 2.40:
m
i
kjkjUa
1
2.39
32
1)( Uβa 2.40
After determining the parameters, ) (2aa is evaluated. If )() ( 22
aaa , the
value of is increased by a factor of 10 to follow the gradient more closely and Equation 2.40 is
solved and the procedure is repeated. On the other hand, if )() ( 22aaa , the value of
is decreased by a factor of 10 to reduce the influence of gradient and the trial solution for the
parameters is updated, ( aaa currentnext ), and the procedure is repeated by solving the
Equation 2.40. The iterative procedure stops when 2 decreases by a negligible amount, say10
-6.
When the minimum of 2 was determined, the inverse of matrix U is computed by
setting 0 . As approaches zero, UU and 1
U is the estimated covariance matrix of the
errors in the fitted parameters a :
1UC 2.41
After determining the parameters and covariance matrix of the errors for the fitted
parameters, the squared uncertainty associated with the calculated dependent variable is given
by Equation 2.21.
2.4.2 Results and Discussion
Table 2.2 shows the fitted parameters for critical temperature, critical pressure, acentric factor,
their uncertainties, and their covariance matrices for the re-evaluated Riazi–Daubert correlations.
Tables 2.3 to 2.5 show the estimated parameters and the uncertainties for the re-parameterized
Lee–Kesler correlations and the covariance matrices for the critical temperature, critical
pressure, and acentric factor, respectively.
33
Table 2.2. Fitted parameters and covariance matrices for new Riazi–Daubert correlations
obtained from nonlinear regression.
32
1);(aa
b SGTaaf
Critical Temperature (K)
Parameters 0006.03249.00005.06114.005.042.16 a
Covariance matrix
775
775
553
103350.3 105933.1106977.1
105933.1105991.2 106077.2
106977.1 106077.2106209.2
C
Critical Pressure (kPa)
Parameters ]009.0259.2007.0176.210)1.04.2([ 9 a
Covariance matrix
555
555
5515
106218.7 103286.3103041.5
103286.3103968.4 105273.6
103041.5 105273.6107184.9
C
Table 2.3. Fitted parameters and covariance matrix for the re-parameterized Lee–Kesler
correlation for critical temperature.
Critical Temperature (R)
Parameters, a :
57.074.543.030.212.073.009.005.104.16512.178800.12531.377
Covariance matrix, C :
112211
112211
222211
222311
111144
111144
102242.3 104108.2106928.6 100468.5103411.9100143.7
104108.2108203.1 100438.5108387.3 100132.7 103167.5
106928.6 100438.5104249.1 100825.1109638.1104860.1
100468.5108387.3 100825.1102964.8 104866.1 101350.1
103411.9100132.7 109638.1104866.1 107238.2 100535.2
100143.7 103167.5104860.1 101350.1100535.2105625.1
34
Table 2.4. Fitted parameters and covariance matrix for the re-parameterized Lee–Kesler correlation for critical pressure.
Critical Pressure (psia)
Parameters, a :
4.2306.2219.3714.3508.3567.4342.4605.2425.10985.9895.149.2003.6399.545.1019.986.233.275.309.25
Covariance matrix, C :
4444534433
4555534434
4555534434
4555534434
5555644544
3333422322
4444423333
4444533433
3333423322
3444423322
103081.5 104684.8108054.7109061.9105023.2 108482.2 104412.1 102835.2102059.5106075.6
104684.8103829.1 102723.1 105555.1 100774.4107312.4103155.2107490.3 105349.8 100970.1
108054.7102723.1 102729.1 102824.1 107232.3100310.5101071.2105262.3 103627.8 100733.1
109061.9105555.1 102824.1 101177.2 106722.4101391.4107449.2100981.4 108646.8 101143.1
105023.2 100774.4107232.3106722.4102066.1 103729.1 108823.6 101075.1105137.2102277.3
108482.2 107312.4100310.5101391.4103729.1 101030.2 106370.7 103389.1102734.3102371.4
104412.1 103155.2101071.2107449.2108823.6 106370.7 109729.3 102893.6104260.1108172.1
102835.2107490.3 105262.3 100981.4 101075.1103389.1102893.6100307.1 103736.2 100594.3
102059.5105349.8 103627.8 108646.8 105137.2102734.3104260.1103736.2 105803.5 101847.7
106075.6 100970.1100733.1101143.1102277.3 102371.4 108172.1 100594.3101847.7103089.9
35
Table 2.5. Fitted parameters and covariance matrix for the re-parameterized Lee–Kesler correlation for acentric factor.
Acentric Factor
For Tbr ≤ 0.8
Parameters, a :
1.1014.34.9279.884.5930.655.5078.541.1519.185.3933.544.2097.349.1417.25
Covariance matrix, C :
44443332
45555554
45554544
45554544
35444444
35554544
35444444
24444444
100225.1 107987.7108802.4100693.4 109865.3100198.5 108325.1 106115.3
107987.7106014.8 104999.5 106952.4100465.1 103704.2102000.1104655.7
108802.4104999.5 105212.3 100097.3108613.6 105694.1109689.7109851.4
100693.4 106952.4100097.3105758.2 100162.6103891.1 100750.7 104496.4
109865.3100465.1 108613.6 100162.6102836.2 108667.5100978.3100667.2
100198.5 103704.2105694.1103891.1 108667.5105483.1 102299.8 105504.5
108325.1 102000.1109689.7100750.7 100978.3102299.8 103826.4 109640.2
106115.3104655.7 109851.4 104496.4100667.2 105504.5109640.2100135.2
For Tbr > 0.8
Parameters, a :
2.33039.217.337245.2170.143472.941.2926.12.115515.686.789371.458
Covariance matrix, C :
787578
898689
788689
566467
788688
899789
100911.1 101109.1105935.4 104617.9 107820.3105656.2
101109.1101374.1 105885.4107048.9108698.3 106397.2
105935.4 105885.4100584.2 108838.3 105657.1100414.1
104617.9 107048.9108838.3 105350.8 103674.3103000.2
107820.3108698.3 105657.1103674.3103343.1 100931.9
105656.2 106397.2100414.1 103000.2 100931.9102312.6
36
Table 2.6 shows the values of the weighted bias, the percentage weighted average bias,
the weighted absolute deviation (AD), and the percentage weighted average absolute deviation
(AAD %) for each model and each property when the uncertainties of the independent variables
(functions of normal boiling point and specific gravity) are taken into account.
Table 2.6. Comparison of re-evaluated correlations using weighted deviation.
Property Model Bias AD Bias% AAD%
Critical temperature (Tc)
RD 8.62 (K) 15.29 (K) 1.56 2.63
New RD 2.39 (K) 13.37 (K) 0.47 2.28
LK –7.84 (K) 13.70 (K) –1.35 2.32
New LK 3.76 (K) 13.62 (K) 0.67 2.27
Critical pressure (Pc)
RD 0.16 (kPa) 3.42 (kPa) 0.03 0.14
New RD 0.04 (kPa) 3.25 (kPa) 0.04 0.14
LK 1.53 (kPa) 3.23 (kPa) 0.07 0.13
New LK 2.03 (kPa) 3.52 (kPa) 0.11 0.16
Acentric factor () LK –0.18 0.37 –78.18 110.29
New LK 0.03 0.13 7.88 34.58
For critical temperature, the new and original versions of the Riazi–Daubert and Lee–
Kesler models show approximately similar AAD% values while for critical pressure, the re-
parameterized Lee–Kesler model presents a greater AAD% value in comparison with the other
correlations. Since the uncertainties of critical pressures and also independent variables, which
are functions of normal boiling point and specific gravity, are taken into account in the least
square method as weighting factors, the greater uncertainty leads to smaller weighting factor for
some points. This data with greater uncertainty had a decreased importance in model parameters
determination.
37
For the acentric factor, the new Lee–Kesler model had a smaller AAD% value than the
original Lee–Kesler correlation. The re-parameterized models for critical properties and acentric
factors are of similar quality when compared to the original models but now encode the
uncertainty of each independent variable and can be used for more advanced statistical analyses.
It is useful to note the uncertainties of specific gravity and normal boiling point as shown
in Figure 2.5. This figure shows the uncertainties in specific gravity using vertical error bars and
uncertainties in the normal boiling point using horizontal error bars for each hydrocarbon.
Figure 2.5. Uncertainties of normal boiling point and specific gravity.
As shown in Figure 2.5, 1,1-methylenebis[(1-methylethyl)benzene] (C19H24) with
Tb=592±35 K and SG=0.99±0.03, 1-nonadecene (C19H38) with Tb=604±16 K and
SG=0.795±0.004, and 1-eicosene (C20H40) with Tb=620±15 K and SG=0.827±0.025 have the
highest uncertainties in the experimental values of the normal boiling point, respectively. On the
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
250 300 350 400 450 500 550 600 650 700 750 800
Sp
ecif
ic G
rav
ity
Normal Boilng Point (K)
38
other hand, p-terphenyl (C18H14) with SG=1.15±0.11 and Tb=657.01±1.39 K, hexamethylbenzene
(C12H18) with SG=0.911±0.091 and Tb=540.9±3.2 K, and 1,2,4,5-tetrakis(1-methylethyl)benzene
(C18H30) with SG=0.856±0.085 and Tb=533.95±0.02 K have the highest uncertainties in the
experimental values of their standard specific gravity. This figure is a complementary figure for
Figures 2.6 to 2.8.
The comparison between the results of the re-evaluated Riazi and Daubert’s model with
original one is shown in Figure 2.6 for the critical temperatures and in Figure 2.7 for the critical
pressures. The figures show the critical properties as a function of the normal boiling point. The
horizontal error bar refers to normal boiling point uncertainty and the vertical error bar indicates
the uncertainty of experimental value of the critical property. As shown in the figures for both
critical temperatures and pressures, the new correlations estimate the critical values very well,
specifically for hydrocarbons with normal boiling points up to about 660 K. Note, the results of
the updated correlation are similar to Riazi–Daubert’s model and the deviation between predicted
value and experimental data are negligible for the entire range, but for high boiling point
components the deviation did increase. This occurs because a more rigorous regression takes
place for the lower boiling point compounds due to the smaller uncertainties.
Figure 2.6 shows that 3,7,7-trimethyl-bicyclo[4.1.0]hep-3-ene (C10H16) with
Tc=660.0±21.7 K and Tb=445.0±2.5 K and p-terphenyl (C18H14) with Tc=912.9±21.6 K and
Tb=657.01±1.39 K have the largest uncertainties for experimental values of critical temperature.
Similarly, as shown in Figure 2.7, 3,7,7-trimethyl-bicyclo[4.1.0]hep-3-ene (C10H16) with
Pc=2967±876 kPa, and 2,3-dimethyl-1-pentene (C7H14) with Pc=2856±795 kPa and
Tb=357.4±1.5 K have the highest values of critical pressure uncertainties.
39
Figure 2.6. Critical temperature versus normal boiling point.
Figure 2.7. Critical pressure versus normal boiling point.
400
500
600
700
800
900
1000
250 300 350 400 450 500 550 600 650 700 750 800
Cri
tica
l T
emp
era
ture
(K
)
Normal Boilng Point (K)
Exp.
New RD
RD
0
1000
2000
3000
4000
5000
6000
250 300 350 400 450 500 550 600 650 700 750 800
Cri
tica
l P
ress
ure
(k
Pa
)
Normal Boilng Point (K)
Exp.
New RD
RD
40
To compare the re-evaluated Lee and Kesler correlation for acentric factor with the
original one, the calculated acentric factor values were plotted against the normal boiling points
in Figure 2.8. Figure 2.8 shows that the updated model and original one predict approximately
the same values for compounds with lower boiling points, while the updated model estimates the
acentric factor of higher boilers more accurately than the original Lee–Kesler correlation. As
shown in Figure 2.8, 1,1-methylenebis[(1-methylethyl)benzene] (C19H24) with =0.500±0.435,
1-eicosene (C20H40) with =0.881±0.222, and 1-nonadecene (C19H38) with =0.877±0.217 have
the largest acentric factor uncertainties.
Figure 2.8. Acentric factor versus normal boiling point.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
250 300 350 400 450 500 550 600 650 700 750 800
Ace
ntr
ic F
act
or
Normal Boilng Point (K)
Exp.
New LK
LK
41
2.4.2.1 Examples
The application of the updated correlations for the calculation of critical properties and acentric
factors and their associated uncertainties is illustrated through two examples. In the first
example, the critical temperature of a C7+ fraction with a normal boiling point of 508.2±0.5 K
and specific gravity of 0.8259±0.001 was calculated using the Riazi and Daubert nonlinear
model, as described in Table 2.1, with the fitted parameters from Table 2.2. The critical
temperature uncertainty can be determined by using the covariance matrix for this model, also
shown in Table 2.2 and in Equation 2.21. The critical temperature for this oil fraction was
calculated to be 696.5±0.5 K.
In the second example, the critical temperatures, critical pressures and acentric factors of
n-hexane and n-dodecane and their associated uncertainties were calculated using the updated
Lee and Kesler correlations. Note, for components with available experimental data and
uncertainties, the experimental values of critical properties were used to calculate the acentric
factor using the Lee and Kesler correlation, Table 2.1. For oil fractions usually only the normal
boiling point and standard gravity are available from oil characterization and all properties are
functions of these two parameters. The results were summarized in Table 2.7.
Table 2.7. Comparison of the experimental and calculated critical properties and acentric
factors of n-hexane and n-dodecane.
Compound Tc (K) Pc (kPa)
n-Hexane
Experimental
Calculated
507.53 ± 0.14
509.08 ± 0.33
3031.18 ± 32.39
2861.50 ± 182.27
0.301 ± 0.005
0.301 ± 0.020
n-Dodecane
Experimental
Calculated
658.28 ± 0.59
658.69 ± 0.47
1812.40 ± 87.20
1787.4 ± 186.0
0.572 ± 0.021
0.571 ± 0.072
42
2.5 Effect of Uncertainties in Thermodynamic Data on Calculated Thermo-physical
Properties
The uncertainties associated with basic thermo-physical properties such as critical temperatures,
critical pressures, and acentric factors affect the quality of results calculated using equations of
state for boiling points, densities, enthalpies, and phase equilibria. To quantify thermodynamic
model uncertainties, Monte Carlo type techniques [16] can be used to propagate input
uncertainties into result uncertainties. In order to use this method and estimate the effect of input
uncertainties on the thermo-physical property of interest, specifying uncertain inputs, the Monte
Carlo technique and sampling method will be explained in following sections. Then, one simple
example was chosen to illustrate how this method works.
2.5.1 Notes on the Uncertainty of Input Variables
Critical temperature, critical pressure and vapour pressure data and their associated uncertainties
are taken from the developed database. It is assumed that the pure component uncertainty values
used in the model development are characterized by normal (Gaussian) distributions, the same
assumption that is used to develop the uncertainties in TDE. The mean values () for a certain
physical property would be equal to the evaluated true values in TDE and the 95% confidence
level used in TDE corresponds to a confidence interval bound by ± 2S. The standard deviation
(S) is then one half of calculated uncertainties.
2.5.2 The Monte Carlo Technique and Sampling
Monte Carlo methods are useful tools for uncertainty propagation analysis by performing
random sampling from probability distributions [5] for complex models. In our case we are
interested in determining the uncertainties of variables derived from process flow sheeting
43
calculations. The use of Taylor based linearization is cumbersome to say the least, except for the
simplest models.
Thus a “lottery” is constructed where the values of the physical property of interest are
calculated using the process model, for example saturation temperatures or compressor horse
power are repeatedly calculated using different values for the input thermodynamic model such
as critical temperature, critical pressure, and acentric factor, assuming that they are randomly
distributed over the range of their uncertainties. This is a significant assumption since the
distribution of experimental data does not necessarily have to be Gaussian, but it is the most
reasonable assumption that can be made.
Commonly used sampling techniques associated with the Monte Carlo method are
random Monte Carlo sampling (MCS) [5], Latin Hypercube sampling (LHS) [6], Shifted
Hammersley sampling (SHS) [49], and Equal Probability sampling (EPS) [16]. In this study, the
traditional random Monte Carlo sampling, in which the samples are taken at random within the
whole range of distribution of variables, was used to determine the effect of sample size on the
estimation of property uncertainty. The development of guidelines for sampling size is an
important feature for the use of this technique in process quality assurance and is explored
briefly using the calculation of the normal boiling point of simple compounds using the Peng–
Robinson equation of state [12].
Three sets of numerical experiments were selected, with 100, 1,000, and 10,000 samples.
For each sample set, the results for the calculated normal boiling point are the mean value and
the quantified uncertainty. Finally, the results obtained for each set are compared.
The Peng–Robinson equation of state was used for the calculation of normal boiling point
for n-hexane and n-dodecane with physical properties and uncertainties shown in Table 2.7. The
44
input variables are critical temperature, critical pressure, and acentric factor for each component
and the assumed random values based on the Gaussian distribution with defined average and
standard deviation (half of the associated uncertainty) as shown in Table 2.7. Figure 2.9 shows
the assumed probability distributions for the critical temperature for the two components. The
same distribution was also used for the critical pressures and acentric factors.
Figure 2.9. Critical temperature normal distributions for (a) n-hexane (b) n-dodecane.
Three sets of samples with 100, 1,000, and 10,000 random values for each of the three
parameters were prepared. For each of these values the normal boiling point is calculated using
the Peng–Robinson equation of state and Soave’s iteration free saturation calculation method
[50]. The mean value and the standard deviation were calculated using Equations 2.42 and 2.43
where n' is the size of the sample.
n
i
bb iT
nT
1
1 2.42
n
i
bbT TTn
Sib
1
2
1
1 2.43
0.0
2.0
4.0
6.0
507.35 507.45 507.55 507.65
Pro
ba
bil
ity
Den
sity
(1
/K)
Critical Temperature (K)
(a)
0.0
0.4
0.8
1.2
1.6
657.6 658 658.4 658.8
Pro
ba
bil
ity
Den
sity
(1
/K)
Critical Temperature (K)
(b)
45
Results for each sample size and each component are presented in Table 2.8 along with
their standard deviations. The results can be analyzed in two different views: compare with the
experimental values and compare the results calculated for each sample size. Firstly, comparison
the results with the values from the prepared database (Appendix A) shows that the calculated
mean values using the Peng–Robinson equation of state are close to the experimental values
(341.85±0.11 K for n-Hexane and 489.45±0.12 K for n-Dodecane), but the estimated
uncertainties are greater than the experimental ones.
Secondly, based on this sampling study it seems that relatively small samples can be used
to provide acceptable uncertainty estimates for values calculated from complex models. This
issue is can be further investigated to develop a more comprehensive set of sample size criteria
using a rigorous statistical approach.
Table 2.8. Monte Carlo sampling for normal boiling point.
No. of Samples Mean Value
(K)
Range (Min-Max)
(K)
Standard Deviation
(K)
Uncertainty
(K)
n-Hexane
n' = 100 341.930 341.253 – 342.529 0.297 0.594
n' = 1,000 341.938 341.146 – 342.710 0.318 0.636
n' = 10,000 341.940 341.136 – 342.779 0.316 0.633
n-Dodecane
n' = 100 488.886 485.648 – 491.610 1.366 2.732
n' = 1,000 488.930 485.291 – 492.445 1.476 2.952
n' = 10,000 488.946 485.233 – 492.824 1.476 2.952
46
2.6 Re-parameterization of the Peng–Robinson Equation of State
With available uncertainties for the necessary input parameters required to develop a cubic
equation of state, namely critical temperature, critical pressure, vapour pressure and acentric
factor combined with the necessary uncertainties for the physical properties, it is now possible to
re-parameterize an equation of state that will then provide uncertainty information when used in
process simulations. Note, in order to develop an equation of state suitable for technical
applications in the hydrocarbon industry, it is necessary to provide estimation methods for
pseudo-components or “plus” fractions as presented earlier in this study. Due to its popularity,
the Peng–Robinson equation of state was chosen, although any other equation could have been
selected.
The most important physical property from a process simulation point of view is the
vapour pressure, and the same formulation as proposed in the original Peng and Robinson paper
was used, based on Soave’s form, Equation 2.37:
211 rPR Tf 2.44
where fω was defined as an equation with three parameters in the original form of Peng–
Robinson equation of state and presented as equation with four parameters in their improved
version of the equation of state for a pure component with acentric factor above 0.491 in 1978
[51].
The parameters were determined by minimizing the error between the estimated and
experimental vapour pressures, calculated by Soave method [50] from the normal boiling point
to the critical temperature using twenty equally spaced points. All input parameters – critical
temperature, critical pressure and vapour pressure are used in the regression taking into account
47
their uncertainties. The same procedure was used to find the parameters and their uncertainties
for both of these equations for all the hydrocarbons in the database. Table 2.9 shows the
equations and results for the original and the improved Peng–Robinson equations of state.
Table 2.9. Peng-Robinson equation of state refitted parameters and covariance matrix.
2
321 aaaf
Parameters 0273.00369.00199.04153.10035.03908.0 a
Covariance matrix
445
445
555
104825.7 103088.5108538.8
103088.5109476.3 108363.6
108538.8 108363.6102268.1
C
3
4
2
321 aaaaf
Parameters 0807.01876.01117.02895.00482.05178.10065.03780.0 a
Covariance matrix
3334
3234
3334
4445
105048.6 107336.8105364.3 103979.4
107336.8102486.1 102871.5108038.6
105364.3 102871.5103231.2 100845.3
103979.4108038.6 100845.3102173.4
C
In order to compare the three-parameter equation and four-parameter equation with the
original PR equation of state, the values of the weighted bias, the percentage weighted average
bias, the weighted absolute deviation (AD), and the percentage weighted average absolute
deviation (AAD %) for vapour pressure were calculated. The results are presented in Table 2.10.
Table 2.10. Comparison results of the original PR equation and the refitted equations.
Model Bias (kPa) AD (kPa) Bias% AAD%
Original three-parameter PR 0.51 1.59 0.12 0.68
Re-parameterized three-parameter PR –0.15 1.26 –0.31 0.52
Re-parameterized four-parameter PR –0.14 1.26 –0.30 0.52
48
The original and the improved Peng–Robinson equation of state show similar values for
the calculated vapour pressures; however, the AAD% values for the re-parameterized equations
are lower than those of the original equation. Both of the re-parameterized equations can be used
to calculate the vapour pressure and associated uncertainty. Figure 2.10 demonstrates the quality
of the predictions of the vapour pressure.
Figure 2.10. Comparison of vapour pressure calculated using the original and the
improved Peng–Robinson equations of state.
The uncertainty values calculated using the covariance-based approach for the 3-
parameter Peng–Robinson equation of state are shown in Appendix B for methane, n-hexane,
and n-dodecane. Typical uncertainty values calculated using both versions of the re-
parameterized equation of state for each of the components at one temperature are shown in
Table 2.11.
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
100 200 300 400 500 600 700
Va
po
r P
ress
ure
(k
Pa
)
Temperature (K)
Experimental
From three-parameter equation
From four-parameter equation
Methane
= 0.0116
n-Hexane
= 0.3009
n-Dodecane
= 0.5715
49
In order to compare the covariance-based approach with the Monte Carlo simulation, the
vapour pressure calculation was performed using the Monte Carlo method with a sample of
1,000 points which are selected using Latin Hypercube Sampling (LHS) technique [6]. Each data
set contains critical temperature, critical pressure, acentric factor and the calculated parameters
of the re-parameterized Peng–Robinson equation of state. Table 2.11 shows that the results of
covariance-based approach and the Monte Carlo simulation are basically the same. So, the
Monte Carlo simulation was selected for this study as an efficient error propagation algorithm to
evaluate the uncertainty in physical properties calculations.
Table 2.11. Comparison of vapour pressure and its uncertainty calculated using the
covariance-based approach and the Monte Carlo simulation.
Model Component @ T(K) Vapour Pressure (kPa)
Covariance-Based Monte Carlo
Three-parameter PR
Methane @ 153 K 1183.3 ± 4.8 1183.3 ± 4.4
n-Hexane @ 432.4 K 902.1 ± 10.6 902.1 ± 10.2
n-Dodecane @ 580.94 K 601.3 ± 30.5 601.4 ± 27.5
Four-parameter PR
Methane @ 153 K 1195.1 ± 7.1 1195.1 ± 6.6
n-Hexane @ 432.4 K 901.9 ± 10.6 901.9 ± 13.0
n-Dodecane @ 580.94 K 601.8 ± 30.5 601.9 ± 28.7
2.7 Natural Gas Processing Examples
In this section we examine the use of the data and techniques developed in this study using two
simple but representative examples in natural gas processing. In both cases we wish to compress
10 Million Standard Cubic Feet per Day (MMSCFD) lean natural gas from 2068.43 kPa (300
50
psia) to 6205.28 kPa (900 psia) using a single stage ideal compressor. The gas enters the
compressor at 25 C and the intercooler has a pressure drop of 68.95 kPa (10 psia) with a
specified outlet temperature of 48.9 C.
Two gas compositions were used, one with 0.999 methane and 0.001 n-hexane
(Composition 1) and another with 0.99999 methane and 0.00001 n-dodecane (Composition 2).
The results are summarized in Tables 2.12 and show the locations of the critical point,
cricondenbar, and cricondentherm on the pressure-temperature envelope for each composition.
The cricondenbar indicates the maximum pressure on the two-phase boundary at which liquid
and vapour can coexist in equilibrium and is located on the on the highest point of the phase
envelope. The farthest point to the right on the pressure-temperature envelope indicates the
cricondentherm that is the lowest temperature above which hydrocarbon can exist in a vapour
phase alone.
Table 2.12. Critical point, cricondenbar and cricondentherm coordinates when
compressing lean natural gas prototype mixtures.
Critical Point Cricondenbar Cricondentherm
Composition 1 192.16 ± 0.01 (K)
4777 ± 9 (kPa)
217.2 ± 0.3 (K)
6530 ± 43 (kPa)
243.5 ± 0.5 (K)
2710 ± 11 (kPa)
Composition 2 190.6 ± 0.1 (K)
4620 ± 20 (kPa)
234.0 ± 1.0 (K)
8100 ± 200 (kPa)
285.0 ± 2.0 (K)
1930 ± 40 (kPa)
Table 2.13 shows the basic information for the compressor and intercooler and
corresponding uncertainties propagated from the uncertainties of pure component critical
properties and acentric factors.
51
Table 2.13. Basic equipment performance data estimated using uncertainty information.
Temperature after
Compressor, K
Compressor Horse
Power, HP
Intercooler Duty,
kJ/h
Composition 1 383.55 ± 0.01 553.2 ± 0.1 1,325,500 ± 400
Composition 2 383.78 ± 0.01 553.7 ± 0.1 1,325,800 ± 400
The gas compositions were chosen to illustrate the effect of a heavy trace component that
may be present in the natural gas. In this case, the uncertainty introduced in the basic energy and
entropy balances as demonstrated by the intercooler duty, compressor horse power and
compressor outlet temperature are negligible, while the position of the cricondenbar and
cricondentherm present significantly more uncertainty as the model component for the natural
gas heavy fraction moves from n-hexane to n-dodecane. Of particular importance is the
uncertainty of the position of the cricondentherm and its effect on the selection of intercooler
operating temperatures to ensure proper compressor operation. It is important to stress that these
uncertainties are related to thermodynamics alone and a complete picture of the problem can
only be established by taking into account the uncertainties of measured inputs such as flows,
temperatures and pressures as well as uncertainties related to equipment such as heat transfer
coefficients and compressor efficiencies.
Figures 2.11 and 2.12 show the pressure–temperature envelopes for two compositions
(Composition 1 and Composition 2) with the uncertainty region calculated using the Monte
Carlo technique. The position of the cricondenbar and cricondentherm present significantly more
uncertainty as the natural gas heavy fraction moves from n-hexane to n-dodecane. The presented
uncertainties are related only the critical pressure, critical temperature and acentric factor
uncertainties.
52
Figure 2.11. Pressure–temperature envelope for Composition 1 (methane and n-hexane).
Figure 2.12. Pressure–temperature envelope for Composition 2 (methane and n-dodecane).
0
1000
2000
3000
4000
5000
6000
7000
120 140 160 180 200 220 240
Pre
ssu
re (
kP
a)
Temperature (K)
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
120 140 160 180 200 220 240 260 280 300
Pre
ssu
re (
kP
a)
Temperature (K)
53
2.8 Conclusions
A comprehensive database for physical properties of simple hydrocarbons with uncertainties was
developed based on NIST’s TDE dynamic evaluation system. The database includes data and
uncertainties for critical properties and acentric factors, fundamental to the development of
models important for the hydrocarbon industry and the estimation of uncertainties related to oil
fractions.
The Riazi and Daubert, and Lee and Kesler correlations were re-evaluated using the
database developed in this work and new correlations were presented to predict the critical
temperature, critical pressure, and acentric factor of pure hydrocarbons in the C5–C36 range.
Model parameters were determined using linear and nonlinear regression methods. The main
advantage of these correlations over the original correlations is their ability to estimate the
uncertainties of critical properties and acentric factors based on the uncertainty of the input
variables (normal boiling point and specific gravity). The parameters of these models were
obtained using the weighted least squares method taking into account the uncertainties of both
dependent and independent variables. The covariance matrix or error matrix was reported for
each model. Using this matrix and the standard error propagation procedures, the critical
properties and acentric factor uncertainties can be predicted for individual compounds or oil
fractions. These re-parameterized correlations can be used in thermodynamic models developed
for the natural gas and refining industries with minimal modifications to existing computer code
and provide estimates for the uncertainty of calculated physical properties. In turn these
uncertainties can be used to estimate uncertainties in process equipment hardware such as sizes
of separators, number or diameter of distillation trays, heat exchanger areas, or compressor horse
power.
54
The Monte Carlo technique can be used to evaluate the error propagation from
uncertainties in input variables of an equation of state to an estimated thermo-physical property
of interest. The Monte Carlo method is general and although computer intensive, can be used to
estimate the uncertainty derived from physical properties and corresponding thermodynamic
models for simulations of unlimited complexity. This was illustrated through estimates of the
uncertainties for boiling point calculations using the Peng–Robinson equation of state, a PT
envelope for a model natural gas mixture and compressor horse power.
A brief study of sample size in the Monte Carlo method was conducted, suggesting that
relatively small sample sizes in the order of 100 randomly distributed inputs may be adequate,
thus placing the method in a favorable light for use in the analysis of uncertainty of chemical
plant simulations. The thermodynamic model developed in this chapter is extended in Chapter 3
to include the uncertainties in the original vapour–liquid equilibrium data and a database of
interaction parameters with uncertainties will be developed.
55
Chapter Three: Uncertainty Analysis Applied to Thermodynamic Models and Process
Design – 2. Binary Mixtures 2
3.1 Abstract
A simple procedure is proposed to evaluate the uncertainty of the binary interaction parameters
from the uncertainties present in the physical properties and equation parameters used for their
calculation using a cubic equation of state. A small but useful database containing 87 binary
mixtures present in natural gas was constructed through collection of available experimental
vapour–liquid equilibrium (VLE) data and associated uncertainties.
A thermodynamic consistency test was performed on each isothermal dataset to
determine the quality of the VLE data. Upon acceptance of the VLE data based on its quality and
consistency, binary interaction parameters and associated uncertainties were determined using a
combination of nonlinear regression and Monte Carlo simulation, taking into account the
uncertainties of the pure components, equation of state parameters, and VLE data. The Monte
Carlo simulation was also used for the error propagation to estimate the uncertainty in the
calculated VLE. Sample calculations were presented illustrating the effect of uncertainties in the
PXY and TXY diagrams of ethane/propane and methane/hydrogen sulfide binary mixtures. The
required minimum number of stages for a simplified de-ethanizer was calculated taking into
account uncertainties in the basic input parameters. In addition the effect of uncertainties in the
position of calculated cricondenbar and cricondentherm was evaluated.
2 Reprinted from Fluid Phase Equilibria, Vol. 364, S. Hajipour, M.A. Satyro, M.W. Foley, Uncertainty Analysis
Applied to Thermodynamic Models and Process Design – 2. Binary Mixtures, 15-30, Copyright (2013), with
permission from Elsevier. It should be noted that the content of this chapter includes the extensions beyond the
cited journal paper.
56
3.2 Introduction
In Chapter 2, a comprehensive database for physical properties of pure hydrocarbons commonly
present in natural gas and their associated uncertainties was developed. Two correlations were
re-parameterized to estimate the critical properties and acentric factors of pseudo-components,
and the Peng–Robinson (PR EOS) [12] equation of state was re-parameterized to reflect the
uncertainties of the underlining physical property data in its structure as represented by the
generalized alpha parameter. The covariance matrices of model parameters were also presented
which can then be used to provide uncertainty information when used in process simulation and
make it possible to estimate the uncertainties of physical properties and thermo-physical
properties of pure components through error propagation calculations. In this Chapter, the
applicability of the re-parameterized PR equation of state is extended to binary mixtures,
required in the evaluation of the uncertainties in process simulation.
Several studies in the area of uncertainty effect on process design emphasize the
importance of uncertainty analysis in the field of process engineering and most of them were
reviewed in Chapter 2. Recently, uncertainty analysis in design and operation of biochemical
processes has been performed by Sin and co-workers on a case study basis; for example,
antibiotic production [52] and biofuel production [53]. They analyzed the uncertainty of model
predictions for a cellulose hydrolysis process and calculated the mean values of the estimated
parameters as well as their variance-covariance matrix based on a 95% confidence interval [53]
using the Monte Carlo technique combined with Latin Hypercube Sampling (LHS) method.
Cinnella et al. [54] also presented error propagation from some common thermodynamic models,
such as the PR EOS, to the pure dense gas flow fields predicted by a computational fluid
dynamics (CFD) solver. They quantified the impact of such uncertainties on aerodynamic data
57
and concluded more complex models may be more sensitive to uncertainties of the fluid physical
properties due to the larger number of input parameters involved.
Most processes in the petroleum and chemical industries operate in liquid and/or vapour
phases. Therefore the proper modelling of VLE is a significant step for the proper design and
simulation of processes. Cubic equations of state are extensively used in process simulators as
reliable models for the prediction of thermodynamic properties and phase equilibria for
hydrocarbon systems due to their simplicity, robustness, and computational efficiency.
Mixing rules and binary interaction parameters are used to generalize the equations of
state from pure fluids to mixtures and to improve the quality of VLE predictions. There is a large
body of work on mixing rules for cubic equations of state and will not be reviewed here because
it is beyond the scope of this study. Suffice it to say that commonly used mixing rules to
represent the behaviour of hydrocarbon mixtures will be used, and a more advanced mixing rule
for thermodynamic consistency calculations will be used when necessary. The procedures and
methods proposed in this Chapter are completely general and can be used with any equation of
state and associated mixing rules.
Optimum values of binary interaction parameters are determined by regression using
screened VLE data. Critical properties and acentric factors of pure components, thermodynamic
model parameters and the VLE data including pressure (P), temperature (T) and both liquid and
vapour phase compositions (x, y), are considered as independent variables used to estimate
binary interaction parameters.
As previously discussed in Chapter 2, uncertainties in input variables propagate through
the model and do affect the accuracy of the final results. Therefore, the quality of the adjusted
model parameters depends on the quality of all dependent and independent variables. Presently
58
binary interaction parameters are used in process simulators without statistical uncertainty
information. This lack of uncertainty information precludes the critical analysis of processes and
associated equipment and consequently further process development is hampered by the use of
empirical rules of thumb for determination of an appropriate overdesign factor.
The experimental uncertainties in VLE data available in the ThermoData Engine (TDE)
[2] developed by the National Institute of Standards and Technology (NIST) are used for the
estimation of uncertainties in the binary interaction parameters together with the pure component
database and EOS developed in Chapter 1. NIST’s ThermoData Engine (TDE) software and the
SOURCE database [3, 4] implement the concept of a dynamic data evaluation to thermo-physical
property data and this fact enables the development of databases and correlations not only for
pure components but also for binary mixtures that reflect the associated uncertainties. TDE
evaluates the uncertainties through the examination of the quality of experimental data originated
from the SOURCE database which combines the knowledge embodied in the scientific literature
with the expert knowledge of NIST’s scientists about error propagation and statistical analysis
for the development of recommended uncertainties for the collected data [55]. In this study, the
uncertainties in physical properties of pure components and binary vapour–liquid equilibrium
data evaluated by TDE are used for estimation of binary interaction parameters and their
uncertainties. The uncertainties of the VLE data were taken into account as weighting factors in
the objective function used in the parameter estimation and they are also considered, along with
the uncertainties related to pure components for the uncertainty estimation of the interaction
parameters using Monte Carlo simulation. This approach maximizes the chance that the best set
of model parameters are calculated for binary mixtures from a data quality point of view.
59
Small uncertainties in phase equilibrium may have a significant impact on process
simulation, design, and performance. This is illustrated by a simple example where uncertainties
in the vapour and liquid compositions are introduced in the design of a simplified de-ethanizer
used to stabilize liquefied petroleum gas (LPG). It is assumed that the de-ethanizer operates at
2758 kPa with a distillate purity specification of 0.99 ± 0.005 (C2) mole fraction and a bottoms
product with a purity specification of 0.98 ± 0.005 (C3) mole fraction.
Figure 3.1 shows the TXY diagram for ethane/propane using the experimental VLE data
[56] and uncertainties of 0.11 K for temperature, 0.005 for liquid phase composition (x), and
0.005 for vapour phase composition (y).
Figure 3.1. Temperature-composition diagram for ethane/propane system at 2758 kPa.
Note that the thickness of the TXY “curves” actually represents the uncertainties
associated with the bubble and dew points curves.
Figure 3.2(a) shows the K-value for each component (Ki=yi / xi) versus mole fraction of
ethane and the error bars indicate the effects of uncertainties in compositions on the calculated
270
290
310
330
350
0.0 0.2 0.4 0.6 0.8 1.0
Tem
per
atu
re (
K)
Ethane (Mole Fraction)
Ethane/Propane System
P= 2758 kPa [56]
60
equilibrium constants at each experimental point using the standard error propagation equation,
Equation 3.1:
22
i
y
i
x
i
K
yxK
iii
3.1
where iK ,
ix , and iy are the uncertainties of the K-value, liquid phase composition and
vapour phase composition for component i, respectively. In this calculation, it was assumed that
there was no correlation between uncertainties in the measured compositions; therefore, the off-
diagonal terms of the covariance matrix are zero. Note the error in equilibrium constant increases
as either ethane or propane approaches high dilution due to the inverse relation between the
uncertainty of K-value and the phase compositions.
The error bars in Figure 3.2(b) indicate the errors in relative volatility propagated from
the uncertainties in compositions at each experimental point against mole fraction of ethane. This
case study provides a compelling example of the effect of uncertainties in physical properties on
equipment design, in this case through the relative volatility. The effect of uncertainties on the
design of the de-ethanizer can be simply illustrated when the Fenske equation [57], Equation 3.2,
is used to calculate the minimum number of stages (Nmin) required for the separation.
.
minln
)1(
)1(ln
avg
DB
BD
xx
xx
N
3.2
where xD and xB are liquid mole fractions of ethane in distillate and bottoms products, and .avg
refers to the geometric average relative volatility of top-stage and bottom-stage values.
61
Figure 3.2. Effect of uncertainties in compositions on (a) vapour–liquid equilibrium
constant (Ki), and (b) relative volatility (α) for ethane/propane system at pressure of 2758
kPa.
0.0
0.5
1.0
1.5
2.0
2.5
0.0 0.2 0.4 0.6 0.8 1.0
Va
po
ur-L
iqu
id E
qu
ilib
riu
m C
on
sta
nt
Ethane (Mole Fraction)
Ethane
Propane
(a)
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
0.0 0.2 0.4 0.6 0.8 1.0
Rel
ati
ve
Vo
lati
lity
fo
r C
2/C
3
Ethane (Mole Fraction)
(b)
62
Without considering the uncertainties in equilibrium data, the minimum number of stages
would be 12, while taking uncertainties into account the estimated number of minimum stages is
12 ± 10 stages. The uncertainty in the minimum number of stages (minN ) is calculated by
Equation 3.3:
22
min
min
22
min
min
.min )1()1(
2
)1()1(
2
ln2
1min
BB
y
BB
x
DD
y
DD
x
avg
N
yyNxx
N
yyNxx
N
N
BBDD
3.3
where yD and yB are vapour mole fractions of ethane in the distillate and bottom products.
This variability in the number of stages has a significant effect on the equipment capital
cost, its operational cost, and for its intended purpose. The results obtained in this example are in
agreement with results presented by Dohrn and Pfohl [58] and will be revisited at the end of this
chapter using a complete thermodynamic model with uncertainties.
The quality of the existing VLE data is important for development of binary interaction
parameters and it is important to have a general method, based on thermodynamic principles, to
screen the experimental data quality. When complete TPXY data is collected there is an intrinsic
redundancy as shown by the Gibbs–Duhem equation written for a binary mixture, Equation 3.4:
)ln()ln( 22112 dddP
RT
VdT
RT
H EE
3.4
where i is the mole fraction of component i in the reference phase, and i is the activity
coefficient of component i in the reference phase. EH and EV are the molar excess enthalpy and
the molar excess volume, respectively. Unfortunately, these data are not always available nor are
they usually measured simultaneously and consistently with the VLE data.
It is well known that the Gibbs–Duhem can be used as the basis for thermodynamic
consistency tests [59]. Since most of the VLE data used in this study is at high pressure and
63
constant temperature, a high-pressure isothermal VLE consistency test is necessary.
Thermodynamic consistency tests derived from the Gibbs–Duhem equation for high pressure
VLE data [60-66] use different equilibrium data and thermodynamic functions. In some cases
[63-65], the liquid phase composition is not used, and these methods can be used only if PTY
data are available. In this study, the Valderrama and Faúndez [66] method was selected for
testing the VLE data because it not only uses all available PTXY data but also does not have the
complexity of the Won and Prausnitz [61] method which requires the definition of arbitrary
functions for activity coefficients and molar volumes of mixtures, or the Christiansen and
Fredenslund [62] approach which requires the calculation of several thermodynamic properties
such as standard state fugacity and Gibbs free energy. Although the Valderrama and Faúndez do
also provide model parameters as part of the thermodynamic consistency procedure, this feature
of the method was not used in this study. In the Valderrama and Faúndez method, the Gibbs–
Duhem equation is expressed in terms of fugacity coefficients calculated in the VLE data
modelling procedure using the PR EOS [12] combined with the Wong–Sandler mixing rules
[67]. The choice of mixing rules is somewhat arbitrary and is essentially determined by its
flexibility to fit data. This method fulfills our requirements for testing all the available PTXY
data for any range of concentration and pressure with calculated properties using the PR EOS.
Whenever possible, thermodynamically consistent sets are used for the determination of
binary interaction parameters and their associated uncertainties. It should be stressed that not all
VLE data is thermodynamically consistent and therefore the construction of a binary interaction
parameter database will almost certainly contain parameters that are based on tentative data.
Details of the data included in the parameter estimation are discussed in the following
section. The binary interaction parameter determination is a function of the objective function
64
[68-73] used in the data regression. Among the various types of objective functions reviewed by
Ashour and Aly [74], the one suggested by Anderson and Prausnitz [75] was used in this study.
This method was chosen because all the variables from binary VLE measurements and their
associated uncertainties are considered in evaluating parameters in Equation 3.5:
PN
n ny
nxnPnT
objσ
yy
σ
xx
σ
PP
σ
TTkF
1
2exp
1
cal
1
2exp
1
cal
1
2expcal
2expcal
12
11
)( 3.5
where ‘s are the uncertainties of the measured variables and may vary from point to point. Pcal
and ycal
are calculated using a bubble-point pressure calculation. Tcal
and xcal
are calculated using
a dew-point temperature calculation.
Arguably the most valuable contribution of this work is the uncertainty estimation of
binary interaction parameters obtained by minimization of the objective function, Equation 3.5.
Two useful approaches for the estimation of best guesses for the interaction parameters and their
uncertainties are described in the statistical literature Frequentist and Bayesian [76] and briefly
reviewed here.
Using the Frequentist approach, the unknown parameters are viewed as fixed values
which are usually determined using the maximum likelihood estimation method by minimizing
an objective function and their uncertainties are quantified under hypothetical repetition of
sampling of the observed data. In contrast, the Bayesian approach regards the unknown model
parameters as random variables with the prior postulated probability distributions based on the
prior knowledge of the analyst. The prior distribution represents beliefs for parameters before
observing the data. The Bayesian approach is based on the posterior probability distribution of
model parameters after observing data which is calculated using the prior distributions along
65
with a given set of observations based on Bayes theorem [76]. The Markov Chain Monte Carlo
method is often used for summarizing the posterior distribution in order to derive statistics such
as posterior mean and standard deviation. This approach is therefore useful when there is plenty
of prior information about the nature of the errors and their correlations and especially valuable
when the number of observations is limited [77].
The bootstrap technique proposed by Efron [78] is the main Frequentist tool used to deal
with complicated sampling distributions. It is a valuable approach that can be used when there is
a limited or no prior knowledge about the parameters and the nature of the measurement errors
[48]. In this method, the parameters are fitted to the actual observations by minimizing an
objective function and any number of random data sets is generated by replacement from the
residual error distribution. Each of these samples is analyzed to obtain the estimated parameters
and finally the Monte Carlo method is used to evaluate the expected values of the parameters.
Each of these approaches has their advantages and shortcomings. The strength of the
Bayesian method is its unified approach to all problems of uncertainty. However, it could not
solve complex problems due to computational difficulties as shown by Efron [79]. The prior
specification of parameter distribution is the other issue of this method, since improper posterior
may result from an improperly specified prior, especially for nonlinear models. The strength of
Frequentist methods is their ability to provide useful uncertainty estimates for model parameters
without the need to know the prior.
Since the objective of this study is to evaluate the fitted parameter uncertainty by taking
into account not only all the independent and dependent variable uncertainties but also model
input parameter uncertainties, none of the above-mentioned methods is entirely adequate as a
means to estimate the uncertainty of binary interaction parameters while taking into account all
66
the input uncertainties and the lack of detailed knowledge of how experiments were actually
performed, thus effectively making the use of Bayesian techniques very difficult.
Consequently, the Monte Carlo technique used in Chapter 1 is used to estimate the
uncertainties of the binary interaction parameters. This procedure is similar to the procedure used
in the bootstrap approach but now uses the Latin Hypercube Sampling (LHS) method [6] instead
of partial data replacement from the residual error distribution. The simultaneous use of the pure
components, VLE data, and equation of state allows for the development of a binary interaction
parameter database with uncertainties determined in a consistent and systematic manner. The
main four steps in using this technique are (1) specification of probability density functions for
the uncertainty of the input variables involved in the study based on the knowledge of their
uncertainty, (2) probabilistic sampling of the uncertainty space, (3) simulation and calculation of
output parameters by passing each sample set through the model, and (4) statistical analysis of
the results to evaluate the uncertainty of the model outputs.
3.3 Thermodynamic Consistency Test
The Valderrama and Faúndez [66] thermodynamic consistency test, used in this study, is briefly
described in this section. The Gibbs–Duhem equation written in terms of residual properties and
fugacity coefficients, Equation 3.6, overcomes the problem of evaluation of the excess volume
and pressure effects on the activity coefficients for isothermal conditions. The fugacity
coefficients and the residual volume can both be calculated by a single equation of state [65].
)ln()ln( 2211 dddPRT
V R
3.6
67
where i and i are the mole fraction and fugacity coefficient of component i in the given phase
(liquid or vapour) and RV is the residual volume of the corresponding phase. Using
PZRTV R )1( and 21 1 , Equation 3.6 becomes Equation 3.7 [66]:
)ln()ln()1()1(
2212 dddPP
Z
3.7
where Z is the compressibility factor of the given phase.
For calculation of the fugacity coefficients and compressibility factor, a suitable
thermodynamic model must be selected to fit the given NP experimental VLE data points within
acceptable deviations. As suggested by Valderrama and Faúndez [66], a model is appropriate if
the average absolute deviations of pressure and gas phase mole fractions of Component 1 defined
by Equations 3.8 and 3.9 are below 10%.
If the bubble point pressure and gas phase composition are not well correlated, it may
signify that the model is not appropriate for the binary system being studied or the data has
errors, and therefore the consistency test cannot be applied. Therefore, before using the
consistency test, these two criteria must be met.
n
N
nexp
expcal
P
P
P
P
PP
NAAD
1
100 3.8
n
N
nexp
expcal
P
y
P
y
yy
NAAD
1 1
11100 3.9
After the model is found acceptable, the thermodynamic consistency test is performed
based on the Equation 3.7 integrated over an interval from data point r to an adjacent data point s
as expressed in Equation 3.10:
68
s
r
s
r
s
rd
Zd
ZdP
P2
2
1
12
2
2 )1(
1
)1(
11
3.10
For a set of VLE data with NP data points, both sides of the Equation 3.10 must be
calculated for each )( 1NP intervals for both liquid and vapour phases. The left hand side of
Equation 3.10 is denoted by AP, Equation 3.11, and the right hand side by A , Equation 3.12:
s
rP dP
PA
2
1
3.11
s
r
s
rd
Zd
ZAAA 2
2
1
12
2
)1(
1
)1(
121
3.12
AP is determined using the experimental values of PX for liquid phase and PY for vapour
phase for each interval l using the trapezoidal rule and is considered as the experimental area,
since it is obtained from experimental data. Figure 3.3 shows a schematic view of the integrated
function in Equation 3.11 plotted against pressure at each data point. The hatched area indicates
AP for interval l.
Figure 3.3. Illustration for the calculation of AP between two consecutive points of r and s.
1
r
s
NP
1/(
P
2)
P
(AP)
rs
rs
lP PPPP
A
22
11
2
1)(
69
A is based on the calculated values of pressure, vapour phase composition, fugacity
coefficients and compressibility factor from the thermodynamic model. The integration
procedure for evaluation of 1
A and 2
A is similar to that for AP. Since A is obtained from
calculated properties, it is considered to be the calculated area.
As an index for testing the consistency of VLE data, the individual absolute deviation
between experimental area (AP) and calculated area (A) is defined by Equation 3.13 for each
interval l ( )1( to1 PNl ) and for a consistent dataset it should be within acceptable defined
deviations for both phases.
lP
P
lA
AAA
100|%| 3.13
According to Valderrama and Faúndez [66], an isothermal VLE dataset is considered
thermodynamically consistent (TC), if all )( 1NP individual area deviations are below 20% for
both phases, and considered as not fully consistent (NFC) if some of the area deviations (equal or
less than 25% of the areas) are more than 20%. The area test can conclude that the dataset is
thermodynamically inconsistent (TI), if most of the area deviations (more than 75% of the areas)
are outside the 20% limit. The defined margins of accepted errors applied in this method for
selection of the appropriate model (10% for average absolute deviations of pressure and vapour
phase composition) as well as for consistency test (20% for area deviations) were discussed in
detail by Valderrama and Alvarez [80].
In this study, if the model is inappropriate (IM) for modelling of an isothermal VLE
dataset or the dataset is TI, the dataset will be removed from the accepted VLE database used for
evaluation of binary interaction parameters. Only VLE data which are TC and NFC are used to
70
calculate the binary interaction parameters. However, in cases where there is no consistent and
not fully consistent data or there is no isothermal dataset, the available data are used for
estimation of binary interaction parameters. These data should be clearly marked as tentative
data and revisited when new data or new correlation methods become available to foster
permanent improvements in the thermodynamic database and associated process simulations and
equipment design.
3.3.1 Computational Approach for Modelling of VLE Data
Similar to the proposed thermodynamic consistency test method [66], the PR equation of state
[12] with the Wong–Sandler mixing rule [67] along with the van Laar activity model was used in
this study as the default thermodynamic model to correlate VLE data for the system being
studied, although any other flexible and accurate thermodynamic model could be used.
As discussed by Brandani et al. [81] the Wong–Sandler mixing rules have enough
flexibility and accuracy to fit high-pressure VLE data. These mixing rules can be accurately
applied for simple mixtures containing hydrocarbons and inorganic gases and mixtures
containing polar, aromatic, and associating species [67] over a wide range of temperature and
pressure using just a minimal number of adjustable binary parameters.
When using the Wong–Sandler mixing rules, the PR EOS parameters for a mixture am
and bm, are calculated using Equation 3.14 through Equation 3.16:
CN
i
i
E
i
iimm
A
b
aba
1
)(
3.14
71
C
C C
N
i i
ii
E
N
i
N
j ij
ji
m
RTb
a
RT
A
RT
ab
b
1
1 1
1
3.15
with the combining rule of:
)1()(2
1ij
ji
ji
ij RT
aabb
RT
ab
3.16
where NC is the number of components and 62322.02)]12[ln( for the PR equation
of state, ai and bi are the PR equation of state parameters for pure component i, ij is a symmetric
binary interaction parameter, and EA is the molar excess Helmholtz energy at infinite pressure
obtained from the van Laar activity coefficient model, Equation 3.17, using the Wong–Sandler
approximation ),pressurehigh ,(),bar 1,( i
E
i
E TAPTG .
2
21
121
2112
RT
A
RT
G EE
3.17
Three adjustable parameters, 12, 12, and 21, were determined by fitting the bubble
point data from NP experimental VLE dataset using Equation 3.18:
PN
n n
n
n
objy
yyq
P
PPF
1
2
exp
1
exp
1
cal
1
2
exp
expcal
211212 ),,( 3.18
where y1 is the mole fraction of Component 1, superscript 'cal' and 'exp' denote the calculated
and experimental values, respectively, and qn are the weighting factors that are chosen such that
both terms are of the same order of magnitude at each point n. Therefore, it can be set as a power
of ten where the exponent is the order of magnitude difference between two terms at each point.
72
For instance, if the order of magnitude of the first term is –3 and one of the second term is –5,
the weighting factor would be 100. Since the order of magnitude difference may vary in different
points, the weighting factor may be different from one point to another one.
The error represented by Equation 3.18 is minimized using the Levenberg–Marquardt
[45] nonlinear regression method to determine the thermodynamic model parameters. Since the
number of data points must be greater than the number of adjustable model parameters, only the
consistency of the isothermal VLE datasets containing more than three data points (NP > 3) can
be used in this method.
3.4 Binary VLE Database Development
The binary VLE database contains temperature (T), pressure (P), liquid phase composition (x),
and vapour phase composition (y) for all experimentally available binary mixtures based on a
pure component set comprised of 18 components including hydrocarbons from C1 to nC10,
nitrogen, oxygen, argon, helium, hydrogen sulfide, and carbon dioxide. The components were
selected based on their importance for natural gas processing. The experimental values of the
VLE and their relevant uncertainties are taken from TDE Version 5.0 [2]. Among 153 possible
binary combinations from the components of interest, experimental data for 87 binaries are
available in TDE.
Table 3.1 shows a small sample of the developed binary vapour–liquid equilibrium
database for the ethane/propane binary mixture selected from 581 VLE data points available for
this mixture. The VLE information for the database of the 87 binaries is shown in Table C.1.
73
Table 3.1. Sample of developed VLE database for the ethane/propane mixture.
There are instances where uncertainties associated with VLE data are not fully reported
in TDE, for example, there are no uncertainties reported for the VLE data of n-decane/hydrogen
sulfide, methane/helium, and propane/n-pentane. Only uncertainties in pressure are reported for
n-octane/nitrogen, nitrogen/hydrogen sulfide, and n-hexane/carbon dioxide, and only
uncertainties in temperature and pressure are reported for n-heptane/n-octane, n-decane/n-octane,
and n-hexane/n-heptane. In order to access the missing uncertainties, the original literature cited
by TDE was reviewed which was a challenging and a time consuming effort. If the uncertainty
information of a binary system was not available, default uncertainties of 0.5 K, 50 kPa, 0.003
Pure Components Properties
Component Tc (K) Pc (kPa)
Ethane 305.36 ± 0.03 4879.4 ± 22.3 0.100 ± 0.002
Propane 364.95 ± 0.26 4594.1 ± 49.0 0.142 ± 0.005
Vapour–liquid equilibrium data
T (K) P (kPa) x1 y1 T (K) P (kPa) x y Ref.
283.15 689.48 0.0236 0.0720 0.05 6.89 0.001 0.001 [82]
283.15 1379.0 0.3570 0.6120 0.05 13.8 0.001 0.001 [82]
283.15 2068.4 0.6510 0.8480 0.05 20.7 0.001 0.001 [82]
283.15 2757.9 0.9066 0.9616 0.05 27.6 0.001 0.001 [82]
344.26 2757.9 0.0253 0.0480 0.11 13.8 0.005 0.005 [56]
344.26 3102.6 0.1040 0.1760 0.11 13.8 0.005 0.005 [56]
344.26 3447.4 0.1740 0.2700 0.11 13.8 0.005 0.005 [56]
344.26 3792.1 0.2410 0.3430 0.11 13.8 0.005 0.005 [56]
344.26 4136.9 0.3050 0.4030 0.11 13.8 0.005 0.005 [56]
344.26 4481.6 0.3690 0.4570 0.11 13.8 0.005 0.005 [56]
344.26 4826.3 0.4320 0.5020 0.11 13.8 0.005 0.005 [56]
74
and 0.01 were assumed for temperature, pressure, liquid phase composition and vapour phase
composition, respectively [75].
To help ensure the accuracy of the data entered in the database, the VLE data reported by
TDE was checked against the data actually reported to eliminate typos and data duplication. For
instance, the composition of propane reported by Kahre [83] for the ethane/propane system was
reported as the composition of ethane in TDE. Both liquid and vapour phase composition of n-
pentane/n-heptane published by Burova et al. [84] in mass fractions were reported as mole
fractions in TDE, and the isothermal VLE data at 169.81 K reported by Heck and Hiza [85] for
methane/helium mixture was reported in TDE at 124.85 K. There are also VLE data repetitions
for some mixtures such as carbon dioxide/nitrogen, n-decane/carbon dioxide, and n-
heptane/nitrogen. All errors were noted and communicated to NIST.
3.4.1 Application of the Selected Consistency Test in This Study
The application of the consistency test method explained in section 3.3 is illustrated for
ethane/propane and methane/hydrogen sulfide (H2S) mixtures.
Table 3.2. Range of VLE data used for the consistency test.
Mixture T (K) NP Range P (kPa) Range x1 Range y1 Ref.
Ethane/Propane
270.00 23 496.00 – 1972.0 0.0469 – 0.8957 0.1864 – 0.9653 [86]
310.93 12 1379.0 – 4998.7 0.0313 – 0.9190 0.0789 – 0.9350 [56]
322.04 11 1723.7 – 4998.7 0.0130 – 0.7730 0.0300 – 0.826 [56]
Methane/H2S
273.20 12 1240.0 – 11820 0.0030 – 0.3560 0.1350 – 0.7380 [87]
277.59 23 1379.0 – 13100 0.0057 – 0.4401 0.1371 – 0.7321 [88]
310.93 21 2757.9 – 13100 0.0007 – 0.3578 0.0117 – 0.5255 [88]
344.26 12 5515.8 – 11376 0.0031 – 0.1830 0.0196 – 0.2811 [88]
75
Three isotherms for the mixture of ethane (1)/propane (2) and four isotherms for the
mixture of methane (1)/H2S (2) from low to high pressures were selected from the VLE database
developed in this study to illustrate the use of the consistency test. The ranges of experimental
VLE data for isotherms are summarized in Table 3.2 and pure component properties are shown
in Table 3.1 for ethane and propane and in Table 3.3 for methane and H2S.
Table 3.3. Critical properties and acentric factors of pure components.
Component Tc (K) Pc (kPa)
Methane 190.56 ± 0.01 4606.8 ± 9.1 0.0116 ± 0.0019
H2S 373.14 ± 0.45 8950.0 ± 21 0.0975 ± 0.0013
The adjusted model parameters for thermodynamic consistency (12, 12 and 21),
average absolute deviations in pressure (AADP) and vapour phase compositions (AADy), average
absolute deviations of area in both liquid and vapour phases, and the consistency test results are
summarized in Table 3.4.
Before performing the consistency test, AADP and AADy must be checked to verify if they
are within the accepted error margins defined by the thermodynamic consistency method as
discussed above. Table 3.4 shows that these values are less than 10% for all isotherms except for
methane/H2S system at 344.26 K where AADy is greater than 10%. This means that the pressure
and vapour phase composition are not well correlated at this temperature due to either the model
limitations when fitting the VLE data or the VLE data may be of poor quality. In this case, the
model is classified as inappropriate (IM) and this particular consistency test cannot be applied for
this mixture at 344.26 K.
76
The ethane/propane mixture at 310.93 K was found to be TC, meaning that the individual
absolute deviations for all intervals (l=1 to11) in this dataset are less than 20%. At 322.04 K the
relevant VLE data was found NFC, meaning that the individual absolute deviations are greater
than 20% for less than 25% of all intervals (l=1 to10) in the dataset, and at 270 K it was found
TI, meaning that the individual absolute deviations are greater than 20% for more than 25% of
all intervals (l=1 to 22). For methane/H2S system, the VLE data was found to be TC at 310.93 K,
NFC at 277.59 K, TI at 273.20 K, hence the consistency test cannot be applied at 344.26 K since
the data were not well correlated (IM).
Table 3.4. Thermodynamic consistency data for ethane/propane and methane/H2S.
Mixture T (K) 12 12 21 %AADP %AADy |%AL| |%A
V| Results
Ethane/Propane
270.00 –0.0508 –16214 –0.3179 0.36 4.85 15.99 16.35 TI
310.93 –0.0343 –49.923 –0.3179 0.81 4.54 6.00 5.60 TC
322.04 –0.0426 –93.844 –0.2823 0.99 7.67 6.05 5.33 NFC
Methane/H2S
273.20 0.0613 1.7291 1.1824 2.66 5.88 33.89 32.44 TI
277.59 0.0730 1.4752 1.2026 3.19 3.95 9.67 9.57 NFC
310.93 0.1010 1.5713 0.8347 0.32 2.13 3.07 3.27 TC
344.26 0.0975 1.4145 0.7824 0.40 10.85 – – IM
The results are also shown in Figure 3.4 for ethane/propane and Figure 3.5 for
methane/H2S. The pressure-composition PXY diagrams for the binary systems are shown in
Figure 3.4(a) to (c) and Figure 3.5(a) to (c), and the calculated individual area deviations at each
interval are shown in Figure 3.4(d) to (f) and Figure 3.5(d) to (f).
77
Figure 3.4. System ethane/propane, (a-c) Pressure-composition diagrams at 270.00, 310.93,
and 273.20 K, (d-f) Deviations in the individual areas ( iA% ) for liquid phase () and
vapour phase ( ).
0
400
800
1200
1600
2000
2400
0.0 0.2 0.4 0.6 0.8 1.0
Pre
ssu
re (
kP
a)
Ethane Mole Fraction (x, y)
Experimental
Calculated
T = 270.00 K (a)
-80
-60
-40
-20
0
20
40
60
0 2 4 6 8 10 12 14 16 18 20 22
Are
a d
evia
tion
(%
)
Interval (l)
(d) T = 270.00 K
1000
2000
3000
4000
5000
6000
0.0 0.2 0.4 0.6 0.8 1.0
Pre
ssu
re (
kP
a)
Ethane Mole Fraction (x, y)
T = 310.93 K (b)
-20
-15
-10
-5
0
5
10
0 1 2 3 4 5 6 7 8 9 10 11 12
Are
a d
evia
tion
(%
)
Interval (l)
(e) T = 310.93 K
1000
2000
3000
4000
5000
6000
0.0 0.2 0.4 0.6 0.8 1.0
Pre
ssu
re (
kP
a)
Ethane Mole Fraction (x, y)
T = 322.04 K (c)
-40
-30
-20
-10
0
10
0 1 2 3 4 5 6 7 8 9 10 11
Are
a d
evia
tion
(%
)
Interval (l)
(f) T = 322.04 K
78
Figure 3.5. System methane/H2S, (a-c) Pressure-composition diagrams at 273.20, 277.59,
and 310.93 K, (d-f) Deviations in the individual areas ( iA% ) for liquid phase () and
vapour phase ( ).
0
2000
4000
6000
8000
10000
12000
14000
0.0 0.2 0.4 0.6 0.8 1.0
Pre
ssu
re (
kP
a)
Methane Mole Fraction (x, y)
T = 273.20 K (a)
-60
-40
-20
0
20
40
60
80
0 1 2 3 4 5 6 7 8 9 10 11 12
Are
a d
evia
tion
(%
)
Interval (l)
(d) T = 273.20 K
0
2000
4000
6000
8000
10000
12000
14000
0.0 0.2 0.4 0.6 0.8 1.0
Pre
ssu
re (
kP
a)
Methane Mole Fraction (x, y)
T = 277.59 K (b)
-20
-10
0
10
20
30
0 2 4 6 8 10 12 14 16 18 20 22
Are
a d
evia
tion
(%
)
Interval (l)
(e) T = 277.59 K
2000
4000
6000
8000
10000
12000
14000
0.0 0.2 0.4 0.6 0.8 1.0
Pre
ssu
re (
kP
a)
Methane Mole Fraction (x, y)
Experimental
Calculated
T = 310.93 K (c)
-10
-5
0
5
10
15
0 2 4 6 8 10 12 14 16 18 20
Are
a d
evia
tion
(%
)
Interval (l)
(f) T = 310.93 K
79
As shown in Figures 3.4(e) and 3.5(f), all the calculated individual area deviations at 310
K for systems ethane/propane and methane/H2S in both liquid and vapour phases are within the –
20% to +20% and does indicate that the VLE datasets are TC.
Figure 3.4(f) at 322.04 K and Figure 3.5(e) at 277.59 K show only one area in the liquid
phase with high area deviation and outside the defined range. For the ethane/propane mixture the
high deviation occurred in the first interval (l=1) represents 10% of the total number of intervals
(one out of ten intervals), while for the methane/H2S mixture the high deviation interval seen in
l=19 represents 4.5% of the total number of intervals (one out of 22 intervals), so both systems at
the studied temperatures are determined to be NFC.
At 270.00 K the mixture of ethane/propane, considered to be TI, the area deviations are
not within the accepted range for five intervals (22.7% of the total intervals) in the liquid phase
and six intervals (27.2% of the total intervals) in the vapour phase, Figure 3.4(d). The same
pattern can be seen for methane/H2S at 273.20 K, Figure 3.5(d), where seven areas (63.6%) in
the liquid phase and six areas (54.6%) in the vapour phase represent high area deviations.
Among the isotherms studied in this section for these mixtures, isotherms at 310.93 and
322.04 K for the ethane/propane and isotherms at 310.93 and 277.59 K for the methane/H2S with
TC and NFC datasets are considered as acceptable VLE data for evaluation of the binary
interaction parameters. This same procedure was used to verify the thermodynamic consistency
for 87 binary mixtures included in the VLE database that was used for the determination of
binary interaction parameters.
80
3.5 Estimation of Binary Interaction Parameters Associated with Uncertainties
Binary interaction parameters are commonly used in equations of state to improve the prediction
of vapour–liquid equilibrium data, particularly in non-ideal fluid systems, and critical to the
ability of a thermodynamic model to accurately represent fluid phase behaviour. To quantify the
binary interaction parameter uncertainties, the Monte Carlo technique previously described in
Chapter 2 is used to propagate independent pure component, equation of state and VLE
uncertainties to the estimated parameter uncertainty.
In this chapter, the re-parameterized PR EOS with quadratic mixing rules, Equations 3.19
and 3.20, is used to model the phase behaviour of binary systems. Although more complicated
mixing rules are available, such as the Wong–Sandler mixing rules used for the thermodynamic
consistency tests, quadratic mixing rules are widely used for modelling of natural gas mixtures
and for the simulation of many important industrial processes and as such is used in this study to
estimate the binary interaction parameter (kij).
)1(2
1
2
1
ij
i j
jijim kaaa
3.19
2
1i
iim bb 3.20
3.5.1 Input Variables and Their Uncertainties
In addition to experimental VLE data, the pure components properties including critical
temperatures (Tc), critical pressures (Pc), and acentric factors () are used as independent
variables in the calculation of the binary interaction parameters. Parameters of the
thermodynamic model also affect the quality of the estimated binary interaction parameter,
therefore, the total number of variables considered in this study for evaluation of a binary
81
interaction parameter and its uncertainty is thirteen, Table 3.5. Pure component properties and
the model parameters and their uncertainties are available in the pure component database
developed in Chapter 2, and the mixture data are taken from the database developed in this
chapter.
As in the TDE, it is assumed that the uncertainties are normally distributed. It should be
noted, distribution of the experimental data is not necessarily Gaussian, but it is the best
assumption that can be made at this time.
Table 3.5. Input variables for estimation of a binary interaction parameter.
Input Variables
Pure compounds iTccT
iPccP i
(i = 1,2)
Thermodynamic model [8] 0273.00369.00199.04153.10035.03908.0 m
( 2
321 mmmf )
Vapour–liquid equilibrium data nTT
nPP nxx
11 nyy
11
(n=1 to NP)
3.5.2 The Monte Carlo Technique and Sampling Method
Monte Carlo techniques are useful statistical methods for uncertainty analysis and error
propagation in complex models. The objective of this study is to determine the uncertainties of
binary interaction parameters estimated from phase equilibrium calculations. Figure 3.6 shows
the conceptual scheme of the approach used for uncertainty evaluation of the estimated
parameters. Incorporating the mean values of all model input variables and uncertainty
information of VLE data shown in Table 3.5, the binary interaction parameter is estimated by
82
minimizing the Anderson–Prausnitz objective function, Equation 3.5. In order to evaluate the
uncertainty of the fitted parameter, a sample set with size n' is randomly generated for each input
uncertain variable. Instead of replacement of the actual data based on the residual error
distribution which is used in the bootstrap approach to construct the sample dataset, due to lack
of uncertainty information of the observations, the LHS sampling method is used in this study.
Since the uncertainty of the experimental VLE data and also model parameters are available, the
LHS method was selected to select random data points from the normal distribution defined for
each input variable based on their uncertainty information.
Figure 3.6. Conceptual scheme of the approach used for uncertainty estimation of the fitted
parameter.
In this sampling method, the range of each input variable is divided into n' non-
overlapping intervals based on the equal probability of 1/ n' and one value from each interval is
taken at random. These obtained n' values for each variable are grouped in a random manner.
The LHS method ensures that the variables have been sampled over the full range of their
uncertainties and decreases the number of runs necessary to stabilize the Monte Carlo simulation,
when compared to the random sampling method.
Equation 3.5
Min.
Dataset (n')
True mean values
Dataset (1)
Dataset (2)
Dataset of input variables
and their uncertainties
(Table 3.5)
Fitted parameter (k12)0
Fitted parameter (k12)1
Fitted parameter (k12)2
Fitted parameter (k12)n'
Mo
nte C
arlo S
imu
lation
83
For a mixture of interest, the binary interaction parameter is estimated for each sample set
with exactly the same procedure used for true mean values, and the resulting binary interaction
parameters are filed as data. The mean value and standard deviation of n' estimated binary
interaction parameters are then calculated and the associated uncertainty is estimated as two
times the standard deviation based on the selected 95% confidence level, consistent with the
uncertainty definition used in TDE. Finally, the resulting binary interaction parameter is
represented as an average value and a quantified uncertainty.
Since sample size determination is an important part of the Monte Carlo simulation a
brief study on the effect of sample size (n') on the estimated uncertainty of the binary interaction
parameter was performed. Three sets of binary interaction parameters are generated with 100,
1000, and 10,000 samples for two binary mixtures of ethane/propane (hydrocarbon/hydrocarbon)
and methane/H2S (hydrocarbon/polar compound).
The pure compounds properties and their uncertainties are shown in Table 3.1 and Table
3.3. Three parameters of the thermodynamic model (m) used for vapour–liquid equilibrium
calculations and their uncertainties are presented in Table 3.5 and the number of VLE data point
used for the binary interaction parameter estimation and the range of variations in temperature
and pressure values are shown in Table 3.6.
Table 3.6. Temperature and pressure ranges of consistent VLE data for ethane/propane
and methane/H2S binary mixtures.
Binary NP T (K) P (kPa)
Ethane/Propane 129 172.04 ± 0.05 – 355.37± 0.11 4.00 ± 0.07 – 5184.9 ± 13.8
Methane/H2S 61 252.00 ± 0.50 – 310.93 ± 0.05 1379.0 ± 0.7 – 13100 ± 7
84
Figure 3.7. Histogram of calculated binary interaction parameters by different sample sizes
(a) ethane/propane, (b) methane /H2S.
0
10
20
30
40
50
Fre
qu
ency
(%
)
Binary Interaction Parameter (k12)
n' = 100
n' = 1000
n' = 10,000
(a)
0
10
20
30
40
50
60
Fre
qu
ency
(%
)
Binary Interaction Parameter (k12)
n' = 100
n' = 1000
n' = 10,000
(b)
85
As shown in Figures 3.7(a) and 3.7(b) the distribution of the estimated parameter values
are approximately the same for sample sets with 100, 1000, and 10,000 points, for both mixtures.
The distribution of the binary interaction parameters for the ethane/propane system estimated for
a sample size of 100 is shown in Figure 3.8. The dashed line in this figure shows the normal
(Gaussian) distribution with mean value of –0.0739 and standard deviation of 0.00335. The
values of estimated parameters are approximately distributed as a normal distribution and the
histograms are centred about the true mean value ( 0739.012 k ), indicated by the arrow. The
hatched area in this figure indicates the range of uncertainty of the binary interaction parameter
)0067.0(12k which is equal to two times the standard deviation based on the 95% confidence
level.
Figure 3.8. Binary interaction parameter distribution for ethane/propane with sample size
100.
0
10
20
30
Fre
qu
ency
(%
)
Binary Interaction Parameter (k12)
86
True mean values of the estimated binary interaction parameters for the mixtures of
interest and associated uncertainties along with the standard deviations are listed in Table 3.7. In
order to measure the accuracy of the results estimated by the Monte Carlo simulation, the Monte
Carlo standard error (MCSE), nSn
, was calculated for each sample set where nS is the
standard deviation estimated by the Monte Carlo technique for sample size of n'. Clearly if n'
increases, the MCSE goes to zero but the estimate is not necessarily more accurate as illustrated
by the uncertainty. Therefore, there is a practical limit for the determination of the sample size,
Table 3.7.
Based on this study, it appears that a relatively small sample size (n' = 100) can be used
to provide good uncertainty estimates for binary interaction parameters. Therefore, the sample
sets with 100 points generated by Latin Hypercube Sampling (LHS) were used. The study of
sample size can be explored in greater detail and could be the subject of future research studies.
Table 3.7. Monte Carlo simulation results for binary interaction parameters (k12) with
different sample sizes.
Sample Size Mean Value Standard Deviation Uncertainty MCSE
( 12k ) ( nS ) (12k ) (%)
Ethane (1)/Propane (2)
n' = 100 –0.0739 0.00335 0.0067 0.033
n' = 1000 –0.0739 0.00341 0.0068 0.011
n' = 10,000 –0.0739 0.00335 0.0067 0.003
Methane (1)/H2S (2)
n' = 100 0.0503 0.00225 0.0045 0.022
n' = 1000 0.0503 0.00224 0.0045 0.007
n' = 10,000 0.0503 0.00228 0.0046 0.002
87
3.6 Results and Discussion
The binary interaction parameters and their uncertainties for 87 binary mixtures are listed in
Table D.1 along with the number of VLE data points used to find the parameters and their
respective temperature and pressure ranges. The importance of the input uncertainties associated
with pure component physical properties (critical properties and acentric factors), re-
parameterized PR model parameters (m) and binary interaction parameters (kij) estimated in this
chapter is shown through two examples. In all the cases studied in this work, the uncertainty of
the property of interest was estimated using the Monte Carlo technique for an input sample
consisting of all these uncertain variables generated by the LHS method with the sampling size
of 100.
3.6.1 Saturation Point Calculation
In this example, the dew-points and bubble-points and associated uncertainties for two
binary mixtures of ethane/propane and methane/H2S are calculated using the VMGSim
simulation software [7] linked with the Monte Carlo technique. For the dew-point calculations at
known and fixed pressure and vapour composition, temperatures and liquid phase compositions
were calculated for each input sample set and their mean values associated with uncertainties
were estimated using the Monte Carlo technique. The same procedure was used for the bubble-
point calculation for the uncertainty calculation of the pressure and vapour phase composition at
a known temperature and liquid phase composition. Figure 3.9 shows the TXY diagrams for
ethane/propane system at pressure of 2758 kPa and methane/H2S system at 6894.8 kPa based on
the dew-point temperature calculations.
88
Figure 3.9. Temperature-composition diagram for (a) ethane/propane at 2758 kPa, and (b)
methane/H2S at 6894.8 kPa.
270
290
310
330
350
0 0.2 0.4 0.6 0.8 1
Tem
per
atu
re (
K)
Ethane Mole Fraction (x, y)
P = 2758 kPa
T-y
T-x
(a)
210
250
290
330
370
0 0.2 0.4 0.6 0.8 1
Tem
per
atu
re (
K)
Methane Mole Fraction (x, y)
P = 6894.8 kPa (b)
T-y
T-x
89
Figure 3.9(a) shows the uncertainty propagation in both bubble curve and dew curve is
similar and the uncertainties of temperatures and vapour phase compositions on the dew curve
can be calculated based on the bubble-point temperature calculations at a known pressure and
corresponding liquid phase compositions. The calculated values of temperature and ethane
composition in both liquid and vapour phases and associated uncertainties are shown in Table
3.8. The uncertainty of the bubble point temperature is of the same order of magnitude as of the
dew point temperature at any equilibrium point; however, at temperatures below the critical
temperature of ethane (305.36 K) the uncertainty of bubble temperatures is slightly greater than
of dew temperatures whereas at higher temperatures this trend is inverted.
Table 3.8. Calculated VLE data and their uncertainties for ethane/propane mixture at
P=2758 kPa using the technique developed in this work.
Bubble Curve Dew Curve
T x T y
336.61 ± 0.52 0.0200 ± 0.0003 336.61 ± 0.52 0.0286 ± 0.0005
334.42 ± 0.52 0.0700 ± 0.0010 334.42 ± 0.53 0.1000 ± 0.0015
331.11 ± 0.52 0.1404 ± 0.0017 331.11 ± 0.56 0.2000 ± 0.0025
327.49 ± 0.54 0.2120 ± 0.0021 327.49 ± 0.60 0.3000 ± 0.0029
323.50 ± 0.56 0.2856 ± 0.0022 323.50 ± 0.64 0.4000 ± 0.0030
319.04 ± 0.58 0.3625 ± 0.0021 319.04 ± 0.66 0.5000 ± 0.0026
314.00 ± 0.59 0.4447 ± 0.0017 314.00 ± 0.65 0.6000 ± 0.0020
310.48 ± 0.59 0.5000 ± 0.0014 310.48 ± 0.62 0.6623 ± 0.0016
308.17 ± 0.59 0.5356 ± 0.0013 308.17 ± 0.59 0.7000 ± 0.0014
301.18 ± 0.55 0.6415 ± 0.0016 301.18 ± 0.49 0.8000 ± 0.0014
292.25 ± 0.44 0.7776 ± 0.0025 292.25 ± 0.32 0.9000 ± 0.0016
279.59 ± 0.18 0.9900 ± 0.0003 279.59 ± 0.18 0.9968 ± 0.0001
90
The same pattern can be seen between uncertainties in the liquid phase compositions and
vapour phase compositions. Moreover, the bubble point temperature at a liquid composition of
0.5 and the dew temperature at a vapour composition of 0.5 have the highest uncertainties. As
presented in Figure 3.9(a), both the bubble and dew curves are narrow near the boiling
temperature of ethane, which is indicates a smaller uncertainty in this region. On the other hand,
near the boiling point temperature of propane, the uncertainty in compositions is low while the
uncertainty of temperature is still high. Therefore, it can be concluded that the uncertainty
propagation for this binary mixture depends on temperature and compositions in both phases, as
one would expect.
The uncertainty propagation for methane/H2S TXY diagram, Figure 3.9(b), shows a
different pattern when compared to the ethane/propane mixture. The higher uncertainty can be
seen in bubble point curve in comparison with the dew point curve and does increase as
temperature is decreased. The PXY diagrams are also constructed using the bubble-point
pressure calculation at a known temperature and liquid phase compositions for these mixtures.
Figure 3.10 shows PXY diagrams for ethane/propane at 310 K and methane/H2S at 320 K. The
maximum uncertainty of the dew point pressure for methane/H2S mixture occurred around a
methane liquid composition of 0.33 which is the cricondenbar at this temperature. The
magnitude of the propagated uncertainty depends on the phase behaviour of the mixture at the
condition being examined and the specific location in the thermodynamic space.
91
Figure 3.10. Pressure-composition diagram for (a) ethane/propane at 310 K, and (b)
methane/H2S at 320 K.
1000
2000
3000
4000
5000
0 0.2 0.4 0.6 0.8 1
Pre
ssu
re (
kP
a)
Ethane Mole Fraction (x, y)
T = 310 K (a)
P-x
P-y
2000
4000
6000
8000
10000
12000
14000
0 0.1 0.2 0.3 0.4 0.5
Pre
ssu
re (
kP
a)
Methane Mole Fraction (x, y)
T = 320 K (b)
P-x
P-y
92
3.6.2 De-ethanizer Example
At the beginning of this chapter, the minimum number of stages required for a de-
ethanizer was calculated using the Fenske equation, Eqution 3.2, and its uncertainty was
evaluated by the error propagation equation, Equation 3.3, using the experimental VLE data and
their uncertainties taken directly from the literature, labeled “Approach I”. In this section, the
minimum number of stages and their uncertainty for the de-ethanizer with the same
specifications are now calculated using the model and binary interaction parameters developed
for the ethane/propane mixture taking into account their uncertainties. It should be stressed that
since in this case study the product purity (x1) as an input variable required for calculation is
uncertain, its associated uncertainty must be considered for uncertainty evaluation of T and y1 for
both distillate and bottom products. Therefore, it is necessary to generate a sample set for x1D (or
x1B) over its range of uncertainty in addition to the other uncertainties for input variables. The
equilibrium temperatures and vapour phase compositions and their uncertainties for both top and
bottom products at known pressure and liquid phase compositions were calculated by the Monte
Carlo technique for each sample using the bubble-point temperature calculation as previously
discussed. Table 3.9 shows the equilibrium data for both the distillate and bottom products and
their uncertainties determined using the Monte Carlo technique.
Table 3.9. The de-ethanizer product specifications (ethane(1)/propane(2)) at P=2758 kPa.
Calculated Based on the Monte Carlo Technique
Product x1 Temperature y1
Distillate (D) 0.9900 ± 0.0050 279.59 ± 0.30 0.9968 ± 0.0014
Bottom (B) 0.0200 ± 0.0050 336.61 ± 0.56 0.0286 ± 0.0064
93
Two approaches, “Approach II” and “Approach III”, were employed to calculate the
required minimum number of equilibrium stages and their uncertainty using the calculated VLE
shown in Table 3.9. In “Approach II”, the required minimum number of equilibrium stages is
calculated by Fenske equation, Equation 3.2, and its uncertainty is estimated by Equation 3.3 as
in “Approach 1” but now using the calculated VLE data and their uncertainties. The result based
on this approach is 12 ± 6.
Note, the error propagation equations, Equations 3.1 and 3.3, were derived using the
simplifying assumption that there is no correlation between compositions in both phases, while
any one of the compositions (in liquid or vapour phase) can be determined from the
thermodynamic model at a given pressure. Therefore, “Approach III” is proposed wherein the
Monte Carlo technique along with the LHS method is used to evaluate the uncertainty in the
minimum number of stages for the de-ethanizer. In this approach, an input sample consisting of
the pure component physical properties, model parameters, binary interaction parameters, and
both top and bottom product purities (x1) are generated and the corresponding vapour phase
compositions and the minimum number of stages are calculated by the thermodynamic model
and Equation 3.2 for each sample set. Finally, the mean value and the uncertainty of the Nmin are
evaluated as the Monte Carlo output variables. Based on this approach, the correlation between
liquid phase composition and vapour phase composition is part of the calculation.
The results of the three calculation approaches are summarized in Table 3.10. The results
show that ignoring the correlation between equilibrium compositions can significantly affect the
quality of the final results. It is interesting to note that the propagation calculations neglecting the
correlation between equilibrium compositions overestimate the uncertainty by a significant
number of stages, underlining the usefulness of the Monte Carlo technique.
94
It can be concluded that that “Approach III” is reasonable and the best in terms of the
uncertainty in the minimum number of stages, since the correlation between the liquid and
vapour phase compositions are taken into account in a natural and straightforward way. These
results suggest that if one is to obtain reliable estimates for the uncertainty of process variables
calculated using unit operation models that encoded thermodynamic uncertainty information,
Monte Carlo is a powerful technique for simulations of any degree of complexity. The trade-off
compared to a traditional analytical error propagation schemes based on differentials and
repeated use of chain rule is greater computer time.
Table 3.10. Comparison of the minimum number of stages using different approaches
applied in this work.
Calculation Method Nmin
Approach I 12 ± 10
Approach II 12 ± 6
Approach III 12 ± 1
3.6.3 Natural Gas Processing Example
In Chapter 2, a compressor/cooler example was used to estimate the uncertainties in the
process propagated from the uncertainties in pure component properties. The example is
revisited here, now by taking into account the uncertainties of the thermodynamic model and the
binary interaction parameters along with the uncertainties previously considered. Figure 3.11
shows the schematic for the process. Since, VLE data for n-dodecane is not reported in TDE and
it is also not in the database developed in this work, n-decane is considered instead as a heavy
trace component present in natural gas.
95
Compressor
T = 298.15 K
P = 2068.43 kPa P = 6205.28 kPa
W
Q
T = 322.05 K
P = 6136.33 kPa
Cooler
Figure 3.11. Schematic diagram of natural gas processing example.
The results for two gas compositions used, one with 0.999 methane and 0.001 n-hexane
(Composition 1) and another with 0.9999 methane and 0.0001 n-decane (Composition 2), are
summarized in Table 3.11. The pressure-temperature envelope for Composition 2 with
associated uncertainty region calculated using the Monte Carlo technique and the VMGSim
process simulator is shown in Figure 3.12. The uncertainty introduced in intercooler duty,
compressor horse power and compressor outlet temperature is negligible, while the positions of
the cricondenbar and cricondentherm present more uncertainty.
Table 3.11. Basic equipment performance data and their uncertainties revisited in this
work.
Temperature after
Compressor, K
Compressor Horse
Power, HP
Intercooler Duty,
kJ/h
Composition 1 383.53 ± 0.01 549.7 ± 0.2 1,325,900 ± 400
Composition 2 383.72 ± 0.01 550.1 ± 0.2 1,326,100 ± 400
96
Figure 3.12. Pressure-temperature envelope for Composition 2 (methane/n-decane).
The uncertainty in the locations of the cricondenbar, cricondentherm and critical point for
both compositions are shown in Table 3.12. Of particular interest is the cricondentherm
uncertainty, calculated to be in the order of 2 K for Composition 2. The dew point of gas at the
compressor inlet pressure and output pressure is an important factor for the definition of the
compressor inlet temperature and the selection of intercooler operating temperature, which can
affect the operation of the compressor stations.
Table 3.12. Positions of the cricondenbar, cricondentherm and critical point calculated
using the Monte Carlo simulation.
Cricondenbar Cricondentherm Critical Point
Composition 1
Pressure (kPa) 6510.34 ± 57.86 2675.59 ± 14.74 4801.04 ± 11.54
Temperature (K) 216.73 ± 0.49 243.32 ± 0.73 192.32 ± 0.06
Composition 2
Pressure (kPa) 8703.67 ± 215.23 2364.71 ± 42.16 4663.68 ± 8.71
Temperature (K) 236.98 ± 1.05 286.54 ± 1.89 191.04 ± 0.02
0
2000
4000
6000
8000
10000
120 140 160 180 200 220 240 260 280 300
Pre
ssu
re (
kP
a)
Temperature (K)
97
3.7 Conclusions
In Chapter 2, a comprehensive database for pure components basic physical properties and
thermodynamic models was developed by taking into consideration the uncertainty information
of available data. In this chapter, the discussion was extended to mixtures and uncertainty
information of binary interaction parameters (kij) for the re-parameterized PR model along with
the simple quadratic mixing rules. The presented method is generally applicable and could be
employed with any other thermodynamic model and mixing rule.
A comprehensive database for vapour–liquid equilibrium data of 87 binary mixtures with
their uncertainty information was developed based on the NIST’s TDE software and reviewing
the original literature cited, therein. The database includes experimental values and uncertainties
of temperature, pressure, liquid phase compositions and vapour phase compositions which are
fundamental for the estimation of the binary interaction parameter and its uncertainty in a
thermodynamic model.
A thermodynamic consistency test was performed using the method suggested by
Valderrama and Faúndez and the database developed in this study. The method determines the
consistency of an isothermal experimental VLE dataset that is well correlated by the PR/Wong–
Sandler/ van Laar thermodynamic model. The optimal values of model parameters were found
by nonlinear regression. Isothermal VLE datasets which are thermodynamically inconsistent or
were not correlated by the defined model were removed from the accepted VLE database and
were not used for estimation of binary interaction parameters. Available data are considered as
tentative data if there is no consistent data or there is no isothermal dataset for a system. The
application of this method was illustrated through the performed consistency test for two binary
mixtures of ethane/propane and methane/H2S.
98
Binary interaction parameters were optimized using weighted least squares method,
taking into account the uncertainties of all process variables of VLE data through the objective
function suggested by Anderson and Prausnitz, and the uncertainty of the fitted parameters is
evaluated using the procedure similar to the bootstrap approach using the Monte Carlo
simulation along with the LHS method.
The Monte Carlo technique was used in this chapter to evaluate the error propagation in
saturation calculations, minimum number of stages calculations, compression duties, and phase
envelope calculations. A brief study of sample size was conducted through comparison of the
results obtained for three different sample sizes for two binary mixtures of ethane/propane and
methane/H2S. Since the results with different sample sizes are essentially the same, it is
suggested that relatively small sample sizes in the order of 100 are adequate, although no strict
proof is presented.
99
Chapter Four: Uncertainty Analysis Applied to Thermodynamic Models and Fuel
Properties – Natural Gas Dew Points and Gasoline Reid Vapour Pressures 3
4.1 Abstract
A simple, consistent, and self-contained error propagation algorithm was developed using the
uncertainty information of pure component physical properties, binary interaction parameters,
and thermodynamic model parameters combined with the Monte Carlo simulation and the Latin
Hypercube Sampling (LHS) method. This algorithm can be used to simulate the error
propagation in process flow sheets of arbitrary complexity as long as the thermodynamic model
parameters encode uncertainty information. In this chapter, two significant problems related to
hydrocarbon processing are studied using uncertainty analysis. First, the injection of a valuable
liquid hydrocarbon , n-butane, into an existing natural gas pipeline was studied in order to find
the optimum injection rate of liquid n-butane that can be safely added to the flowing gas without
undesired condensation. The main factors considered in this calculation are the hydrocarbon dew
point, the natural gas physical properties, and conformity to pipeline specifications. Second,
uncertainties in Reid vapour pressure (RVP) calculations were taken into account for the
calculation of the optimal rate of liquid n-butane blending into gasoline. Gasoline blending is an
important operation in refineries where gasoline must be produced with enough volatility for the
proper operation of engines in cold climates.
3 Reprinted with permission from S. Hajipour, M.A. Satyro, M.W. Foley, Uncertainty Analysis Applied to
Thermodynamic Models and Fuel Properties – Natural Gas Dew Points and Gasoline Reid Vapour Pressures,
Energy & Fuels, DOI: 10.1021/ef4019838. Copyright (2013) American Chemical Society. It should be noted that
the content of this chapter includes the extensions beyond the cited journal paper.
100
4.2 Introduction
In Chapter 2, a comprehensive database for pure hydrocarbons containing basic physical
properties and associated uncertainties, two correlations for estimation of pseudo-components
properties with required uncertainty information, and the variance-covariance matrix for the re-
parameterized Peng–Robinson equation of state parameters were developed. In Chapter 3, a
database for 87 binary mixtures containing the experimental values of vapour–liquid equilibrium
(VLE) data and their uncertainties was developed and the re-parameterized Peng–Robinson
equation of state [8] was extended to mixtures using simple quadratic mixing rules. Binary
interaction parameters and their uncertainties were evaluated using a thermodynamically
consistent VLE database taking into account the uncertainties of measured binary data.
With the available uncertainty information for pure components, model parameters, and
binary interaction parameters, it is now possible to perform a rigorous uncertainty analysis for
process simulation/design through a self-contained and consistent computational procedure. In
this chapter, a simple error propagation algorithm integrated with an internally consistent
thermodynamic data set and correlations along with associated uncertainties was developed and
coupled with the VMGSim process simulation software
[7] to evaluate the effect of
thermodynamic uncertainties on the simulation results.
4.2.1 Liquid Hydrocarbon Injection into an Existing Natural Gas Pipeline
In this process, liquid hydrocarbon is injected into the natural gas existing pipeline in order to
increase the pipeline capacity and/or heating value of the gas. This approach allows for the
transportation of high value hydrocarbon liquids in a gas pipeline. The amount of liquid
101
hydrocarbon that can be safely injected into the pipeline is controlled by the hydrocarbon dew
point of the gas and the existing pipeline specifications.
Commonly used cryogenic fluids such as liquefied natural gas (LNG) and liquefied
petroleum gas (LPG) or a mixture of hydrocarbons such as propane and butane with air are
added to the gas pipeline in the supply and distribution of natural gas to increase the capacity
during peak demand periods [89, 90]. A liquid hydrocarbon, normally butane, is sometimes
added to fuel gas to increase the heating value of the gas used to provide energy in refining
processes [91].
Stark et al. [89] presented an innovative method and apparatus for adding LNG or a
mixture of hydrocarbon/air to plant fuel gas pipelines using a Venturi jet. Different injection
possibilities were tested in which the kinetic energy of the flowing gas is used to aspire the
cryogenic liquid into the high pressure gas stream and the heat content of the flowing gas stream
is used to supply the latent heat of vaporization. The amount of injected liquid is determined by
the temperature of the pipeline or the gravity specification of the natural gas. They claimed that
using the Venturi system reduces the capital investment and required power by eliminating the
need for pumps or compressors to increase the pressure of the injected liquid and vaporizers to
supply the energy required for liquid vaporization.
On the other hand, Arenson [90] claimed that the liquefied cryogenic fluid must be
preheated, vaporized, and superheated prior to injection into the flowing gas, and therefore he
provided a new method to vaporize and inject the cryogenic gas. The possibility of condensation
and controlling the amount of liquid hydrocarbon added to the gas were not discussed by this
author. While Chin and co-workers [91] invented a device to control the amount of butane
injected into fuel gas that was based on monitoring the butane saturation temperature of the gas.
102
Regardless of the method used for the injection, the main objective of this study is to
investigate the effect of variation in the liquid hydrocarbon injection rate on the calculated dew
point. As previously discussed in Chapters 2 and 3, the uncertainty of input parameters affects
the calculation of the dew point of the natural gas and this effect would be significant in the
presence of a trace amount of heavy components. Therefore, uncertainty analysis using the error
propagation algorithm is performed to estimate the optimum amount of liquid hydrocarbon that
can be safely injected into the existing gas pipeline without liquid formation.
4.2.2 Gasoline Blending
Gasoline is a complex mixture of many different hydrocarbons and its composition varies
depending on the crude oil source, refining process, and additives. There are standard
specifications established by American Society for Testing of Materials (ASTM) for gasoline
dealing with the performance requirements such as volatility [92]. In order to meet the standard
specifications for a product, different gasoline cuts are blended together with additives and
lighter hydrocarbons.
The volatility of the gasoline blend is one of the most important properties affecting the
performance of engines and their ability to function properly independent of the weather.
Volatility is directly related to the Reid vapour pressure (RVP) [93] which is the vapour pressure
of the gasoline blend at 310.93 K (100 °F) measured using a specific apparatus and vapour
fraction. The maximum allowable limit of the RVP varies with seasonal temperature changes and
geographical location.
In the summer and in high altitude regions, it is important to have a lower RVP gasoline
to reduce evaporation losses and prevent vapour lock. On the other hand, in the winter and in low
103
altitude areas, a higher RVP gasoline must be produced to improve engine starting characteristics
[93, 94]. As such, the RVP ranges from 49.64 kPa (7.2 psia) in the summer to 93.08 kPa (13.5
psia) in the winter [94]. Since n-butane is a relatively inexpensive, has a lower sales value than
gasoline on a volume basis, and has a high RVP equal to 358 kPa (52 psia), it is often used in
refineries as a blending component to produce gasolines with required RVP specifications.
In this section, the use of the error propagation algorithm to evaluate the amount of n-
butane to be added to a gasoline to produce a desired RVP is demonstrated. Since the RVP of
the gasoline depends on the quantity and RVP of each component in the blend, the uncertainty in
vapour pressure of each individual component affects the quality of the overall RVP calculation.
As previously shown in Chapter 2, uncertainties of input parameters affect the quality of the
calculated vapour pressure of pure components. Therefore, physical property uncertainties
influence the quality of the overall RVP calculations and must be taken into account for reliable
calculation of the amounts added during blending.
4.3 Development of the Error Propagation Algorithm
When calculating any quantity of interest through mathematical relations, the uncertainties
associated with the independent variables are propagated into the final quantity. Evaluation of
the uncertainty in the final result using the principles of error propagation based on Taylor
linearization is an exceptionally tedious procedure [8] and difficult to generalize from a process
simulation point of view. Therefore, the error propagation equation is rarely used in evaluation of
uncertainties in complex calculations.
In Chapter 2, it was demonstrated that a novel version of the Monte Carlo method can be
used for the error propagation calculations for flow sheets of any complexity. This method is
104
simple, adaptable, and reasonably fast for complex computations. The basic requirements of a
self-contained and consistent error propagation algorithm for physical property calculations are
described in detail in Chapters 2 and 3. A very brief summary of its major points follows.
The sequence for the overall error propagation evaluation is shown in Figure 4.1 as a
block flow diagram. For pure components, critical temperature, critical pressure, and acentric
factor data and their associated uncertainties are taken from the pure component database
developed in Chapter 2. The re-parameterized Riazi–Daubert or Lee–Kesler models [8] and their
variance-covariance matrices of model parameters are used for estimation of critical properties
and acentric factors for any undefined oil fractions or plus fractions and their uncertainties. The
re-parameterized Peng–Robinson equation of state and associated variance-covariance matrix [8]
along with the van der Waals quadratic mixing rules are used for thermodynamic calculations,
and binary interaction parameters and their uncertainties are taken from the database evaluated
for 87 binary mixtures and used for mixture calculations [9].
In order to use the Monte Carlo technique for error propagation, it is assumed that all
input uncertainty values are characterized by normal (Gaussian) distributions with standard
deviations equal to one half of their uncertainty values based on the 95% confidence interval
(CI). It should be stressed that the application of the proposed error propagation method rests
solidly on the applicability of a thermodynamic method for the modelling of the behaviour of
pure components and mixtures. In other words, the thermodynamic models should be devoid of
consistent bias when estimating relevant pure component properties such as vapour pressures and
mixture properties such as saturation pressures or vapour fractions. Thermodynamic models must
thereby be constructed using the best practices related to data regression and should be verified
before use for extensive error propagation calculations.
105
Figure 4.1. Sequence of overall error propagation evaluation process.
The LHS method [6] is used to generate a sample set containing all input parameters
subject to uncertainty. The sample size (100) was determined from studies previously completed
in Chapters 2 and 3, and is used in this chapter without modification. The quantities of interest
Latin Hypercube Sampling Method
Uncertainty Evaluation
Internal Database
Pure Component Database [8]
Critical properties and uncertainties
Acentric factors and uncertainties
Characterization models for undefined oil
fractions and variance-covariance matrices
Binary Mixtures Database [9]
Vapour-liquid equilibrium (VLE) data and
uncertainties
Binary interaction parameters and uncertainties
Thermodynamic Models [8, 9]
Re-parameterized Peng-Robinson equation of
state and variance-covariance matrix
van der Waals quadratic mixing rules
Simulation
Process Simulation
by VMGSim [7]
Monte Carlo
Technique
106
and their associated uncertainties are calculated through multiple simulations. The distribution of
the calculated quantities shows the effect of the input uncertainties. The mean value of the
calculated quantities is reported as the true value with the uncertainty equal to two times of the
standard deviation. This approach is similar to Whiting et al.’s method [15] for studying the
uncertainty of process performance, the main difference being the use of a comprehensive
database of thermodynamic parameters with associated uncertainties.
4.4 Case Study Problems
4.4.1 Injection of Liquid n-Butane into an Existing Natural Gas Pipeline
Although it is desirable to maximize the amount of liquid hydrocarbons in the natural gas in
order to maximize the hydrocarbon transport capacity and heating value of the gas, the injection
of liquid hydrocarbons is limited by the dew point of the components and the specifications for
the existing pipeline. With too much injection of liquid hydrocarbon into the pipeline, the gas
will be oversaturated, and some of the mixture will condense.
One of the essential factors in gas pipeline design is avoiding the condensation of liquid
hydrocarbons. Increased pressure drop, capacity reduction, and pipeline equipment problems
such as compressor damage are the main issues caused by hydrocarbon liquid dropout. In order
to prevent hydrocarbon condensation, the operating temperature of the pipeline must be kept
above the hydrocarbon dew point shown in the pressure-temperature (PT) envelope. Figure 4.2
represents a typical PT envelope of a natural gas that shows the effect of the operating
temperature of the pipeline on hydrocarbon condensation and indicates that the condensation
occurs at the temperatures (T2) lower than the dew point temperature.
107
Figure 4.2. Pressure-temperature (PT) envelope for a natural gas and thermodynamic
positions of the pipeline with temperatures of higher (T1) and lower (T2) than dew point
temperature at pressure of P.
The effect of uncertainty on the dew point calculation is illustrated through the injection
of liquid n-butane into an existing natural gas pipeline designed to transport a maximum natural
gas flow rate of 25.49 MMSCMD (900 MMSCFD) at 288.71 K (60 °F) from a source (A) to a
delivery location (B), 130 km away, with a delivery pressure of 5515.8 kPa (800 psia). The
composition of the gas and the pipeline specifications are given in Table 4.1 and Table 4.2,
respectively.
Critical point
T1,P T2,P
Pre
ssu
re
Temperature
Liquid Phase
Vapour Phase
Vapour-Liquid Phase
Dew Point
108
Table 4.1. Composition of natural gas used in this study.
Component Mole Fraction
Oxygen 0.0002
Nitrogen 0.0149
Carbon dioxide 0.0070
Methane 0.9500
Ethane 0.0250
Propane 0.0020
i-Butane 0.0003
n-Butane 0.0003
i-Pentane 0.0001
n-Pentane 0.0001
n-Hexane 0.0001
Table 4.2. Existing natural gas pipeline specifications used in this work.
Specs
Nominal Pipe Size (NPS) 30 in
Pipe wall thickness 0.5 in
Roughness 1.8×10-5
m (0.0007 in)
Length 130 km
Pressure MAOP a 7584.2 kPa (1100 psia)
Temperature Max. 323.15 K
Gas heating value Min. 36 MJ/m3
Max. 41 MJ/m3
Hydrocarbon dew point Max. 263.15 K at 5515.8 kPa
a Maximum Allowable Operating Pressure.
109
In order to calculate the number of compressor stations, the required mechanical work to
be supplied by the compressors, the compressor discharge temperature, and the intercooler duty,
the existing natural gas pipeline was simulated using VMGSim [7]. To simplify the simulation,
the number and location of the compressor stations required to transport the natural gas were
calculated by neglecting temperature and elevation differences along the pipeline. The re-
parameterized Peng–Robinson equation of state [8] was used for the process calculations, and the
pure component critical properties, acentric factors, and binary interaction parameters are
available in the databases developed in Chapters 2 and 3.
For the design of the pipeline, it was assumed that the first compressor station at point A
has a discharge pressure of 7584.2 kPa (MAOP) and the intercooler outlet temperature is 288.71
K. The intermediate compressor station is located halfway between A and B (65 km) with a
suction and discharge pressure of 5515.8 kPa and 7584.2 kPa, respectively. The simulation
results show that the natural gas can be transported to B using only two compressor stations and
that the gas pressure at the delivery point would be greater than 5515.8 kPa which can be
controlled by a back pressure regulator. The schematic view of the gas pipeline is shown in
Figure 4.3.
Separator
Compressor
Cooler To B
P=5515.8 kPaW
Compressor station II
Compressor station I
(A)
Q
Figure 4.3. Schematic view of the existing natural gas pipeline used in this work.
110
The basic information for the equipment associated with the uncertainties propagated
from the uncertainties in all input variables including pure components properties, binary
interaction parameters, and thermodynamic model parameters are summarized in Table 4.3. The
uncertainty analysis was performed using the error propagation algorithm previously described,
assuming that the inlet and outlet temperatures and pressure drop of the intercooler, and the
adiabatic efficiency and pressure ratio of the compressor are specified with no associated errors.
Note if uncertainty information of these parameters was available, they could easily be used in
the Monte Carlo simulations.
Table 4.3. Existing pipeline equipment performance data.
Equipment Specifications
Compressor
Adiabatic efficiency (%) 80
Pressure ratio 1.375
Inlet temperature (K) 288.71
Adiabatic work (kW) 10,825.1 ± 6.2
Intercooler
Inlet temperature (K) 323.15a
Outlet temperature (K) 288.71
Pressure drop (kPa) 68.95
Duty (MJ/h) 19,803.4 ± 9.2
a Maximum allowable temperature of the pipeline.
Since the hydrocarbon dew point calculation is strongly dependent on the composition of
natural gas, especially its heaviest components, the dew point of the gas will change due to the
injection of n-butane. The maximum allowable hydrocarbon dew point of 263.15 K is the
specification for the existing pipeline. By increasing the amount of n-butane in the gas stream,
the hydrocarbon dew point increases, therefore, the injection rate of liquid n-butane is limited by
111
this specification. In this study, the maximum allowable amount of n-butane was calculated
using VMGSim assuming that the liquid n-butane is injected after the first compression station,
point A, and that the pressure of the pipeline does not change during the mixing process as a
result of the small amounts of liquid n-butane injected into the natural gas. Also, the flow rate of
the final gas mixture was kept constant at the maximum gas flow rate of 25.485 MMSCMD.
Without taking into account the uncertainties, the maximum standard flow rate of n-
butane is 137.52 m3/hr (116,550 ft
3/day) which is added to 24.7 MMSCMD (872.26 MMSCFD)
of natural gas. The ratio of the added liquid n-butane to the transported natural gas is equivalent
to 0.0134% in standard volume (in other words, without taking into account the vaporization of
n-butane that will happen at the actual pipeline condition). As shown in Chapters 2 and 3, the
dew point calculation is also dependent on the uncertainty of the input parameters used in the
thermodynamic model. Since results from a simple dew point calculation are obtained without
taking into account the input uncertainties, the calculation is not complete from uncertainty
analysis point of view and therefore under- or over-estimation has to be applied to the calculated
amount of liquid n-butane.
4.4.2 Gasoline Blending
Blending n-butane into gasoline not only increases the RVP of the gasoline but also increases the
capacity of the gasoline supplies and reduces the gasoline price [95]. The amount of n-butane
that can be added to the blend is limited by the gasoline product specifications such as RVP.
Addition of n-butane increases the RVP and the tendency of gasoline to vaporize at high
temperatures and high altitude areas. The presence of vapour in the fuel line and the combination
of the vapour and liquid feeding the fuel pump interrupts the normal car engine operation and
112
presents a safety hazard. Similarly, gasoline with too low a RVP does not provide enough
volatility to start the engine in cold weather. In order to minimize gasoline evaporation, the
amount of n-butane in gasoline blending process must be controlled [94].
This process is illustrated by blending n-butane into 331.23 m3/hr (50,000 bbl/day) low
RVP gasoline at standard conditions, with the chemical composition given in Table 4.4, in order
to increase RVP from 70.72 kPa (10.26 psia) to 93.08 kPa (13.5 psia). Although a mixture of
pure compounds does not truly represent physical and chemical characteristics of gasoline, this
simplifying assumption is reasonably accurate, and a surrogate for gasoline is represented as a
mixture of pure compounds. The chemical composition proposed by Kreamer and Stetzenbach
[96] as a reference surrogate Low RVP gasoline for environmental research studies is used in
this study.
Table 4.4. Low RVP gasoline blend chemical composition.
Compound Weight Fraction a Standard Volume Fraction
i-Butane 0.030 0.0393
n-Butane 0.030 0.0379
i-Pentane 0.050 0.0589
n-Pentane 0.050 0.0585
n-Hexane 0.050 0.0555
n-Heptane 0.050 0.0535
2,2,4-Trimethylpentane 0.050 0.0528
n-Octane 0.140 0.1463
2-Methyldecane 0.050 0.0499
2-Methyl-2-butene 0.050 0.0553
2,3-Dimethyl-1-butene 0.050 0.0541
Benzene 0.020 0.0167
Toluene 0.150 0.1267
m-Xylene 0.034 0.0289
o-Xylene 0.033 0.0275
p-Xylene 0.033 0.0281
1,2,4-Trimethylbenzene 0.080 0.0671
i-Butylbenzene 0.050 0.0430 a
Data are from ref [96].
113
After selecting the gasoline stock, the quantity of n-butane required to give the desired
RVP was calculated using VMGSim [7]. For the calculation, the input properties of pure
components (critical properties and acentric factors) were taken from the database previously
developed in Chapter 2. The critical properties of xylene(s) and 2-metyldecane which are not
available in the database were taken from ThermoData Engine (TDE) [2], and the acentric
factors associated with uncertainties were calculated using a similar approach to that of the
developed database [8]. The properties of these components are shown in Table 4.5.
Table 4.5. Properties of pure components.
Compound Tc (K) a Pc (kPa)
a
m-Xylene 616.85 ± 0.56 3540 ± 13 0.328 ± 0.002
o-Xylene 630.43 ± 0.61 3745 ± 25 0.312 ± 0.003
p-Xylene 616.19 ± 0.15 3528 ± 16 0.323 ± 0.002
2-Methyldecane 624.10 ± 9.70 1811 ± 23 0.545 ± 0.006
a Data are from ref [2].
Since the database of the binary interaction parameters [9] was developed based on the
components present in natural gas, the uncertainty information of binary interaction parameters
for some of the components considered in this case study are not available. For these binaries,
the calculation was done using the default binary interaction parameters provided by the
VMGSim for the Advanced Peng–Robinson (APR) [51]. The expansion of the database to
include the missing binary interaction parameters and their uncertainties is straightforward as
discussed in Chapter 3. In this example, the uncertainty analysis was conducted by taking into
account only the uncertainties of the critical properties and acentric factors of each of the
114
individual components and using the APR equation of state, and the uncertainties related to
thermodynamic models and binary interaction parameters were not considered.
Without taking into consideration the uncertainties of input parameters, the RVP of the
gasoline with the specified composition is 70.72 kPa (10.26 psia) and the flow rate of n-butane
required to provide a RVP equal to 93.08 kPa is 23.73 m3/hr (3582.44 bbl/day) at standard
conditions. The volume of the blended n-butane is equivalent to 7.17% of the initial volume of
the gasoline at standard conditions. Similarly to the dew point calculation, the RVP also depends
on the composition. Consequently, the uncertainty analysis was carried out and used to provide
an estimate of the under- or over-estimation of the blended n-butane rate.
4.5 Uncertainty Analysis Results and Discussion
In this section, the uncertainty in dew point and Reid vapour pressure calculations was quantified
using the error propagation algorithm developed in section 4.3. The uncertainty analysis results
provide an estimate for the rate of n-butane that can be safely added to the natural gas stream and
gasoline blend to meet the required specification.
For the first case study, depending on the amount of liquid hydrocarbon injected into the
pipeline, the composition of the gas will change and hence the dew point of gas will vary. The
uncertainty analysis was used to estimate the safe flow rate of injected n-butane based on the
uncertainty of dew point of gas at 5515.8 kPa.
In the previous section, a maximum standard flow rate of 137.52 m3/hr was estimated for
n-butane by setting the dew point of final gas mixture to 263.15 K without considering the
uncertainties in the input parameters. In order to find the uncertainty of dew point with the
maximum injection and compare the design parameters of the process after injection with the
115
existing pipeline specifications, the uncertainty analysis was performed using the Monte Carlo
method with the LHS sampling. Figure 4.4 shows the PT envelopes and associated uncertainties
for the natural gas before and after the n-butane injection. Points A and B in this figure indicate
the thermodynamic positions of the source and delivery locations. As shown in Figure 4.4, the
dew point curve is strongly affected by changing the gas composition, and increasing the amount
of n-butane in the gas leads to a significant increase in cricondentherm. This effect was
previously discussed in Chapter 2.
Figure 4.4. Pressure-temperature envelopes for a natural gas before and after the liquid n-
butane injection.
The results of uncertainty analysis for the natural gas before and after the injection are
summarized in Table 4.6. As indicated in this table, there is no dew point reported for the natural
Pipeline Spec
(Max. Dew-point)
A
B
0
2000
4000
6000
8000
10000
130 150 170 190 210 230 250 270 290
Pre
ssu
re (
kP
a)
Temperature (K)
After injection
Before injection
116
gas before injection at 5515.8 kPa, since that pressure is greater than the cricondenbar and there
is no possibility for condensation of the gas at that pressure by temperature reduction. The
calculated dew point of the gas after injecting 137.52 m3/hr liquid n-butane is 263.15± 0.34 K at
5515.8 kPa.
Table 4.6. Results of the phase envelopes uncertainty analysis.
Property Before Injection After Injection
n-butane injection rate (m3/hr) – 137.52
Cricondentherm temperature (K) 218.18 ± 0.77 263.35 ± 0.34
Cricondentherm pressure (kPa) 2529.8 ± 41.3 4978.0 ± 12.5
Cricondenbar temperature (K) 199.65 ± 2.60 239.10 ± 0.20
Cricondenbar pressure (kPa) 5257.3 ± 24.6 9110.5 ± 39.4
Dew temperature at 5515.8 kPa (K) – 263.15 ± 0.34
For clarification purposes, the zoomed-in version of the PT envelope of gas after
injection and the distribution of dew points calculated using the Monte Carlo for 100 sample sets
are shown in Figure 4.5(a) and 4.5(b), respectively. Figure 4.5(a) shows a maximum dew point
located in the retrograde condensation region with the hydrocarbon dew points being greater than
263.15 K at 5515.8 kPa, and therefore gas will be condensed at the specified condition. Figure
4.5(b) also shows that half of the Monte Carlo (MC) results are located above the maximum
allowable dew point line. Two green dashed-dotted lines show the lower and upper limits of the
calculated dew points at 5515.8 kPa which represent the uncertainty of the dew point on the 95%
confidence interval (CI).
117
Figure 4.5. (a) The zoomed-in version of Figure 4.4 for pressure-temperature envelope of
gas after the injection of 137.52 m3/hr, and (b) Monte Carlo simulation results for dew
point calculation at 5515.8 kPa.
4000
4500
5000
5500
6000
261 262 263 264
Pre
ssu
re (
kP
a)
Temperature (K)
Pipeline Spec
(Max. Dew-point)
(a)
262.6
262.8
263.0
263.2
263.4
263.6
263.8
0 10 20 30 40 50 60 70 80 90 100
Dew
Po
int
(K)
Sample Number
MC results
95% CI
Max. allowable dew point
(b)
118
The uncertainty analysis shows that there is a possibility that the existing pipeline
specification cannot be met due to uncertainties in the physical properties and that the amount of
injected butane should be reduced. In order to find the maximum rate of n-butane such that the
upper limit of the calculated dew point at 5515.8 kPa is below 263.15 K, the uncertainty analysis
was performed for different standard volume amounts of the liquid n-butane injection. Figure
4.6(a) shows the results of uncertainty analysis. The intersection of the curve with the maximum
allowable dew point line (T=263.15 K) determines the maximum safe n-butane/gas standard
volume ratio which is shown more clearly in the zoomed-in version of the plot, Figure 4.6(b).
Therefore, the maximum safe standard volume ratio of the injected liquid n-butane to the
natural gas would be 0.0132% which is equivalent to 135.45 m3/hr of liquid n-butane at standard
conditions. The dew point temperature with the new injection rate was calculated using the error
propagation algorithm to ensure that the upper uncertainty of temperature is still below 263.15 K.
The reduction of n-butane injection rate from 137.52 m3/hr to 135.45 m
3/hr leads to a reduction
of target dew point from 263.15 k to 262.71 K. Figure 4.7 shows the oscillation of data points
around 262.71 K with the uncertainty of 0.34 K indicated with two green dashed-dotted lines.
This type of guided determination of safety factors for design is one of the major benefits of
uncertainty analysis.
119
Figure 4.6. (a) Calculated dew point and associated uncertainty at 5515.8 kPa against the
injected liquid/gas standard volume ratio, and (b) zoomed-in version of (a) in the vicinity of
maximum dew point.
220
230
240
250
260
270
0 0.005 0.01 0.015
Ca
lcu
late
d D
ew P
oin
t (K
)
Volume Ratio of the Injected n-Butane to Natural Gas (%)
Max. allowable dew point @ 5515.8 kPa (a)
259
261
263
265
0.012 0.013 0.014
Dew
Po
int
(K)
Volume Ratio of the Injected n-Butane to Natural Gas (%)
Max. allowable dew point
(b)
120
Figure 4.7. Monte Carlo simulation results for the dew point calculation at 5515.8 kPa after
the injection of 135.45 m3/hr n-butane.
Since the aim of this process is to use the existing pipeline to transport the gas after
injection, it is also necessary to check that the current equipment can transport the gas with
different physical properties. In order to do so, the compressor and intercooler performance were
checked using the uncertainty analysis algorithm. Comparing the results shown in Table 4.7 with
those listed in Table 4.3 indicates that the current equipment can be used to transport the gas
without any changes since the adiabatic work of the compressor and duty of the intercooler
required to transport the gas are smaller than those of existing equipment.
Adding n-butane to the gas increases the specific gravity and ratio of specific heats which
are two main parameters in calculating of compressor work. Since the pressure ratio does not
change, increasing the values of these two parameters decreases the required compressor
horsepower. On the other hand, the specific heat of the gas increases after injection and leads to
262.2
262.4
262.6
262.8
263.0
263.2
0 10 20 30 40 50 60 70 80 90 100
Dew
Po
int
(K)
Sample Number
MC results 95% CI
Max. allowable dew point Mean dew point
121
an increase in the duty of the intercooler; however, it is still below the design duty of the existing
intercooler. The compressor outlet temperature was also calculated to ensure that the upper
uncertainty of temperature is below 323.15 K and did meet the pipeline specification.
Table 4.7. Physical properties of the gas and the pipeline equipment performance data
before and after the injection.
Property Before Injection After Injection
Natural gas flow rate (MMSCMD) 25.485 24.711
n-butane injection rate (m3/hr) 0 135.45
Gas flow rate after injection (MMSCMD) 25.485 25.485
Specific gravity 0.583 0.626
Net heating value (MJ/Sm3) 33.98 36.35
Gross heating value (MJ/Sm3) 37.67 40.21
Dew point at 5515.8 kPa (K) – 262.71 ± 0.34
Compressor
Inlet temperature (K) 288.71 288.71
Pressure ratio 1.375 1.375
Adiabatic efficiency (%) 80 80
Adiabatic work (kW) 10,825.1 ± 6.2 10,529.1 ± 8.6
Outlet temperature (K) 316.22 ± 0.01 315.10 ± 0.01
Intercooler
Inlet temperature (K) 316.22 ± 0.01 315.10 ± 0.01
Outlet temperature (K) 288.71 288.71
Pressure drop (kPa) 68.95 68.95
Average specific heat (kJ/kmol-K) 46.29 ± 0.02 50.08 ± 0.03
Duty (kW) 15,858.0 ± 6.4 16,455.5 ± 9.0
122
The physical properties of gas including specific gravity and net and gross heating values
were also calculated, Table 4.7. The specific gravity of the gas increases from 0.583 to 0.626 due
to injection of 135.45 Sm3/hr of liquid n-butane. This significant increase in the gas specific
gravity results from the n-butane vaporization at pipeline conditions and its actual volume
fraction in the final gas is 3.04 actual volume percent corresponding to a volume flow increase of
0.774 MMSCMD. If there is a specified upper limit for the density of the gas, it should be
included in the determination of the safe injection amount. The heating value of the gas also
increases by increasing the amount of n-butane, but is still less than the maximum heating value
specified in Table 4.2.
Since the dew point temperature varies widely depending on the gas composition and
pressure of the pipeline and the model used for the calculation, it may vary by changing any one
of these parameters. Note, the effect of other compounds present in small quantities such as
methanol and ethylene glycol that may be injected as part of a hydrate formation prevention can
be quantified in the same way.
For the second case study, the uncertainty analysis was performed to estimate the safe
amount of blended n-butane into gasoline based on the uncertainty of RVP of the gasoline blend.
The uncertainty in the RVP calculation propagated from uncertainties of critical temperatures,
critical pressures, and acentric factors of each of 18 components of gasoline listed in Table 4.4
was estimated using VMGSim, the APR equation of state, and the Monte Carlo technique.
The simple calculation of RVP shows that 23.73 m3/hr of n-butane is required to increase
the RVP of the gasoline blend from 70.72 kPa to 93.08 kPa. The uncertainty analysis results
indicate that by blending of this amount of n-butane into the gasoline, there is a possibility that
123
the gasoline RVP will be greater than the maximum allowable value of 93.08 kPa for the final
blend.
The saturation pressure curves as a function of temperature and their associated
uncertainties for the gasoline blend before and after n-butane blending are shown in Figure 4.8
for the temperature range of 290–330 K. The envelopes were constructed using the results of
Monte Carlo simulation for a sample size of 100 generated by the LHS method for 54 uncertain
input parameters. The uncertainty analysis shows that the RVP of the initial gasoline is 70.72 ±
0.76 kPa (10.26 ± 0.11 psia), and after blending of 23.73 Sm3/hr n-butane it will increase to
93.08 ± 0.72 kPa (13.5 ± 0.1 psia). The true mean values of RVP at initial and final conditions
are also indicated by dashed lines in this figure.
Figure 4.8. Pressure-temperature envelopes for the gasoline before and after n-butane
blending.
15
40
65
90
115
140
165
290 300 310 320 330
Pre
ssu
re (
kP
a)
Temperature (K)
Before blending
After blending
RVP final
RVP initial
124
As shown in Figure 4.8, increasing the n-butane composition causes the bubble point
pressure and consequently the RVP to increase. Table 4.8 shows the values of calculated vapour
pressures and their estimated uncertainties at each temperature. The results would indicate that
the uncertainty of vapour pressure increases as temperature increases.
Table 4.8. Vapour pressures and uncertainties calculated using the Monte Carlo simulation
for the gasoline before and after n-butane blending at different temperatures.
Temperature (K) Vapour Pressure (kPa)
Before Blending After Blending
270 18.02 ± 0.26 24.33 ± 0.25
280 26.46 ± 0.36 35.51 ± 0.33
290 37.79 ± 0.47 50.39 ± 0.44
300 52.64 ± 0.61 69.73 ± 0.57
310 71.73 ± 0.77 94.35 ± 0.73
320 95.79 ± 0.97 125.1 ± 0.9
330 125.6 ± 1.2 162.9 ± 1.1
Figure 4.9 shows the result of Monte Carlo simulation as a function of the sample
number. The results are distributed around the maximum allowable RVP of 93.08 kPa (red
dashed line) with the maximum amount of 93.95 kPa at sample number 72 and the minimum
amount of 92.34 kPa at sample number 20. Two green dashed-dotted lines show the lower and
upper limits of RVP based on a 95% confidence interval (CI). For 50 points out of 100 points in
the sample set, the calculated RVPs are greater than 93.08 kPa, which represents excessive
amount of the blended n-butane.
125
Figure 4.9. Monte Carlo simulation results for the RVP calculation of the final gasoline
blend with 7.17 volume percent of blended n-butane.
In order to find the rate of n-butane based on the uncertainty analysis, the RVP of the
gasoline blend was calculated at different volume flow rates of n-butane using the Monte Carlo
technique. Figure 4.10 shows the RVP of the gasoline blend against the standard volume ratio of
the blended n-butane into 331.23 Sm3/hr of gasoline. The RVP at a volume ratio of zero
represents the RVP of the initial gasoline with an average value of 70.72 kPa. The red dashed
line indicates the maximum allowable RVP of the final blend (93.08 kPa). The numerical values
of the RVP and associated uncertainty estimated based on 95% confidence interval along with
the range of variations of Monte Carlo results of RVP are summarized in Table 4.9.
92.2
92.6
93.0
93.4
93.8
94.2
0 10 20 30 40 50 60 70 80 90 100
RV
P (
kP
a)
Sample Number
MC results
95% CI
Max. allowable RVP
126
Figure 4.10. Calculated RVP and associated uncertainty against the blended n-
butane/gasoline standard volume ratio.
Table 4.9. Results of uncertainty analysis of RVP calculation depending on the volume
ratio of the blended n-butane to gasoline at standard conditions.
Volume Ratio (%) RVP (kPa) Range (Min-Max) (kPa)
0.00 70.72 ± 0.76 69.94 – 71.61
1.00 74.09 ± 0.75 73.30 – 74.95
2.00 77.36 ± 0.75 76.25 – 78.22
3.00 80.55 ± 0.74 79.78 – 81.42
4.00 83.67 ± 0.73 82.90 – 84.54
5.00 86.72 ± 0.73 85.96 – 87.58
6.86 92.19 ± 0.72 91.45 – 93.06
7.17 93.08 ± 0.72 92.34 – 93.95
8.00 95.47 ± 0.71 94.72 – 96.32
69
73
77
81
85
89
93
97
0 1 2 3 4 5 6 7 8
RV
P (
kP
a)
Volume Ratio of the Blended n-Butane to Gasoline (%)
Max. allowable RVP
127
The maximum volume ratio of the blended n-butane can be estimated using the results of
the uncertainty analysis such that the calculated RVP is always smaller than the specification
value of 93.08 kPa. The green arrow in Figure 4.10 shows that a value of 6.86 volume percent,
which is equivalent to 22.71 m3/hr of n-butane can be added to the 331.23 m
3/hr gasoline at
standard conditions. The results of the uncertainty analysis at this volume ratio were listed in
Table 4.9. As shown in this table, the calculated RVP is 93.06 kPa which is smaller than the
maximum allowable RVP.
Figure 4.11 was developed for the clarity purposes to show the distribution of RVP
calculated for the gasoline blend with 6.86 volume percent using the Monte Carlo method. The
green solid line indicates the mean value of the calculated vapour pressures that would be the
true mean value of the reported RVP (92.19 kPa) and will be used as a new specification in the
gasoline blending process studied in this work, and the two green dashed-dotted lines indicate
the lower and upper limits of RVP variations with a 95% CI and represent the uncertainty of the
calculated RVP (0.72 kPa). The new specification for RVP can be determined rigorously using
the results of the uncertainty analysis such that the RVP does not exceed the defined
specification. Note, this type of analysis allows for the development of realistic safety parameters
for blending based on the uncertainty of the calculations instead of simply basing operational
guidelines on rules of thumb.
128
Figure 4.11. Monte Carlo simulation results for RVP calculation of the final gasoline blend
with 6.86 volume percent of blended n-butane.
4.6 Conclusions
A consistent and self-contained error propagation using uncertainties in physical properties and
thermodynamic model parameters was used on two typical hydrocarbon processing problems.
This approach can provide valuable understanding about a process thanks to the calculation of
uncertainties of key operating or product specification parameters resulting from uncertainties in
the thermodynamic model and can assist engineers in providing realistic operational conditions
for safe equipment operation or reliable product production. Although not done here,
uncertainties resulting from process parameters such as temperatures, pressures, flows, and even
equipment performance can be easily implemented due to the process simulator modular
structure and the Monte Carlo algorithm.
91.3
91.7
92.1
92.5
92.9
93.3
0 10 20 30 40 50 60 70 80 90 100
RV
P (
kP
a)
Sample Number
MC results 95% CI
Max. allowable RVP Mean RVP
129
One of the major advantages of the proposed uncertainty analysis is its ability to better
estimate safety factors in the design of processes or product specifications. This method was
used for the process of injection of liquid n-butane into an existing gas pipeline and also for the
process of blending of liquid n-butane to gasoline, to evaluate the error propagation in the dew
point calculation of natural gas and Reid vapour pressure calculation of gasoline.
The safe amount of liquid hydrocarbon that can be added to the existing natural gas
pipeline without resulting in hydrocarbon liquid dropout was found by taking into account the
uncertainties of the pure components properties, binary interaction parameters, and
thermodynamic model. The uncertainty in specifications for design of the existing pipeline and
equipment performance were also evaluated using the error propagation algorithm to ensure that
the injection of n-butane does not result in off-specification gas.
For the gasoline blending case, the safe rate of n-butane that can be added to gasoline,
without exceeding the RVP specification of the final blend, was calculated using the APR EOS
and default values of the VMGSim for the binary interaction parameters, by taking into
consideration only the uncertainties of pure components properties. The quality of the estimated
uncertainties for mixtures containing heavier hydrocarbons can be bettered through data
regression as discussed in Chapter 3. This self-contained and consistent computational procedure
can be used by modular process simulation systems such as VMGSim.
130
Chapter Five: Conclusions and Recommendations
5.1 Conclusions
In this study, the reliability and accuracy of physical properties was identified as a main
parameter that has a significant impact on the quality of the process simulation and design.
However, this effect has been overlooked in commercial process simulators. This study
developed a new comprehensive mathematical procedure for error propagation calculations
based on uncertainties associated with the physical properties, binary interaction parameters, and
thermodynamic models commonly used for chemical process simulation and design.
Uncertainties related to process parameters such as process temperatures and pressures were not
considered in this study although these additional uncertainties could be easily included in the
developed mathematical procedure.
Two comprehensive databases for physical properties consisting of 176 pure components
and VLE data for 87 binary mixtures with their associated uncertainties were developed using
the uncertainty information provided by TDE. The uncertainty of relevant pure component
physical and thermo-physical properties to this research was evaluated using the standard error
propagation equation, Equation 2.21. Two generalized correlations for the estimation of critical
properties and acentric factors necessary to model oil fractions were updated. The Peng–
Robinson equations of state was re-parameterized by taking into account the uncertainties in both
dependent and independent variables using a weighted least squares method and the uncertainty
information of the model parameters was presented as the variance-covariance matrix. The
regression technique used in this study is based on the maximum likelihood principle in which, at
each point, the weighting factor is inversely proportional to the square of the uncertainty in that
131
point, and it involves the recalculation of the uncertainty at each iteration of the optimization
procedure. The results obtained from the comparison of the new models to the original ones
showed that the re-parameterized models not only provide better results for pure component
properties, Tables 2.6 and 2.10, but can also be used for more advanced statistical analyses due
to the ability to calculate uncertainty information for physical properties associated with the
pseudo-components.
Since the quality of the experimental VLE data affects the quality of the binary
interaction parameters and hence process simulation and design, the accuracy of the data
reported by TDE was vetted and the data quality was tested using the high-pressure
thermodynamic consistency test. A list of typographical errors, unit-conversion errors, report
interpretation errors, and data repetition errors present in TDE were prepared by reviewing the
original literature cited by TDE, reported to NIST, and resolved during the process of the
database development.
The consistency test was performed using the Valderrama–Faúndez method which is
based on the Gibbs–Duhem equation and the consistent datasets were used for determination of
the binary interaction parameters. The uncertainties of the binary interaction parameters were
evaluated using the Monte Carlo technique coupled with the nonlinear optimization method used
for estimation of the binary interaction parameters. The objective function used for the
evaluation of the parameters was a function of all process variables of the VLE data including
pressure, temperature, and both liquid and vapour phases compositions and their uncertainties.
It was shown in this study that the Monte Carlo method is a useful tool for uncertainty
analysis and can be successfully used to study and analyze the effect of input uncertainties on the
calculated values of variables of interest for flow sheets of any complexity. The Monte Carlo
132
technique coupled with Latin Hypercube Sampling (LHS) method, the developed databases for
pure component physical properties and binary interaction parameters, and the re-parameterized
Peng–Robinson equation of state along with the van der Waals quadratic mixing rules formed the
basis of a consistent and self-contained error propagation algorithm. The application of this
algorithm coupled with the VMGSim process simulator was illustrated through the uncertainty
evaluation for normal boiling point calculations, natural gas pressure-temperature envelopes, and
gasoline Reid vapour pressure estimations in realistic industry-based case studies.
In general, development of this error propagation algorithm with available uncertainty
information through its internal databases reduces guesswork in process simulation and design
and allows the designers to perform uncertainty analysis to critically evaluate design decisions
such as over-design for a given equipment. There is no standard practice to establish an over-
design value but rather this is currently done based on empirical rules of thumb and accumulated
experience of designer or design company. The amount of under- or over-design is a function of
what is not known by the designers and different from company to company and even from
engineer to engineer in the same company.
By having an assessment of the accuracy of a simulation or design, a company may
design a tighter piece of equipment, for example a smaller compressor, that will still work and
meet the design specifications, but may be considerably cheaper. This is, of course, a significant
advantage to the company with the knowledge of uncertainties over other companies when
bidding for a project. It should be noted that the turn-up/turn-down design factors that are
routinely applied to plants depending on the degree of flexibility required by the client, are
entirely separated from the analysis developed in this study. It should be noted that the analysis
133
of uncertainties associated with chemical processes provides a rich area for the targeting of
relevant research.
This work is one of the fundamental building blocks necessary to allow the users of
process simulators to know the expected errors in the design. It also mitigates the unknown or
little understood design risks by providing an analytical way of knowing uncertainty based on the
underlying thermodynamics. The process equipment design equations such as equations for the
calculation of heat transfer coefficients and pressure drops are the next fundamental building
blocks. In order to move uncertainty evaluation from belief to science, it is necessary to revise
the thermodynamic models and design equations that have been used for decades through
analysis of the physical property data and associated uncertainties and finally determine the
uncertainties in the model parameters.
After these two building blocks –thermodynamics and design equations- are constructed
and integrated into the process simulator, the errors in the thermodynamic properties of a stream,
the area of a heat exchanger, the make-up stream required for operation, and everything in
between can be scientifically obtained using the error propagation algorithm developed in this
study. It can also be easily incorporated into plant wide flow sheeting and control by taking into
consideration the uncertainties in independent process parameters such as temperatures,
pressures, and flows.
5.2 Recommendations
One of the main features of the error propagation algorithm developed in this study is the
capability to be linked with process simulators. In order to extend the application of the error
134
propagation algorithm to quantify uncertainties related to any chemical processes, the following
recommendations are suggested for future study.
1. Extend the databases to include heavier hydrocarbons, non-hydrocarbons and water. For
instance, the uncertainty information for amines and glycols should be added to the databases
in order to use the algorithm for sweetening and dehydration plants.
2. Add other popular models used to estimate temperature-dependent physical properties such
the Antoine equation for the estimation of vapour pressures and the Rackett model for the
estimation of liquid density. The other thermodynamic models, such as the SRK and the RK
equations of state, should be re-parameterized as was done for the Peng–Robinson equation
of state and their variance-covariance matrix of parameters should be determined.
3. Evaluate more complex mixing rules such as the Wong-Sandler mixing rule or the Huron-
Vidal mixing rule for the modelling of more complex phase behaviour such as liquid–liquid
equilibrium (LLE) and evaluate new interaction parameters and their uncertainties.
4. Finally, the most long lasting contribution of this study would be the implementation of error
analysis and error propagation into the engineer’s thought process and its further usage into
other important models used for process design which are related to physical and transport
properties. For example, the availability of heat transfer coefficients with their associated
uncertainty, combined with the techniques developed in this work would allow the design of
better heat exchangers, which would effectively improve the overall quality of plant designs.
135
APPENDIX A: DATABASE FOR PURE HYDROCARBONS FROM C5 TO C364
No Compound Formula Tb (K) (K) SG Tc (K) (K) Pc (kPa) (kPa)
1 3-Methyl-1-butene C5H10 293.20 0.02 0.6343 0.0014 452.69 0.50 3510.4 47.2 0.206 0.006
2 2-Methylbutane C5H12 301.02 0.48 0.6246 0.0005 460.37 0.09 3342.9 156.8 0.224 0.022
3 1-Pentene C5H10 303.10 0.03 0.6510 0.0063 464.74 0.03 3550.2 30.7 0.236 0.004
4 Pentane C5H12 309.21 0.09 0.6311 0.0002 469.71 0.10 3367.0 19.0 0.251 0.003
5 cis-2-Pentene C5H10 310.08 0.02 0.6594 0.0017 474.89 0.45 3691.7 27.1 0.254 0.003
6 2-Methyl-2-butene C5H10 311.59 0.14 0.6643 0.0016 470.40 1.00 3415.2 137.9 0.281 0.018
7 3,3-Dimethyl-1-butene C6H12 314.38 0.03 0.6581 0.0005 477.40 0.90 3025.1 154.2 0.212 0.022
8 Cyclopentene C5H8 317.37 0.05 0.7771 0.0004 506.08 0.21 4783.6 107.0 0.199 0.010
9 Cyclopentane C5H10 322.39 0.02 0.7502 0.0002 511.74 0.18 4515.1 83.5 0.195 0.008
10 2,2-Dimethylbutane C6H14 322.87 0.02 0.6539 0.0003 489.09 0.47 3101.8 13.4 0.232 0.002
11 4-Methyl-1-pentene C6H12 326.84 0.05 0.6687 0.0004 493.10 0.50 3178.4 9.4 0.256 0.001
12 2,3-Dimethyl-1-butene C6H12 328.74 0.02 0.6830 0.0004 497.70 0.90 3044.2 120.5 0.221 0.017
13 cis-4-Methyl-2-pentene C6H12 329.51 0.02 0.6741 0.0004 496.30 0.70 3360.6 79.9 0.284 0.010
14 2,3-Dimethylbutane C6H14 331.12 0.06 0.6660 0.0002 500.16 0.31 3132.8 25.6 0.246 0.004
15 2-Methylpentane C6H14 333.39 0.03 0.6576 0.0002 497.85 0.23 3032.0 13.4 0.278 0.002
16 2-Methyl-1-pentene C6H12 335.24 0.01 0.6855 0.0011 501.90 5.08 3162.8 9.9 0.283 0.001
17 3-Methylpentane C6H14 336.40 0.03 0.6690 0.0003 504.56 0.15 3123.1 15.3 0.272 0.002
18 1-Hexene C6H12 336.58 0.09 0.6780 0.0004 504.07 0.89 3206.4 89.3 0.287 0.012
19 cis-3-Hexene C6H12 339.57 0.01 0.6847 0.0004 510.20 1.40 3299.4 12.6 0.285 0.002
4 Reprinted from Fluid Phase Equilibria, Vol. 307, S. Hajipour, M.A. Satyro, Uncertainty Analysis Applied to Thermodynamic Models and Process Design – 1.
Pure Components, 78-94, Copyright (2011), with permission from Elsevier.
136
No Compound Formula Tb (K) (K) SG Tc (K) (K) Pc (kPa) (kPa)
20 trans-3-Hexene C6H12 340.21 0.01 0.6820 0.0003 507.40 2.00 3333.7 87.5 0.320 0.011
21 2-Methyl-2-pentene C6H12 340.43 0.01 0.6914 0.0004 509.30 0.50 3258.3 12.1 0.297 0.002
22 trans-2-Hexene C6H12 341.00 0.02 0.6828 0.0004 509.00 0.70 3306.3 90.1 0.313 0.012
23 Hexane C6H14 341.85 0.11 0.6642 0.0002 507.53 0.14 3031.2 32.4 0.301 0.005
24 cis-2-Hexene C6H12 342.04 0.14 0.6920 0.0003 513.40 0.90 3233.9 133.8 0.280 0.018
25 Methylcyclopentane C6H12 344.94 0.04 0.7537 0.0002 532.78 0.05 3779.5 80.5 0.229 0.009
26 4,4-Dimethyl-1-pentene C7H14 345.63 0.03 0.6872 0.0003 516.00 4.00 3047.3 83.7 0.284 0.012
27 2,3-Dimethyl-2-butene C6H12 346.33 0.03 0.7128 0.0006 521.00 0.90 3404.0 141.0 0.291 0.018
28 2,2-Dimethylpentane C7H16 352.32 0.02 0.6784 0.0002 520.62 0.55 2767.8 44.9 0.285 0.007
29 Benzene C6H6 353.23 0.03 0.8846 0.0002 562.02 0.14 4896.9 18.6 0.210 0.002
30 2,4-Dimethylpentane C7H16 353.56 0.21 0.6773 0.0003 519.95 0.66 2732.4 57.3 0.299 0.010
31 Cyclohexane C6H12 353.86 0.04 0.7835 0.0001 553.35 0.29 4064.0 14.0 0.210 0.002
32 2,2,3-Trimethylbutane C7H16 354.00 0.02 0.6946 0.0001 531.27 0.49 2944.3 17.2 0.248 0.003
33 Cyclohexene C6H10 356.06 0.06 0.8157 0.0007 560.45 0.02 4421.4 148.4 0.217 0.015
34 2,3-Dimethyl-1-pentene C7H14 357.40 1.50 0.7098 0.0008 533.60 4.30 2856.0 795.3 0.257 0.123
35 5-Methyl-1-hexene C7H14 358.30 1.00 0.6972 0.0010 528.70 0.40 2862.1 183.7 0.304 0.031
36 3,3-Dimethylpentane C7H16 359.19 0.02 0.6983 0.0008 536.37 0.40 2938.8 14.2 0.266 0.002
37 2,3-Dimethylpentane C7H16 362.95 0.24 0.6994 0.0004 537.47 0.59 2911.1 155.8 0.295 0.024
38 2-Methylhexane C7H16 363.12 0.13 0.6833 0.0003 530.37 0.15 2734.1 24.9 0.329 0.004
39 2-Methyl-1-hexene C7H14 364.90 1.30 0.7069 0.0012 541.80 1.10 2901.6 279.6 0.284 0.045
40 3-Methylhexane C7H16 364.97 0.04 0.6915 0.0003 535.36 0.51 2817.0 21.1 0.322 0.003
41 3-Ethylpentane C7H16 366.59 0.02 0.7031 0.0004 540.67 0.37 2900.3 15.5 0.311 0.002
42 1-Heptene C7H14 366.77 0.22 0.7017 0.0006 537.34 0.28 2851.8 32.0 0.332 0.006
137
No Compound Formula Tb (K) (K) SG Tc (K) (K) Pc (kPa) (kPa)
43 trans-3-Heptene C7H14 369.00 1.20 0.7027 0.0015 538.60 0.70 2959.6 225.9 0.363 0.037
44 cis-2-Heptene C7H14 370.50 1.30 0.7103 0.0018 548.50 0.60 2975.1 257.8 0.306 0.042
45 trans-2-Heptene C7H14 371.20 1.10 0.7054 0.0013 542.80 0.40 2959.2 151.2 0.355 0.026
46 Heptane C7H16 371.53 0.10 0.6883 0.0002 540.06 0.19 2735.8 28.4 0.350 0.005
47 2,2,4-Trimethylpentane C8H18 372.36 0.21 0.6963 0.0002 543.91 0.44 2567.9 59.8 0.304 0.011
48 Methylcyclohexane C7H14 374.04 0.07 0.7739 0.0002 572.31 0.05 3481.0 175.0 0.235 0.022
49 Ethylcyclopentane C7H14 376.61 0.07 0.7710 0.0002 569.48 0.05 3396.9 40.1 0.271 0.005
50 2,2-Dimethylhexane C8H18 379.96 0.02 0.6997 0.0003 549.88 0.40 2527.3 30.4 0.337 0.005
51 2,5-Dimethylhexane C8H18 382.23 0.02 0.6983 0.0003 550.01 0.34 2487.7 13.5 0.356 0.002
52 2,4-Dimethylhexane C8H18 382.55 0.02 0.7037 0.0014 553.00 3.29 2540.3 32.1 0.344 0.006
53 2,2,3-Trimethylpentane C8H18 382.96 0.02 0.7199 0.0003 563.46 0.40 2726.0 24.5 0.297 0.004
54 Toluene C7H8 383.73 0.11 0.8718 0.0002 591.89 0.07 4132.5 52.8 0.265 0.006
55 3,3-Dimethylhexane C8H18 385.10 0.01 0.7140 0.0005 561.98 0.40 2655.4 8.7 0.321 0.001
56 2,3,4-Trimethylpentane C8H18 386.60 0.02 0.7230 0.0003 566.37 0.40 2706.2 32.4 0.312 0.005
57 2,3,3-Trimethylpentane C8H18 387.89 0.02 0.7301 0.0004 573.52 0.40 2823.7 31.3 0.291 0.005
58 2,3-Dimethylhexane C8H18 388.73 0.02 0.7168 0.0004 563.45 0.40 2628.3 19.2 0.346 0.003
59 2-Methyl-3-ethylpentane C8H18 388.77 0.02 0.7235 0.0003 567.05 0.40 2698.1 16.5 0.329 0.003
60 2-Methylheptane C8H18 390.78 0.04 0.7019 0.0003 559.63 0.14 2502.1 30.5 0.381 0.005
61 4-Methylheptane C8H18 390.83 0.02 0.7089 0.0002 561.70 0.40 2541.2 17.7 0.371 0.003
62 3,4-Dimethylhexane C8H18 390.85 0.02 0.7232 0.0005 568.81 0.40 2691.8 17.8 0.338 0.003
63 3-Ethyl-3-methylpentane C8H18 391.39 0.01 0.7316 0.0001 576.54 0.40 2773.4 12.4 0.299 0.002
64 3-Ethylhexane C8H18 391.66 0.02 0.7175 0.0004 565.45 0.40 2605.1 16.1 0.360 0.003
65 Cycloheptane C7H14 391.94 0.05 0.8159 0.0001 604.26 0.10 3858.9 86.8 0.243 0.010
138
No Compound Formula Tb (K) (K) SG Tc (K) (K) Pc (kPa) (kPa)
66 3-Methylheptane C8H18 392.04 0.03 0.7099 0.0004 563.68 0.40 2542.6 25.8 0.369 0.004
67 1-Heptene, 2-methyl- C8H16 392.40 1.30 0.7248 0.0007 567.50 0.90 2629.8 231.6 0.357 0.042
68 1,trans-4-Dimethylcyclohexane C8H16 392.47 0.02 0.7672 0.0003 587.66 1.60 3037.0 12.5 0.268 0.002
69 1-Octene C8H16 394.42 0.07 0.7196 0.0003 566.58 0.05 2675.7 35.0 0.395 0.006
70 2-Heptene, 2-methyl- C8H16 395.20 1.50 0.7287 0.0010 568.90 1.10 2638.2 367.3 0.379 0.063
71 2,2,4,4-Tetramethylpentane C9H20 395.39 0.04 0.7237 0.0005 574.61 0.50 2485.1 3.4 0.312 0.001
72 trans-4-Octene C8H16 395.54 0.08 0.7188 0.0008 566.30 1.10 2546.9 55.3 0.390 0.009
73 2,2,5-Trimethylhexane C9H20 397.16 0.11 0.7114 0.0005 569.88 2.00 2457.3 33.0 0.364 0.006
74 1,cis-4-Dimethylcyclohexane C8H16 397.43 0.04 0.7872 0.0003 603.19 0.33 3431.7 30.3 0.260 0.004
75 cis-1,3-Dimethylcyclohexane C8H16 397.57 0.02 0.7711 0.0010 587.67 0.50 2879.1 12.9 0.299 0.002
76 trans-2-Octene C8H16 398.02 0.17 0.7238 0.0011 569.80 0.40 2687.1 140.0 0.413 0.023
77 Octane C8H18 398.81 0.04 0.7070 0.0001 568.78 0.14 2484.9 9.6 0.398 0.002
78 Ethylcyclohexane C8H16 404.82 0.19 0.7925 0.0002 606.90 0.40 3277.8 100.9 0.292 0.014
79 Heptane, 2,2-dimethyl- C9H20 405.76 0.53 0.7147 0.0008 576.65 0.50 2349.5 68.2 0.390 0.014
80 2,2,3,4-Tetramethylpentane C9H20 406.14 0.12 0.7431 0.0005 592.61 0.50 2602.1 34.8 0.313 0.006
81 Ethylbenzene C8H10 409.32 0.05 0.8724 0.0004 617.12 0.11 3615.9 12.0 0.304 0.002
82 2,2,5,5-Tetramethylhexane C10H22 410.30 1.10 0.7228 0.0013 581.41 0.50 2186.0 3.0 0.372 0.013
83 1,4-Dimethylbenzene C8H10 411.46 0.06 0.8657 0.0004 616.19 0.15 3527.0 16.0 0.323 0.002
84 1,3-Dimethylbenzene C8H10 412.22 0.05 0.8687 0.0004 616.85 0.56 3539.7 13.6 0.327 0.002
85 2,2,3,3-Tetramethylpentane C9H20 413.39 0.05 0.7607 0.0005 607.51 0.50 2742.3 15.9 0.304 0.003
86 r-1, c-3, t-5-Trimethylcyclohexane C9H18 413.80 1.40 0.7835 0.0012 602.16 1.60 2648.9 273.2 0.331 0.047
87 2,3,3,4-Tetramethylpentane C9H20 414.67 0.12 0.7588 0.0005 607.51 0.50 2715.8 36.3 0.313 0.006
88 Octane, 2-methyl- C9H20 416.12 0.43 0.7173 0.0007 582.83 0.15 2302.5 21.0 0.455 0.007
139
No Compound Formula Tb (K) (K) SG Tc (K) (K) Pc (kPa) (kPa)
89 1,2-Dimethylbenzene C8H10 417.54 0.07 0.8844 0.0002 630.43 0.61 3745.3 24.6 0.312 0.003
90 Styrene C8H8 418.40 0.43 0.9111 0.0005 635.20 2.00 3880.9 207.0 0.302 0.024
91 1-Nonene C9H18 420.01 0.02 0.7335 0.0003 593.70 1.20 2378.3 12.8 0.422 0.002
92 Nonane C9H20 423.78 0.17 0.7221 0.0002 594.12 0.67 2294.4 81.5 0.448 0.016
93 Cyclooctane C8H16 424.29 0.03 0.8408 0.0003 647.36 0.49 3551.0 80.6 0.250 0.010
94 Isopropylbenzene C9H12 425.51 0.13 0.8667 0.0001 631.16 1.42 3185.2 282.3 0.322 0.039
95 Isopropylcyclohexane C9H18 427.56 0.21 0.8063 0.0004 632.20 0.40 3062.8 192.0 0.322 0.027
96 (1S)-(-)-.alpha.-pinene C10H16 429.10 0.44 0.8623 0.0020 644.00 5.00 3353.9 205.0 0.295 0.027
97 Propylcyclohexane C9H18 429.85 0.03 0.7977 0.0003 630.80 0.90 2868.6 40.2 0.327 0.006
98 3,3,5-Trimethylheptane C10H22 429.90 1.90 0.7472 0.0022 609.51 0.50 2317.0 48.4 0.395 0.024
99 Propylbenzene C9H12 432.34 0.09 0.8665 0.0003 638.29 0.14 3201.4 47.1 0.345 0.006
100 Hexane, 2,2,3,3-tetramethyl- C10H22 432.70 1.30 0.7686 0.0017 623.01 0.50 2510.0 55.5 0.357 0.017
101 1-Ethyl-4-methylbenzene C9H12 435.13 0.24 0.8659 0.0008 640.20 0.45 3234.1 41.9 0.365 0.006
102 1,3,5-Trimethylbenzene C9H12 437.91 0.23 0.8695 0.0005 637.31 0.10 3127.8 53.9 0.399 0.008
103 Tert-butylbenzene C10H14 442.26 0.03 0.8710 0.0003 648.08 1.08 2998.4 34.0 0.351 0.005
104 1,2,4-Trimethylbenzene C9H12 442.52 0.31 0.8803 0.0003 649.12 0.11 3261.0 218.0 0.381 0.029
105 1-Decene C10H20 443.71 0.01 0.7449 0.0004 616.00 0.30 2157.2 5.1 0.472 0.001
106 Isobutylcyclohexane C10H20 444.45 0.13 0.7993 0.0006 642.10 0.60 2610.0 67.0 0.358 0.011
107 Tert-butylcyclohexane C10H20 444.72 0.16 0.8168 0.0006 652.00 0.40 2824.7 94.3 0.326 0.015
108 Bicyclo[4.1.0]hept-3-ene, 3,7,7-trimethyl- C10H16 445.00 2.50 0.8700 0.0043 660.00 21.68 2967.3 876.3 0.297 0.131
109 Isobutylbenzene C10H14 445.87 0.31 0.8574 0.0005 650.28 3.00 3047.0 192.0 0.379 0.028
110 Sec-butylbenzene C10H14 446.45 0.03 0.8666 0.0004 652.50 1.10 2940.5 24.5 0.355 0.004
111 Decane C10H22 447.27 0.06 0.7341 0.0003 618.07 0.95 2102.0 51.0 0.485 0.011
140
No Compound Formula Tb (K) (K) SG Tc (K) (K) Pc (kPa) (kPa)
112 1,2,3-Trimethylbenzene C9H12 449.20 0.30 0.8987 0.0003 664.42 0.10 3455.1 60.6 0.367 0.009
113 p-Cymene C10H14 450.27 0.08 0.8614 0.0005 653.80 8.06 2799.0 100.0 0.363 0.016
114 (R)-(+)-Limonene C10H16 450.79 0.02 0.8479 0.0017 653.00 5.00 2809.0 19.1 0.376 0.003
115 Indane C9H10 450.92 0.04 0.9672 0.0023 684.85 0.40 3965.5 49.6 0.309 0.005
116 Butylcyclohexane C10H20 454.10 0.11 0.8033 0.0004 653.10 0.40 2556.1 54.7 0.370 0.009
117 Butylbenzene C10H14 456.43 0.07 0.8645 0.0003 660.48 0.10 2887.1 58.5 0.393 0.009
118 1,4-Diethylbenzene C10H14 456.89 0.17 0.8663 0.0006 657.90 0.03 2798.1 85.8 0.403 0.013
119 trans-Decalin C10H18 460.43 0.20 0.8740 0.0002 687.02 1.00 3119.0 245.0 0.291 0.034
120 1,1'-Bicyclopentyl C10H18 463.61 0.03 0.8686 0.0006 690.00 2.00 3268.7 31.0 0.319 0.004
121 Decahydronaphthalene C10H18 464.90 1.00 0.8873 0.0032 645.13 10.00 2090.0 354.0 0.452 0.074
122 cis-Decalin C10H18 468.96 0.28 0.9009 0.0004 702.22 1.00 3207.0 576.0 0.287 0.078
123 Undecane C11H24 469.03 0.15 0.7445 0.0002 638.82 0.19 2009.0 68.0 0.545 0.015
124 1,2,4,5-Tetramethylbenzene C10H14 470.50 1.70 0.8909 0.0200 675.68 2.00 2854.0 266.0 0.424 0.044
125 Pentylbenzene C11H16 476.00 2.80 0.8635 0.0006 675.00 7.00 2578.2 260.0 0.443 0.053
126 1,3-Dimethyladamantane C12H20 476.43 0.01 0.8885 0.0030 708.00 2.00 2862.7 5.5 0.276 0.001
127 Tetralin C10H12 480.32 0.34 0.9736 0.0007 720.28 1.45 3522.7 244.4 0.313 0.030
128 p-Diisopropylbenzene C12H18 483.42 0.04 0.8618 0.0016 675.00 1.00 2297.0 37.0 0.470 0.007
129 1-Dodecene C12H24 486.51 0.02 0.7623 0.0003 657.60 0.60 1875.0 5.0 0.558 0.001
130 Benzene, 1,3,5-triethyl- C12H18 488.92 0.03 0.8665 0.0009 679.00 2.00 2321.9 15.0 0.506 0.003
131 Dodecane C12H26 489.45 0.12 0.7529 0.0004 658.28 0.59 1812.4 87.2 0.572 0.021
132 Naphthalene C10H8 491.14 0.02 1.0303 0.0072 748.33 0.26 4050.0 44.0 0.304 0.005
133 Hexylbenzene C12H18 498.90 1.30 0.8633 0.0009 695.00 7.00 2372.2 272.4 0.497 0.051
134 1-Tridecene C13H26 505.91 0.11 0.7694 0.0004 673.00 7.00 1736.0 54.3 0.617 0.014
141
No Compound Formula Tb (K) (K) SG Tc (K) (K) Pc (kPa) (kPa)
135 Tridecane C13H28 508.60 0.12 0.7603 0.0004 675.89 0.40 1659.6 62.2 0.605 0.016
136 1,4-Di-tert-butylbenzene C14H22 510.45 0.02 0.8679 0.0160 708.00 2.00 2233.8 11.6 0.494 0.002
137 2-Methylnaphthalene C11H10 514.24 0.12 1.0108 0.0005 761.15 3.00 3376.9 119.6 0.354 0.015
138 Heptylbenzene C13H20 515.40 4.40 0.8625 0.0011 708.00 7.00 2135.2 231.4 0.531 0.067
139 1-Methylnaphthalene C11H10 517.56 0.29 1.0248 0.0011 770.71 4.64 3510.0 112.0 0.341 0.014
140 2,2,4,4,6,8,8-Heptamethylnonane C16H34 519.44 0.03 0.7879 0.0014 691.95 4.00 1526.9 7.0 0.536 0.002
141 1-Tetradecene C14H28 524.29 0.11 0.7757 0.0006 691.00 7.00 1584.1 74.9 0.634 0.021
142 Tetradecane C14H30 526.70 0.25 0.7671 0.0004 692.49 1.15 1563.6 143.7 0.645 0.040
143 1,1'-Biphenyl C12H10 528.38 0.04 1.0388 0.0066 772.16 5.22 3407.0 60.0 0.415 0.008
144 Benzene, 1,2,4,5-tetrakis(1-methylethyl)- C18H30 533.95 0.02 0.8558 0.0851 703.00 1.00 1650.3 10.6 0.661 0.003
145 Naphthalene, 2,7-dimethyl- C12H12 535.52 0.04 0.9960 0.0180 775.00 2.00 2948.0 31.0 0.400 0.005
146 Octylbenzene C14H22 536.50 1.30 0.8609 0.0007 725.00 7.00 1984.2 210.0 0.589 0.047
147 Diphenylmethane C13H12 537.45 0.07 1.0120 0.0017 775.83 9.32 3020.0 77.0 0.422 0.011
148 Hexamethylbenzene C12H18 540.90 3.20 0.9109 0.0911 757.93 2.00 2585.7 430.7 0.508 0.078
149 1-Pentadecene C15H30 541.55 0.07 0.7804 0.0010 705.00 7.00 1555.6 46.9 0.705 0.013
150 Pentadecane C15H32 543.77 0.34 0.7721 0.0007 706.88 2.17 1424.0 195.0 0.678 0.060
151 1-Hexadecene C16H32 558.00 1.40 0.7853 0.0008 718.00 7.00 1377.7 362.0 0.728 0.115
152 Hexadecane C16H34 560.10 1.10 0.7774 0.0005 722.25 0.83 1411.9 319.9 0.726 0.099
153 Decylbenzene C16H26 570.96 0.35 0.8595 0.0006 752.00 8.00 1724.5 95.6 0.681 0.024
154 1-Heptadecene C17H34 573.90 1.30 0.7898 0.0011 734.00 7.00 1344.2 81.8 0.758 0.028
155 Heptadecane C17H36 575.80 1.20 0.7818 0.0005 735.71 0.95 1318.9 102.7 0.756 0.035
156 Benzene, undecyl- C17H28 584.80 1.40 0.8607 0.0019 763.00 8.00 1627.5 129.5 0.732 0.036
157 1-Octadecene C18H36 588.70 1.00 0.7933 0.0018 748.00 8.00 1286.5 140.2 0.792 0.048
142
No Compound Formula Tb (K) (K) SG Tc (K) (K) Pc (kPa) (kPa)
158 Octadecane C18H38 589.50 1.90 0.7868 0.0100 747.63 1.00 1204.6 224.9 0.778 0.082
159 1,1'-Methylenebis[(1-methylethyl)benzene] C19H24 592.00 35.00 0.9913 0.0292 795.00 8.00 1550.0 111.7 0.500 0.435
160 Nonadecane C19H40 602.81 0.73 0.7912 0.0077 755.68 5.20 1156.0 74.0 0.835 0.028
161 1-Nonadecene C19H38 604.00 16.00 0.7949 0.0042 755.00 8.00 1190.0 131.1 0.877 0.217
162 1,1':2',1''-Terphenyl C18H14 610.70 1.40 1.0851 0.0120 857.12 6.00 2884.0 97.0 0.548 0.019
163 Tridecylbenzene C19H32 613.20 2.00 0.8558 0.0280 790.00 8.00 1530.5 116.2 0.798 0.037
164 Eicosane C20H42 617.25 0.24 0.7930 0.0012 768.22 5.77 1078.0 49.0 0.869 0.020
165 1-Eicosene C20H40 620.00 15.00 0.8275 0.0250 772.00 15.00 1140.0 266.9 0.881 0.222
166 Heneicosane C21H44 632.10 5.90 0.7944 0.0086 777.60 7.80 1030.0 109.0 0.923 0.073
167 Docosane C22H46 641.90 3.50 0.7994 0.0031 785.52 5.75 991.9 129.7 0.966 0.061
168 m-Terphenyl C18H14 647.80 2.20 1.0901 0.0360 882.51 6.80 2115.0 191.0 0.581 0.043
169 Tricosane C23H48 653.80 9.10 0.8033 0.0045 789.70 7.90 914.3 133.2 1.047 0.110
170 p-Terphenyl C18H14 657.01 1.39 1.1461 0.1141 912.94 21.58 2406.5 533.3 0.526 0.097
171 Tetracosane C24H50 664.20 2.50 0.8007 0.0059 799.56 5.24 868.7 114.3 1.058 0.058
172 Hexacosane C26H54 688.00 11.00 0.8071 0.0037 816.00 8.00 787.2 189.2 1.125 0.142
173 2,6,10,15,19,23-Hexamethyltetracosane C30H62 693.50 2.60 0.8138 0.0011 795.90 2.00 595.5 37.6 1.264 0.038
174 Octacosane C28H58 707.00 3.20 0.8092 0.0032 824.00 8.00 751.5 117.9 1.264 0.068
175 Triacontane C30H62 725.70 8.40 0.8088 0.0130 843.00 8.00 645.1 140.7 1.181 0.117
176 Hexatriacontane C36H74 777.30 6.70 0.8217 0.0070 872.00 9.00 471.4 71.9 1.500 0.094
143
APPENDIX B: CALCULATED UNCERTAINTY OF VAPOUR PRESSURE USING NEW 3-PARAMETER PENG-
ROBINSON EQUATION OF STATE BY COVARIANCE APPROACH5
Methane (= 0.0116) n-Hexane (= 0.3009) n-Dodecane (= 0.5715)
T
(K)
Pexp
(kPa)
(kPa)
Pcal
(kPa)
(kPa)
T
(K)
P exp
(kPa)
(kPa)
P cal
(kPa)
(kPa)
T
(K)
P exp
(kPa)
(kPa)
P cal
(kPa)
(kPa)
111.69 101.32 0.29 98.25 1.16 341.85 101.31 0.33 101.23 1.74 489.45 101.33 0.28 99.91 6.26
112.50 108.29 0.31 105.07 1.21 349.60 128.42 0.46 128.21 2.10 496.61 119.78 0.39 118.22 7.22
117.00 153.77 0.46 149.67 1.54 358.80 167.58 0.66 167.21 2.60 505.98 147.80 0.61 146.11 8.65
121.50 212.51 0.69 207.45 1.91 368.00 215.37 0.90 214.87 3.18 515.35 180.70 0.95 178.94 10.30
126.00 286.70 1.00 280.68 2.31 377.20 273.00 1.20 272.41 3.85 524.72 219.00 1.40 217.31 12.19
130.50 378.80 1.40 371.73 2.73 386.40 341.60 1.60 341.11 4.62 534.09 263.40 2.10 261.84 14.36
135.00 491.00 1.90 483.08 3.17 395.60 422.50 2.00 422.34 5.51 543.46 314.40 3.10 313.19 16.82
139.50 625.80 2.60 617.28 3.60 404.80 517.10 2.50 517.53 6.53 552.83 372.70 4.40 372.06 19.63
144.00 785.60 3.20 776.92 4.04 414.00 626.80 3.00 628.16 7.70 562.20 439.00 6.10 439.18 22.81
148.50 973.00 4.00 964.70 4.45 423.20 753.00 3.70 755.80 9.04 571.57 514.10 8.30 515.33 26.41
153.00 1190.60 4.80 1183.30 4.85 432.40 897.30 4.40 902.09 10.55 580.94 599.00 11.00 601.32 30.46
157.50 1440.90 5.60 1435.60 5.22 441.60 1061.50 5.40 1068.72 12.27 590.31 694.00 15.00 698.01 35.01
162.00 1726.80 6.30 1724.40 5.56 450.80 1247.20 6.60 1257.49 14.22 599.68 800.00 19.00 806.32 40.11
5 Reprinted from Fluid Phase Equilibria, Vol. 307, S. Hajipour, M.A. Satyro, Uncertainty Analysis Applied to Thermodynamic Models and Process Design – 1.
Pure Components, 78-94, Copyright (2011), with permission from Elsevier.
144
Methane (= 0.0116) n-Hexane (= 0.3009) n-Dodecane (= 0.5715)
T
(K)
Pexp
(kPa)
(kPa)
Pcal
(kPa)
(kPa)
T
(K)
P exp
(kPa)
(kPa)
P cal
(kPa)
(kPa)
T
(K)
P exp
(kPa)
(kPa)
P cal
(kPa)
(kPa)
166.50 2051.20 6.80 2052.60 5.90 460.00 1456.70 8.20 1470.26 16.42 609.05 919.00 25.00 927.21 45.82
171.00 2417.30 7.20 2423.30 6.25 469.20 1692.00 10.00 1709.01 18.91 618.42 1052.00 32.00 1061.69 52.20
175.50 2828.70 7.40 2839.50 6.65 478.40 1956.00 13.00 1975.80 21.70 627.79 1199.00 41.00 1210.85 59.29
180.00 3289.70 7.30 3304.50 7.15 487.60 2253.00 18.00 2272.83 24.83 637.16 1364.00 52.00 1375.84 67.18
184.50 3806.00 7.20 3821.60 7.84 496.80 2585.00 23.00 2602.43 28.34 646.53 1548.00 65.00 1557.91 75.93
189.00 4386.40 8.20 4394.30 8.80 506.00 2963.00 31.00 2967.03 32.26 655.90 1755.00 82.00 1758.35 85.62
190.56 4606.80 9.10 4606.80 9.20 507.53 3031.18 32.39 3031.18 32.96 658.28 1812.40 87.19 1812.40 88.24
145
APPENDIX C: DETAILS ON THE DEVELOPED VLE DATABASE 6
The detailed information about the constructed VLE database for 87 binary mixtures of interest
in this study is summarized in Table C.1. The number of data points (NP), number of isothermal
datasets with more than three data points (NT=cte.), and the ranges of variation in temperature and
pressure values are shown in this table. This table shows information for all experimental VLE
data used in this study before performing consistency test.
Table C.1. Detailed information about the developed binary VLE database.
No. Binary NP NT=cte. Range T (K) Range P (kPa)
1 Methane/Ethane 511 49 110.90 – 283.15 15.789 – 6894.8
2 Methane/Propane 604 54 114.10 – 363.15 42.058 – 10163
3 Methane/i-Butane 93 9 191.50 – 377.55 490.33 – 11768
4 Methane/n-Butane 422 40 144.26 – 410.95 137.90 – 13135
5 Methane/i-Pentane 21 3 344.26 – 410.93 3440.5 – 15106
6 Methane/n-Pentane 1140 64 176.21 – 460.93 60.795 – 16719
7 Methane/n-Hexane 170 16 190.50 – 444.25 137.21 – 19783
8 Methane/n-Heptane 165 14 199.82 – 510.93 689.48 – 24883
9 Methane/n-Octane 71 9 273.15 – 423.15 1013.3 – 28878
10 Methane/n-Nonane 127 8 223.15 – 423.15 1013.2 – 31917
11 Methane/n-Decane 234 19 277.59 – 583.05 137.90 – 36198
12 Methane/Nitrogen 678 54 91.600 – 199.82 21.198 – 5061.5
13 Methane/Argon 181 17 91.600 – 178.00 16.132 – 5098.7
14 Methane/Helium 484 44 91.100 – 290.00 481.29 – 1013250
15 Methane/H2S 145 9 188.71 – 366.48 860.00 – 13100
16 Methane/CO2 339 35 153.15 – 301.00 1078.0 – 8519.4
17 Ethane/Propane 581 35 127.59 – 369.18 0.0180 – 5184.9
18 Ethane/i-Butane 36 4 311.26 – 394.04 1068.7 – 5371.0
6 Reprinted from Fluid Phase Equilibria, Vol. 364, S. Hajipour, M.A. Satyro, M.W. Foley, Uncertainty Analysis
Applied to Thermodynamic Models and Process Design – 2. Binary Mixtures, 15-30, Copyright (2013), with
permission from Elsevier.
146
No. Binary NP NT=cte. Range T (K) Range P (kPa)
19 Ethane/n-Butane 112 13 260.00 – 366.45 161.30 – 5571.0
20 Ethane/n-Pentane 91 10 277.59 – 449.82 344.74 – 6550.0
21 Ethane/n-Hexane 60 5 298.15 – 449.82 172.37 – 7901.4
22 Ethane/n-Heptane 89 10 338.71 – 449.82 2757.9 – 8618.4
23 Ethane/n-Octane 64 7 313.15 – 373.15 405.30 – 6800.0
24 Ethane/Nitrogen 332 28 110.93 – 290.00 196.50 – 13466
25 Ethane/Argon 4 – 84.760 – 113.52 64.636 – 690.83
26 Ethane/Helium 73 10 133.15 – 273.15 490.30 – 11768
27 Ethane/H2S 45 4 199.93 – 283.15 65.155 – 3052.3
28 Ethane/CO2 382 30 207.00 – 298.15 329.20 – 6629.7
29 Propane/i-Butane 230 32 237.15 – 395.01 40.797 – 4171.3
30 Propane/n-Butane 241 18 236.53 – 414.35 26.265 – 4357.5
31 Propane/n-Pentane 149 14 336.56 – 460.93 334.37 – 4481.6
32 Propane/n-Octane 13 2 473.10 – 523.10 2540.0 – 5140.0
33 Propane/n-Nonane 5 1 376.75 – 377.15 938.00 – 3468.0
34 Propane/n-Decane 6 1 376.85 – 377.15 945.00 – 3516.0
35 Propane/Nitrogen 178 23 114.10 – 353.15 150.31 – 21919
36 Propane/H2S 54 – 217.59 – 367.04 137.90 – 4143.7
37 Propane/CO2 422 43 230.00 – 361.15 194.00 – 6894.8
38 n-Butane/i-Butane 24 1 273.15 108.72 – 153.45
39 i-Butane/Nitrogen 90 7 120.00 – 310.87 232.35 – 20774
40 i-Butane/H2S 84 10 277.65 – 398.15 206.84 – 8887.0
41 i-Butane/CO2 129 11 273.15 – 398.15 273.58 – 7400.0
42 n-Butane/n-Pentane 89 1 270.00 – 409.96 34.000 – 2484.0
43 n-Butane/n-Heptane 36 – 339.26 – 525.93 689.48 – 2757.9
44 n-Butane/n-Decane 61 6 310.93 – 510.93 172.37 – 4826.3
45 n-Butane/Nitrogen 152 16 250.00 – 422.04 452.00 – 28751
46 n-Butane/Argon 43 3 339.67 – 380.25 1393.0 – 18485
47 n-Butane/H2S 51 6 366.45 – 418.15 1482.0 – 7894.0
48 n-Butane/CO2 510 35 227.98 – 418.48 33.095 – 8087.6
49 i-Pentane/n-Pentane 13 – 328.15 – 384.82 234.38 – 785.93
147
No. Binary NP NT=cte. Range T (K) Range P (kPa)
50 i-Pentane/n-Hexane 32 – 306.89 – 335.25 101.33
51 i-Pentane/H2S 35 4 323.15 – 413.15 310.00 – 8377.0
52 i-Pentane/CO2 171 19 253.15 – 453.15 151.68 – 9230.0
53 n-Pentane/n-Hexane 32 3 298.70 – 308.70 25.80 – 94.66
54 n-Pentane/n-Heptane 36 – 293.15 – 521.95 101.33 – 3060.0
55 n-Pentane/n-Octane 34 3 303.70 – 313.70 5.700 – 97.00
56 n-Pentane/n-Decane 17 2 317.70 – 333.70 48.180 – 146.41
57 n-Pentane/Nitrogen 113 9 277.43 – 447.90 250.28 – 35470
58 n-Pentane/H2S 53 6 277.59 – 444.26 137.90 – 8963.2
59 n-Pentane/CO2 231 21 252.67 – 463.15 159.00 – 9671.0
60 n-Hexane/n-Heptane 31 – 339.70 – 369.45 94.00 – 101.00
61 n-Hexane/n-Octane 11 1 328.15 10.839 – 61.275
62 n-Hexane/Nitrogen 124 10 310.93 – 488.40 960.00 – 51470
63 n-Hexane/H2S 25 3 322.95 – 422.65 430.00 – 7545.0
64 n-Hexane/CO2 134 15 298.15 – 393.15 443.70 – 12630
65 n-Heptane/n-Octane 20 – 369.55 – 394.45 94.00
66 n-Heptane/Nitrogen 332 25 305.37 – 523.70 1200.0 – 99850
67 n-Heptane/H2S 49 6 293.25 – 477.59 397.90 – 8363.3
68 CO2/n-Heptane 93 8 310.65 – 502.00 186.16 – 13314
69 n-Octane/n-Decane 27 – 349.15 – 392.25 20.00
70 n-Octane/Nitrogen 73 5 344.50 – 543.50 2050.0 – 50140
71 n-Octane/CO2 167 18 298.20 – 393.20 600.00 – 14440
72 n-Nonane/Nitrogen 70 5 344.30 – 543.40 1970.0 – 49750
73 n-Nonane/CO2 77 8 298.20 – 418.82 520.00 – 16773
74 n-Decane/Nitrogen 232 11 310.93 – 563.10 1040.0 – 50320
75 n-Decane/H2S 44 6 277.59 – 444.26 137.90 – 12411
76 n-Decane/CO2 212 22 277.59 – 594.20 344.74 – 17990
77 Nitrogen/Oxygen 318 9 76.79 – 136.17 60.440 – 2966.6
78 Nitrogen/H2S 126 12 200.15 – 344.26 140.65 – 20705
79 Argon/Nitrogen 271 11 95.02 – 133.72 253.90 – 2839.1
80 Helium/Nitrogen 519 32 76.50 – 136.05 1379.0 – 413710
148
No. Binary NP NT=cte. Range T (K) Range P (kPa)
81 CO2/Nitrogen 234 22 218.15 – 303.30 1276.7 – 16706
82 Argon/Oxygen 343 3 84.81 – 138.67 63.995 – 2634.5
83 CO2/Oxygen 112 12 218.15 – 298.15 1013.2 – 14297
84 Argon/Helium 217 21 91.34 – 159.90 1418.6 – 413710
85 CO2/Argon 77 8 233.15 – 299.21 1520.0 – 14034
86 CO2/Helium 59 8 219.90 – 293.13 2979.0 – 20123
87 CO2/H2S 152 – 224.82 – 363.71 689.48 – 8273.7
149
APPENDIX D: BINARY INTERACTION PARAMETERS AND THEIR
UNCERTAINTIES 7
The evaluated binary interaction parameters associated with their uncertainties for 87 binary
mixtures are listed in Table D.1. The number of VLE data points used to estimate the binary
interaction parameters, the ranges of variation in temperature and pressure for these data and the
Monte Carlo standard error (MCSE) for sample size of 100 are also indicated in this table.
Table D.1. Binary interaction parameters associated uncertainties.
No. Binary NP Range T (K) Range P (kPa) 12k12 σk MCSE (%)
1 Methane/Ethane 277 130.37 – 280.00 123.30 – 6894.8 0.0106 ± 0.0025 0.013
2 Methane/Propane 148 187.54 – 343.15 282.69 – 9625.9 –0.0316 ± 0.0048 0.024
3 Methane/i-Butane 72 233.15 – 377.55 490.33 – 11768 –0.0479 ± 0.0111 0.055
4 Methane/n-Butane 250 166.48 – 394.26 137.90 – 13100 0.0146 ± 0.0023 0.011
5 Methane/i-Pentane 15 344.26 – 377.59 3440.5 – 15106 0.0307 ± 0.0172 0.086
6 Methane/n-Pentane 883 299.82 – 460.00 101.33 – 16719 –0.0091 ± 0.0072 0.036
7 Methane/n-Hexane 70 190.50 – 310.95 137.90 – 18271 –0.0025 ± 0.0171 0.085
8 Methane/n-Heptane 75 310.93 – 510.93 1379.0 – 24132 –0.0066 ± 0.0052 0.026
9 Methane/n-Octane 13 373.15 – 423.15 1013.3 – 7092.8 0.0721 ± 0.0061 0.031
10 Methane/n-Nonane 20 373.15 – 423.15 1013.2 – 10133 0.0495 ± 0.0160 0.080
11 Methane/n-Decane 98 348.15 – 583.05 137.90 – 27579 0.0168 ± 0.0143 0.071
12 Methane/Nitrogen 366 95.00 – 180.00 172.37 – 4964.2 0.0387 ± 0.0011 0.005
13 Methane/Argon 109 112.55 – 178.00 141.30 – 4976.1 0.0328 ± 0.0019 0.009
14 Methane/Helium 46 113.15 – 188.15 1134.8 – 20528 0.6780 ± 0.0122 0.061
15 Methane/H2S 61 252.00 – 310.93 1379.0 – 13100 0.0503 ± 0.0045 0.022
16 Methane/CO2 104 219.26 – 301.00 1398.0 – 8519.4 0.0965 ± 0.0016 0.008
17 Ethane/Propane 129 172.04 – 355.37 4.00 – 4998.7 –0.0739 ± 0.0067 0.033
7 Reprinted from Fluid Phase Equilibria, Vol. 364, S. Hajipour, M.A. Satyro, M.W. Foley, Uncertainty Analysis
Applied to Thermodynamic Models and Process Design – 2. Binary Mixtures, 15-30, Copyright (2013), with
permission from Elsevier.
150
No. Binary NP Range T (K) Range P (kPa) 12k12 σk MCSE (%)
18 Ethane/i-Butane 22 311.26 – 344.48 1068.7 – 5371.0 –0.0302 ± 0.0072 0.036
19 Ethane/n-Butane 61 260.00 – 363.40 161.30 – 5326.5 0.0110 ± 0.0039 0.019
20 Ethane/n-Pentane 78 310.93 – 444.26 344.74 – 6550.0 0.0083 ± 0.0026 0.013
21 Ethane/n-Hexane 12 338.71 172.37 – 5515.8 –0.0696 ± 0.0298 0.149
22 Ethane/n-Heptane 87 338.71 – 449.82 2757.9 – 8518.0 –0.0371 ± 0.0076 0.038
23 Ethane/n-Octane 12 318.15 – 338.15 1500.0 – 6800.0 0.0177 ± 0.0017 0.009
24 Ethane/Nitrogen 106 125.00 – 290.00 644.0 –13195 0.0335 ± 0.0073 0.037
25 Ethane/Argon 4 84.760 – 113.52 64.636 – 690.83 0.0578 ± 0.0022 0.011
26 Ethane/Helium 62 173.15 – 273.15 490.30 – 11768 1.1624 ± 0.0433 0.216
27 Ethane/H2S 20 255.32 – 283.15 642.59 – 3052.3 0.0869 ± 0.0029 0.014
28 Ethane/CO2 180 207.00 – 283.15 329.20 – 4994.3 0.1330 ± 0.0022 0.011
29 Propane/i-Butane 131 260.00 – 395.01 132.38 – 3659.1 –0.0593 ± 0.0057 0.028
30 Propane/n-Butane 109 260.00 – 363.38 75.900 – 3413.6 –0.0498 ± 0.0046 0.023
31 Propane/n-Pentane 114 336.56 – 444.26 334.37 – 4481.6 –0.0369 ± 0.0042 0.021
32 Propane/n-Octane 13 473.10 – 523.10 2540.0 – 5140.0 –0.0079 ± 0.0036 0.018
33 Propane/n-Nonane 5 376.75 – 377.15 938.00 – 3468.0 –0.0505 ± 0.0046 0.023
34 Propane/n-Decane 6 376.85 – 377.15 945.00 – 3516.0 –0.0419 ± 0.0036 0.018
35 Propane/Nitrogen 58 173.15 – 330.00 1379.0 – 15903 0.0744 ± 0.0077 0.038
36 Propane/H2S 52 217.59 – 355.37 137.90 – 4136.9 0.0228 ± 0.0047 0.023
37 Propane/CO2 194 263.15 – 327.75 423.10 – 6677.7 0.0685 ± 0.0053 0.026
38 n-Butane/i-Butane 24 273.15 108.72 – 153.45 –0.0015 ± 0.0037 0.018
39 i-Butane/Nitrogen 30 255.37 – 310.87 418.51 – 20774 0.0377 ± 0.0119 0.060
40 i-Butane/H2S 64 310.93 – 398.15 740.50 – 7412.0 0.0352 ± 0.0045 0.023
41 i-Butane/CO2 114 273.15 – 394.26 273.58 – 7196.0 0.1097 ± 0.0043 0.021
42 n-Butane/n-Pentane 89 270.00 – 409.96 34.000 – 2484.0 –0.0154 ± 0.0061 0.030
43 n-Butane/n-Heptane 36 339.26 – 525.93 689.48 – 2757.9 0.0009 ± 0.0043 0.021
44 n-Butane/n-Decane 55 377.59 – 510.93 172.37 – 4826.3 0.0067 ± 0.0053 0.027
45 n-Butane/Nitrogen 60 250.00 – 411.10 452.00 – 15785 0.0509 ± 0.0147 0.073
46 n-Butane/Argon 23 339.67 – 380.25 1393.0 – 18485 –0.0079 ± 0.0194 0.097
47 n-Butane/H2S 41 366.45 – 418.15 1482.0 – 7894.0 0.0468 ± 0.0027 0.013
48 n-Butane/CO2 437 250.00 – 418.48 104.80 – 7929.0 0.1175 ± 0.0039 0.020
151
No. Binary NP Range T (K) Range P (kPa) 12k12 σk MCSE (%)
49 i-Pentane/n-Pentane 13 328.15 – 384.82 234.38 – 785.93 –0.0469 ± 0.0108 0.054
50 i-Pentane/n-Hexane 32 306.89 – 335.25 101.33 –0.0003 ± 0.0072 0.036
51 i-Pentane/H2S 23 323.15 – 413.15 310.00 – 7632.0 0.0524 ± 0.0044 0.022
52 i-Pentane/CO2 127 272.43 – 443.55 151.68 – 8984.0 0.0953 ± 0.0216 0.108
53 n-Pentane/n-Hexane 32 298.70 – 308.70 25.80 – 94.66 0.0075 ± 0.0035 0.018
54 n-Pentane/n-Heptane 36 293.15 – 521.95 101.33 – 3060.0 0.0077 ± 0.0026 0.013
55 n-Pentane/n-Octane 34 303.70 – 313.70 5.700 – 97.00 –0.0098 ± 0.0083 0.042
56 n-Pentane/n-Decane 17 317.70 – 333.70 48.180 – 146.41 –0.0139 ± 0.0141 0.071
57 n-Pentane/Nitrogen 70 310.71 – 447.90 250.28 – 21230 0.0665 ± 0.0082 0.041
58 n-Pentane/H2S 35 277.59 – 410.93 137.90 – 8273.7 0.0432 ± 0.0055 0.028
59 n-Pentane/CO2 156 273.41 – 438.15 172.00 – 9651.0 0.1077 ± 0.0204 0.102
60 n-Hexane/n-Heptane 31 339.70 – 369.45 94.00 – 101.00 0.0008 ± 0.0010 0.005
61 n-Hexane/n-Octane 11 328.15 10.839 – 61.275 –0.0084 ± 0.0048 0.024
62 n-Hexane/Nitrogen 42 377.90 – 488.40 960.00 – 20930 0.0823 ± 0.0091 0.045
63 n-Hexane/H2S 25 322.95 – 422.65 430.00 – 7545.0 0.0570 ± 0.0033 0.016
64 n-Hexane/CO2 36 313.15 – 393.15 779.11 – 11597 0.1086 ± 0.0069 0.034
65 n-Heptane/n-Octane 20 369.55 – 394.45 94.00 0.0029 ± 0.0019 0.010
66 n-Heptane/Nitrogen 149 305.45 – 523.70 1520.0 – 99850 0.0816 ± 0.0052 0.026
67 n-Heptane/H2S 31 333.28 – 477.59 449.50 – 8363.3 0.0653 ± 0.0092 0.046
68 CO2/n-Heptane 73 310.65 – 502.00 186.16 – 11611 0.1082 ± 0.0245 0.123
69 n-Octane/n-Decane 27 349.15 – 392.25 20.00 –0.0011 ± 0.0018 0.009
70 n-Octane/Nitrogen 54 344.50 – 543.50 2050.0 – 50140 0.1341 ± 0.0074 0.037
71 n-Octane/CO2 29 322.39 – 372.53 2000.0 – 13772 0.1117 ± 0.0131 0.065
72 n-Nonane/Nitrogen 70 344.30 – 543.40 1970.0 – 49750 0.1136 ± 0.0194 0.097
73 n-Nonane/CO2 77 298.20 – 418.82 520.00 – 16773 0.1273 ± 0.0067 0.033
74 n-Decane/Nitrogen 232 310.93 – 563.10 1040.0 – 50320 0.1227 ± 0.0145 0.073
75 n-Decane/H2S 41 277.59 – 444.26 137.90 – 9652.7 –0.0228 ± 0.0046 0.023
76 n-Decane/CO2 51 344.30 – 542.95 1379.0 – 16060 0.1161 ± 0.0072 0.036
77 Nitrogen/Oxygen 84 90.34 – 125.06 119.26 – 2966.6 –0.0087 ± 0.0046 0.023
78 Nitrogen/H2S 93 227.98 – 344.26 334.40 – 20684 0.1555 ± 0.0063 0.031
79 Argon/Nitrogen 62 100.00 – 122.89 394.15 – 2839.1 –0.0075 ± 0.0023 0.012
152
No. Binary NP Range T (K) Range P (kPa) 12k12 σk MCSE (%)
80 Helium/Nitrogen 156 85.00 – 120.00 1379.0 – 13824 0.4581 ± 0.0122 0.061
81 CO2/Nitrogen 121 220.00 – 293.15 1373.0 – 16706 –0.0315 ± 0.0035 0.017
82 Argon/Oxygen 343 84.81 – 138.67 63.995 – 2634.5 0.0134 ± 0.0002 0.002
83 CO2/Oxygen 61 218.15 – 273.15 1013.2 – 14297 0.0496 ± 0.0069 0.035
84 Argon/Helium 24 91.34 – 148.03 1418.6 – 27358 0.4507 ± 0.0112 0.056
85 CO2/Argon 66 233.32 – 299.21 1520.0 – 14034 0.0922 ± 0.0041 0.020
86 CO2/Helium 59 219.90 – 293.13 2979.0 – 20123 0.8765 ± 0.0090 0.045
87 CO2/H2S 151 224.82 – 363.71 689.48 – 8273.7 0.0962 ± 0.0025 0.012
153
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