THERMODYNAMIC DATA OF WORKING FLUIDS FOR

Transcription

THERMODYNAMIC DATA OF WORKING FLUIDS FOR
ENERGY ENGINEERING
Dissertation zur Erlangung des akademischen Grades /
Dissertation for the obtainment of the academic degree
Doktor der Bodenkultur/ Doctor rerum naturalium technicarum
(Dr. nat.techn.)
angefertigt am /worked out at
Institut für Verfahrens- und Energietechnik,
Department für Materialwissenschaften und Prozesstechnik
Universität für Bodenkultur Wien /
Institute of Chemical and Energy Engineering
Department of Material Sciences and Process Engineering
University of Natural Resources and Applied Life Sciences, Vienna
von / by
LAI Ngoc Anh
September 2009
Betreuer / Supervisor: O. Univ. Professor Dr. Johann FISCHER
Abstract
Presently there are strong efforts to develop new environmentally friendly processes
for energy conversion. Interesting processes for that purpose are Clausius-Rankine cycles
for conversion of heat to power and heat-pump cycles. In these cycles, a crucial problem is
to find suitable working fluids, which requires knowledge of thermodynamic data, for
optimization of the processes in certain temperature ranges. For medium-temperature
processes in which the working fluids reach temperatures higher than 200°C, there are still
considerable needs for study on thermodynamic data of potential working fluids.
Potential working fluids for the medium-temperature processes are siloxanes and
aromates. However, experimental data of many interesting fluids are mostly not sufficient
to set up empirical multi-parameter equations of state. Thus, it is necessary to use
physically based equations of state which have few physically meaningful parameters, need
only vapour pressures and saturated liquid densities for fitting and still give reliable results
for thermal and caloric data over a wide range.
Saturated liquid densities and vapour pressures of siloxanes and aromates are mostly
not available in high-reduced temperature ranges and the descriptions of the equations of
state can be improved if there are data for filling the gaps. Thus extrapolations of vapour
pressures and saturated liquid densities are studied. The study points out the most suitable
equations for extrapolations with available experimental data set.
Before applying physically based BACKONE and PC-SAFT equations of state for
aromates and siloxanes, we review theory studies of available equations of state and
relations between Helmholtz energy and thermodynamic quantities from different
approaches for systems of hard molecules like hard spheres, hard convex bodies, hard fused
spheres and hard chain molecules. The studied equations and relations are the cores for the
developments of physically based equations of state. The review contains also some minor
own contributions.
From experimental vapour pressures, saturated liquid densities and data from
equations for extrapolation of vapour pressures and saturated liquid densities, parameters of
BACKONE equation of state for seven aromates such as benzene, toluene, ethylbenzene,
butylbenzene, o-xylene, m-xylene, p-xylene are determined. The results show that
BACKONE can describe thermodynamic properties of these substances from good to
ii
excellent. For chain-like molecules as linear siloxanes, PC-SAFT outperforms BACKONE.
Thus parameters of PC-SAFT equation of state are determined for five linear siloxanes
such as MM (Hexamethyldisiloxane, C6H18OSi2), MDM (Octamethyltrisiloxane,
C8H24O2Si3),
MD2M
(Decamethytetrasiloxane,
C10H30O3Si4),
MD3M
(Dodecamethylpentasiloxane, C12H36O4Si5), and MD4M (Tetradecamethylhexasiloxane,
C14H42O5Si6).
Thermodynamic data from BACKONE and PC-SAFT equations of state are used to
study medium-temperature Organic Rankine cycles (ORC). Our study shows that ORC
plants have higher thermal efficiency than conventional steam power plants. Furthermore,
working fluid yields the highest thermal efficiency may not yields highest power output and
normally yields large size of the turbine, consequently large ORC or high investment cost.
Thus, the selection of working fluid should be based not only on cycle efficiency but also
the size of equipment and further on supply and processing of heat carrier fluid.
If heat carrier is heated up and circulated in a loop, the fluid will regain its
temperature after the external heat exchanger (EHE). In this case, selection of working fluid
should be based on cycle efficiency and size of equipment. For this case, aromates prove to
be the most potential working fluids.
If heat carrier isn’t circulated in a loop, the criteria for selection of working fluid are
the size of equipment and the total efficiency, not the cycle efficiency. The study shows
that investigated alkanes, MM, MDM, toluene and o-xylene are the most potential working
fluids, depending on the working temperature range.
iii
Abstrakt
Zur Zeit gibt es starke Bemühungen zur Entwicklung neuer umweltfreundlicher
Verfahren zur Energiewandlung. Interessante Prozesse dafür sind Clausius-RankineKreisprozesse zur Umwandlung von Wärme in Strom und Wärme-Pumpen-Kreisprozesse.
Ein entscheidendes Problem bei diesen Kreisprozessen ist die Suche nach geeigneten
Arbeitsmedien. Diese erfordert die Kenntnis der thermodynamische Daten dieser Stoffe für
die Optimierung der Prozesse in bestimmten Temperaturbereichen. Für Mittel-TemperaturProzesse, in denen die Arbeitsmedien bei Temperaturen oberhalb von 200°C eingesetzt
werden, sind organische Stoffe als Arbeitsmedien besser geeignet sind als Wasser. Die mit
organischen Stoffen betriebenen Clausius-Rankine-Kreisprozesse werden ORC (Organic
Rankine Cycle)-Prozesse genannt. Für diese organischen Arbeitsmedien gibt es aber noch
erheblichen Bedarf an thermodynamischen Daten.
Potenzielle Arbeitsmedien für die Mittel-Temperatur-Prozesse sind Siloxane und
Aromate. Für viele dieser Stoffe liegen jedoch keine ausreichenden experimentellen
Datensätze
vor,
um
empirische
Multi-Parameter-Fundamentalzustandsgleichungen
aufzustellen. Daher ist es notwendig, physikalisch begründete Zustandsgleichungen zu
verwenden, die nur wenige physikalisch sinnvolle Parameter benötigen. Dampfdrücke und
Siededichten reichen für die Bestimmung dieser Parameter aus, wenn sie über den
gesamten Temperaturbereich verfügbar sind. Zu ergänzen sind diese noch durch die
isobaren Idealgas-Wärmekapazitäten.
Dampfdrücke und Siededichten der Siloxane und Aromaten sind jedoch meist nicht
für höhere Temperaturen verfügbar, so dass Extrapolationsgleichungen für diese
Sättigungs-eigenschaften benötigt werden. In dieser Arbeit werden daher zunächst die am
besten geeigneten Gleichungen für aufwärts-Extrapolationen von Dampfdrücken und
Siededichten aus den verfügbaren experimentellen Daten untersucht.
Als
physikalisch
begründeten
Fundamentalzustandsgleichungen
werden
die
BACKONE-Gleichung für kompakte nichtkugelförmige Moleküle (Aromate) und die PCSAFT - Gleichung für Kettenmoleküle (Siloxane) verwendet. Diese physikalisch
begründeten Gleichungen folgen den Gedanken von van der Waals und setzen die
Helmholtz-Energie aus einem Anteil von der Wechselwirkung harter Körper und einem
iv
Anteil der anziehenden zwischenmolekularen Kräfte zusammen. Für die Bestimmung der
Parameter von PC-SAFT durch nichtlineare Regression mussten eigene Programme
geschrieben werden, wozu ein Nacharbeiten der Theorie erforderlich war. Hier wurde
großes Augenmerk auf die Helmholtz-Energie von harten Ketten gelegt, die in der
Anwendung für PC-SAFT zu ziemlich komplexen mathematischen Ausdrücken führt, die
in der Literatur nicht angegeben sind. Diese Nacharbeit beinhaltet auch einige kleinere
eigene Beiträge.
Ausgehend von experimentellen oder extrapolierten Dampfdrücken und Siedichten
wurden BACKONE Parameter für die sieben Aromate Benzol, Toluol, Ethylbenzol,
Butylbenzol, o-Xylol, m-Xylol, und p -Xylol bestimmt. Die Ergebnisse zeigen, dass
BACKONE die thermodynamischen Eigenschaften dieser Stoffe von gut bis sehr gut
beschreiben kann. Für die linearen Siloxane erwies sich PC-SAFT besser als BACKONE.
Es wurden daher für die fünf linearen Siloxane wie MM (Hexamethyldisiloxane,
C6H18OSi2),
MDM
(Octamethyltrisiloxane,
C10H30O3Si4),
MD3M
C8H24O2Si3),
MD2M
(Dodecamethylpentasiloxane,
C12H36O4Si5), und MD4M (Tetradecamethylhexasiloxane, C14H42O5Si6) die Parameter
der PC-SAFT-Gleichung bestimmt.
Thermodynamische Daten, die mit den Zustandsgleichungen BACKONE und PCSAFT gewonnen werden, wurden schließlich dafür eingesetzt, Mitteltemperatur-ORCProzesse zu beschreiben. Als Stoffe werden Alkane, Cyclopentan, Aromate und Siloxane
betrachtet. Berechnet wurden zunächst thermische Wirkungsgrade und Volumenströme.
Verwendet man innere Wärmeübertrager, dann steigt der Wirkungsgrad im allgemeinen mit
der kritischen Temperatur des Stoffes, gleichzeitig wird aber der Volumenstrom beim
Austritt aus der Turbine sehr groß. Weiters wurde auch das Pinch-Point- Problem in die
Analyse einbezogen, das mit der Wärmeübertragung auf das Arbeitsmedium verbunden ist.
In diesem Fall sind, abhängig vom Temperaturbereich, Alkane, Hexamethyldisiloxan,
Octamethyltrisiloxan, Toluol und o-Xylol vielversprechende Arbeitsmedien.
v
Acknowledgments
The author gratefully acknowledges financial support by a Technologiestipendium of
Österreichischer Austauschdienst.
I gratefully acknowledge the assistance and supervision of Prof. Johann FISCHER. I
have learned a lot from his lectures, suggestions and introductions on different topics. His
insight and continual support have been important factors in the completion of this
dissertation. I also acknowledge the supports, suggestions and introductions of Prof. Martin
WENDLAND for the BACKONE package, the setting up of the experimental apparatus
and the review of some parts of my thesis.
I thank Prof. Herbert WEINGARTMANN and Prof. Martin WENDLAND for their
participation in my advisory team and their willingness to write the report on the thesis. I
also appreciate the support of Prof. Gerd MAURER, University of Kaiserslautern,
Germany, for literature supply.
I am grateful to my colleagues, Dr. VU Hong-Thang, Dr. Rupert TSCHELIEßNIG,
Dr. Werner BILLES, Dr. Gerald KOGLBAUER, MSc. Emmerich HAIMER, Mr. Krapf
CHRISTIAN, Mr. Lukas GEYRHOFER, Mr. Karl BELER, Mrs. Verena WETTER, Mrs.
Sabine EISENSCHENK, and MSc. NGUYEN Viet Cuong, for their friendship and
assistance during our study and stay in Vienna.
vi
Publications, presentations from this thesis
Papers in refereed SCI journals:
1.
N. A. LAI, M. Wendland, J. Fischer, Description of linear siloxanes with PC-SAFT
equation, Fluid Phase Equilibria, 283 (2009) 22-30. See also in chapter 7.
2.
N. A. LAI, M. Wendland, J. Fischer, upward extrapolation of saturated liquid density,
Fluid Phase Equilibria, 280 (2009) 30-34. See also in chapter 4.
Papers in preparation:
1.
N. A. LAI, J. Fischer, M. Wendland, Description of aromates with BACKONE
equations of state. See also in chapter 8.
2.
N. A. LAI, M. Wendland, J. Fischer, Working fluids for medium-temperature Organic
Rankine cycles. See also in chapter 10.
Oral presentations in scientific conferences:
1.
N. A. LAI, M. Wendland, J. Fischer, Development of equations of state for siloxanes
as working fluids for ORC Processes, Proceeding of 24th European symposium on
applied thermodynamics, June 27 – July 1, 2009, 200-205, ISBN: 978-84-692-26643, Santiago de Compostela, Spain. See in chapter 9.
2.
N. A. LAI, M. Wendland, J. Fischer, Siloxanes: equations of state and ORC Cycle
efficiencies, „Thermodynamik-Kolloquium“ und „Ingenieurdaten“, 5. – 7. October
2009, Berlin, Germany
3.
M. Wendland, N. A. LAI, G. Koglbauer and J. Fischer, Electricity Generation from
Geothermal Heat and other Sustainable Energy Technology. Austria-Russian Science
Day: Alternative and Renewable Energy Sources, 15. October 2007, Wien, Austria
Poster presentations in scientific conferences:
1.
N. A. LAI, G. Koglbauer, M. Wendland, J. Fischer: Working Fluids for Organic Rankine
Cycles. Thermodynamics 2007, 26.- 28. September 2007, Rueil-Malmaison, France
2.
J. Fischer , N. A. LAI, G. Koglbauer, M. Wendland: Arbeitsmedien für ORCProzesse,
ProcessNet
2007,
16.-
18.
Oktober
2007,
or [abstract] in Chemie Ingenieur Technik 79 (2007),1342.
vii
Aachen,
Germany
Table of contents
ABSTRACT ...................................................................................................................................................... II
ABSTRAKT .....................................................................................................................................................IV
ACKNOWLEDGMENTS ....................................................................................................................................VI
PUBLICATIONS, PRESENTATIONS FROM THIS THESIS ......................................................................................VII
TABLE OF CONTENTS .................................................................................................................................. VIII
NOTATION .....................................................................................................................................................XI
1
INTRODUCTION................................................................................................................................... 1
REFERENCES ...................................................................................................................................................6
2
EXTRAPOLATION OF VAPOUR PRESSURES ............................................................................... 9
ABSTRACT ......................................................................................................................................................9
2.1
INTRODUCTION .................................................................................................................................9
2.2
INVESTIGATED EQUATIONS .............................................................................................................10
2.3
SUBSTANCES AND DATA SOURCES ..................................................................................................12
2.4
RESULTS AND DISCUSSIONS ............................................................................................................14
2.5
SUMMARY AND CONCLUSIONS........................................................................................................20
REFERENCES .................................................................................................................................................21
3
MEASUREMENT OF VAPOUR PRESSURES ................................................................................ 24
ABSTRACT ....................................................................................................................................................24
3.1
INTRODUCTION ...............................................................................................................................24
3.2
EXPERIMENTAL SET-UPS .................................................................................................................25
3.2.1
Pressure sensors and indicators ............................................................................................... 25
3.2.2
Temperature sensor and indicator ............................................................................................ 26
3.2.3
Experimental set-ups................................................................................................................. 27
3.3
PRESSURE MEASUREMENT AND CALIBRATION ................................................................................28
3.3.1
Pressure calibration.................................................................................................................. 28
3.3.2
Vapour pressure of water.......................................................................................................... 33
3.4
REFERENCES .................................................................................................................................................35
4
UPWARD EXTRAPOLATION OF SATURATED LIQUID DENSITIES* ................................... 37
ABSTRACT ....................................................................................................................................................37
4.1
INTRODUCTION ...............................................................................................................................37
4.2
EQUATIONS AND OPTIMIZATION .....................................................................................................38
4.3
SUBSTANCES AND DATA SOURCES ..................................................................................................40
4.4
RESULTS AND DISCUSSIONS ............................................................................................................40
4.5
REFERENCES .................................................................................................................................................50
5
HELMHOLTZ ENERGY OF HARD CONVEX BODIES AND HARD CHAIN SYSTEMS ....... 54
ABSTRACT ....................................................................................................................................................54
5.1
INTRODUCTION ...............................................................................................................................54
5.2
HARD SPHERES ...............................................................................................................................55
5.2.1
Background............................................................................................................................... 55
5.2.2
Equation of state for hard spheres ............................................................................................ 59
5.2.3
Helmholtz energy for hard spheres ........................................................................................... 60
5.3
HARD CHAIN SYSTEMS....................................................................................................................60
5.3.1
Results from Wertheim SAFT theory......................................................................................... 60
5.3.2
Hard chain equation using Carnahan- Starling equation......................................................... 62
5.3.3
Hard chain equation using Kolafa-Boublik-Nezbeda equation ................................................ 62
5.3.4
Comparison among simulation data, results from CS and KBN equations .............................. 63
viii
5.4
HARD CONVEX BODIES AND HARD DUMBBELLS ..............................................................................66
5.4.1
Hard convex bodies (HCB) ....................................................................................................... 66
5.4.2
Hard dumbbells (HD) ............................................................................................................... 67
5.4.3
Helmholtz energy derived from equation for hard convex bodies and hard dumbbells............ 67
5.4.4
Hard convex bodies approach to hard chain molecules ........................................................... 68
5.5
COMPARISON OF RESULTS DERIVED FROM EQUATIONS FOR HARD CONVEX BODIES AND HARD
CHAIN SYSTEMS ............................................................................................................................................68
5.6
REFERENCES .................................................................................................................................................72
6
PC-SAFT EQUATION OF STATE..................................................................................................... 75
ABSTRACT ....................................................................................................................................................75
6.1
INTRODUCTION ...............................................................................................................................75
6.2
BARKER-HENDERSON PERTURBATION THEORY ..............................................................................75
6.2.1
Characterization of the reference system by Barker and Henderson........................................ 76
6.2.2
Barker and Henderson perturbation theory.............................................................................. 79
6.3
PC-SAFT EQUATION FOR PURE FLUIDS ..........................................................................................80
6.3.1
The potential model................................................................................................................... 80
6.3.2
Residual Helmholtz energy for hard chains.............................................................................. 81
6.3.3
First and second order perturbation terms ............................................................................... 82
6.3.4
Complete Helmholtz energy equation ....................................................................................... 85
6.4
THERMODYNAMIC PROPERTIES OF PURE FLUIDS DERIVED FROM THE HELMHOLTZ ENERGY ...........86
6.4.1
Thermodynamic properties derived from the Helmholtz energy............................................... 86
6.4.2
Derivatives of Helmholtz energy............................................................................................... 87
6.5
REFERENCES .................................................................................................................................................94
7
DESCRIPTION OF LINEAR SILOXANES WITH PC-SAFT EQUATION*................................ 96
ABSTRACT ....................................................................................................................................................96
7.1
INTRODUCTION ...............................................................................................................................96
7.2
EXPERIMENTAL DATA .....................................................................................................................98
7.3
EQUATIONS .................................................................................................................................. 101
7.3.1
Extrapolation equations.......................................................................................................... 101
7.3.2
PC-SAFT equation .................................................................................................................. 103
7.4
FITTING MODES FOR PC-SAFT..................................................................................................... 105
7.5
RESULTS AND DISCUSSION ............................................................................................................ 108
7.5.1
Hexamathyldisiloxane (MM)................................................................................................... 109
7.5.2
Octamethyltrisiloxane (MDM) ................................................................................................ 112
7.5.3
Decamethyltetrasiloxane (MD2M), dodecamethylpentasiloxane (MD3M), and
tetradecamethylhexasiloxane (MD4M) ................................................................................................. 115
7.6
TABLES OF SATURATION PROPERTIES AND T,S-DIAGRAMS ........................................................... 118
7.7
SUMMARY AND CONCLUSIONS...................................................................................................... 125
REFERENCES ............................................................................................................................................... 127
8
DESCRIPTION OF AROMATES WITH BACKONE EQUATION OF STATE ........................ 132
ABSTRACT .................................................................................................................................................. 132
8.1
INTRODUCTION ............................................................................................................................. 132
8.2
EXPERIMENTAL DATA AND AUXILIARY EQUATIONS...................................................................... 134
8.3
EQUATION OF STATE ..................................................................................................................... 139
8.4
RESULTS AND DISCUSSION ............................................................................................................ 140
8.4.1
Benzene and toluene ............................................................................................................... 141
8.4.2
Ethylbenzene, butylbenzene .................................................................................................... 144
8.4.3
o-xylene, m-xylene, p-xylene ................................................................................................... 146
8.5
THERMODYNAMIC PROPERTIES FROM BACKONE FOR ETHYLBENZENE, BUTYLBENZENE, OXYLENE, M-XYLENE, AND P-XYLENE ........................................................................................................... 148
8.5.1
Ideal gas heat capacity ........................................................................................................... 148
ix
8.5.2
Tables of saturation properties and T,s-diagrams .................................................................. 151
8.6
ORC CYCLE WITH BENZENE ......................................................................................................... 159
8.7
REFERENCES ............................................................................................................................................... 163
9
COMPARISON BETWEEN BACKONE AND PC-SAFT.............................................................. 168
ABSTRACT .................................................................................................................................................. 168
9.1
DEVELOPMENT OF BACKONE EQUATIONS OF STATE FOR SILOXANES ....................................... 168
9.1.1
Introduction ............................................................................................................................ 168
9.1.2
Availability of experimental data ............................................................................................ 169
9.1.3
BACKONE equation of state................................................................................................... 171
9.1.4
ORC cycle with MDM ............................................................................................................. 173
9.2
COMPARISON BETWEEN BACKONE AND PC-SAFT ................................................................... 175
9.2.1
Benzene ................................................................................................................................... 175
9.2.2
MM.......................................................................................................................................... 178
9.3
REFERENCES ............................................................................................................................................... 181
10
WORKING FLUIDS FOR MEDIUM-TEMPERATURE ORGANIC RANKINE CYCLES...... 184
ABSTRACT .................................................................................................................................................. 184
10.1
INTRODUCTION ............................................................................................................................. 184
10.2
CYCLE DESCRIPTION ..................................................................................................................... 186
10.2.1
Organic Rankine cycles...................................................................................................... 186
10.2.2
Water cycle with extraction................................................................................................ 191
10.3
SCREENING OF FLUIDS AND THERMODYNAMIC DATA ................................................................... 193
10.3.1
Selection of fluids ............................................................................................................... 193
10.3.2
Equation of state and caloric properties of selected fluids ................................................ 194
10.3.3
Comparison of BACKONE data with those from reference EOS....................................... 196
10.4
THERMAL EFFICIENCIES ................................................................................................................ 198
10.4.1
Cycle efficiencies of all considered substances.................................................................. 198
10.4.2
Comparison of cycle efficiencies from different equations of state .................................... 210
10.4.3
Efficiency of medium-temperature Rankine cycle using water as working fluid................ 211
10.5
HEAT TRANSFER FROM THE HEAT CARRIER TO THE WORKING FLUIDS .......................................... 211
10.6
REFERENCES ............................................................................................................................................... 218
11
SUMMARY AND CONCLUSIONS.................................................................................................. 222
x
Notation
List of symbols
A
A0
A1
Helmholtz energy, cross-section area of piston
Coefficients of vapour pressure, saturated liquid density,
and ideal gas heat capacity equations
Helmholtz energy of reference system
First order perturbation term
A2
AAD
AHC
Second order perturbation term
Average absolute deviation
Residual Helmholtz energy of the hard chain system
ai, aij
Coefficients of A1
Aid
Helmholtz energy of ideal gas
Ar
Residual Helmholtz energy
Ares
Residual Helmholtz energy
bi, bij
c0 p
ci, ji, ki,li,mi, ni, oi
Coefficients of A2
Isobaric heat capacity of the ideal gas
Coefficients of FA, FD, FQ
cp,Res
CS
cv,Res
d
F
FA
Residual isobaric heat capacity
Carnahan and Starling
Residual isochoric heat capacity
Hard sphere diameter
Helmholtz energy, force
Attractive dispersion force contribution
FD
Dipolar contribution
FH
Hard body contribution
FQ
g
G
g0(r)
GRes
h
h0
Quadrupolar contribution
Pair correlation function
Gibbs energy
Pair correlation function of the soft reference system
Residual Gibbs energy
Enthalpy
Reference enthalpy
h1, h2, h2a, h3, h4, h4a
HCB
HCM
HD
hRef
Specific enthalpies at the respective state points
Hard convex bodies
Hard chain molecule
Hard dummbells
Reference enthalpy
A, A', B, B', C, C', D, E
xi
hRes
HS
I1, I2
IHE
k
k
KBN
L
LJ
m
ْm
ْmc
MC
MD2M
Residual enthalpy
Hard sphere
Integral function
Internal heat exchanger
Boltzmann constant
Parameter vector
Kolafa, Boublik and Nezbeda
Elongation
Lennard-Jones
Number of segments
Flow rate of working fluid
Flow rate of heat carrier
Monte-Carlo
Decamethytetrasiloxane, C10H30O3Si4
MD3M
Dodecamethylpentasiloxane, C12H36O4Si5
MD4M
Tetradecamethylhexasiloxane, C14H42O5Si6
MDM
Octamethyltrisiloxane, C8H24O2Si3
meff
Correlation parameter
MM
p
p0
Hexamethyldisiloxane, C6H18OSi2
Number of chain molecules, Number of experimental
data point
Pressure
Residual temperature
pc
Critical pressure
pmax
Maximum pressure of ORC
pmin
Minimum pressure of ORC
pr
Reduced vapour pressure, pr = ps/pc
pRes
Residual temperature
ps
PY
Q*
ْq56
R
r, ri,j
Vapour pressure
Percus and Yevick
Reduced quadrupole moment
Specific heat received in evaporator
Gas constant
Distance between spheres i and j
ri
RMS
s
S
s0
Coordinate of hard sphere i, i=1-N
Root-mean-square deviation
Entropy
Surface of hard convex body
Reference entropy
N
xii
SAFT
SDV
sRef
Statistical association fluid theory
Standard deviation
Reference entropy
sRes
T
Tc
Residual entropy
Temperature
Characteristic critical temperature, Reference
temperature
Critical temperature
Tmin
Minimum temperature of ORC
Tp
Pseudo critical temperature
Tr
Reduced temperature, Tr = T/Tc
TRef
Reference temperature
u, uij, u(rij)
u0(r)
u1(r)
uLJ(r)
uRef
Potential energy between the two spheres i and j
Reference potential
Perturbation potential
Lennard-Jones potential
Reference internal energy
uRes
V
VD
Residual internal energy
Volume of a system of hard spheres
Volume hard dummbells
Vm
Volume of a hard sphere molecule
v3
Flow rate at the turbine inlet
v4
W
w
wp
wr
Xcal
Flow rate at the turbine outlet
Virial
Hard sphere potential with coupling parameters
Weight for vapour pressure
Weight for saturated liquid density
Calculated data from either equation (3.1) or (3.2)
Xdata
Reading data from pressure indicator
xi
y
Z0
Mole fraction of component i of mixtures
Packing fraction
Partition function of hard spheres
Zc
Critical compression factor
Zp
Pseudo-compression factor
T0
xiii
Greek letters
α, γ
Δp
Δpbaro
Coupling parameters
Total uncertainty of measurement
Uncertainty of the mercury barometer
Δpcal
Uncertainty of calibration of pressure
Δpdrift
Uncertainty of pressure due to drift during a certain time
Δpref
Uncertainty of the piston-cylinder manometer
Δpsys
Systematic errors of pressure
Δptrans,MKS
Uncertainty of pressure transducer MKS
Δptrans,paro
Uncertainty of pressure transducer Paro
ΔX
ε
η
ηs,P
ηs,T
ηth
μ*
ρ
ρ’
Xdata - Xcal
Well depth
Packing fraction
Isentropic turbine efficiency
Isentropic pump efficiency
Thermal efficiency
Reduced dipole moment
Number density of a chain
Saturated liquid density
Characteristic critical density, number density of hard
spheres
Critical density
Pseudo-critical density
Reference density
Residual density
Segment diameter
Packing fraction
Anisotropy parameter
ρ0
ρc
ρp
ρRef
ρRes
σ
ξ
α
xiv
1
Introduction
Engineers can not design a car, a building, a bridge without knowledge of strength
and stress of materials. Process and chemical engineers need chemical physical properties
of materials to design and optimize equipment and processes. Among different properties,
thermodynamic properties of fluids and their mixtures calculated from equations of state
play important roles for different purposes such as designing distillation columns,
extraction equipment, designing and optimization of different energy conversion cycles and
so on.
Recently, energy and environment have become hot issues for our world. Energy
consumption in the world has increased with an annual rate of about 2.3% whilst fossil
fuels, the primary sources of energy accounting for about 86% [1.1] of primary energy
production in the world, are limited and will be exhausted in this century. Thus using other
energy sources and applying energy saving solutions are of vital challenges for engineers,
scientists and authorities.
The production and use of fossil fuels raise environment concerns. One of major
problems from burning fossil fuels to generate energy is that the burning releases a large
amount of carbon dioxide to the atmosphere, around over 30 billion tons of carbon dioxide
per years [1.2]. However, natural processes can absorb about half of that amount. The left
amount of carbon dioxide still exists in the atmosphere and has great contribution in global
warming which causes major adverse effects.
Renewable energy sources like solar energy, wind energy, geothermal energy,
ocean energy, or biomass energy can be used instead of fossil fuels. Renewable energy is
abundant for human being. The key point for replacing fossil fuels by renewable energy is
in economical aspect. Currently, using renewable energy is not as competitive as using
fossil fuels. Hence further researches for making feasible and competitive use of renewable
sources are necessary and urgent.
One of feasible and effective ways for exploiting geothermal energy, solar energy,
waste heat, and biomass energy is to use Organic Rankine cycles (ORC) to generate
electricity [1.3]. Another possible way is to use heat-pump cycles (HPC) to generate
thermal sources from available renewable energy sources for different applications. In the
1
1980s and 1990s, some attentions were paid for optimization of the Kalina processes [1.4]
and some publications were published [1.5-1.9]. Recently more attentions have been paid
for ORC processes. The number of publications for optimization of ORC with different
working fluids has increased significantly from 2005 [1.10-1.20].
In order to optimize ORC and/or HPC processes, accurate thermodynamic
properties of working fluids must be known. Experiment is very important because it can
provide thermodynamic properties accurately. However, only experiment is not really
practical because too many data in large spaces have to be measured and connections
between different pvT data and caloric properties are not easily handled, especially entropy
can not be measured. Thus, in order to have different thermodynamic properties in large
spaces, fundamental equations of state must be used.
Most practical fundamental equations of state are explicit in the Helmholtz energy
[1.21-1.25]. These equations allow accurate calculation of different properties and
quantities as density, pressure, temperature, enthalpy, entropy, heat capacity, speed of
sound, Joule Thomson coefficient and so on.
One typical form of the fundamental equations of state in the Helmholtz energy is in
form of multi-parameter equations of state developed by Wagner and co-workers. Multiparameter equations of state for some substances have recently been accepted as reference
equations of state [1.21, 1.22]. Characteristic of multi-parameter equations is that they
require a large number of accurate experimental data set in large space for construction of
the equations. The extension from multi-parameter equations of state for pure fluids to
mixtures is a big problem. Up to now, the extension to mixtures has only been applied for
natural gas which has a large number of experimental data for mixtures. In case
experimental data do not cover large space of pvT data, other types of equations of state
based on molecular theory should be used instead of multi-parameter equations of state.
A typical fundamental equation of state in the Helmholtz energy based on molecular
theory is BACKONE equation of state. BACKONE equation of state has been developed
by Fischer and co-workers [1.23, 1.24]. BACKONE is a family of physically based EOS
which is developed for nonpolar, dipolar and quadrupolar fluids of compact molecules. The
Helmholtz energy is written in term of a sum of molecular hard-body contribution AH,
2
attractive dispersion force contribution AA, dipolar contribution AD and quadrupolar
contribution AQ: A = AH + AA + AD + AQ.
One typical characteristic of BACKONE is that it has only 3 to 5 parameters
depending on type of molecules. These parameters are found by fitting to experimental
vapour pressures and saturated densities. Another typical characteristic of BACKONE is
that it has only one more fitted parameter, a constant, for each binary. Thus extension from
equations of state for pure fluids to equations of state for mixtures can be easily handled if
experimental data for binaries are available.
BACKONE equation of state has successfully been applied for different substances
and their mixtures [1.23-1.29]. The descriptions of BACKONE for natural gas, alternative
refrigerants, and other substances are from good to excellent. Furthermore, BACKONE
equation of state for many substances has been used to calculate and optimize cycle
efficiencies of low-temperature Organic Rankine cycles. The study of low-temperature
organic Rankine cycles for electric generation from solar energy and geothermal energy
shown that supercritical organic Rankine cycles using R143a as working fluid yields about
20% more power than subcritical cycles if the pinch problem in the evaporator is also taken
into consideration.
Another typical fundamental equation of state in the Helmholtz energy based on
molecular theory is PC-SAFT equation of state. PC-SAFT equation of state was developed
by Gross and Sadowski for chain-like molecules [1.30]. In PC-SAFT, residual Helmholtz
energy is written in term of a sum of the hard chain system AHC, first and second order
perturbation terms, A1 and A2. For pure fluids, PC-SAFT has only 3 parameters which are
found by fitting to experimental vapour pressures and saturated densities.
Similar to BACKONE, PC-SAFT has only one fitted parameter for each binary. The
extension from PC-SAFT equation of state for pure fluids to equation of state for mixtures,
similar to BACKONE, can be implemented more easily than those from multi-parameter
equation of state.
As mentioned above, multi-parameter equations of state are constructed by using a
large number of experimental data set in large space. Few substances having a large
number of experimental data have been described by accurate multi-parameter equations of
3
state. However, many substances have only limited experimental data. Thus, physically
based equations of state like BACKONE and/or PC-SAFT should be used because these
equations have few physically meaningful parameters and need only vapour pressures and
saturated liquid densities for fitting.
Returning to energy conversion processes, thermodynamic properties of different
fluids and optimization for low-temperature organic Rankine cycles have been thoroughly
studied. Thus, in this study, we pay attention to medium-temperature organic Rankine
cycles, in which the maximum temperature of ORC is higher than 200°C. Smaller alkanes
might be used in supercritical cycles. However, if chain length of molecules increases, the
auto-ignition temperature decreases to about 200°C. Thus longer alkanes which are
environmentally friendly and yield good thermal efficiencies can not be used for safety
reasons. Fluorinated alkanes have a strong global warming potential and extremely long
atmospheric lifetimes and hence should not be used for environmental reasons. Hence,
siloxanes and aromates have been suggested as potential working fluids for the medium
temperature range and these fluids are considered in this study [1.26, 1.27, and 1.31].
Experimental saturated liquid densities and vapour pressures of siloxanes, except
MM (Hexamethyldisiloxane, C6H18OSi2) and MDM (Octamethyltrisiloxane, C8H24O2Si3),
and saturated liquid densities of aromates are available only in low reduced temperature
ranges. In fitting the equations of state to vapour pressures and saturated liquid densities,
the quality of description of equations of state for the case with experimental data in full or
in both low and high reduced temperature ranges are better than the case with only
experimental data in low reduced temperature ranges. Thus, in order to improve quality of
description of BACKONE and/or PC-SAFT equations of state for substances with
experimental vapour pressures and saturated liquid densities in low reduced temperature
ranges, equations for correlation and extrapolation of vapour pressures and saturated liquid
densities are to be studied.
This thesis has 11 chapters. Chapter 2 is titled “Extrapolation of vapour pressures”.
This chapter presents investigations of different equations for upward and downward
extrapolations of vapour pressures. Chapter 3 is titled “Measurement of vapour pressures”.
The chapter 3 presents a new apparatus for measuring vapour pressures. “Upward
extrapolation of saturated liquid densities” is given in chapter 4.
4
Chapter 5 presents an overview on “Helmholtz energy of hard convex bodies and
hard chain systems” which is the core for developments of different physically based
equations of state such as BACKONE and PC-SAFT.
Chapter 6 presents an overview on “PC-SAFT equation of state”. Chapter 7 is titled
“Description of linear siloxanes with PC-SAFT equation of state”. Parameters for 5 linear
siloxanes as MM (Hexamethyldisiloxane, C6H18OSi2), MDM (Octamethyltrisiloxane,
C8H24O2Si3),
MD2M
C10H30O3Si4),
MD3M
(Dodecamethylpentasiloxane, C12H36O4Si5), and MD4M (Tetradecamethylhexasiloxane,
C14H42O5Si6) are determined. Before fitting PC-SAFT equation of state, we use
extrapolation equations from Chapters 2 and 4 to generate input data.
“Description of aromates with BACKONE equation” is chapter 8 of this thesis.
Parameters of BACKONE equation of state for seven aromates as benzene, toluene,
ethylbenzene, butylbenzene, o-xylene, m-xylene, and p-xylene have been found.
Thermodynamic properties of these fluids from BACKONE equation of state are used to
calculate efficiencies of organic Rankine cycles in chapter 10.
In chapter 9, a “Comparison between BACKONE and PC-SAFT” is given. The first
part of this chapter presents BACKONE equation of state for small siloxanes. The second
part presents PC-SAFT equation of state for benzene and compares the possibilities of
correlation and prediction of BACKONE and PC-SAFT for benzene and MM.
Chapter 10 is titled “Working fluids for medium-temperature Organic Rankine
cycles”. In this chapter, thermodynamic properties of potential working fluids from
BACKONE equation of state and PC-SAFT equation of state are used to calculate
efficiencies of medium-temperature organic Rankine cycles. Our study shows that ORC
plants have higher thermal efficiencies than conventional steam power plants. Furthermore,
working fluid yields the highest thermal efficiency may not yields highest power output and
normally yields large size of the turbine, consequently large ORC or high investment cost.
Thus, selection of working fluids should be based on power output and sizes of equipment.
The study points out that toluene, o-xylene and MDM are potential working fluids for the
medium temperature range.
5
Finally, summary and conclusions for this thesis are given in chapter 11 titled
“Summaries and conclusions”
References
[1.1] U.S. Energy Information Administration, World Consumption of Primary Energy by
Energy Type and Selected Country Groups, International Energy Annual 2006,
http://www.eia.doe.gov/pub/international/iealf/table18.xls
[1.2] U.S. Energy Information Administration, Energy-Related Carbon Dioxide Emissions,
International Energy Outlook 2009, http://www.eia.doe.gov/oiaf/ieo/emissions.html
[1.3] Sachverständigenrat für Umweltfragen (H.-J. Koch et al.), Sondergutachen
„Klimaschutz
durch
Biomasse“,
Hausdruck,
Berlin
2007.
(http://www.umweltrat.de/02gutach/
downlo02/sonderg/SG_Biomasse_2007_Hausdruck.pdf)
[1.4] A.I. Kalina, Combined-cycle system with novel bottoming cycle, Journal of
Engineering for Gas Turbines and Power 106 (1984), 737-742
[1.5] A.I. Kalina, H.M. Leibowitz, Application of the Kalina cycle technology to
geothermal power generation, Transactions - Geothermal Resources Council 13 (1989)
605-611
[1.6] C.H. Marston, Parametric analysis of the Kalina Cycle, Journal of Engineering for
Gas Turbines and Power 112 (1990) 107-116
[1.7] P.K. Nag, A.V.S.S.K.S. Gupta, Exergy analysis of the Kalina cycle, Applied Thermal
Engineering 18 (1998), 427-439
[1.8] Y.M. El-Sayed, M. Tribus, Theoretical comparison of the Rankine and Kalina cycles,
American Society of Mechanical Engineers, Advanced Energy Systems Division
(Publication) AES 1, (1985) 97-102
[1.9] E.D. Rogdakis, K.A. Antonopoulos, A high efficiency NH3/H2O absorption power
cycle, Heat Recovery Systems and CHP 11 (1991) 263-275
6
[1.10] D. Manolakos, G. Papadakis, S. Kyritsis, K. Bouzianas, Experimental evaluation of
an autonomous low-temperature solar Rankine cycle system for reverse osmosis
desalination, Desalination 203 (2007) 366-374
[1.11] D. Manolakos, G. Papadakis, E.Sh. Mohamed, S. Kyritsis, K. Bouzianas, Design of
an autonomous low-temperature solar Rankine cycle system for reverse osmosis
desalination, Desalination 183 (2005) 73-80
[1.12] D. Wei, X. Lu, Z. Lu, J. Gu, Performance analysis and optimization of organic
Rankine cycle (ORC) for waste heat recovery, Energy Conversion and Management 48
(2007) 1113-1119
[1.13] H.D. Madhawa Hettiarachchi, M. Golubovic, W.M. Worek, Y. Ikegami, Optimum
design criteria for an Organic Rankine cycle using low-temperature geothermal heat
sources, Energy 32 (2007) 1698-1706
[1.14] B. Saleh, G. Koglbauer, M. Wendland, J. Fischer, Working fluids for lowtemperature organic Rankine cycles, Energy 32 (2007) 1210-1221
[1.15] B.P. Brown, B.M. Argrow, Application of Bethe-Zel'dovich-Thompson fluids in
organic Rankine cycle engines, Journal of Propulsion and Power 16 (2000) 1118-1124
[1.16] T. Yamamoto, T. Furuhata, N. Arai, K. Mori, Design and testing of the organic
rankine cycle, Energy 26 (2001) 239-251
[1.17] B.T. Liu, K.H. Chien, C.C. Wang, Effect of working fluids on organic Rankine cycle
for waste heat recovery, Energy 29 (2004) 1207-1217
[1.18] T.C. Hung, T.Y. Shai, S.K. Wang, A review of organic rankine cycles (ORCs) for
the recovery of low-grade waste heat, Energy 22 (1997) 661-667
[1.19] T.C. Hung, Waste heat recovery of organic Rankine cycle using dry fluids, Energy
Conversion and Management 42 (2001) 539-553
[1.20] D. Mills, Advances in solar thermal electricity technology, Solar Energy 76 (2004)
19-31
7
[1.21] W. Wagner, A. Pruß, The IAPWS formulation 1995 for the thermodynamic
properties of ordinary water substance for general and scientific use, J. Phys. Chem. Ref.
Data, 31 (2002) 387 - 535.
[1.22] R. Span, W. Wagner, A new equation of state for carbon dioxide covering the fluid
region from the triple-point temperature to 1100 K at pressures up to 800 MPa. J. Phys.
Chem. Ref. Data, 25 (1996) 1509-1596.
[1.23] A. Mueller, J. Winkelmann, J. Fischer, Backone family of equations of state: 1.
Nonpolar and polar pure fluids, AIChE J, 42 (1996) 1116–1126.
[1.24] U. Weingerl, M. Wendland, J. Fischer, A. Mueller, J. Winkelmann, Backone family
of equations of state: 2. Nonpolar and polar fluid mixtures. AIChE, 47 (2001) 705–717.
[1.25] B. Saleh, M. Wendland, Screening of pure fluids as alternative refrigerants, Int J
Refrig, 29 (2006) 260–269.
[1.26] N. A. LAI, M. Wendland, J. Fischer, Development of equations of state for
siloxanes as working fluids for ORC Processes, Proceeding of 24th European symposium
on applied thermodynamics, June 27 – July 1, 2009, 200-205, ISBN: 978-84-692-2664-3,
Santiago de Compostela, Spain
[1.27] N. A. LAI, J. Fischer, M. Wendland, Description of aromates with BACKONE
equations of state, to be submitted in refereed journal
[1.28] S. Calero, M. Wendland, J. Fischer, Description of alternative refrigerants with
BACKONE equations, Fluid Phase Equilibria, 152 (1998) 1 – 22.
[1.29] M. Wendland, B. Saleh, J. Fischer, Accurate thermodynamic properties from the
BACKONE equation for the processing of natural gas, Energy Fuels, 18 (2004) 938–951
[1.30] J. Gross and G. Sadowski, Perturbed-Chain SAFT: An Equation of State Based on a
Perturbation Theory for Chain Molecules, Ind. Eng. Chem. Res., 40 (2001) 1244-1260.
[1.31] N. A. Lai, M. Wendland, J. Fischer, Description of linear siloxanes with PC-SAFT
equation, Fluid Phase Equilib., 283 (2009) 22-30.
8
2
Extrapolation of vapour pressures
Abstract
Downward and upward extrapolations of vapour pressures with Antoine equation,
Val der Waals equation, Korsten equation, Wagner equation and a new one-parameter
equation are investigated for fluids from different molecular classes such as argon,
ethylene, ethane, sulfur hexafluoride, benzene, and water. It is shown that the new oneparameter equation outperforms other equations for downward extrapolation from different
input data ranges. For upward extrapolation, we find out that Antoine equation gives the
best performance if experimental data range from 0.5Tc to 0.6Tc. However, if experimental
data range from 0.7Tc to 0.8Tc Wagner equation outperforms other equations.
2.1
Introduction
In construction of equations of state [2.1–2.7] one needs experimental saturated
liquid densities, vapour pressures, ideal gas heat capacities, and eventually also of other
properties. Among them, values of vapour pressures are in a large range, normally from
mPa to MPa. Because of the variety of fluids and because each substance has certain
temperature and vapour pressure ranges, one needs different types of pressure sensors as
well as different methods for measuring vapour pressures. There have been some published
vapour pressures in nearly full fluid region with less accuracy in low reduced temperature
[2.8, 2.9]. Vapour pressures of almost all substances have been measured and published in
limited temperature and pressure ranges, mostly in moderate or high reduced temperature
ranges [2.10, 2.11]. Thus, continuation of experiment and using equations for correlation
and extrapolation of vapour pressure are necessary. It should be mentioned that the
measurements of very low vapour pressures usually contain large errors, maybe far more
than 50%. Therefore, low-pressure data presented in the literature are often the results of
smooth extrapolation from equations that were thought to be appropriate for extrapolations.
Many equations for vapour pressures have been published. Most of them have
multi-parameters [2.12]. The multi-parameter equations allow high accuracy for
correlations. However, these equations require lots of experimental data for determining
parameters and these equations are not really reliable for extrapolation outside of maximum
and minimum available data ranges. Methods with fewer parameters show sufficient
9
accuracies within relative small ranges. For outward extrapolation in large temperature and
pressure ranges, equations with few parameters are preferred. In this study, we concentrate
on one-parameter equations and two most popular equations, Antoine equation and Wagner
equation.
2.2
Investigated equations
The first approximation to the exact Clausius-Clapeyron equation assumes an ideal
gas phase and a constant enthalpy of vaporization. With these assumptions one gets
ln ps = A + B/T,
(2.1)
where A and B are fitted parameters.
Using the condition that at Tr = 1 we have pr =1, one gets
ln pr = A(1 – 1/Tr).
(2.2)
The equation (2.1) and equation (2.2), called Wan der Waals equation, mean that
ln(ps) is a linear function of 1/T. The assumptions of ideal gas phase and constant enthalpy
of vaporization, however, become worse in approaching the critical point and hence the real
vapour pressures become with increasing T increasingly smaller than the results from the
linear function in 1/T. Antoine [2.13] realized these deficiencies and improved it by his
equation.
ln ps = A + B/(C +T),
(2.3)
where A, B, and C are fitted parameters.
Equation (2.3) can also be written as
ln pr= A’ + B’/(C’ +Tr).
(2.4)
Eq. (2.4) does not fulfil the condition that pr is equal to one at Tr = 1. If one enforces
that condition, one reduces the three parameters to two ones by 0 = A’ + B’/(C’ +1).
The next remark on Van der Waals concerns the value of A. In the sense of the
corresponding-states principle, A in equation (2.2) should be a universal constant. As a
matter of fact, the slope of ln pr vs 1/Tr becomes steeper with increasing deviation of the
molecules from noble gas molecules [2.14]. This fact is usually expressed by the acentric
factor ω in engineering thermodynamics.
10
Korsten proposed a one-parameter equation for hydrocarbons and other species,
[2.15]. The equation has a common reference point (T0, p0) with T0 = 1994.49 K, p0 =
1867.68 bar and is written as
ln ps= ln p0 + A(1/T1.3 – 1/T01.3),
(2.5)
where A is fitted parameter.
According to our investigation, using common reference point is not as accurate as
using critical point for each substance as a reference point. Furthermore, in order to make a
consistence for comparison with other equations using critical data, the reference point is
replaced by critical point and equation (2.5) becomes
ln ps = ln pc + A(1/T1.3 – 1/Tc1.3),
or
ln pr = A(1 – 1/Tr1.3).
(2.6)
Obviously, the difference between equation (2.2) and equation (2.6) is the exponent
1.3. In Van der Waal equation (2.2) and Kosten equation (2.6), vapour pressure equals to
zero when T = 0. However, for many substances, vapour pressure at triple point or below
triple point is very small and it nearly equals to zero at certain low temperature whilst
temperature is still relatively high, from 0.3Tc to 0.5Tc for almost all substances. Thus,
starting from Val der Waals equation (2.2), we carry out intensive analysis and get an
equation:
⎞
⎛
T − Tc
⎟⎟ .
ln pr = A⎜⎜
⎝ T − 28.012 * LnTc + 154.71 ⎠
(2.7)
Similar to equation (2.2) and equation (2.6), equation (2.7) has only one parameter,
A, found by fitting to experimental data. All these three equations can be used for both
upward and downward extrapolations. If critical data are not available, the critical data can
be replaced by maximum available values.
In this study, we also investigate Wagner equation, [2.16]:
A.(1 − Tr ) + B.(1 − Tr )1.5 + C.(1 − Tr ) 3 + D.(1 − Tr ) 6
ln pr =
,
Tr
where A, B, C, D are fitted parameters.
11
(2.8)
Wagner equation is well known for accurate correlation. However, possibilities of
extrapolation of this equation are not clear. Thus in this study, we investigate the
possibilities of extrapolation of this equation.
Recently Velasco et. al. have recommended a predictive vapour pressure equation
with one unknown parameter [2.17]. The unknown parameter is found by fitting to
experimental data. The equation is verified by using 53 fluids with an overall average
deviation of 0.55%. This equation needs not only data at critical point but also data at triple
and normal boiling points. Because all our interested siloxanes do not have data of triple
points and other investigated equations in this study do not use data of triple and boiling
points so we do not investigate this equation.
In order to test the possibilities of extrapolations of Antoine equation, Val der Waals
equation, Korsten equation, Wagner equation and the new one-parameter equation, we use
different typical fluids from various species such as argon, ethylene, ethane, sulfur
hexafluoride, benzene, and water.
2.3
Substances and data sources
We use 6 substances from different molecular classes for which experimental data
are available in full or nearly full fluid region to study the equations for extrapolation of
vapour pressures. The first considered substance is argon. Argon is an ideal substance and
has only one atom. Vapour pressures of argon in low reduced temperature range are quite
high and accurate values have been measured [2.18].
The second considered substance is water. Water has 2 hydrogen atoms and one
single oxygen atom. Water is one of the most important fluids for life and science. In
comparison with other substances, water has the greatest number of experimental data.
Nearly twenty thousands of experimental data points have been measured. In this
investigation, we use only experimental data which are used for construction of reference
equation of state for normal water (IAPWS-95), [2.5].
The third investigated substance is ethane. Ethane is one of paraffin or alkanes.
Ethane has two carbon atoms and six hydrogen atoms. Ethane is the second-largest
component of natural gas, after methane. This substance has a long fluid range and accurate
12
vapour pressures have been measured in nearly full fluid region [2.8]. Thus, we decide to
use this substance for testing the above equations.
Next considered fluid is ethylene or ethene. Ethylene is one of alkenes. Comparison
with ethane, ethylene has two hydrogen atoms fewer than those of ethane. Ethylene, the
most produced organic compound in the world, has long fluid range. Accurate vapour
pressures of ethylene have been measured in nearly full fluid region [2.9] and reference
equation of state for this substance has been constructed [2.19].
Next substance in our investigation is benzene. Benzene is one of aromatic
hydrocarbons. Accurate vapour pressures of benzene have been published by Ambrose
[2.20], [2.21] in nearly full fluid region.
Molecular characterizations of the selected substances are given in Table 2.1. Their
critical temperatures Tc, critical pressures pc, triple point temperature Tt, triple point
pressure pt are given in Table 2.2 together with sources. Table 2.3 gives available
experimental data ranges and sources of selected substances.
Table 2.1: Molecular characterization of the substances
No
1
Substance/
Formula
argon/
Ar
Structure
M [g/mol]
Ar
39.948
2
ethylene/
C2H4
28.053
3
ethane/
C2H6
30.069
4
sulfur hexafluoride/
SF6
146.056
5
benzene/
C6H6
78.112
6
water/
H2O
18.015
13
Table 2.2: Properties at triple point, critical point and sources
No
Substance
Tt [K]
Pt [kPa]
Tc [K]
pc [MPa]
1
2
3
argon
ethylene
ethane
sulfur
hexafluoride
benzene
water
83.798
103.986
90.348
68.89
0.1225
0.0011308
150.687
282.35
305.322
4.8630
5.0418
4.8722
Ref.
triple
point
[2.24]
[2.25]
[2.26]
223.555
231.43
318.723
3.755
[2.22]
[2.22]
278.68
273.16
4.785
0.61165
562.16
647.1
4.898
22.0640
[2.27]
[2.23]
[2.20]
[2.23]
4
5
6
Ref.
critical
point
[2.18]
[2.9]
[2.8]
Table 2.3. Experimental data ranges and sources of the selected substances
No
Substance
1
2
3
5
argon
ethylene
ethane
sulfur
hexafluoride
benzene
6
water
4
2.4
Texp,min
[K]
Texp,max [K]
Tr,min
Tr,max
Tr,t
Ref. pressure
84
104
91
150.65
280
303
0.56
0.37
0.30
1.00
0.99
0.99
0.56
0.37
0.30
[2.18]
[2.9]
[2.8]
224
314.6
0.70
0.99
0.70
[2.22]
285.957
562.16
0.51
1.00
0.50
273.16
647.0834
0.42
1.00
[2.20], [2.21]
[2.28], [2.29], [2.30],
0.42
[2.31], [2.32]
Results and discussions
In this investigation of the upward and downward extrapolations of vapour pressure
equations for various substances from different molecular classes, we fit equations (2.2),
(2.3), (2.6), (2.7) and (2.8) to experimental vapour pressures in temperature ranges from
0.5Tc to 0.6Tc, from 0.7Tc to 0.8Tc and from 0.9Tc to around critical point.
In order to find parameters of studied equations, we take both Tc and pc from Table
2.2 and the vapour pressures from the sources listed in Table 2.3. The fit criterion is to
minimize Σi[(ps,exp,i - ps,cal,i )/ps,exp,i]2. Results for extrapolation at different reduced
temperatures are given in tables 2.4 to 2.20.
Tables 2.4 and 2.5 present results for downward extrapolations with different
equations using experimental data from 0.5Tc to 0.6Tc. The results show that Antoine
14
equation (2.3) is the most accurate equation. Following is the new equation (2.7). The worst
equation is Van der Waal equation (2.2).
Table 2.4. Deviations between experimental data and predicted values of vapour pressures
at triple point based on experimental data in reduced temperature range of 0.5 to 0.6.
Substance
ethylene
ethane
benzene
water
Average absolute
deviation (AAD)
Equ. (2.2)
63.3%
192.2%
14.3%
50.9%
80.2%
Equ. (2.3) Equ. (2.6)
-0.7%
-24.8%
-6.5%
-51.6%
-0.1%
-0.3%
-2.2%
-13.1%
2.4%
12.7%
Equ. (2.7)
6.6%
-13.2%
4.8%
4.9%
Equ. (2.8)
35.5%
-18.1%
-0.2%
0.7%
7.4%
13.6%
at Tr = 0.4 based on experimental data in reduced temperature range of 0.5 to 0.6.
Substance
ethylene
ethane
(AAD)
Equ. (2.2)
35.4%
24.4%
29.9%
Equ. (2.3)
-0.9%
-5.1%
3.0%
Equ. (2.6)
-19.5%
-26.0%
22.8%
Equ. (2.7)
2.8%
-8.1%
5.5%
Equ. (2.8)
13.9%
-7.2%
10.6%
Results for upward extrapolations based on experimental data from 0.5Tc to 0.6Tc of
different equations are given in Table 2.6, 2.7 and 2.8. The worst equation for upward
extrapolation for this case is equation (2.6) of Korsten. With an offset of reduced
temperature of 0.1 and 0.2, Antoine equation is the best one. Whilst, Van der Waals
equation (2.2) is the best one for the case with offset of reduced temperature of 0.3.
Substance
ethylene
ethane
benzene
water
(AAD)
Equ. (2.2)
-3.1%
-2.7%
-7.6%
-5.7%
4.8%
Equ. (2.3)
0.0%
1.0%
0.0%
0.7%
0.4%
15
Equ. (2.6)
8.9%
9.6%
4.0%
6.7%
7.3%
Equ. (2.7)
2.1%
3.1%
-1.1%
1.1%
1.9%
Equ. (2.8)
-3.1%
1.8%
0.5%
-0.2%
1.4%
Substance
ethylene
ethane
benzene
water
(AAD)
Equ. (2.2)
-0.9%
-0.4%
-4.6%
-3.2%
2.3%
Equ. (2.3)
-0.7%
1.3%
-0.6%
0.8%
0.9%
Equ. (2.6)
9.4%
10.1%
5.9%
8.2%
8.4%
Equ. (2.7)
3.3%
4.4%
1.1%
2.8%
2.9%
Equ. (2.8)
-8.2%
3.9%
1.7%
-1.1%
3.7%
Substance
ethylene
ethane
benzene
water
(AAD)
Equ. (2.2)
0.9%
1.0%
-1.2%
-0.3%
0.9%
Equ. (2.3)
-2.6%
-0.1%
-3.1%
-0.6%
1.6%
Equ. (2.6)
6.6%
6.8%
4.9%
6.3%
6.2%
Equ. (2.7)
3.2%
3.6%
2.0%
3.1%
3.0%
Equ. (2.8)
-9.9%
4.2%
2.0%
-1.5%
4.4%
Results for upward and downward extrapolations based on input experimental data
from 0.7Tc to 0.8Tc of different equations are given in Table 2.9 to Table 2.13. From this
temperature range, Wagner equation is the best one for upward extrapolation whilst the
worst equation is the Antoine equation, Table 2.13.
For downward extrapolations using data from 0.7Tc to 0.8Tc the new equation (2.7)
is the best one, except for the case with an offset reduced temperature of 0.1, Table 2.9 to
Table 2.12. The worst equation is the equation of Antoine.
From these analyses, Antoine equation is the worst one for both upward and
downward extrapolations. Thus, Antoine equation should not be used if experimental
vapour pressures range from 0.7Tc to 0.8Tc. In this case, Wagner equation should be used
for upward extrapolation and downward extrapolation within an offset of reduced
temperature of 0.1. The new equation should be used for downward extrapolation when
offset of reduced temperature is larger than 0.1.
16
at triple point based on experimental data in reduced temperature range of 0.7 to 0.8.
Substance
Equ. (2.2)
Equ. (2.3)
Equ. (2.6) Equ. (2.7)
Equ. (2.8)
argon
-4.4%
3.6%
-23.3%
1.8%
0.7%
ethylene
80.9%
16524.2%
-54.6%
-7.9%
-23.2%
ethane
228.4%
136.9%
-77.5%
-34.3%
26538.4%
SF6
0.2%
0.0%
-1.8%
-0.7%
0.1%
benzene
38.7%
7.1%
-14.7%
5.1%
-32.6%
water
84.3%
11.9%
-35.6%
-1.6%
-19.3%
(AAD)
72.8%
2780.6%
18.9%
8.6%
4435.7%
Substance
Equ. (2.2)
Equ. (2.3) Equ. (2.6) Equ. (2.7) Equ. (2.8)
ethylene
48.0%
4946.4%
-47.8%
-9.4%
-15.2%
ethane
33.9%
25.5%
-53.0%
-22.2%
272.2%
(AAD)
41.0%
2486.0%
50.4%
15.8%
143.7%
Substance
Equ. (2.2) Equ. (2.3) Equ. (2.6) Equ. (2.7) Equ. (2.8)
ethylene
10.2%
382.4%
-27.4%
-7.9%
-1.6%
ethane
7.3%
7.5%
-29.2%
-11.8%
31.4%
water
29.0%
5.8%
-22.6%
-2.8%
-5.2%
(AAD)
15.5%
131.9%
26.4%
7.5%
12.7%
Substance
Equ. (2.2)
Equ. (2.3)
Equ. (2.6) Equ. (2.7) Equ. (2.8)
argon
-3.8%
1.4%
-15.8%
-0.2%
0.2%
ethylene
0.5%
57.0%
-12.6%
-5.0%
0.1%
ethane
-0.7%
0.7%
-13.4%
-6.5%
2.5%
benzene
7.1%
1.0%
-8.6%
-1.5%
-3.6%
water
8.3%
3.5%
-8.7%
-0.8%
1.9%
(AAD)
4.1%
12.7%
11.8%
2.8%
1.7%
Substance
Equ. (2.2) Equ. (2.3) Equ. (2.6) Equ. (2.7) Equ. (2.8)
argon
1.9%
-0.5%
3.8%
1.4%
0.0%
ethylene
1.6%
33.6%
3.7%
2.3%
0.3%
ethane
1.5%
-0.9%
3.8%
2.4%
-0.5%
SF6
1.3%
-0.6%
3.7%
2.3%
-1.0%
benzene
0.9%
-1.4%
3.3%
2.0%
0.3%
water
1.4%
-0.9%
4.1%
2.6%
0.2%
(AAD)
1.4%
6.3%
3.7%
2.2%
0.4%
17
The downward extrapolations of the equations are studied for the case with input
experimental data in reduced temperature range of 0.9 to around critical point. Deviations
between experimental data and predicted values of vapour pressures at different
temperatures are given in Table 2.14 to Table 2.19. From these tables, we observe that both
Antoine equation and Wagner equation are not reliable. The best equation for downward
extrapolations in this case is Van der Waals equation. The second-best equation for
downward extrapolations in this case is the new equation.
Table 2.14. Deviations between experimental data and predicted vapour pressures at triple
point based on experimental data in reduced temperature range of 0.9 to around critical
point.
Substance
Equ. (2.2) Equ. (2.3) Equ. (2.6)
Equ. (2.7)
Equ. (2.8)
argon
-18.7%
68.4%
-45.1%
-9.9%
1678.0%
ethylene
31.2%
2834.9%
-79.7%
-43.3%
#
ethane
127.1%
11614.4%
-92.1%
-65.0%
-100.0%
SF6
-5.9%
8.9%
-17.1%
-10.5%
3338.5%
benzene
25.4%
207.0%
-39.6%
-14.9%
-64.1%
water
46.0%
1307.6%
-67.2%
-35.7%
#
(AAD)
42.4%
2673.5%
49.7%
29.9%
1295.2%
#: The values are too large and will not be considered for calculation and comparison
Table 2.15. Deviations between experimental data and predicted vapour pressures at Tr =
0.4 based on experimental data in reduced temperature range of 0.9 to around critical point.
Substance
Equ. (2.2)
Equ. (2.3)
Equ. (2.6)
Equ. (2.7)
Equ. (2.8)
ethylene
11.8%
1206.2%
-73.9%
-40.5%
#
ethane
6.1%
490.5%
-74.7%
-46.7%
-100.0%
(AAD)
9.0%
848.4%
74.3%
43.6%
100.0%
at Tr = 0.5 based on experimental data in reduced temperature range of 0.9 to around
critical point.
Substance
Equ. (2.2)
Equ. (2.3)
Equ. (2.6)
Equ. (2.7)
Equ. (2.8)
ethylene
-8.6%
196.4%
-53.3%
-29.9%
#
ethane
-8.1%
103.7%
-52.3%
-31.0%
-100.0%
water
8.9%
286.2%
-51.9%
-28.2%
#
(AAD)
8.5%
195.4%
52.5%
29.7%
100.0%
18
critical point.
Substance
Equ. (2.2) Equ. (2.3) Equ. (2.6)
Equ. (2.7)
Equ. (2.8)
argon
-16.0%
40.1%
-36.1%
-9.9%
307.2%
ethylene
-11.3%
52.3%
-34.2%
-20.6%
409.8%
ethane
-10.4%
27.9%
-33.0%
-20.4%
-100.0%
benzene
0.2%
47.7%
-26.5%
-13.9%
-25.6%
water
-3.3%
72.3%
-32.8%
-18.6%
903.3%
(AAD)
8.2%
48.1%
32.5%
16.7%
349.2%
critical point.
Substance
Equ. (2.2)
Equ. (2.3)
Equ. (2.6)
Equ. (2.7) Equ. (2.8)
argon
-9.4%
11.3%
-18.6%
-6.9%
22.0%
ethylene
-8.2%
14.3%
-18.5%
-12.0%
25.4%
ethane
-7.0%
7.0%
-16.9%
-11.0%
-95.0%
benzene
-3.9%
12.0%
-14.7%
-9.1%
-6.8%
water
-6.7%
17.9%
-19.2%
-12.5%
36.8%
(AAD)
7.0%
12.5%
17.6%
10.3%
37.2%
critical point.
Substance
Equ. (2.2)
Equ. (2.3) Equ. (2.6) Equ. (2.7) Equ. (2.8)
argon
-3.9%
2.1%
-6.7%
-3.2%
1.2%
ethylene
-4.0%
2.5%
-7.3%
-5.2%
1.1%
ethane
-3.0%
1.0%
-6.0%
-4.1%
-17.2%
SF6
-3.5%
0.9%
-7.0%
-4.9%
23.7%
benzene
-2.4%
2.0%
-5.4%
-3.7%
-0.7%
water
-3.7%
3.3%
-7.6%
-5.4%
1.9%
(AAD)
3.4%
2.0%
6.7%
4.4%
7.6%
Previous analyses of extrapolations of different equations with input data ranges
from 0.5Tc to 0.6Tc, from 0.7Tc to 0.8Tc, and from 0.9Tc to around critical point show the
best equation for upward and/or downward extrapolations from certain input data range. In
order to find the best equation for all input data ranges, we make an average of all average
absolute deviations from Table 2.4 to Table 2.19 and put in Table 2.20. Table 2.20 shows
that the new equation (2.7) is the best equation for downward extrapolation. The second
best equation for downward extrapolation is the Van der Waals equation. This equation
gives the best performance for input data range from 0.9Tc up to around critical point.
19
Table 2.20 also shows that both Wagner equation and Antoine equation are the
worst equations for downward extrapolation. The results in Table 2.20 can explain why
predicted vapour pressures of triple point in references [2.10], [2.11] based on Wagner
equation are not reliable.
Table 2.20. Absolute deviations for downward extrapolations from different input data
ranges
Offset of Tr
Fitting range,
from minimal Equ. (2.2) Equ. (2.3) Equ. (2.6) Equ. (2.7) Equ. (2.8)
Tr
input Tr
0.5-0.6
0.1
29.9%
3.0%
22.8%
5.5%
10.6%
0.7-0.8
0.1
4.1%
12.7%
11.8%
2.8%
1.7%
0.9-critical
0.1
3.4%
2.0%
6.7%
4.4%
7.6%
0.7-0.8
0.2
15.5%
131.9%
26.4%
7.5%
12.7%
0.9-critical
0.2
7.0%
12.5%
17.6%
10.3%
37.2%
0.7-0.8
0.3
41.0%
2486.0%
50.4%
15.8%
143.7%
0.9-critical
0.3
8.2%
48.1%
32.5%
16.7%
349.2%
0.9-critical
0.4
8.5%
195.4%
52.5%
29.7%
100.0%
0.9-critical
0.5
9.0%
848.4%
74.3%
43.6%
100.0%
0.5-0.6
*
80.2%
2.4%
12.7%
7.4%
13.6%
0.7-0.8
*
72.8%
2780.6%
18.9%
8.6%
4435.7%
0.9-critical
*
42.4%
2673.5%
49.7%
29.9%
1295.2%
Average of all
26.8%
766.4%
31.4%
15.2%
542.3%
AADs
*: The offsets of reduced temperature are at various values depending on
substances. The calculation and comparison are carried out at triple points directly.
2.5
Summary and conclusions
This chapter investigates possibilities of extrapolation of vapour pressure with
different equations. In the investigation, we use data of substances from various molecular
classes such as argon, ethylene, ethane, sulfur hexafluoride, benzene, and water. For
downward extrapolation, the results show that new equation (2.7) is the most stable one and
outperforms other equations for downward extrapolation from different temperatures.
For upward extrapolation, we found from experimental data in range from 0.5Tc to
0.6Tc that Antoine equation gives the best performance with offset reduced temperature of
0.1, 0.2 and 0.3. However, if experimental data is in range from 0.7Tc to 0.8Tc Wagner
equation outperforms Antoine equation.
20
References
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[2.2] N. A. Lai, M. Wendland, J. Fischer, Description of aromates with BACKONE
equations of state, Fluid Phase Equilibria, MS in preparation
[2.3] N. A. Lai, M. Wendland, J. Fischer, Development of equations of state for siloxanes
as working fluids for ORC Processes, proceeding of 24th European symposium on applied
thermodynamics, 2009, 200-205
[2.4] R. Span, Multiparameter Equation of State - An Accurate Source of Thermodynamic
Property Data, Springer, Berlin, 2000
[2.5] W. Wagner, A. Pruß, The IAPWS formulation 1995 for the thermodynamic properties
of ordinary water substance for general and scientific use. J. Phys. Chem. Ref. Data, 31
(2002) 387 - 535
BACKONE equations. Fluid Phase Equilibria, 152 (1998) 1–22.
BACKONE equation for the processing of natural gas. Energy Fuels, 18 (2004) 938–951.
[2.8] M. Funke, R. Kleinrahm, W. Wagner, Measurement and correlation of the (p,ρ,T)
relation of ethane II. Saturated-liquid and saturated-vapour densities and vapour pressures
along the entire coexistence curve, J. Chem. Thermodyn. 34 (2002) 2017–2039
[2.9] P. Nowak, R. Kleinrahm and W. Wagner, Measurement and correlation of the (p,ρ,T)
relation of ethylene: II. Saturated-liquid and saturated-vapour densities and vapour
pressures along the entire coexistence curve, J. Chem. Thermodyn. 28 (1996) 1441–1460
[2.10] L. A. Weber, vapour Pressure of Heptane from the Triple Point to the Critical Point,
J. Chem. Eng. Data, 45 (2000) 173-176
[2.11] L. A. Weber, vapour pressures and gas-phase PVT data for 1,1-dichloro-2,2,2trifluoroethane, J. Chem. Eng. Data, 35 (1990) 237-240
21
[2.12] J. M. Prausnitz, B. E. Poling, J. P. O’Connell, The properties of gases and liquids,
5th ed. New York: McGraw-Hill; 2001.
[2.13] C. Antoine, Tensions des vapeurs; nouvelle relation entre les tensions et les
températures, Comptes Rendus des Séances de l'Académie des Science 107 (1888) 681684, 778-780, 836-837
[2.14] J. Fischer, R. Lustig, H. Breitenfelder-Manske and W. Lemming, Influence of
intermolecular potential parameters on orthobaric properties of fluids consisting of
spherical and linear molecules, Molecular Physics 52 (1984) 485-497
[2.15] H. Korsten, Internally Consistent Prediction of vapour Pressure and Related
Properties, Ind. Eng. Chem. Res., 39 (2000) 813-820
[2.16] W. Wagner, New vapour pressure measurements for argon and nitrogen and a new
method for establishing rational vapour pressure equations, Cryogenics 13 (1973) 470-482.
[2.17] S. Velasco, F.L. Roman, J.A. White, A. Mulero, A predictive vapor-pressure
equation, The Journal of Chemical Thermodynamics, 40 (2008) 789-797
[2.18] R. Gilgen, R. Kleinrahm, W. Wagner, Measurement and correlation of the (pressure,
density, temperature) relation of argon. II. saturated-liquid and saturated-vapour densities
and vapour pressures along the entire coexistence curve, J. Chem. Thermodyn. 26 (1994)
399-413
[2.19] J. Smukala, R. Span, W. Wagner, A New Equation of State for Ethylene Covering
the Fluid Region for Temperatures from the Melting Line to 450 K at Pressures up to 300
MPa, J. Phys. Chem. Ref. Data, 29 (2000) 1053-1122
[2.20] D. AMBROSE, Vapour pressures of some aromatic hydrocarbons, J. Chem.
Thermodyn.,19 (1987) 1007-1008
[2.21] D. AMBROSE, Reference values of vapour pressure the vapour pressures of
benzene and hexafluorobenzene, J. Chem. Thermodynamics, 13 (1981) 1161-1167
[2.22] M. Funke, R. Kleinrahm, and W. Wagner, Measurement and correlation of the (p, ρ,
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densities and vapour pressures along the entire coexistence curve, J. Chem. Thermodyn. 34
(2001) 735–754
22
[2.23] H. Sato, K. Watanabe, J.M.H. Levelt Sengers, J.S. Gallagher, P.G. Hill, J. Straub, W.
Wagner, , Sixteen Thousand Evaluated Experimental Thermodynamic Property Data for
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[2.27] R.D. Goodwin, Benzene thermophysical properties from 279 to 900 K at pressures to
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23
3
Measurement of vapour pressures
Abstract
Experimental apparatus for measurement of vapour pressures is constructed with a
total uncertainty of 0.17 kPa for pressure up to 100 kPa. This apparatus can be used to
measure vapour pressures up to 2.67 MPa for temperature higher than 107°C and up to 6.9
MPa for temperature lower than 107°C. Full temperature range of this apparatus is from
- 54°C to +200°C.
3.1
Introduction
Vapour pressure of a pure liquid or of a solution is an important property in
chemical and engineering applications. It is used, for examples, in the designs of distillation
columns, storage tanks, pipelines, and in the construction of equation of state, [3.1-3.9].
Due to the need of the knowledge of vapour pressures, vapour pressures of many
substances have been measured and published [3.10-3.14].
Many measurements were made a long time ago when apparatus was poor by the
present standards. Beside that, in many cases, apparatuses were not carefully calibrated or
measured fluids were not highly purified. For these reasons, experimental data for many
substances from different sources have large disagreements. The available experimental
vapour pressures are mostly not covered in full temperature range, generally in moderate or
high reduced temperatures [3.13, 3.14]. Furthermore, hundreds of new chemicals are
discovered annually. Thus accurate experiments are still needed forever.
We have currently needed to measure vapour pressures of different fluids hence
apparatus for measurement of vapour pressure has been constructed. This apparatus uses
different accurate pressure sensors/transducers which are available in our institute.
Following sections will present specifications as well as assessments for accuracy and
uncertainty of the sensors and apparatus.
24
3.2
3.2.1
Experimental set-ups
Pressure sensors and indicators
Experimental set-up has three pressure sensors or transducers. The first one, product
of Desgranges & Huot, Aubervilliers, France, model 21000M, is used to calibrate other
pressure sensors such as Paroscientific, Model 31K-101, and MKS Baratron®, type 615.
Absolute Pressure of Desgranges & Huot model 21000M
The Absolute Pressure of Desgranges & Huot model 21000M has 2 blocks:
measuring block (pressure-block) and dynamometer block. In measuring block, force F is
calculated from pressure p and cross-section area A of a piston as F = p.A. During
measurement, the piston is rotated to avoid affection of static viscosity on the measurement.
The dynamometer block is a digital balance for indicating pressure values. Calibration of
indicating values of this apparatus is based on reference dead weights.
The method using a reference dead weight for measurement of pressure is a
fundamental method. This method is used to calibrate pressure gauges and other pressure
sensitive instruments periodically.
A pressure-block 410 of the Desgranges & Huot (D&H) model 21000M is used to
calibrate vapour pressure of Paroscientific and MKS sensors. The digital piston manometer
or pressure sensor of D&H works in room temperature with uncertainty of 0.1 mbar (0.075
mmHg) and 0.2 mmHg for pressure range up to 0.6 bar and 2.4 bar, respectively. Because
the digital piston manometer measures only gauge pressures so we use a mercury barometer
with an uncertainty of 30 Pa, to indicate atmospheric pressure.
Absolute Pressure Transducer Paroscientific Model 31K-101 (Paro)
Principle of the Absolute Pressure Transducer Paroscientific Model 31K-101 is
based on characteristics of Quartz crystal resonators of which resonance frequency changes
by external force or pressure. Based on this principle, pressure signal is transferred into
frequency which is used to indicate pressure.
The Absolute Pressure Transducer Paroscientific Model 31K-101 has following
specifications:
- Pressure Range: 0-1000 psia (0-6.9 MPa)
25
- Operating Temperature Range: - 54°C to +107 °C (-65 °F to 224 °F)
- Accuracy: 0.01%
- Repeatability: Better than ±0.01% Full Scale
Vapour pressures are indicated in Digiquartz® display model 710 which connects to
the pressure sensor.
Absolute pressure sensor MKS Baratron® type 615.
Absolute pressure sensor MKS Baratron® type 615, product of MKS manufacturer,
uses capacitance diaphragm sensor technology. In capacitive technology, the pressure
diaphragm is one plate of a capacitor that changes its value under pressure-induced
displacement. Based on this principle, pressure signal is transferred into an electrical
quantity which is used to indicate pressure.
The Absolute pressure sensor MKS 615 has following specifications:
- Pressure Range: 0-20000 mmHg (0- 2.67 Mpa)
- Operating Temperature Range: 15°C to +200 °C (59 °F to 392 °F)
- Accuracy: 0.25% and 0.12% for the range up to 10 mmHg and 20000 mmHg
respectively.
The pressure sensor is connected to “electronics equipment 670” for indicating
experimental values.
3.2.2
Temperature sensor and indicator
Platinum thermometer PT100 (Serkal, Austria) or temperature sensor is used to
measure temperature of thermal fluids in controlled bath. Temperature is indicated by a
digital resistance bridge called “precision thermometer bridge F300” of Automatic system
laboratory, UK. Calibration of the thermometer was done according to the International
Temperature Scale of 1990 (ITS-90). The specifications of the precision thermometer
bridge F300 are:
- Precision: better than ±5mK full range, ±1mK at 0°C
- Resolution: 0.25ppm (0.1mK)
26
- Range: 0 – 1500 Ω (0.25 – 1000 ohm thermometers/ 13K to 1064°C)
- Temperature displayed according to ITS90
3.2.3
Experimental set-ups
Apparatus for measuring vapour pressures has one sample cell for containing fluids.
The three mentioned sensors are connected to the sample cell 1 via different pipes, figure
3.1. The apparatus has one pipe to be connected to argon tank. Argon is used to clean
connecting pipes and pressurizing for calibration of pressure above 1 bar. One channel of
MKS pressure sensor is connected to absolute vacuum pump which can maintains pressure
down to 0.029 mbar. We use 7 needle valves for connecting and/or disconnecting the
sensors, sample cell, vacuum pump, and argon tank, figure 3.1.
In order to measure vapour pressures at different temperatures, the measuring cell 1
is submerged in thermal fluids of thermal bath Julabo F32 where temperature is controlled
by Julabo HE. The temperature can be set from -0°C to 320°C and maintained with a
fluctuation of 0.01°C. When measuring temperature of fluids is about or higher than room
temperature, the connecting pipes are equipped with a heating wire and both of them are
insulated. In order to avoid vapour condenses in the connecting pipes, temperature of the
heating wire is maintained at a value higher than measuring temperature. The temperature
of heating wire is controlled by temperature controller LC6 of Julabo.
With this set-up, the apparatus has following specifications:
- Pressure Range: 0-6.9 MPa (if temperature above 107 °C, the maximum pressure
is 2.67 MPa)
- Operating Temperature Range: 0°C to +200 °C (If measuring cell is put in other
bath where temperature can be as low as - 54°C, this apparatus can measure vapour
pressure at temperature from - 54°C)
27
MKS
2
3
Paro
v5
v4
vacuum
pump
4
v6
v7
D&H
Argon
v3
v2
v1
1
Figure 3.1. Experimental set-up. 1: sample cell, 2: MKS sensor, 3: Paroscientific sensor, 4:
D&H sensor, vi (I = 1-7) is valve i.
3.3
3.3.1
Pressure measurement and calibration
Pressure calibration
Experimental data and relations
D&H sensor is used and valve 6 is opened. During the calibration, valves 1, 3 and 7
are closed. For calibration of pressure smaller than the atmospheric pressure, valve 3 is
opened and vacuum pump operates to create vacuum pressure in the system. Valve 3 is
closed to maintain vacuum pressure of one channel of the MKS sensor. When vacuum
pressure is 0.029 mbar we start to read data from indicators. After that, we open and close
valve 1 swiftly. Waiting for about 4 minute till indicated value is stable; we start to pick up
data. We continue to open and close valve 1 and pick up data again and again.
Experimental data for calibrations of pressure sensors are given in table 3.1.
Table 3.1. Experimental data for calibration of the sensors.
MKS
D&H
Paro
MKS
No
No
[mBar]
[Bar]
[Bar]
[mBar]
1
-40.39 0.0123 0.0396
62
132.73
2
-40.46 0.0123 0.0397
63
132.69
3
-40.46 0.0122 0.0398
64
132.68
4
-40.45 0.0126 0.0400
65
132.67
5
-40.46 0.0127 0.0400
66
132.69
6
-24.63 0.0277 0.0553
67
158.32
7
-24.63 0.0278 0.0553
68
158.30
28
D&H
[Bar]
0.1808
0.1808
0.1808
0.1808
0.1808
0.2058
0.2058
Paro
[Bar]
0.2089
0.2088
0.2089
0.2088
0.2089
0.2339
0.2339
No
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
MKS
[mBar]
-24.62
-24.62
-24.61
-10.44
-10.48
-10.47
-10.48
-10.48
-10.49
4.40
4.42
4.39
4.39
4.40
15.23
15.26
15.25
15.24
15.24
34.22
34.26
34.28
34.29
34.29
49.18
49.15
49.15
49.15
63.57
63.57
63.58
63.57
63.55
77.87
77.86
77.86
77.85
77.84
92.48
92.47
D&H
[Bar]
0.0278
0.0278
0.0278
0.0405
0.0406
0.0406
0.0406
0.0406
0.0406
0.0550
0.0550
0.0550
0.0550
0.0550
0.0658
0.0658
0.0658
0.0658
0.0658
0.0853
0.0852
0.0852
0.0852
0.0852
0.0993
0.0993
0.0993
0.0993
0.1135
0.1135
0.1135
0.1135
0.1135
0.1271
0.1271
0.1271
0.1271
0.127
0.1413
0.1412
Paro
[Bar]
0.0553
0.0553
0.0556
0.0691
0.0692
0.0692
0.0692
0.0693
0.0693
0.0837
0.0837
0.0838
0.0837
0.0837
0.0943
0.0943
0.0943
0.0943
0.0944
0.1129
0.1129
0.1128
0.1129
0.1128
0.1274
0.1273
0.1273
0.1274
0.1414
0.1415
0.1415
0.1414
0.1415
0.1554
0.1553
0.1553
0.1553
0.1554
0.1697
0.1697
No
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
29
MKS
[mBar]
158.26
158.26
158.24
183.35
183.33
183.31
183.28
183.25
243.87
243.77
243.67
243.66
243.68
243.69
243.63
348.50
348.42
348.35
348.27
348.25
468.3
468.15
468.04
467.92
467.84
467.67
552.87
552.88
552.77
552.66
552.60
668.97
668.76
668.63
668.36
668.34
803.22
803.04
802.92
802.73
D&H
[Bar]
0.2057
0.2057
0.2057
0.2303
0.2302
0.2302
0.2302
0.2302
0.2896
0.2895
0.2893
0.2893
0.2893
0.2893
0.2892
0.3914
0.3913
0.3912
0.3911
0.3911
0.5104
0.5102
0.5101
0.5100
0.5099
0.5097
0.5937
0.5936
0.5934
0.5932
0.5931
0.7085
0.7081
0.7079
0.7076
0.7076
0.8410
0.8408
0.8406
0.8404
Paro
[Bar]
0.2338
0.2338
0.2338
0.2584
0.2584
0.2583
0.2583
0.2582
0.3175
0.3175
0.3174
0.3174
0.3174
0.3175
0.3174
0.42
0.4199
0.4198
0.4197
0.4197
0.5376
0.5374
0.5373
0.5371
0.5371
0.5369
0.6207
0.6207
0.6207
0.6205
0.6204
0.7352
0.735
0.7348
0.7345
0.7345
0.8679
0.8678
0.8676
0.8673
No
48
49
50
51
52
53
54
55
56
57
58
59
60
61
MKS
[mBar]
92.47
92.48
92.45
105.99
105.99
105.99
105.98
105.99
105.91
119.55
119.54
119.51
119.51
119.51
D&H
[Bar]
0.1413
0.1413
0.1413
0.1546
0.1545
0.1545
0.1545
0.1545
0.1544
0.1678
0.1678
0.1678
0.1678
0.1677
Paro
[Bar]
0.1696
0.1696
0.1696
0.1827
0.1828
0.1829
0.1828
0.1829
0.1828
0.1961
0.1961
0.1961
0.1960
0.1960
No
109
110
111
112
113
114
115
116
117
118
119
120
121
122
MKS
[mBar]
802.59
880.95
880.84
880.65
880.52
880.38
879.35
954.49
954.31
954.16
954.09
953.51
954.41
953.79
D&H
[Bar]
0.8402
0.9181
0.9179
0.9177
0.9175
0.9174
0.9162
0.9912
0.9909
0.9907
0.9906
0.9900
0.9907
0.9902
Paro
[Bar]
0.8672
0.9449
0.9447
0.9446
0.9444
0.9442
0.9431
1.0177
1.0175
1.0173
1.0172
1.0165
1.0175
1.0167
Measurement of vapour pressures of different fluids is conducted with MKS or/and
Paro sensors. Thus, relation between reading values from MKS or/and Paro and D&H
should be made. From the data in table 3.1 we have relation between reading values of
D&H and MKS, equation (3.1), and between reading values of D&H and Paro, equation
(3.2).
pD&H, cal, MKS [Bar] = 1.06730E-08*pMKS,read*pMKS,read + 9.74513E-04*pMKS,read + 0.051251
(3.1)
pD&H, cal, paro [Bar] = 1.45801E-03*pparo,read*pparo,read + 1.00022E+00*pMKS,read – 0.0282029
(3.2)
In order to evaluate the calculated data from equation (3.1) and equation (3.2), we
plot the differences between calculated data and experimental data in figure 3.2 and relative
deviations between calculated data and experimental data in figure 3.3. Figure 3.2 shows
that the differences between indicated values of D&H and calculated values from MKS
sensor and Paro sensor are within ± 1.0 mBar for pressure up to 1 Bar. The relative
deviations between indicated values of D&H and calculated values from MKS sensor and
Paro sensor, figure 3.3, are within ± 0.2% for pressure higher than 0.15 Bar. For pressure
30
lower than 0.15 Bar the relative deviations can be reached 7.1% for Paro sensor and 6.8%
for MKS sensor.
0.10
0.08
100(PD&H-Pcal ) [Bar]
0.06
0.04
0.02
0.00
-0.02
-0.04
-0.06
-0.08
-0.10
0
0.2
0.4
0.6
0.8
1
PD&H [Bar]
Figure 3.2. Difference between indicated values of D&H and calculated values from (×)
MKS sensor and (Ο) Paro sensor.
0.6
100(PD&H-Pcal )/PD&H
0.4
0.2
0.0
-0.2
-0.4
-0.6
0
0.2
0.4
0.6
0.8
1
PD&H [Bar]
Figure 3.3. Relative deviations between indicated values of D&H and calculated values
from (×) MKS sensor and (Ο) Paro sensor.
31
Accuracy assessment for the calibration
Statistical analyses for N (N=122) experimental data are used to determine the
overall estimated accuracy of the calibration from equation (3.1) and (3.2). We define Xdata
as the reading data from D&H indicator and Xcal is calculated data from either equation
(3.1) or (3.2).
The difference between Xdata and Xcal is named as: ΔX = Xdata - Xcal
The deviation between Xdata and Xcal is:
%ΔX = 100(Xdata - Xcal)/ Xdata
The average absolute deviation (AAD) is:
AAD =
The bias or average deviation is:
The standard deviation (SDV) is:
The root-mean-square (RMS) deviation is:
Bias =
SDV =
1
N
1
N
N
∑ %ΔX
i =1
i
N
∑ (%ΔX )
i
i =1
1 N
∑ (%ΔX i − Bias )
N − 1 i =1
RMS =
1
N
N
∑ (%ΔX i )
2
2
i =1
The results for statistical analyses of the calibration between D&H sensor and other
sensors are given in table 3.2.
Table 3.2. Statistical analyses of the calibration
For reading data For reading data from
Quantity
from MKS sensor
Paro sensor
0.000860
0.000872
Max |ΔX | [Bar]
AAD
0.454
0.547
Bias
0.169
0.244
SDV
1.095
1.411
RMS
1.103
1.426
Uncertainty of the measurements:
To estimate uncertainty in the pressure measurement, one should consider five
sources of errors, namely the uncertainty of the piston-cylinder manometer (Δpref ≤
±26.66Pa = 0.2 mmHg), the uncertainty of the mercury barometer (Δpbaro ≤ ±30Pa), the
uncertainty of pressure transducers (Δptrans,paro ≤ ±10 Pa for the range up to 1 bar and 690Pa
32
for the range up to 6.9MPa; Δptrans,MKS ≤ ±3.33 Pa for the range up to 1.33 kPa and 3199Pa
for the range up to 2.67 MPa). The uncertainty of pressure due to the drift after a certain
time of calibration should be considered. We assume Δpdrift ≤ ±100Pa after 6 months after
calibration. The four errors are systematic errors so they can be added together. Δpsys =
Δpref + Δpbaro + Δptrans + Δpdrift. The uncertainty of calibration, Δpcal, is 0.86 Pa for MKS
sensor and 0.87 Pa for Paro sensor. The total uncertainty of the apparatus can be written as:
Δp2 = Δpsys2 + Δpcal2
If the apparatus is used to measure vapour pressures of fluids within 6 months of
calibration, and if maximum vapour pressure is 1 Bar and Paro sensor is used to read
experimental data, the total uncertainty of measurement is:
Δp =
3.3.2
(26.26 + 30 + 10 + 100)2 + (0.87 )2
= 166.7 Pa
Vapour pressure of water
The apparatus is used to measure vapour pressures of some industrial products. In
this part we present our test for measuring vapour pressures of water at about 35°C and
compare them with calculated values from IAPWS-95 [3.1].
In our measurement, we set temperature of thermal bath of Julabo at 35°C, the
connecting pipes are maintained at 45°C. Reading values from the thermometer bridge
F300 and calculated values from Paro sensor are presented in table 3.3. The results show
good agreement with reference data from IAPWS-95 and are within the uncertainty of
calibration.
Table 3.3. Experimental results and comparisons with calculated data from IAPWS-95
3.4
t [oC]
Pexp [Bar]
PIAPWS95 [Bar]
100(pexp-pEOS)/ pexp
35.02
35.04
35.02
0.05652
0.05688
0.05715
0.056352
0.056415
0.056352
0.3
0.8
1.4
Experimental apparatus for measurement of vapour pressures is constructed. This
apparatus can be used to measure vapour pressures up to 2.67 MPa for temperature higher
than 107°C and up to 6.9 MPa for temperature lower than 107°C. The temperature range of
33
this apparatus is from -54°C to +200 °C. This apparatus has total uncertainty of 0.17 kPa
for pressure up to 100 kPa.
Two pressure sensors of MKS and Paro have been calibrated by using D&H sensor
for the pressure up to about 1 Bar. Statistical analyses of the calibration show that average
absolute deviations of MKS sensor and Paro sensor compared to reference sensor are
0.454% and 0.547%, respectively. The standard deviations of MKS sensor and Paro sensor
compared to reference sensor are 1.095 and 1.411, respectively.
34
References
properties of ordinary water substance for general and scientific use. J. Phys. Chem. Ref.
Data, 31 (2002) 387 - 535.
[3.2] R. Span, W. Wagner, A new equation of state for carbon dioxide covering the fluid
region from the triple-point temperature to 1100 K at pressures up to 800 MPa. J. Phys.
Chem. Ref. Data, 25 (1996) 1509-1596.
[3.3] A. Mueller, J. Winkelmann, J. Fischer. Backone family of equations of state: 1.
[3.4] U. Weingerl, M. Wendland, J. Fischer, A. Mueller, J. Winkelmann, Backone family of
equations of state: 2. Nonpolar and polar fluid mixtures. AIChE, 47 (2001) 705–717.
equation, Fluid Phase Equilibria, 283 (2009) 22-30.
equations of state, MS in preparation
[3.7] J. Fischer , N. A. LAI, G. Koglbauer, M. Wendland: Arbeitsmedien für ORCProzesse, Chemie Ingenieur Technik, 79 (2007) 1342.
[3.8] N. A. LAI, M. Wendland, J. Fischer, Development of equations of state for siloxanes
as working fluids for ORC Processes, Proceeding of 24th European symposium on applied
thermodynamics, June 27 – July 1, 2009, 200-205, ISBN: 978-84-692-2664-3, Santiago de
Compostela, Spain
[3.10] B.Saleh, M.Wendland, Measurents of vapor pressures and saturates liquid densities
of pure fluids with a new apparatus. J. Chemical Engineering Data, 50 (2005) 429-437
along the entire coexistence curve, J. Chem. Thermodyn. 34 (2002) 2017–2039
35
[3.12] P. Nowak, R. Kleinrahm and W. Wagner, Measurement and correlation of the
(p,ρ,T) relation of ethylene: II. Saturated-liquid and saturated-vapour densities and vapour
pressures along the entire coexistence curve, J. Chem. Thermodyn. 28 (1996) 1441–1460
[3.13] L. A. Weber, vapour Pressure of Heptane from the Triple Point to the Critical Point,
J. Chem. Eng. Data, 45 (2000) 173-176
[3.14] L. A. Weber, vapour pressures and gas-phase PVT data for 1,1-dichloro-2,2,2trifluoroethane, J. Chem. Eng. Data, 35 (1990) 237-240
36
4
Upward extrapolation of saturated liquid densities*
Abstract
The design of new energy conversion processes requires equations of state for the
working fluids. For their construction saturated liquid densities are needed which are not
available for some potential working fluids at higher temperatures. Hence we investigate
how Racket type equations behave in extrapolations from saturated liquid densities in the
temperature range 0.5 ≤ T/Tc ≤ 0.75 up to the critical temperature Tc. Extrapolation
methods with different inputs from the critical point are used: a) no critical point data, b)
critical temperature, c) critical temperature and pressure, and d) critical temperature and
compression factor. It is found that upward extrapolations of the saturated liquid densities
without using critical point data can be done with some care and that the additional use of
the critical temperature improves the quality of the predictions substantially.
4.1
Introduction
The knowledge of saturated liquid densities is important for direct practical
applications like sizing a pump. Moreover, saturated liquid densities are also an important
auxiliary tool in fitting parameters of physically based equations of state [4.1], or in
determining parameters of molecular interaction models [4.2]. Hence, the correlation and
extrapolation of saturated liquid densities has been the subject of numerous studies over the
decades [4.3-4.20]. Whilst previous work mainly concentrated on correlations of saturated
liquid densities or on predictions based on the knowledge of critical point data, we are
interested here mainly on predictions of saturated liquid densities at higher temperatures
from data measured at lower temperatures. The motivation for this work is the intention to
apply physically based equation of state like BACKONE [4.1] for studies of ClausiusRankine [4.21] and heat pump cycles [4.22] also at higher temperatures. As saturated liquid
densities are less frequently available at higher temperatures, an extrapolation from low to
high temperatures is requested.
*
See also: N. A. Lai, M. Wendland, J. Fischer, Upward extrapolation of saturated liquid densities, Fluid phase
equilibria, 280 (2009) 30-34
37
4.2
Equations and optimization
Starting point for our investigations is the Rackett equation [4.13]
ln ρ’ = ln ρc - (ln Zc)(1- T/Tc)2/7,
(4.1)
where ρ´ is the saturated liquid density at temperature T, ρc the critical density, Tc the
critical temperature, and Zc is the critical compression factor. The latter is defined as Zc =
pc/(ρcRTc) with pc being the critical pressure and R the gas constant. More recently
Daubert [4.18] suggested a generalized Racket equation which can be written in the form
ln ρ’ = ln ρp - (ln Zp)(1-T/Tp)D,
(4.2)
where ρp, Tp, D, and Zp are four adjustable parameters. Contrary to Eq. (4.1), Eq. (4.2) does
not require any critical point data. Both these equations have advantages and disadvantages
depending on their intended use and the available data. The original Rackett equation does
not require any adjustable parameters but knowledge of the full set of critical point data ρc,
Tc and for Zc also pc. The Daubert equation, on the other hand does not require any critical
point data but requires the fit of four adjustable parameters. If the intended use of an
equation is correlation of data, in general more adjustable parameters may be helpful. On
the other hand, an increasing number of adjustable parameters may lead to inaccuracies for
extended extrapolations. It is obvious that the number of adjustable parameters in the
Daubert equation can be restricted by equating one or more of them to the corresponding
expressions in the Rackett equation, i.e. ρp = ρc, Tp = Tc, D = 2/7, and/or Zp = Zc, which
paves the way for our considerations.
Here, we consider the following five Rackett-type equations for upward
extrapolation of saturated liquid densities. The first equation is
ln ρ’ = ln ρp - A(1- T /Tp)2/7,
(4.3)
which does not require any experimental critical point data and contains the three fit
parameters ρp, A, and Tp. Here and in the following we call ρp pseudo-critical density and
Tp pseudo-critical temperature. The second equation is
ln ρ’ = ln ρp – A(1- T /Tc)2/7,
(4.4)
38
which requires the experimental critical temperature Tc and contains the two fit parameters
ρp and A. The third equation is that given by Spencer and Danner [4.14]
ln ρ’ = ln ρp – (lnZp)(1-T/Tc)2/7,
(4.5)
which requires the experimental critical temperature Tc and the experimental critical
pressure pc and contains only the pseudo-critical density ρp as fit parameter. The pseudocompression factor Zp is related to Tc, pc and ρp by Zp = pc/ρpRTc. In the original work
[4.14] the present Zp was called ZRA and considered as fit parameter which, however, is
equivalent to the present procedure. Whilst the above three equations, Eq. (4.3) to Eq. (4.5),
all use the exponent (2/7) suggested by Rackett, one may alternatively also try a
compression factor as exponent. Then, one arrives from Eq. (4.5) directly at
ln ρ’ = ln ρp – (lnZp)(1-T/Tc)Zp,
(4.6)
which again requires the experimental critical temperature Tc and the experimental critical
pressure pc and contains only the pseudo-critical density ρp as fit parameter. Finally,
starting from Eq. (4.4) and replacing (2/7) by Zc yields
ln ρ’ = ln ρp - A (1- T /Tc)Zc
(4.7)
which requires all experimental critical data Tc, pc, ρc, as Zc = pc/ρcRTc and contains the
two fit parameters ρp and A.
In order to find the parameters of Eqs. (4.3) to (4.7), we use the objective function
Σi (ln ρ’cal,i - ln ρ’exp,i)2 → Min.
(4.8)
where the ρ’cal,i are the calculated and the ρ’exp,i are the experimental densities in the
temperature range where the fit is performed. By rewriting the ln terms as ln [1 +(ρ’cal,i ρ’exp,i)/ ρ’exp,i] and performing a series expansion one finds that (ln ρ’cal,i - ln ρ’exp,i )2 ≈ [(
ρ’cal,i - ρ’exp,i)/ ρ’exp,i]2 which are the usual terms in the objective function. For Eq. (4.3), we
guess some value for Tp and perform therewith a linear regression to find A and ln ρp.
Then we search for that Tp which minimizes the objective function for Eq. (4.3). For Eqs.
(4.4) and (4.7) a linear regression is made to find A and ln ρp. For Eq. (4.5) and (4.6) only
ρp has to be found which can be done iteratively.
39
4.3
Substances and data sources
In order to study the extrapolation of saturated liquid densities from low to high
temperatures we consider 18 substances from different molecular classes for which
experimental data or reference equations of state (EOS) are available up to the critical
point. The critical temperatures Tc, the critical pressures pc, the critical densities ρc, the
critical compression factors Zc as well as the temperature ranges of experimental or EOS
data together with the references for the saturated liquid densities and the critical point data
are given in Table 4.1.
For the discussion of the results in the subsequent Section a rough molecular
classification of the fluids could be: a) anorganic substances (argon, nitrogen, carbon
dioxide, sulfur hexafluoride), b) alkanes (ethane, propane, n-butane, n-pentane, n-hexane),
c) alkenes (ethylene), d) siloxanes (octamethyltrisiloxane), e) aromates (benzene, toluene),
f) refrigerants (R134a, R143a, R152a), and g) alcohols (methanol, trifluoroethanol).
4.4
Results and discussions
In order to get a consistent picture, we fitted Eqs. (4.3) to (4.7) to data from
experiments or reference EOS. In principle this was done in the reduced temperature range
0.50 ≤ Tr ≤ 0.75 with Tr = T/Tc. In some cases, however, the lowest fit temperature had to
be higher than 0.50 Tr. One example is carbon dioxide where the reduced triple point
temperature is known to be rather high. Another example is benzene, for which the
reference EOS is based on saturated liquid density data only above 290 K [4.36]. The
fitting range and the number of data points used for fitting are given in the last two columns
of Table 4.1. In case that the fit was made directly to experimental data, the strategy was to
use all data points from the referenced source in the given temperature range. In case that
the fit was made to data from a reference EOS, temperature intervals of 10 K were taken
within the given temperature range. We found that using smaller temperature intervals (5
K or 2.5 K) had negligible influence on the results.
40
Table 4.1: Critical temperatures Tc, pressures pc, densities ρc, and compression factors Zc,
the temperature ranges of experimental or reference-EOS data, and references for the
densities and the critical point (CP) data. In the last two columns the fitting range and the
number of data points used for fitting are given. (Zc = pc/(ρcRTc) with R = 8.314472 J/mol
K)
Fitting
ρc
Exp. or EOS
Ref.
range
Tc
No. of
Ref. CP
Substance
pc [MPa]
Zc
T-range [K] densities
Tmin- fit points
[K]
[mol/l]
Tmax
argon
150.687 4.863 13.407 0.2895 84.0-150.7
[4.23] [4.23]
84-115
9
nitrogen 126.192 3.3958 11.184 0.2894 64.0-125.0
[4.24] [4.24]
64-93
16
1
CO2
304.134 7.3783 10.625 0.2746 217.0 – 304.0 [4.25] [4.26] 217-230
7
2
SF6
318.723 3.755 5.082 0.2788 224.0-314.6
[4.27] [4.27] 224-240
6
ethane 305.322 4.8722 6.857 0.2799 91.0-303.0
[4.28] [4.28] 150-230
10
[4.29],
propane 369.825 4.24709 4.955 0.2788 85.5 - 369.8 [4.29]*
190-280
10
[4.30]
n-butane 425.125 3.796 3.9200 0.2740 134.9 - 425.1 [4.31]* [4.32] 210-320
12
*
n-pentane 469.7 3.370 3.2156 0.2684 143.5 - 469.7 [4.33]
[4.33] 240-350
12
263.2-428.3
[4.11]
[4.11] 263-373
10
n-hexane 507.9 3.035 2.7282 0.2634
428.3- 507.8 [4.33]*
ethylene 282.35 5.0418 7.637 0.2812 104.0-280.0
[4.34] [4.34] 140-210
19
3
MDM
564.13 1.415 1.134 0.2660 273.2 - 563.4 [4.35] [4.35] 287-426
10
**
benzene 562.05 4.894 3.9561 0.2647
290-562
[4.36]* [4.36]
290-420
14
*
**
toluene 591.75 4.1263 3.169 0.2646
290- 591
[4.36] [4.36]
295-445
16
R134a4 374.21 4.059 5.0176 0.2600 169.8 - 374.2 [4.37]* [4.37] 190-280
10
5
*
R143a
345.857 3.7610 5.12845 0.2550 161.3 - 345.8 [4.38]
[4.38] 170-260
10
6
*
R152a
386.411 4.51675 5.57145 0.2523 154.6 - 386.4 [4.39]
[4.40] 190-290
11
methanol 512.6 8.1035 8.60 0.2211 175.6 - 512.6 [4.41]* [4.41] 260-380
13
7
TFE
499.29 4.87
4.838 0.2425 263.1 - 473.1 [4.42] [4.42] 263-376
10
* The saturated liquid densities are taken from NIST thermophysical properties [4.43]
which are based on the references given.
** The critical point data are taken from NIST thermophysical properties [4.43] which are
based on the reference given.
1
Carbon Dioxide, 2Sulfur Hexafluoride, 3Octamethyltrisiloxane,
4
1,1,1,2-tetrafluoroethane, 51,1,1-trifluoro-ethane, 61,1-difluoroethane,
7
2,2,2-trifluoroethanol
Upward extrapolation results from Eqs (4.3) to (4.7) for the 18 substances are
shown in Tables 4.2 to 4.4. Table 4.2 and 4.3 show the deviations Δρ = (ρ’cal - ρ’exp)/ρ’exp
of the extrapolated saturated densities from the underlying experimental or reference EOS
data, both denoted as ρ’exp, for the reduced temperature Tr = 0.90 and Tr = 0.95. Table 4.4
shows the pseudo-critical temperatures Tp from Eq. (4.3) and the deviations Δρp = (ρp 41
ρc)/ρc of the extrapolated densities at the pseudo-critical temperature Tp in case of Eq. (4.3)
and at the critical temperature Tc in case of Eqs. (4.4) to (4.7). Table 4.5 shows average
absolute deviations (AAD) over all substances at the temperatures 0.90 Tc, 0.95 Tc and Tp.
The latter agrees with Tc for Eqs. (4.4) to (4.7), for Eq. (4.3) the Tp values are given in
Table 4.4. Moreover, deviations between the densities calculated from Eqs. (4.3) to (4.5)
and the experimental or reference EOS data are shown as function of the temperature in
Figure 4.1 for carbon dioxide, in Figure 4.2 for hexane, and in Figure 4.3 for R134a.
Table 4.2: Relative deviations Δρ = (ρ’cal - ρ’exp)/ρ’exp of extrapolated saturated liquid
densities at T = 0.9Tc from experimental or reference EOS data. The extrapolation was
made by Eq. (4.3) to Eq. (4.7) from the experimental or reference EOS data in the range of
reduced temperatures Tr = T/Tc given in column 2. The last line shows the average absolute
deviations (AAD).
Substance
Tr range
for fit
ρ’exp
[mol/l]
Eq. (4.3)
Δρ [%]
Eq. (4.4)
Δρ [%]
Eq. (4.5)
Δρ [%]
Eq. (4.6)
Δρ [%]
Eq. (4.7)
Δρ [%]
argon
nitrogen
CO2
SF6
ethane
propane
n-butane
n-pentane
n-hexane
ethylene
MDM
benzene
toluene
R134a
R143a
R152a
methanol
TFE
0.56-0.76
0.51-0.74
0.71-0.76
0.70-0.75
0.49-0.75
0.51-0.76
0.49-0.75
0.51-0.75
0.52-0.73
0.50-0.74
0.51-0.76
0.52-0.75
0.50-0.75
0.51-0.75
0.49-0.75
0.49-0.75
0.51-0.74
0.53-0.75
25.128
21.126
20.995
9.923
13.180
9.723
7.724
6.388
5.442
14.649
2.328
7.648
6.320
10.127
10.351
11.311
18.015
10.074
0.35
0.60
0.38
0.26
0.89
0.87
1.16
1.53
0.89
1.14
2.66
1.89
1.83
1.72
1.73
1.59
4.14
-0.61
0.15
0.07
0.10
0.37
0.06
0.04
-0.03
0.16
-0.42
-0.12
-1.13
-0.66
0.02
-0.11
-0.23
0.00
1.40
2.40
0.46
0.59
0.50
-0.16
0.03
-0.14
-0.36
-0.68
-0.93
0.39
0.10
0.06
-0.56
-0.47
-0.67
-0.68
-5.79
-2.97
0.16
0.36
1.11
0.19
0.32
0.40
0.42
0.40
0.43
0.64
2.07
1.06
0.77
1.22
1.28
1.49
-2.28
-0.49
0.20
0.13
-0.02
0.30
-0.04
-0.07
-0.23
-0.13
-0.82
-0.20
-1.46
-1.03
-0.35
-0.56
-0.78
-0.61
0.38
1.76
42
Table 4.3: Relative deviations Δρ = (ρ’cal - ρ’exp)/ρ’exp of extrapolated saturated liquid
densities at T = 0.95Tc from experimental or reference EOS data. The extrapolation was
made by Eq. (4.3) to Eq. (4.7) from the experimental or reference EOS data in the range of
reduced temperatures Tr = T/Tc given in column 2. The last line shows the average absolute
deviations (AAD).
Substance
argon
nitrogen
CO2
SF6
ethane
propane
n-butane
n-pentane
n-hexane
ethylene
MDM
benzene
toluene
R134a
R143a
R152a
methanol
TFE
Tr range
for fit
0.56-0.76
0.51-0.74
0.71-0.76
0.70-0.75
0.49-0.75
0.51-0.76
0.49-0.75
0.51-0.75
0.52-0.73
0.50-0.74
0.51-0.76
0.52-0.75
0.50-0.75
0.51-0.75
0.49-0.75
0.49-0.75
0.51-0.74
0.53-0.75
ρ’exp
[mol/l]
22.308
18.738
18.506
8.795
11.677
8.608
6.838
5.642
4.820
12.973
2.054
6.758
5.567
8.916
9.113
9.947
15.864
8.838
Eq. (4.3)
Δρx100
1.04
1.76
1.31
0.45
2.58
2.59
3.33
4.23
2.96
3.31
7.49
5.47
5.21
4.90
4.99
4.46
10.27
-9.73
43
Eq. (4.4)
Δρx100
0.44
0.32
0.37
0.80
0.37
0.34
0.23
0.69
-0.43
0.04
-1.56
-0.78
0.59
0.24
0.05
0.37
3.66
4.85
Eq. (4.5)
Δρx100
0.89
1.05
0.99
-0.01
0.32
0.08
-0.24
-0.49
-1.13
0.75
0.09
0.23
-0.21
-0.27
-0.57
-0.57
-6.45
-2.87
Eq. (4.6)
Δρx100
0.51
0.77
1.82
0.45
0.68
0.75
0.72
0.84
0.53
1.06
2.46
1.47
1.42
1.83
1.84
2.10
-2.11
0.25
Eq. (4.7)
Δρx100
0.57
0.46
0.08
0.62
0.16
0.09
-0.20
0.05
-1.27
-0.13
-2.29
-1.57
-0.20
-0.73
-1.14
-0.93
1.42
3.40
Table 4.4: Relative deviations Δρp = (ρp - ρc)/ρc of the pseudo-critical densities ρp from
experimental or reference EOS critical densities ρc. The extrapolation was made by Eq.
(4.3) to Eq. (4.7) on the basis of experimental or reference EOS data in the range of reduced
temperatures Tr = T/Tc given in Table 4.3. For Eq. (4.3) ρp is taken at the pseudo-critical
temperature Tp, whilst for the other equations ρp is taken at the experimental critical
temperature Tc used in these equations. The last line shows the average absolute deviations
(AAD) of the pseudo-critical densities.
Subst.
argon
nitrogen
CO2
SF6
ethane
propane
n-butane
n-pentane
n-hexane
ethylene
MDM
benzene
toluene
R134a
R143a
R152a
methanol
TFE
Tc
[K]
150.687
126.192
304.134
318.723
305.322
369.825
425.125
469.7
507.9
282.35
564.13
562.05
591.75
374.21
345.857
386.411
512.6
499.29
ρc
[mol/l]
13.407
11.184
10.625
5.082
6.857
4.955
3.9200
3.2156
2.7282
7.637
1.134
3.9561
3.169
5.0176
5.12845
5.57145
8.60
4.838
Eq. (4.3)
Tp [K]
151.354
127.526
306.479
317.869
310.589
376.426
435.985
483.876
521.976
290.010
632.519
597.907
616.446
389.675
361.180
399.554
556.062
476.759
Eq. (4.3)
Δρpx100
-2.84
-4.35
-3.11
3.39
-4.76
-3.52
-5.39
-5.37
-6.36
-8.67
-31.44
-19.99
-9.38
-9.90
-11.15
-7.83
-2.20
30.75
Eq. (4.4)
Δρpx100
-1.63
-1.66
-0.51
2.51
-0.28
1.32
1.36
2.63
0.97
-1.72
-1.37
-4.14
1.61
1.19
0.61
1.39
18.28
19.39
Eq. (4.5)
Δρpx100
-0.61
-0.02
1.10
0.40
-0.38
0.73
0.27
-0.11
-0.63
-0.12
2.28
-1.87
-0.25
0.00
-0.79
-0.78
-6.49
-5.73
Eq. (4.6)
Δρpx100
-0.35
0.15
0.28
-0.06
-0.60
0.29
-0.32
-0.94
-1.62
-0.31
1.11
-2.63
-1.25
-1.28
-2.20
-2.34
-8.91
-8.19
Table 4.5: Average absolute deviations (AAD) of the extrapolated densities of all 18
substances at the temperatures 0.90 Tc, 0.95 Tc and Tp. The latter agrees with Tc for Eq.
(4.4) to (4.7), for Eq. (4.3) the Tp values are given in Table 4.4.
T
0.90 Tc
0.95 Tc
Tp
Eq. (4.3)
Δρx100/N
1.35
4.23
9.47
Eq. (4.4)
Δρx100/N
0.42
0.90
3.48
Eq. (4.5)
Δρx100/N
0.86
0.96
1.25
44
Eq. (4.6)
Δρx100/N
0.84
1.20
1.82
Eq. (4.7)
Δρx100/N
0.50
0.85
5.28
Eq. (4.7)
Δρpx100
-0.45
-0.47
-4.09
0.35
-2.22
-1.04
-2.72
-3.50
-7.03
-3.22
-8.59
-11.43
-5.95
-8.23
-10.79
-11.19
-7.60
-6.08
100x(ρ'cal-ρ'exp )/ρ'exp
5
4
3
2
1
0
217
-1
210
230
230
304.134
250
270
290
310
T [K]
Figure 4.1. Deviation plot of calculated from experimental saturated liquid densities [4.25]
for CO2. The fit was made to experimental data between the first (217 K) and the second
vertical line (230 K). The results between the second and the third vertical line (critical
temperature at 304 K) are obtained from extrapolation. Results are shown from: - o - Eq.
(4.3); ⎯■⎯ Eq. (4.4); ⎯•⎯ Eq. (4.5).
4
3
2
1
0
-1
-2
263
-3
240
507.9
372
290
340
390
440
490
540
T [K]
Figure 4.2. Deviation plot of calculated saturated liquid densities from experimental [4.11]
and reference EOS [4.33] values for n-hexane. The fit was made to experimental data
between the first (263 K) and the second vertical line (372 K). The results between the
second and the third vertical line (critical temperature at 507.9 K) are obtained from
extrapolation. Up to 428 K comparison is made with experimental values [4.11], for higher
temperatures comparison is made with the reference EOS [4.33]. Results are shown from: o - Eq. (4.3); ⎯■⎯ Eq. (4.4); ⎯•⎯ Eq. (4.5).
45
3
2
1
0
-1
-2
150
280
190
200
250
374.21
300
350
400
T [K]
Figure 4.3. Deviation plot of calculated saturated liquid densities from reference EOS
[4.37] values for R134a. The fit was made to experimental data between the first (190 K)
and the second vertical line (280 K). The results between the second and the third vertical
line (critical temperature at 374.2 K) are obtained from extrapolation. Results are shown
from: - o - Eq. (4.3); ⎯■⎯ Eq. (4.4); ⎯•⎯ Eq. (4.5).
Let us start with a discussion of the results from Eq. (4.3), which needs only
saturated liquid densities up to 0.75 Tr and no critical point data for the upward
extrapolation. From Tables 4.2 to 4.4 we make the following observations for the different
molecular classes.
a) For the anorganic substances argon, nitrogen, CO2, and SF6 the density deviations
are very small and far below the AADs. Also the values of the predicted pseudo-critical
temperatures are quite good. We should note that in these cases the extrapolation was made
from very accurate experimental data [4.23-4.27].
b) For the alkanes ethane, propane, n-butane, n-pentane, and n-hexane the density
deviations are small and in general smaller than the AADs. The pseudo-critical
temperatures Tp are larger than the critical temperatures Tc by 1.7 to 3.0%. Whilst we
would have expected a systematic tendency in the deviations in the series from ethane to
hexane, this tendency (mostly increasing deviations with increasing chain length) holds
only from ethane to n-pentane and then turns for hexane. Regarding the underlying data
sources, ethane was fitted to very accurate experimental data [4.28], propane to n-pentane
to reference EOS data [4.29-4.33], whilst hexane was fitted to the most accurate
46
experimental data available [4.11]. For hexane, we also made a fit to the densities of a
reference EOS [4.33] at the same temperatures. Whilst the difference between the
experimental and the reference EOS densities is rather small, the fit to the reference EOS
densities gave larger deviations in the extrapolation.
c) For the alkene ethylene the extrapolation was made from very accurate
experimental data [4.34]. The density deviations are smaller than the AADs but in general
larger than those of the alkanes. The pseudo-critical temperature Tp is larger than the
critical temperature Tc by 2.7%. This is somewhat surprising as we would have expected
smaller deviations like for ethane or carbon dioxide.
d) From the siloxanes we investigated octamathlytrisiloxane (MDM), for which
densities have been measured with two different experimental devices in two separate
temperature ranges [4.35]. As the density at 361.82 K seemed to be inconsistent with the
other data it was eliminated from the fit. Nevertheless the deviations of the extrapolated
from the experimental densities are rather large.
e) For the aromates benzene and toluene the extrapolations are based on saturated
liquid densities from the Bender equation [4.36]. The deviations of the extrapolated from
the EOS densities are larger than the AADs and are larger for benzene than for toluene.
f) For the refrigerants R134a, R143a, and R152a the extrapolations are again based
on saturated liquid densities resulting from reference EOSs [4.37 – 4.39]. The deviations
correspond to the AADs and are very similar for all three fluids.
g) Finally, the extrapolations for the alcohols show in general for methanol large
deviations from the reference EOS [4.41] and for trifluoroethanol large deviations from the
experimental data [4.42]. Moreover, the deviations go into opposite directions.
Next, we consider the results from Eq. (4.4) which needs besides saturated liquid
densities up to 0.75 Tr also the critical temperature Tc for the upward extrapolation. We see
from Tables 4.2 to 4.4 that the prediction of the densities is substantially improved in
comparison with Eq. (4.3) by using the experimental critical temperature as additional
information. The AADs of the relative density deviations decrease at Tr = 0.90 to 0.42 %
(1.35%),
at Tr
= 0.95 to 0.90% (4.23%), and for the pseudo-critical densities the
deviations decrease to 3.48 % (9.47%), where the numbers in brackets denote the
47
corresponding AADs from Eq. (4.3). Regarding the different molecular classes we see that
the predictions are nearly equally good for the anorganic substances, the alkanes, ethylene,
the refrigerants, and toluene. The predictions are less good for benzene, MDM and the
alcohols.
It seems appropriate to consider now Eq. (4.7). We remind that the difference
between Eq. (4.7) and Eq. (4.4) is in the exponent of (1- T/Tc), which is 2/7 in Eq. (4.4) and
Zc in Eq. (4.7). This means, that Eq. (4.4) requires as critical data input only Tc, whilst Eq.
(7) requires Tc, ρc, and pc. Looking now on the AADs given in Table 4.5 we see that there
is practically no difference in the predictions of the densities at Tr = 0.90 and 0.95, whilst
the predicted critical densities are worse from Eq. (4.7) in comparison with Eq. (4.4).
Finally, we consider Eqs. (4.5) and (4.6) which both use in addition to the saturated
liquid densities at lower temperatures the experimental critical temperature Tc and the
critical pressure pc. We observe from Table 4.5 that the overall deviations from these
extrapolation methods are larger for Tr = 0.90 than the results from Eq. (4.4). They become,
however, nearly equally good for Tr = 0.95. Moreover Eqs. (4.5) and (4.6) allow good
predictions of the critical density except for the alcohols as can be seen from Table 4.4.
Deviation plots of calculated from experimental saturated liquid densities are shown
in Fig 4.1 for CO2, in Fig 4.2 for hexane, and in Figure 4.3 for R134a. The plots show the
results from Eq. (4.3), Eq. (4.4) and Eq. (4.5). The fit to experimental data was made
between the first and the second vertical line which indicates Tr ≈ 0.75. The results from
extrapolation are shown between the second and the third vertical line which indicates the
critical temperature.
4.5
In Eq. (4.3) only saturated liquid densities in a temperature range up to T/Tc = 0.75
are used for upward extrapolation. It yields very good extrapolations for the anorganic
substances, good extrapolations for the alkanes, reasonable results for ethylene, systematic
tendencies for the refrigerants and to some extent also for the aromates. Unfortunately, the
results for MDM and the alcohols are less encouraging. In Eq. (4.4) the critical temperature
Tc is used as additional experimental information. This decreases the deviations of the
extrapolations substantially to about 30% of those from Eq. (4.3). Replacement of the
48
exponent 2/7 of (1- T/Tc) in Eq. (4.4) by Zc in Eq. (4.7) requires additional knowledge of
the critical pressure pc and the critical density ρc. The use of Zc does not improve the
extrapolation results but makes them alltoghether even slightly worse in comparison with
Eq. (4.4). Finally, Eqs. (4.5) and (4.6) which use the experimental critical temperature Tc
and the critical pressure pc as additional information have merits in the predictions of the
density close to the critical temperature.
The conclusions of the present investigation are that upward extrapolations of the
saturated liquid densities can be done by Eq. (4.3) with some care and that the additional
use of the critical temperature according to Eq. (4.4) improves the quality of the predictions
substantially. Good estimates of the critical density can be obtained by Eqs. (4.5) and (4.6).
49
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models based on the corresponding states principle, Phys. Chem. Liq. 46 (2008) 263-277.
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density, temperature) relation of argon II. Saturated-liquid and saturated-vapour densities
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51
[4.25] W. Duschek, R. Kleinrahm, W. Wagner, Measurement and correlation of the
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(2001) 735–754.
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property data, Springer, Berlin 2000.
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(p,ρ,T) relation of ethylene: II. Saturated-liquid and saturated-vapour densities and vapour
pressures along the entire coexistence curve, J. Chem. Thermodyn. 28 (1996) 1441–1460.
[4.35] D.L. Lindley, H.C. Hershey, The orthobaric region of octamethyltrisiloxane, Fluid
Phase Equilibria 55 (1990) 109-124.
52
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Bender fuer 14 mehratomige reine Stoffe, Chem. Tech. (Leipzig) 44 (1992) 216-224.
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thermodynamic properties of 1,1,1,2-tetrafluoroethane (HFC-134a) covering temperatures
from 170 K to 455 K at pressures up to 70 MPa, J. Phys. Chem. Ref. Data 23 (1994) 657729.
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thermodynamic properties of 1,1,1-trifluoroethane (HFC-143a) for temperatures from 161
to 450 K and pressures to 50 MPa, J. Phys. Chem. Ref. Data 29 (2000) 521-540.
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for the thermodynamic properties of R152a (1,1-difluoroethane), J. Phys. Chem. Ref. Data
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refrigerant 152a, JSME International Journal 30 (1987) 1106-1112.
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[4.43]
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Chemistry
WebBook,
Thermophysical
http://webbook.nist.gov/chemistry/fluid/
53
properties
of
fluids,
5
Helmholtz energy of hard convex bodies and hard chain systems
Abstract
In this chapter, we review theory studies of available equations of state and relations
between Helmholtz energy and thermodynamic quantities from different approaches for
systems of hard molecules like hard spheres, hard convex bodies, hard fused spheres and
hard chain molecules. The chapter contains also some minor own contributions.
5.1
Introduction
Van der Waals type equations of state are well-known for allowing a highly
accurate description of real fluids with few substance-specific parameters. The Helmholtz
energy A in the form of a generalized van der Waals equation can be written for non-polar
fluids as A = AH + AA, where AH accounts for contribution of the short-range repulsion,
and AA for long-range attractive dispersion. In this form, Helmholtz energy of attractive
dispersion contribution AA can be found by fitting to experimental data of real fluids
provided that Helmholtz energy of hard body contribution AH is known. The Helmholtz
energy of hard body contribution can be derived from equations of state for hard bodies like
hard spheres, hard convex bodies, hard fused spheres and hard chain molecules.
The hard bodies are the cores for simulations and constructions of equations of
state. For instance, computer simulation is carried out for square well potential [5.1 - 5.3].
Constructions of BACK (Boublik-Alder-Chen-Kreglewski) equation of state [5.4, 5.5] and
BACKONE [5.6, 5.7] equation of state were based on equation for hard convex bodies. PCSAFT [5.8] equation of state was based on equation for hard chain molecules. Because
BACKONE and PC-SAFT equations of state are used in this thesis and because of the
importance of the hard bodies, it is necessary to review theory studies of available
equations for hard bodies and relations between Helmholtz energy and thermodynamic
quantities from different approaches. Following sections will review and compare, if
possible, the equations of state and Helmholtz energy for hard spheres, hard chain systems,
hard convex bodies and hard dumbbells.
54
5.2
5.2.1
Hard spheres
Background
Volume Vm of a hard sphere molecule with diameter σ is Vm = 4π/3(σ/2)3. Let
consider a system with volume V and N molecules. The packing fraction of the system η is
η= N.Vm/V. From number density ρ = N/V, one has relation η = ρπσ3/6. Geometrically, the
most dense packing is ρmaxd3 = √2 and hence ηmax = 0.74048.
Coordinates of N hard spheres are denoted as r1, r2…rN. The distance between
spheres i and j is calculated by ri,j = |ri – rj|. Potential energy between the two spheres i and j
is uij = u(rij). If distance between centres of two spheres is denoted as r, equation for hard
sphere potential is written as:
⎧∞
⎪
u (r ) = ⎨
⎪⎩ 0
r <σ
r ≥σ
(5.1)
The pressure is a macroscopic quantity and hence by definition an average value on
the microscopic scale (pressure is the average of the transferred molecular momentum) we
have:
pV/NkT = 1 + (1/3NkT) <W>
(5. 2)
where <W> is average value of virial W, W= - Σi<jrijdu(rij)/drij. The average value of virial
is calculated with the help of the pair correlation function g(r1, r2) as:
<W> = - (Nρ/2) ∫ [r12 ∂u(r1, r2)/ ∂r12] g(r1, r2)dr12
(5. 3)
The pair correlation function for a real fluid of spherical molecules can be
experimented by either Roentgen or neutron scattering method. The results from neutron
scattering experiment of Yarnell et al. [5.9] are in good agreement with the predictions
from Monte Carlo simulation [5.10] for Argon at 85 K, figure 5.1.
55
Figure 5.1. The pair correlation function of liquid argon at 85K from neutron scattering
experiment (line) of Yarnell et al. [5.9] and Monte Carlo calculation (dot) of Barker et al.
[5.10].
In this section, we focus on spherical molecules for which Eq. (5.3) writes as
<W>= - (Nρ/2) ∫ [r du(r)/ dr] g(r) dr
(5.4)
With the help of the background correlation function y(r) which is defined as
g(r) = e-ßu y(r).
(5.5)
one obtains
[du(r)/ dr] g(r) = [du(r)/ dr] e-ßu y(r).
(5.6)
Moreover the relation holds
[du(r)/ dr] e-ßu = - (1/ß) de-ßu/ dr
(5.7)
which yields by insertion into Eq. (5.4)
<W>= (1/ ß)(Nρ/2) ∫ r [de-ßu/ dr]y(r) dr
(5.8)
For the particular case of the hard sphere potential Eq. (5.1) the exponential of the
potential is just the Heaviside (or unit step) function
e-ßu = θ (r-d)
(5.9)
which by differentiation yields the Dirac delta function
56
dθ (r-d)/dr = δ(r-d)
(5.10)
Inserting Eqs. (5.9) and (5.10) into Eq. (5.8) yields
<W>= (1/ ß)(Nρ/2) ∫ r δ(r-d) y(r) dr
(5.11)
By using the spherical symmetry one has for the volume element dr = 4πr2dr, and
hence
<W>= (2π/ ß)(Nρ) ∫ r3δ(r-d) y(r) dr.
(5.12)
According to the properties of the Dirac delta function the integral yields
∫ r3δ(r-d) y(r) dr = d3y(d),
(5.13)
and as for hard spheres according to Eqs. (5.5) and (5.9) y(d) = g(d), one obtains
from Eqs. (5.12) and (5.13)
<W> = (2π/ ß)(Nρ) d3 g(d).
(5.14)
Finally, by insertion of <W> from Eq. (5.14) into Eq. (5.2) and using the packing
fraction η = ρπd3/6 one obtains
pV/NkT = p/ρkT =1 + 4ηg(d)
(5.15)
By inversion of Eq. (5.15) one can also obtain the pair correlation function at
contact g(d) from the compression factor p/ρkT as
g(d) = (1/4η)[( p/ρkT) – 1]
(5.16)
The pair correlation at contact for a hard sphere system can only be determined by
either theory (approximately) or by simulations (with simulation uncertainty). Figure 5.2
presents relation between pair correlation function and reduced distance of hard sphere
from Percus-Yevick integral equation and simulation data.
57
Figure 5.2. Relation between pair correlation function and reduced distance of hard sphere:
− Percus-Yevick integral equation, • simulation data.
Hoover and Ree [5.11] carried out Monte-Carlo (MC) simulation of system of hard
spheres for solid and liquid to study melting transition and to discover the densities of the
coexisting phases. According to Hoover and Ree [5.11], the relative close-packing of
coexisting phases for fluid and solid are 0.667 and 0.736, figure 5.3. From the most dense
packing ηmax = 0.74048, one has the packing fraction at freezing η = 0.494 and packing
fraction at melting η = 0.545. Hence, for hard spheres the liquid is restricted to 0< η <
0.494.
Figure 5.3. Experimental results from Monte-Carlo simulation for system of hard spheres.
58
5.2.2
Equation of state for hard spheres
There are two approaches for calculation of equations of state when the radial
distribution function is known. The first approach derived from virial theorem refers to
pressure equation. The second approach based on Ornstein-Zernike relation [5.12] refers to
compressibility equation. The two approaches must give the same equation of state if radial
distribution function is exact. In 1957, Percus and Yevick (PY) [5.13] presented
approximate integral equation for determination of the radial distribution function. They
pointed out that their equation lead to good results for the fourth virial coefficient of hard
sphere gas.
An analytic solution for the Percus and Yevick equation for hard spheres [5.13] was
found by Thiele [5.14] in 1963. With this solution, the pressure equation of state (PY-p) is:
p/ρkT = [1+ 2η + 3η2]/(1-η)2,
(5.17)
and the compressibility equation of state (PY-c) is:
p/ρkT = [1+ η + η2]/(1-η)3.
(5.18)
In 1969, Carnahan and Starling [5.15] (CS) carried out analysis of the reduced virial
series and proposed equation (5.19):
p/ρkT = [1+ η + η2 - η3]/(1-η)3.
(5.19)
The equation of Carnahan and Starling has been proved to be accurate and has been
used by many researchers. Another potential equation of state for hard spheres is proposed
by Kolafa, Boublik and Nezbeda [5.16] (KBN):
p/ρkT = [1+ η + η2 – (2/3)η3 – (2/3)η4]/(1-η)3
(5.20)
In order to examine the differences between two proposed models of Carnahan and
Starling (CS) and Kolafa et al. (KBN), we vary packing fraction from 0.1 to 0.5 with a step
of 0.1 and make a comparison, table 5.1. We observe that the differences between CS and
KBN are very small and nearly within MC uncertainties, table 5.1.
59
Table 5.1. Comparison between equation of state for hard spheres of CS and KBN
η
0.1
0.2
0.3
0.4
0.5
5.2.3
CS
1.5213
2.4063
3.9738
6.9259
13.0000
KBN
1.5217
2.4094
3.9843
6.9457
13.0000
Δ = KBN - CS
0.0004
0.0031
0.0105
0.0198
0.0000
(Δ/CS) in %
0.03
0.13
0.26
0.29
0.00
Helmholtz energy for hard spheres
Residual Helmholtz energy can be calculated from compression factor Z as
following:
η
Ar
dη
= ∫ (Z − 1)
η
RT 0
(5.21)
From equation of state for hard spheres of Carnahan and Starling (5.19), and the
relation (5.21), we obtain an equation for Helmholtz energy for system of hard spheres:
Ar , HS −CS
RT
η
= ∫ (Z HS −CS − 1)
0
dη
η
=
η (4 − 3η )
(1 − η )2
(5.22)
Similarly, Helmholtz energy of hard spheres from Kolafa, Boublik and Nezbeda
approach (KBN) can be obtained as:
(
)
Ar , HS − KBN η 34 − 33η + 4η 2 − 10(1 − η )2 ln (1 − η )
=
2
RT
6(1 − η )
5.3
5.3.1
(5.23)
Hard chain systems
Results from Wertheim SAFT theory
A chain-like molecule consists of atoms with different diameters. For simplicity, in
theory study, one usually considers a chain of m hard spheres with the same diameter d,
figure 5.4. The sphere 1 and sphere m are in tangent contact with only one neighbour
sphere 2 and m-1, respectively. The other spheres are in tangent contact with only 2
60
neighbour spheres. The difference between molecular classes can be represented by
differences of the diameter d and number of hard spheres m.
1
2
3
d
m
Figure 5.4. Model of hard chain system
The relation between number density of a chain ρ and number density of hard
spheres ρ0 is ρ0 = mρ. The packing fraction is:
π
π
η = ρ0 . d 3 = mρ . d 3
6
6
The compression factor of the spherical associating molecular system can be
obtained by using the SAFT approach of Wertheim [5.17 - 5.20] and its applications by
Chapman et al. [5.21, 5.22]. Expression in terms of compression factor of hard spheres and
chain contribution is written as:
p/ρ0kT = ZHS + Zchain
or
p/ρkT = mZHS + mZchain
(5.24)
in which
Zchain = - [(m-1)/m]{1 + ρ[∂lng(d)/∂ρ]}
(5.25)
or
mZchain = - [(m-1)]{1 + ρ[∂lng(d)/∂ρ]}
= - [(m-1)]{1 + η[∂lng(d)/∂η]}
Helmholtz energy of a system of hard chains in the WERTHEIM approach can be
calculated as:
Ar , HCM
RT
η
= ∫ ( Z HCM − 1)
0
dη
η
(5.26)
We have
ZHCM – 1 = mZHS + mZchain - 1
= m (ZHS -1) + m Zchain + (m-1)
= m (ZHS -1) – (m -1)η[∂lng(d)/∂η]}
61
Now we have ∫(ZHCM -1)(dη/η) = m∫(ZHS -1)(dη/η) - (m-1)∫η[∂lng(d)/∂η]}(dη/η)
or
5.3.2
∫(ZHCM -1)(dη/η) = m∫(ZHS -1)(dη/η) – (m-1).lng(d)
(5.27)
Hard chain equation using Carnahan- Starling equation
From Carnahan- Starling equation of state (5.19) and equation for pair correlation at
contact (5.15) we have:
1 +η +η 2 −η3
= 1 + 4ηg
(1 − η )3
1
or
1− η
η −2
2
g=
=
3
3
2(η − 1) (1 − η )
we have
ln(g) = ln(1- ½η) – 3 ln (1-η)
(5.28)
or
∂lng(d)/∂η = -1/(2-η) + 3/(1-η)
(5.29)
Inserting (5.29) into Eq. (5.25), we have
Zchain = - [(m-1)/m] {1 + η(-1/(2-η) + 3/(1-η))}
The compression factor of the spherical associating molecular system from
Carnahan-Starling approach can be rewritten as:
ZHCM-CS = p/ρkT = mZHS-CS + mZchain
Z HCM −CS
η
1 + η + η 2 −η 3
3η
=m
− (m − 1)(1 −
+
)
3
2 −η 1 −η
(1 − η )
(5.30)
From equations (5.22), (5.26) and (5.27) and (5.28) we have:
Ar,HCM-CS/RT = m[η(4-3η)/(1-η)2] - (m-1)ln[(1- ½η)/(1-η)3]
5.3.3
Hard chain equation using Kolafa-Boublik-Nezbeda equation
From the equation of state of Kolafa-Boublik-Nezbeda (KBN)
p
=
ρkT
2
2
1+η +η2 − η3 − η4
3
3
(1 − η )3
and the pair correlation at contact
62
(5.31)
p
= 1 + 4ηg
ρkT
We have
2
2
1+η +η2 − η3 − η 4
3
3
1 + 4ηg =
(1 − η )3
or
g=
We have
12 − 6η + η 2 − 2η 3
3
12(1 − η )
(5.32)
∂Ln( g (d ))
5(− 6 + 2η + η 2 )
=−
∂η
12 − 18η + 7η 2 − 3η 3 + 2η 4
As given above, mZchain = - [(m-1)]{1 + η[∂ln(g(d))/∂η]}
we have
and Z HCM − KBN
mZ chain = −(m − 1)
12 + 12η − 3η 2 − 8η 3 + 2η 4
12 − 18η + 7η 2 − 3η 3 + 2η 4
2
2
1+η +η2 − η3 − η4
2
3
4
3
3 − (m − 1) 12 + 12η − 3η − 8η + 2η
=m
12 − 18η + 7η 2 − 3η 3 + 2η 4
(1 − η )3
(5.33)
The equation (5.33) is identical to equation 9 in [5.22]. Thus, equation 3 in [5.23]
has a typing mistake in the last term.
From equations (5.23), (5.26), (5.27) and (5.32) we have:
Ar , HCM − KBN
⎛ 12 − 6η + η 2 − 2η 3 ⎞
η (34 − 33η + 4η 2 ) 5m
⎜⎜
⎟⎟ (5.34)
(
)
(
)
=m
−
−
−
−
ln
1
η
1
ln
m
2
3
3
RT
6(1 − η )
12(1 − η )
⎝
⎠
5.3.4
Comparison among simulation data, results from CS and KBN equations
5.3.4.1 Comparison between results derived from CS and KBN equations
In order to see the difference between results derived from CS and KBN equations
for chain-like hard sphere molecules, we plot relative deviations of compression factor and
residual Helmholtz energy derived from CS and KBN equations, figure 5.5 and 5.6. The
relative deviation of compression factor and deviation of residual Helmholtz energy are
within 0.6% and 0.4%, respectively for the case m=5, m=10, and m=50.
63
100(ZHCM-KBN-ZHCM-CS)/ZHCM-KBN
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
0
0.1
0.2
η
0.3
0.4
0.5
100(AHCM-KBN-AHCM-CS)/AHCM-KBN
Figure 5.5. Relative deviation between ZHCM-CS and ZHCM-KBN: Δ (m=5), Ο (m=10), Χ
(m=50)
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
0
0.1
0.2
η
0.3
0.4
0.5
Figure 5.6. Relative deviation between AHCM-CS and AHCM-KBN: Δ (m=5), Ο (m=10), Χ
(m=50)
5.3.4.2 Comparison between simulation data and results derived from KBN equation
Monte-Carlo simulation data of Honnell and Hall [5.24] are used to investigate the
compression factors derived from KBN equation. Comparisons between the compression
factors from simulation, Zsim, for the case m = 4, m = 8, and m = 16 and those derived from
KBN equation, ZHCM-KBN, for hard chain molecules are given in figure 5.7. The figure 5.7
shows that there is a large disagreement between compression factors from simulation and
those derived from KBN equation. The discrepancy between compression factors from
simulation and those derived from KBN equation increases with m and decreases with η
generally.
64
100(Zsim-ZHCM-KNB )/Zsim
5
0
-5
-10
-15
-20
0.0
0.1
0.2
0.3
0.4
0.5
η
Figure 5.7. Relative deviation of ZHCM-KBN to simulation data for: Δ (m = 4), Ο (m = 8), Χ
(m = 16).
5.3.4.3 Comparison for the case with correlation parameter
The figures 5.5 and 5.6 show that results derived from CS and KBN equations are
nearly the same. The figure 5.7 shows the large discrepancy between compression factors
from simulation and those derived from KBN equation. The discrepancy increases from
about 0% to 19% when the packing fraction decreases from 0.44 to 0.07. Thus, it is
interesting to introduce and test with a correlation parameter to get a fit equation for the
simulation data of hard chains.
In this study, we start from CS equation with a correlation parameter meff:
Z HCM − CS , meff = meff
η
1 +η +η 2 −η 3
3η
)
− (meff − 1)(1 −
+
3
2 −η 1 −η
(1 − η )
(5.35)
The criterion of optimization for finding meff is a minimization of the following
summation:
2
⎛ Z sim ,i − Z HCM − CS , meff ,i ⎞
⎟ → Min
⎜
∑
⎟
⎜
Z sim ,i
i =1 ⎝
⎠
N
where N is number of simulation data points, Zsim is compression factor of simulation data.
After fitting to simulation data of Honnell and Hall [5.24] we have values of meff,
table 5.2. We find out in table 5.2 that meff is always smaller than number of hard spheres in
a chain. Comparison of compression factors between simulation data and results with
65
correlation parameter meff from CS equation, ZHCM-CS,meff, is shown in figure 5.8. With the
use of meff, almost all deviations are within ±5%.
Table 5.2. Correlation parameter meff from CS equation
m
4
8
16
Data sources
[5.24]
[5.24]
[5.24]
meff
3.8530
7.1591
14.0763
100(Zsim.-ZHCM-CS,meff)/Zsim.
15
10
5
0
-5
-10
-15
0.0
0.1
0.2
0.3
0.4
0.5
η
Figure 5.8. Relative deviation between compression factor from simulation and ZHCM-CS,meff
for: Χ (meff = 3.8530), Δ (meff = 7.1591), Ο (meff = 14.0763).
5.4
5.4.1
Hard convex bodies and hard dumbbells
Hard convex bodies (HCB)
Boublik proposed an equation of state for a system of one component hard convex
bodies [5.25]:
p
1 + (3α − 2)η + (3α 2 − 3α + 1)η 2 − α 2η 3
=
ρkT
(1 − η )3
(5.36)
where, α = R.S/(3V) is the nonsphericity parameter. V, S, and R stand for a volume, a
surface, and the (1/4π) multiple of the mean curvature integral. η = ρV is packing fraction.
It is obvious that in the limited case of hard spheres (with α = 1), the equation of
Boublik is identical with equation of CS.
For mixture of hard convex bodies, the Boublik equation is written as:
66
p
1
kv
k v 2 (3 − v )
=
+ 1 2+ 2
ρkT 1 − v (1 − v )
(1 − v )3
(5.37)
where v = ρV, k1 and k2 are dimensionless quantities, k1 = RmSm/Vm, k2 = QmSm2/(9Vm2).
Qm=Rm2. The geometric quantities of mixture are calculated from mole fraction and
geometric quantities of individual component, Xm = Σi(xi.Xi). In which, X stands for either
of V or S or R or Q. xi is mole fraction of component i of mixtures.
5.4.2
Hard dumbbells (HD)
After proposing equation of state for hard convex bodies, Boublik and Nezbeda
[5.26] show that the Boublik equation of state for hard convex bodies yields good results
for a system of hard dumbbells if the mean curvature integral is taken as that of prolate
spherocylinder. Required geometric quantities of hard dumbbells with sphere diameter d
and a center-to-center distance L are:
VD= (4π/3)(d/2)3[1+ 3/2 L – 1/2L3]
η = (π/6)ρd3 [1+ 3/2 L – 1/2L3]
α = (1+L)(2+L)/(2+3L-L3); 0 ≤ L ≤ 1 or 1 ≤ α ≤ 1.5
Where, VD is volume of hard dumbbells.
5.4.3
Helmholtz energy derived from equation for hard convex bodies and hard
dumbbells
From equation of state for hard convex bodies and hard dumbbells (5.36):
Z HCB _ HD =
p
1 + (3α − 2)η + (3α 2 − 3α + 1)η 2 − α 2η 3
=
ρkT
(1 − η )3
we obtain a relation for residual Helmholtz energy as following:
Ar , HCB _ HD
RT
Ar ,HCB _ HD
RT
η
= ∫ ( Z HCB _ HD − 1)
0
dη
η
η
dη
1 + (3α − 2)η + (3α 2 − 3α + 1)η 2 − α 2η 3
= ∫(
− 1)
3
η
(1 − η )
0
67
(5.38)
(
)
η
⎛
⎛
⎞⎞
3α − 2
3α 2 − 3α + 1 (1 − 2η )
1
2
1
1
η
= ⎜⎜
+
+
Ln
[
1
−
]
+
−
− α 2 ⎜⎜
−
+ Ln[1 − η ] ⎟⎟ ⎟⎟
2
2
2
2
2(1 − η )
2(1 − η )
⎝ 2(1 − η ) 1 − η
⎠⎠ 0
⎝ 2(1 − η ) 1 − η
(
)
= α 2 − 1 ln(1 − η ) +
Ar , HCB _ HD
RT
5.4.4
(
η
α (3 + α − 3η )
(1 − η )2
)
= α 2 − 1 ln(1 − η ) +
η
α (3 + α − 3η )
(1 − η )2
(5.39)
Hard convex bodies approach to hard chain molecules
According to Boublik et al. [5.23], the hard convex body equation (5.38) can be
extended to hard chain molecules of overlapping hard spheres (0.5 < L < 1) or tangent hard
spheres (L = 1) with formulations of packing fraction and parameter of nonsphericity as:
η = ρ Vc = ρ .
α=
πd 3 ⎛
(
)
1
3 ⎞
⎜ 1 + (m − 1 ) 3 L − L ⎟
6 ⎝
2
⎠
[1 + (m − 1)L].[2 + (m − 1)L]
2 + (m − 1)(3L − L3 )
(5.40)
(5.41)
Equation of residual Helmholtz energy for this case is identical to equation (5.39)
5.5
Comparison of results derived from equations for hard convex bodies and hard
chain systems
In this part we investigate the differences of compression factors and residual
Helmholtz energies derived from equation for hard convex bodies, equation for hard chain
of Boublik and equation for hard chain of Carnahan- Starling (CS).
When hard spheres are tangent, L=1, equations for packing fraction (5.40) and
parameter of nonsphericity (5.41) become:
η = mρ .
α=
πd 3
6
m +1
2
Comparisons of compression factors and residual Helmholtz energies derived from
equations for hard convex bodies and hard chain systems are given in figures 5.9, 5.10, and
68
5.11. The relative deviations of compression factor and residual Helmholtz energy for the
case m = 2 are within 1.1%, figure 5.9, 5.10. For compact molecules, results derived from
equations for hard convex bodies and hard chain systems are similar.
For chain-like molecules with large number of m, the relative deviations of
compression factors between simulation data of Honnell and Hall and results derived from
KBN equation are not exceeded 20%, figure 5.7, whilst the deviations of compression
factors between simulation data of Honnell and Hall and results derived from equation for
hard convex bodies are much higher, even over 100%, figure 5.11. Comparison of residual
Helmholtz energy derived from equations of CS and KBN, figure 5.12, shows strong
disagreements, especially for the case with high values of m.
From these analyses we observe that results derived from equation of hard convex
bodies for hard chain systems do not agree well with results derived from equations of CS
and KBN.
100(ZHCM-CS-ZHCM-HCB)/ZHCM-CS
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0
0.1
0.2
0.3
0.4
0.5
η
Figure 5.9. Deviation of compression factor derived from equation of CS and equation for
hard convex bodies for the case L=1 and m=2
69
100(Ar,HCM-CS-Ar,HCM-HCB)/Ar,HCM-CS
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
0
0.1
0.2
0.3
0.4
0.5
η
Figure 5.10. Deviation of residual Helmholtz energy derived from equation of CS and
equation for hard convex bodies for the case L=1 and m=2
100(Zsim-ZHCM-HCB)/Zsim
10
-10
-30
-50
-70
-90
-110
-130
0.0
0.1
0.2
η
0.3
0.4
0.5
Figure 5.11. Deviation of compression factor between simulation data of Honnell and Hall
[5.24] and results derived from equation for hard convex bodies for the case L=1 and:
Χ (m = 4), Δ (m = 8), Ο (m = 16).
70
100(AHCM-CS-AHCM-HCB)/AHCM-CS
0
-10
-20
-30
-40
-50
-60
-70
-80
0.0
0.1
0.2
η
0.3
0.4
0.5
Figure 5.12. Deviation of residual Helmholtz energy between results derived from equation
of CS and equation for hard convex bodies for the case L=1 and Χ (m = 4), Δ (m = 8),
Ο (m = 16).
5.6
Residual Helmholtz energies and compression factors of hard spheres, hard convex
bodies, hard dumbbells and hard chain systems from different approaches have been
reviewed. The results show that the application of hard convex bodies to hard chain
systems is good for compact molecules, but not for large chain-like molecules. Comparison
between simulation data and results from SAFT of Wertheim using hard sphere description
of CS and KBN shows good agreement. Some results from this chapter will be used in next
chapter.
71
References
[5.1] B. J. Alder and C. E. Hecht, Studies in Molecular Dynamics. VII. Hard-Sphere
Distribution Functions and an Augmented van der Waals Theory, J. Chem. Phys. 50 (1969)
2032- 2037.
[5.2] T. Einwohner and B. J. Alder, Molecular Dynamics. VI. Free-Path Distributions and
Collision Rates for Hard-Sphere and Square-Well Molecules, J. Chem. Phys. 49 (1968)
1458-1473.
[5.3] B. J. Alder, D. A. Young, and M. A. Mark, Studies in Molecular Dynamics. X.
Corrections to the Augmented van der Waals Theory for the Square Well Fluid, J. Chem.
Phys. 56 (1972) 3013-3029.
[5.4] Chen, S. S., and A. Kreglewski, Application of the augmented van der Waals Theory
of Fluids: I. Pure Fluids, Ber. Bunsenges. Phys. Chern., 81 (1977) 1048-1052.
[5.5] Kreglewski, A, Equilibrium Properties of Fluids and Fluid Mixtures, Texas A&M
Univ. Press, College Station (1984)
[5.6] A. Müller, J. Winkelmann, J. Fischer, Backone Family of Equations of State: 1.
Nonpolar and Polar Pure Fluids, AIChE J, 42 (1996) 1116-1126.
[5.7] U. Weingerl, M. Wendland, J. Fischer, A. Müller, J. Winkelmann, The BACKONE
family of equations of state: 2. Nonpolar and polar fluid mixtures, AIChE-J, 47 (2001) 705
-717.
[5.9] Yarnell, J.L., Katz, M.J., Wenzel, R.G., Koenig, S.H., Structure factor and radial
distribution function for liquid argon at 85°K, Physical Review A, 7 (1973) 2130-2144
[5.10] J. A. Barker and D. Henderson, R. O. Watts, The Percus-Yevick theory is alive and
well, Physics Letters A, 31 (1970) 48-49
[5.11] W. G. Hoover, F. H. Ree, Melting transition and communal entropy for hard spheres,
The journal of chemical physics, 49 (1968) 3609-3617
72
[5.12] L.S. Ornstein, F. Zernike, Accidental deviations of density and opalescence at the
critical point of a single substance, Proc. Acad. Sci. Amsterdam, 17 (1914) 793-806
[5.13] Percus I. K., Yevlck G. I., Analysis of classical statistical mechanics by means of
collective coordinates. Phys. Rev. 110 (1958) 1-13
[5.14] E. Thiele, Equation of State for Hard Spheres, J. Chem. Phys. 39 (1963) 474-479
[5.15] N. F. Carnahan, K.E. Starling, J. Chem. Phys., 51 (1969) 635-636
[5.16] J. Kolafa according to T. Boublik, I. Nezbeda, Coll. Czech. Chem. Commun., 51
(1986) 2301
[5.17] M. S. Wertheim, Fluids with highly directional attractive forces. I. Statistical
thermodyna-mics, J. Stat. Phys. 35 (1984) 19-34
[5.18] M. S. Wertheim, Fluids with highly directional attractive forces. II. Thermodynamic
perturbation theory and integral equations, J. Stat. Phys. 35 (1984) 35-47.
[5.19] M. S. Wertheim, Fluids with highly directional attractive forces. III. Multiple
attraction sites. J. Stat. Phys. 42 (1986) 459-476.
[5.20] M. S. Wertheim, Fluids with highly directional attractive forces. IV. Equilibrium
polymeri-zation. J. Stat. Phys. 42 (1986) 477-492.
[5.21] W. G. Chapman, G. Jackson, K. E. Gubbins, Phase equilibria of associating fluids.
Chain molecules with multiple bonding sites, Molec. Phys. 65 (1988) 1057-1079.
[5.22] W. G. Chapman, K. E. Gubbins, G. Jackson, M. Radosz, New reference equation of
state for associating liquids, Ind. Eng. Chem. Res. 29 (1990) 1709-1721
[5.22] T. Boublik, Equation of state of linear fused hard-sphere models, Molec. Phys.68
(1989) 191 – 198
[5.23] T.Boublik, C. Vega, M. Diaz-Pena, Equation of state of chain molecules, J. Chem.
Phys., 93 (1990) 730-736
[5.24] K. G. Honnell, C. K. Hall, A new equation of state for athermal chains, J. Chem.
Phys. 90 (1989) 1841-1855
[5.25] T. Boublik, Hard convex body equation of state, J. Chem. Phys., 63 (1975) 4084
73
[5.26] T. Boublik and I. Nezbeda, Equation of state for hard dumbbells, Chem. Phys.
Letters, 46 (1977) 315-316
74
6
PC-SAFT equation of state
Abstract
Methodology and strategy for development of PC-SAFT equation of state have been
studied. Firstly, we review Barker and Henderson perturbation theory. Secondly,
formulation of PC-SAFT equation of state for pure fluids is reviewed. Finally, we derive
derivatives of residual Helmholtz energy for calculations of residual parts of pressure,
enthalpy, entropy, internal energy, isobaric and isochoric heat capacity, and Gibbs energy.
6.1
Introduction
Molecular based equations of state have recently been proved to be one of the most
powerful types of equation of state. The first typical characteristic of molecular based
equations of state is that they have few parameters which can be found by fitting to
experimental data. The second typical characteristic of the molecular based equations of
state is that they can be extended to mixtures easily, normally with one fitted parameter
whilst multi-parameter equations of state for pure fluids are not easily to be extended to
mixtures because many parameters are needed to be found in fitting. Last but not least,
molecular based equations of state are accurate whilst cubic equations of state are not
sufficiently accurate.
BACKONE and PC-SAFT are accurate and reliable molecular based equations of
state [6.1 - 6.3]. The papers for BACKONE equation of state are well written. However, the
paper for PC-SAFT equation of state [6.3] contains some problems. In details, formulation
of PC-SAFT equation of state contains a mixing of compression factor and Helmholtz
energy. Furthermore, formulas of integrals in the first- and second-order perturbation terms
are wrong. Thus, in order to make a clear formulation in term of Helmholtz energy we
review PC-SAFT equation of state and prepare materials for further programming and
calculation.
6.2
Barker-Henderson perturbation theory
Knowledge of the molecular electrostatic potential is critically important when
molecular interactions are to be studied. Molecular interaction resulting from electrostatics
can be separated into attractive and repulsive contributions. At large intermolecular
75
distance, molecular interaction is dispersive or attractive. The original of the attractive
interaction is that fluctuations in the electron cloud produce instantaneous dipoles which in
turn polarize the electron of neighbouring molecules. These dipoles give the rise to
attractive forces. At small distance, molecular interaction is repulsive due to charge cloud
overlaps of interacting particles.
Barker and Henderson perturbation theory is valid for all type of pair potential
function such as square well potential, Lennard-Jones potential. In this subsection,
dispersive and repulsive molecular interactions with the Lennard-Jones pair potential
function at interaction site distance r are considered:
uLJ(r) = 4ε[(σ/r)12 - (σ/r)6]
(6.1)
where the physical r-6 term describes dispersive interaction at long range, the empirical r-12
term describes repulsive interactions at short range. ε is the Lennard-Jones energy
parameter. σ is the Lennard-Jones size parameter.
Detail Barker-Henderson perturbation theory and equation of state for fluids are
given in [6.4] and [6.5]. Reviews of Barker and Henderson perturbation theory and other
theory can be found in the book of Boublik et al. [6.6] and the book McQuarrie [6.7]. In the
following subsections we present characterization of reference system and Helmholtz free
energy from Barker and Henderson perturbation theory.
6.2.1
Characterization of the reference system by Barker and Henderson
In the application of perturbation theories of fluids, the potential energy of a system
is possible to be separated into a reference part accounting for large contribution and a
perturbation part accounting for small contribution. For a dense fluid the repulsive forces
dominate the liquid structure. This means that other forces play minor role in liquid
structure. Thus, if we separate the Lennard-Jones intermolecular potential into a reference
part u0(r) containing all repulsive forces and a perturbation part u1(r) containing all
attractive forces we have an expression:
uLJ(r) = u0(r) + u1(r)
(6.2)
In statistical thermodynamics, the energy A of such a system of N molecules can be
written as:
76
exp[− β . A] = ( N !) −1 λ−3 N ∫ ...∫ exp[− β (U 0 + U 1 )]d {N }
(6.3)
1
⎛
⎞
= ( N!) −1 λ−3 N ∫ ...∫ exp[− β .U 0 ]⎜1 − β .U 1 + β 2 .U 12 + ....⎟d {N }
2
⎝
⎠
where β = 1/kT. k is the Boltzmann-constant
The force determined from equation (6.1) is repulsive for all r < σ. If u0(r)
containing all repulsive forces then u0(r) = 0 for all r > σ. In case there are no attractive
forces, we have:
d[uLJ(r)]/dr = d[u0(r)]/dr
(6.4)
Equation (6.2) and (6.4) then determine u0(r) and u1(r) uniquely as:
u0(r) = uLJ(r),
r<σ
(6.5a)
u0(r) = 0,
r>σ
(6.5b)
u1(r) = uLJ(r0) = 0,
r<σ
(6.6a)
u1(r) = uLJ(r),
r>σ
(6.6b)
and
The potential from equation (6.5) and (6.6) is shown in figure 6.1.
U
U
LJ
0
Repulsive
σ
σ
r
U
1
σ
r
Attractive
Figure 6.1. The separation of the Lennard-Jones potential uLJ(r) into a part u0(r) containing
all repulsive interactions and a part u1(r) containing all attractive interaction
77
For a rational determination of the diameter ‘d’ of representative hard spheres in
systems with soft repulsive interactions, Barker and Henderson modified original pair
potential by introducing two coupling parameters, α and γ:
w(r, α, γ , d) = uLJ(d+ (r-d)/α),
w(r, α, γ , d) = 0,
d+ (r-d)/α< σ
σ<d+ (r-d)/α< d+ (σ-d)/α
w(r, α, γ , d) = uLJ(r),
r>σ
(6.7a)
(6.7b)
(6.7c)
Equation (6.7) appears to be complicated but notice that w(r, α, γ , d) is independent
of d and reduces to u1(r) when α= γ=1. When α= γ=0, w(r, α, γ , d) becomes a hard-sphere
potential of diameter d.
An expansion of the partition function logarithm in powers of α and γ at about α = γ
= 0 can be written as:
ln Z = ln Z0 + (∂lnZ/∂α)0α + (∂lnZ/∂γ)0γ
+....
(6.8)
where Z0 is the partition function of hard spheres with diameter d. The partial derivative of
ln Z with respect to α and γ can be written as:
∞
∂ ln Z
⎛ ∂ exp[− β .w(r )] ⎞
LJ
2
= 2πNρ ∫ ⎜
⎟.g 0 (r ). exp[β .u (r )].r dr (6.9)
∂α
∂α
⎠
0⎝
∞
∂ ln Z
= −2πNρβ ∫ u LJ (r ).g 0 (r ). exp[ β w].r 2 dr
∂γ
σ
(6.10)
Barker and Henderson used Heaviside function H(x) and made use of the
relationship for r < σ as:
⎡
⎡
r−d
r − d ⎞⎤
r−d
⎛
⎞
⎛
⎛
⎞⎤
−σ ⎟
exp[− β .w(r , α , γ , d )] = ⎢1 − H ⎜ d +
− σ ⎟⎥ exp ⎢− β .u LJ ⎜ d +
⎟⎥ + H ⎜ d +
α
α ⎠⎦
α
⎝
⎠
⎝
⎝
⎠⎦
⎣
⎣
(6.11)
Taking derivative of equation (6.11) and substituting of z = d + (r-d)/α then
inserting them into equation (6.9), the following expression is valid for α = 0:
78
⎧σ
⎫
⎛ ∂ ln Z ⎞
2
LJ
LJ
⎜
⎟ = 2πNρ .g 0 (d ).exp[β .u (d )].d ⎨∫ exp[− β .u ( z )].dz − (σ − d )⎬ (6.12)
⎝ ∂α ⎠0
⎩0
⎭
Barker and Henderson chose the diameter d so that the expression in braces of
(6.12) and consequently the whole derivative may vanish; this make d equal to
σ
d = ∫ {1 − exp[− β .u LJ (x )]}dx
(6.13)
0
6.2.2
Barker and Henderson perturbation theory
From knowledge of the potential energy of a system, one can have a relation
between Helmholtz free energy and the potential energy. Detail steps for obtaining the
relations of Helmholtz free energy in Barker and Henderson perturbation theory can be
found in original paper [6.4]. In this subsection we only give out the results of the original
paper.
Based on the assumption that macroscopic expressions can be applied to
microscopic, Barker and Henderson [6.4] found an approximation expression, called
“macroscopic compressibility”:
(A-A0)/RT = (ρ/2)∫ [βu1(r)] g0(r) 4πr2dr - (ρ/4)∫ [βu1(r)] 2 [kT(∂ρ/∂p)0] g0(r)4πr2dr
+O(β3)
(6.14)
where β=1/kT, and g0(r) is pair correlation function of the soft reference system or the
corresponding hard sphere system. O(β3) is the contribution of higher order terms, from
third-order term.
Barker and Henderson [6.4] also suggested another approximation expression,
called “local compressibility”:
(A-A0)/RT = (ρ/2)∫ [βu1(r)] g0(r) 4πr2dr - (ρ/4)∫ [βu1(r)] 2 {kT∂[ρ g0(r)] /∂p}4πr2dr
+O(β3)
(6.15)
The first part of the right hand-side of the expression (6.14) or of the expression
(6.15) is the first-order contribution. Let A1 denote the first-order perturbation term, we
write:
79
A1/RT = (ρ/2)∫ [βu1(r)] g0(r) 4πr2dr
(6.16)
The second part of the right hand-side of the expression (6.14) or expression (6.15)
is second-order contribution. Let A2 denote the second-order perturbation term, from
macroscopic compressibility approach, one has:
A2/RT = - (ρ/4)∫ [βu1(r)] 2 [kT(∂ρ/∂p)0] g0(r)4πr2dr
(6.17)
and from local compressibility approach, one has:
A2/RT = - (ρ/4)∫ [βu1(r)] 2 {kT∂[ρ g0(r)] /∂p}4πr2dr
(6.18)
Expression (6.14) and (6.15) can now be written as:
A = A0 + A1 +A2 + O(β3)
6.3
6.3.1
(6.19)
PC-SAFT equation for pure fluids
The potential model
In PC-SAFT, molecules are conceived to be chains composed of spherical segments
having the same diameter and energy interactions. The pair potential for the segments of a
chain is assumed as square well with step potential according to Chen and Kreglewski
[6.8]. The step ranges from 0.88 σ to σ and has a height of 3ε, the well has the depth ε:
u (r) = ∞,
r < 0.88σ
(6.20a)
u (r) = 3ε,
0.88σ < r < σ
(6.20b)
u (r) = -ε,
σ<r <σ+ε
(6.20c)
u (r) = 0,
r>σ+ε
(6.20d)
If the potential u(r) is divided into reference part u0(r) and perturbation part u1(r) at
distance σ we have:
and
u0 (r) = ∞,
r < 0.88σ
(6.21a)
u0 (r) = 3ε,
0.88σ < r < σ
(6.21b)
u0 (r) = 0,
r>σ
(6.21c)
u1(r) = 0,
r <σ
(6.22b)
80
u1(r) = -ε,
σ<r <σ+ε
(6.22c)
u1(r) = 0,
r>σ+ε
(6.22d)
The full segment-segment potential u(r) as well as the reference u0(r) and the
perturbation u1(r) potential of equations (6.20), (6.21), and (6.22) are shown in figure 6.2.
u0(r)
u(r)
3ε
3ε
0.88σ
0.88σ σ
σ+ε
u1(r)
r
-ε
0.88σ σ
σ
r
σ+ε
r
-ε
Figure 6.2. Square-well potential of molecular model in PC-SAFT
From equations (6.13) and (6.21) we have the temperature dependent hard sphere
diameter d(T) for the segments as:
σ
d (T ) = ∫ {1 − exp[− β .u 0 (r )]}dr =
0
0.88σ
σ
0
0.88
∫ {1 − exp[−∞]}dr + ∫ σ{1 − exp[−3εβ ]}dr
= 0.88σ + 0.12σ[1-exp(-3εβ)]
or
d = σ [1-0.12exp(-3ε/kT)],
6.3.2
(6.23)
Residual Helmholtz energy for hard chains
The hard body reference system of PC-SAFT is based on Carnahan and Starling
equation, [6.9]. We use subscript ‘0’ for reference system, one of elements of residual part.
The Helmholtz energy of hard chain contribution A0 of reference system is given in
applications of Chapman et al. [6.10, 6.11] basing on the SAFT approach of Wertheim
81
[6.12 - 6.15]. A detail steps for obtaining compression factor and residual Helmholtz energy
of PC-SAFT’s reference system of residual part are given in chapter 5. For convenience, we
represent equations for compression factor and residual Helmholtz energy here:
1 + η +η 2 −η 3
3η
η
)
− (m − 1)(1 −
+
3
2 −η 1 −η
(1 − η )
(6.24)
A0/RT = m[η(4-3η)/(1-η)2] - (m-1)ln[(1- ½η)/(1-η)3]
(6.25)
Z0 = m
where η = ρπd3/6 is packing fraction, representing for reduced segment density.
6.3.3
First and second order perturbation terms
Barker and Henderson theory for spherical molecules can be extended to chain
molecules as each segment of a chain is again of spherical shape. The total interaction
between two chain molecules is the sum of all individual segment-segment interactions.
The expression for radial distribution function gαβ(m, rαβ,ρ) for a segment α of one chain
and a segment β of other chain by radial distance rαβ has been given by Chiew [6.16]. He
also introduced an average radial distribution function gHC(m, r, ρ) for a chain having
different non-distinguishable segments. The results of Chiew were used by Gross and
Sadowski [6.17] for the first-order perturbation contribution of chain molecules as:
A1/RT = (ρ/2)∫ [βu1(r)] gHC(m,r,ρ) 4πr2dr
With
= - 2πρσ3 m2(ε/kT) I1(y,m)
(6.26)
I1(y,m) = - (1/4πσ3 m2ε )∫ [u1(r)] gHC(m,r,ρ) 4πr2dr
(6.27)
Neglecting the temperature dependence of gHC(m,r,ρ), the integral I1(η,m) can be
represented by a power series of number of segments and ai as:
6
I1 (η , m ) = ∑ ai (m ).η i
(6.28)
i =0
The dependence of coefficients a0i, a1i, a2i on the segment number m was shown
earlier [6.17] to be well represented by an ansatz of Liu and Hu [6.18]
ai(m) = a0i + [(m-1)/m] a1i + [(m-1)(m-2)/m2] a2i,
82
(6.29)
For the second order perturbation contribution, Gross and Sadowski used the
macroscopic compressibility version in PC-SAFT:
A2/RT = - (ρ/4)∫[βu1(r)]2 [kT(∂ρ/∂p)0] gHC(m,r,ρ) 4πr2dr
= - (ρ/4)[kT(∂ρ/∂p)0]∫[βu1(r)]2 gHC(m,r,ρ) 4πr2dr
From
kT(∂ρ/∂p)0 = [∂ρ/∂(p/kT)]0 = [∂(p/kT)/∂ρ]0-1
and
p/ρkT = Z0 or p/kT = ρZ0
we have
∂(p/kT)/∂ρ = Z0 + ρ∂Z0/∂ρ = Z0 + η∂Z0/∂η
(6.30)
Equation (6.30) becomes:
A2/RT = - (ρ/4) [Z0 + ρ∂Z0/∂ρ]-1 ∫ [βu1(r)]2 gHC(m,r,ρ) 4πr2dr
= - πρσ3 m3(ε/kT)2 [Z0 + ρ∂Z0/∂ρ]-1I2(η,m)
I2(y,m) = (1/4πσ3 m3ε2) ∫ [u1(r)]2 gHC(m,r,ρ) 4πr2dr
Where
(6.31)
(6.32)
Similar to approach for the first-order perturbation of Chiew, neglecting the
temperature dependence of gHC(m,r,ρ), the integral I2(η,m) can be represented by a power
series of number of segments and bi as:
6
I 2 (η , m ) = ∑ bi (m ).η i
(6.33)
i =0
The dependence of coefficients b0i, b1i, b2i on the segment number m was shown
earlier [6.17] to be well represented by an ansatz of Liu and Hu [6.18]
bi(m) = b0i + [(m-1)/m] b1i + [(m-1)(m-2)/m2] b2i,
(6.34)
From equation (6.24), we have:
−1
∂Z ⎞
⎛
20η − 27η 2 + 12η 3 − 2η 4
8η − 2η 2
⎜⎜ Z 0 + ρ 0 ⎟⎟ = 1 + m
m
+
(
1
−
)
∂ρ ⎠
[(1 − η )(2 − η )]2
(1 − η )4
⎝
Equation (6.31) now becomes:
(6.35)
A2/RT = - πρσ3 m3(ε/kT)2 [{1 + m(8η-2η2)/(1-η)4 – (m-1)(20η -27η2 +12η3 –
2η4)/[(2-η)2(1-η)2]}-1] I2(η,m)
(6.36)
83
The parameters of perturbation terms a0i, a1i, a2i, b0i, b1i, and b2i in equation (6.29)
and equation (6.34) can be found by either fitting to the radial distribution functions or
fitting to experimental data. The shortcomings in fitting to the radial distribution functions
are that the assumptions of the molecular model might be oversimplified as chains of
segments, the assumed perturbation potential and approximations of g0 might also contain
uncertainties, and the reference equation of state for hard-sphere of Carnahan-Starling
might not be exactly as it should be.
In order to avoid shortcomings from fitting to radial distribution functions, the
parameters of perturbation terms in PC-SAFT were found by fitting to experimental data of
alkanes. Before determining the parameters of perturbation terms, three pure-component
parameters (m, σ, ε/k) of Alkanes were firstly identified by fitting vapour pressure and PvT
data to the integral expressions of I1 and I2 assumed as a Lennard-Jones perturbation
potential. The parameters of perturbation terms were secondly determined by fitting to
vapour pressures, liquid, vapour, and supercritical volumes. The parameters of perturbation
terms from Gross and Sadowski PC-SAFT equation of state are given in table 6.1 and 6.2.
Table 6.1. Parameters of the first-order perturbation term
a1i
a2i
i
a0i
9.105631445E-01
-3.084016918E-01
-9.061483510E-02
0
6.361281449E-01
1.860531159E-01
4.527842806E-01
1
2.686134789E+00 -2.503004726E+00 5.962700728E-01
2
-2.654736249E+01 2.141979363E+01 -1.724182913E+00
3
9.775920878E+01 -6.525588533E+01 -4.130211253E+00
4
-1.595915409E+02 8.331868048E+01 1.377663187E+01
5
9.129777408E+01 -3.374692293E+01 -8.672847037E+00
6
Table 6.2. Parameters of the second-order perturbation term
b1i
b2i
i
b0i
7.240946941E-01
-5.755498075E-01
9.768831160E-02
0
2.238279186E+00
6.995095521E-01 -2.557574982E-01
1
-4.002584949E+00 3.892567339E+00 -9.155856153E+00
2
-2.100357682E+01 -1.721547165E+01 2.064207597E+01
3
2.685564136E+01 1.926722645E+02 -3.880443005E+01
4
2.065513384E+02 -1.618264617E+02 9.362677408E+01
5
-3.556023561E+02 -1.652076935E+02 -2.966690559E+01
6
84
6.3.4
Complete Helmholtz energy equation
Total Helmholtz energy is written in term of a sum of Helmholtz energy of ideal gas
part, Aid, and Helmholtz energy of residual part, Ares:
A = Aid + Ares
(6.37)
The Helmholtz energy of ideal gas part can be calculated from internal energy,
entropy and temperature as:
Aid(T,ρ) = uid(T) – T.sid(T,ρ)
(6.38)
When introducing the isobaric heat capacity of the ideal gas, c0p, one obtains the
expression:
Aid (T , ρ ) = u Re f − TsRe f +
T
∫c
T
0
p
dT − T
TRe f
∫
TRe f
c 0p
dT
ρT
− R(T − TRe f ) + T .R ln(
)
T
ρ Re f TRe f
(6.39)
where R is the ideal gas constant, R = 8.314472 J/molK. The subscript “Ref” stands for
corresponding quantities at reference point. Values of reference point temperature TRef,
density ρRef, internal energy uRef, and entropy sRef can be chosen randomly. In practical, one
often replaces reference internal energy uRef by reference enthalpy hRef via following
relation:
u Re f = hRe f − RTRe f
(6.40)
With this relation, expression (6.39) becomes:
Aid (T , ρ ) = hRe f − TsRe f +
T
∫ c p,id dT − T
TRe f
T
∫
TRe f
c p,id
dT
ρT
− RT + T .R ln(
)
T
ρ Re f TRe f
(6.41)
Neglecting the contribution from third-order terms of equation (6.19), the
Helmholtz energy of residual part is written in term of a sum of the Helmholtz energy from
reference hard chain contribution A0, the first-order perturbation term A1, and the secondorder perturbation term A2:
Ares = A0 + A1 + A2
(6.42)
All three contributions of residual part are given in previous sections. For
convenience of reader, we represented here the expressions of residual part:
85
A0/RT = m[η(4-3η)/(1-η)2] - (m-1)ln[(1- ½η)/(1-η)3]
A1/RT = - 2πρσ3 m2(ε/kT) I1(η,m)
A2/RT = - πρσ3 m3(ε/kT)2 [{1 + m(8η-2η2)/(1-η)4 – (m-1)(20η -27η2 +12η3 –
2η4)/[(2-η)2(1-η)2]}-1] I2(η,m)
6.4
6.4.1
Thermodynamic properties of pure fluids derived from the Helmholtz energy
Thermodynamic properties derived from the Helmholtz energy
All thermodynamic properties of pure fluids can be calculated from the complete
Helmholtz energy equation (6.37) and input data of temperature and density. In case input
data do not contain temperature and density, iterative procedures must be used.
Similar to total Helmholtz energy, all thermodynamic properties is written as a sum
of ideal gas part and residual part. Example of things is pressure equation:
p/ρRT = ρ∂[A/RT]T/∂ρ=ρ∂[Aid/RT+ Ares/RT]T/∂ρ
= ρ∂[Aid/RT]T/ ∂ρ+ ρ∂[Ares/RT]T/∂ρ = 1+ ρ∂[Ares/RT]T/∂ρ
Or
p = ρRT + ρ2RT∂[Ares/RT]T/∂ρ = ρRT + pres = pid+ pres
∂ ART
= RTρ (
)T
∂ρ
res
Where p res
2
The thermodynamic properties of ideal gas are well-known so we present only the
most common thermodynamic properties of residual part calculated from residual
Helmholtz energy as following:
U res
∂ ART
= RT ( −
)V
∂T
- Enthalpy:
H res
∂ ART
∂ ART
= RT [−T (
)ρ + ρ(
)T ]
∂T
∂ρ
- Entropy:
S res = − R[
res
- Internal energy:
2
res
86
res
∂ ART
+T(
)V ]
∂T
res
Ares
RT
∂ ART
∂ 2 ART
= − RT [2(
)ρ + T (
)ρ ]
∂T
∂T 2
res
- Isochoric heat capacity:
cv ,res
res
- Isobaric heat capacity:
c p ,res = c v ,res
∂ 2 ART
∂ ART
[ ρT (
+ρ
+ 1) 2 ]
∂ρ∂T
∂ρ
+R
2 A
∂ ART
2 ∂
RT
[ρ (
) + 2ρ (
) + 1]
∂ρ
∂ρ 2
- Residual Gibbs energy:
Gres = RT (
Ares
RT
res
6.4.2
res
res
res
∂ ART
+(
)ρ )
∂ρ
res
Derivatives of Helmholtz energy
Calculations of residual part of pressure, enthalpy, isochoric heat capacity, isobaric
heat capacity, and other properties require different derivatives of Helmholtz energy. Thus,
this subsection presents necessary derivatives of residual Helmholtz energy for further
calculation.
Derivatives of reference hard chain contribution:
⎛ 2 −η
A0
η (4 − 3η )
=m
− ( m − 1) ln⎜⎜
2
3
RT
(1 − η )
⎝ 2(1 − η )
⎞
⎟
⎟
⎠
∂ (A0 / RT )
5 − 2η
3 + 4η − 7η 2 + 2η 3
=
+m
(2 − η )(1 − η )
∂η
(2 − η )(1 − η ) 3
∂ 2 (A0 / RT ) 11 − 10η + 2η 2
29 − 24η − 7η 2 + 10η 3 − 2η 4
m
=
+
∂η 2
(2 − η ) 2 (1 − η ) 2
(2 − η ) 2 (1 − η ) 4
c
⎛
⎞
⎜ 3c 2 c 3 Exp[ 3 ] ⎛
⎟
2
⎞
∂ A0 / RT
(
)
(
)
1
1
η
5
2
η
−
−
T η ⎜ − 3mη (1 − η ) − (m − 1)
⎜
⎟⎟
(
)
(
)(
)
2
m
η
4
3
η
m
1
η
4
3
η
= 2
+
−
+
−
−
3
⎟⎟
2 −η
∂T
T (1 − η ) ⎜ 1 − c Exp[ c 3 ] ⎜⎝
⎠⎟
⎜
2
T
⎝
⎠
(
)
c
∂ 2 (A0 / RT )
1
⎞
⎛
= 4
⎜ 3c1 ρ .c3 c 2 Exp[ 3 ](− 1 + c 2 Exp[c3 / T ]).tg 3a ⎟
2
4
T
∂T
T (1 − η ) ⎝
⎠
2
(
)
∂ 2 A0 / RT
=
∂T .∂ρ
− 3c1 c 2 c 3 Exp[
(
c3 ⎛
c
⎞
2
]⎜ c 2 Exp[ 3 ] − 1⎟ − (1 − η ) 10 − 8η + η 2 + m − 6 − 28η + 47η 2 − 22η 3 + 3η 4
T ⎝
T
⎠
(
) (
T 2 (1 − η ) (2 − η )
4
where
87
2
))
3
⎡
⎛ c ⎞⎤
η = c1.ρ .⎢1 − c2 Exp⎜ 3 ⎟⎥ ,
⎝ T ⎠⎦
⎣
⎛ π
⎞
c1 = ⎜ m. σ 03 ⎟ ;
⎝ 6
⎠
c 2 = 0.12 ;
c3 =
− 3ε
k
Tg1a=
5 − 8c1ρ + 3c12 ρ 2 + 117c12 ρ 2c25 Exp[
− 3c1 ρc22 Exp[
5c3
6c
7c
] − 51c12 ρ 2c26 Exp[ 3 ] + 9c12 ρ 2c27 Exp[ 3 ]
T
T
T
2c3
4c
3c
](3c1ρ − 4) − 3c1 ρc24 Exp[ 3 ](45c1ρ − 4) + c1 ρc23 Exp[ 3 ](75c1ρ − 28)
T
T
T
c
− 3c2 Exp[ 3 ].(5 − 4c1 ρ + 3c12 ρ 2 )
T
Tg2a= − 10 + 19c1ρ − 11(c1ρ ) + 2(c1ρ ) + 18c13 ρ 3c28 Exp[
2
− 3c12 ρ 2c27 Exp[
3
8c3
9c
] − 2c13 ρ 3c29 Exp[ 3 ]
T
T
7c3
5c
6c
](24c1ρ − 1) − 3c12 ρ 2c25 Exp[ 3 ](84c1ρ − 37 ) + c12 ρ 2c26 Exp[ 3 ](168c1ρ − 29)
T
T
T
c
− 3c2 Exp[ 3 ](− 10 + 27c1 ρ − 23c12 ρ 2 + 6c13 ρ 3 )
T
− c1 ρ c23 Exp[
3c3
](91 − 265c1 ρ + 168c12 ρ 2 )
T
+ 3c1 ρ c22 Exp[
2c3
4c
](43 − 61c1 ρ + 24c12 ρ 2 ) + 3c1 ρ c24 Exp[ 3 ](8 − 75c1 ρ + 84c12 ρ 2 )
T
T
⎛ ⎛
⎞
c ⎞
Tg3a= mη ⎜⎜ 2T ⎜1 − c 2 Exp[ 3 ] ⎟(5 − 8η + 3η 2 ) + c3 tg1a ⎟⎟
T ⎠
⎝ ⎝
⎠
+
1
(2 − η )
2
⎛
⎛
⎞⎞
c
⎜ (m − 1)(1 − η )2 ⎜⎜ 2T ⎛⎜ c 2 Exp[ 3 ] − 1⎞⎟(10 − 19η + 11η 2 − 2η 3 ) + c3 tg 2a ⎟⎟ ⎟
⎟
⎜
T
⎠
⎝ ⎝
⎠⎠
⎝
88
⎛ ⎛
c
c
c
⎞
⎛
⎞⎞
2
− m⎜⎜ c3 ⎜ 3c 2 Exp[ 3 ] − 1⎟ + 2T ⎜ c 2 Exp[ 3 ] − 1⎟ ⎟⎟(1 − η ) (4 − 3η ) − 6myc 2 c3 Exp[ 3 ](1 − η )(5 − 3η )
T
T
T
⎠
⎝
⎠⎠
⎝ ⎝
Derivatives of first-order perturbation term:
A1
⎞
⎛ ε ⎞ ⎛ 6
= −2πρm 2 ⎜ ⎟σ 3 ⎜ ∑ ai (m )η i ⎟
RT
⎝ kT ⎠ ⎝ i =0
⎠
6
A1
ep
1
= −12m
a η i +1
3 ∑ i
RT
T ⎛
c ⎞ i =0
⎜1 − c2 Exp[ 3 ] ⎟
T ⎠
⎝
Or
(
)
6
∂ A1 / RT
ep
1
(i + 1)aiη i
= −12m
∑
3
∂η
T ⎛
c ⎞ i =0
⎜1 − c2 Exp[ 3 ] ⎟
T ⎠
⎝
(
)
(
)
6
1
∂ 2 A1 / RT
ep
=
−
12
i (i + 1)aiη i −1
m
3 ∑
2
∂η
T ⎛
c ⎞ i =0
⎜1 − c2 Exp[ 3 ] ⎟
T ⎠
⎝
ep ⎛ − 3
c ⎛
c ⎞
∂ 2 A1 / RT
c1ρ .c1c2c3 Exp[ 3 ]⎜1 − c2 Exp[ 3 ] ⎟
= 12m 2 ⎜
T ⎜⎝ T
T ⎝
T ⎠
∂T .∂ρ
(
)
6
⎞
i −1
i⎟
(
)
i
i
1
a
η
c
+
+
∑
i
1 ∑ (i + 1)aiη ⎟
i =0
i =0
⎠
2 6
1
∂ A1 / RT
ep
.
= 12m 2
T ⎛
∂T
c3 ⎞
⎜1 − c2 Exp[ ] ⎟
T ⎠
⎝
⎛
⎜
⎜−3
c3 6
c3 6
1
3
1
i
(
)
+
+
c
ρ
c
c
Exp
a
i
η
c
c
Exp
[
]∑ a iη i +1 +
.
[
]
1
⎜
∑
1
2 3
i
3 2 3
2
T i =0
T ⎛
T i =0
c3 ⎞
c3 ⎞
⎛
⎜ T
⎜1 − c 2 Exp[ ] ⎟
⎜1 − c 2 Exp[ ] ⎟
⎜
T ⎠
T ⎠
⎝
⎝
⎝
(
)
⎞
⎟
i +1 ⎟
a iη ⎟
∑
i =0
⎟
⎟
⎠
6
∂ 2 A1 / RT
1
tg 2b
= tg 3b + tg 4b −
3
2
∂T
c3 ⎞
⎛
⎜1 − c2 Exp[ ] ⎟
T ⎠
⎝
where:
Tg1 = η
⎤
c3 c 2 Exp[c3 / T ] ⎡ c3 c 2 Exp[c3 / T ]
c c Exp[c3 / T ]
Sum3 + 6 3 2
Sum 2 − 3c3 Sum 2 − 6 Sum 2⎥
⎢9η
4
c4T
c4
c4
⎣
⎦
89
Tg2b= 12m
⎤
2
ep ⎡ c3 c 2 Exp[c3 / T ]
Sum 2 + 2 Sum1 + tg1⎥
⎢6η
3
T ⎣
c4T
T
⎦
Tg3b= − 12m
Tg4b= 72m
⎡ c3 c 2 Exp[c3 / T ]
⎤
c3
ep
+
+
c
c
Exp
[
c
/
T
]
12
3
6
⎢
⎥ Sum1
3
2
3
T
c 44T 4
c
T
4
⎣
⎦
⎡ c c Exp[c3 / T ]
⎤
ep
c c Exp[c3 / T ]⎢3η 3 2
Sum2 − Sum1⎥
4 3 2
c T
c4T
⎣
⎦
4
4
6
6
6
i =0
i =0
i =0
Sum1= ∑ aiη i +1 ; sum2= ∑ ai (i + 1)η i ; sum3= ∑ i (i + 1)aiη i −1
ep =
ε
⎛c ⎞
; c4= 1 − c2 Exp⎜ 3 ⎟ ;
k
⎝T ⎠
Derivatives of second-order perturbation term:
A2
8η − 2η 2
20η − 27η 2 + 12η 3 − 2η 4 ⎞
⎛ 6
⎞⎛
⎛ ε ⎞
⎟
(
)
+
−
1
m
= −πρm 3 ⎜ ⎟ σ 3 ⎜ ∑ bi (m )η i ⎟⎜⎜1 + m
⎟
RT
(1 − η )4
[(1 − η )(2 − η )]2
⎝ kT ⎠
⎝ i =1
⎠⎝
⎠
2
−1
A2
1
1
ep 2
sum1
= −6m 2 2
3
RT
T ⎛
c3 ⎞ tgz
⎜1 − c 2 Exp[ ] ⎟
T ⎠
⎝
(
)
2
∂ A2 / RT
tg 5
1
2 ep
= 6m
2
3
2
∂η
T ⎛
c ⎞ tgz
⎜1 − c 2 Exp[ 3 ] ⎟
T ⎠
⎝
= 6m 2
(
6
∑bη
i =0
i +1
i
1
1 ⎛ tg 5
ep 2
⎜⎜
2
3
T ⎛
c3 ⎞ tgz ⎝ tgz
⎜1 − c 2 Exp[ ] ⎟
T ⎠
⎝
ep 2
1
1 6
− 6m
bi (i + 1)η i
∑
2
3
T ⎛
c ⎞ tgz i =0
⎜1 − c 2 Exp[ 3 ] ⎟
T ⎠
⎝
2
6
⎞
i +1
−
b
η
bi (i + 1)η i ⎟⎟
∑
∑
i
i =0
i =0
⎠
6
)
2
∂ 2 A2 / RT
1
1
2 ep
=
−
6
m
2
2
3
∂η
T ⎛
c ⎞ tgz
⎜1 − c 2 Exp[ 3 ] ⎟
T ⎠
⎝
6
⎛ ⎛ 2.tg 5 2 tg 6 ⎞ 6
⎞
2.tg 5 6
i +1
i
i −1
⎜⎜
⎟
⎟
(
)
(
)
−
b
η
b
i
1
η
b
.
i
.
i
1
η
−
+
+
+
∑
∑
∑
i
i
i
⎟
⎜ ⎜ tgz 2
⎟
tgz
tgz
i =0
i =0
⎠ i =0
⎝⎝
⎠
90
(
)
ep 2 m 2 c 2 .c3 .Exp[c3 / T ]
∂ 2 A2 / RT
=
(tg 30 − tg 31 − tg 32)
∂T .∂ρ
c4
(
)
∂ A2 / RT
1
1
ep 2
= 6m 2 3
3
∂T
T ⎛
c ⎞ tgz
⎜ 1 − c 2 Exp [ 3 ] ⎟
T ⎠
⎝
⎞
⎛⎛
⎞
⎟
⎜⎜
⎟ 6
6
[
/
]
c 2 c3 Exp[c3 / T ] ⎟
c
c
Exp
c
T
⎟
⎜⎜
3
biη i +1 − 3η 2 3
bi (i + 1)η i ⎟
∑
∑
⎜⎜ 2 + 3 ⎛
⎟
c ⎞
c ⎞
⎛
⎟
⎜ ⎜⎜
T ⎜1 − c 2 Exp[ 3 ] ⎟ ⎟⎟ i =0
T ⎜1 − c 2 Exp[ 3 ] ⎟ i =0
⎟
⎜
T ⎠
T ⎠⎠
⎝
⎝
⎠
⎝⎝
⎛
⎞
⎜
⎟
1
1 ⎜ T .tg 7 c 2 c3 Exp[c3 / T ] ⎟ 6
2 ep
6
.
biη i +1
+ 6m
∑
3
⎜
⎟
c
tgz
T3 ⎛
tgz
⎛
⎞ i =0
c ⎞
⎜⎜
⎜1 − c 2 Exp[ 3 ] ⎟ ⎟⎟
⎜1 − c 2 Exp[ 3 ] ⎟
T ⎠⎠
⎝
T ⎠
⎝
⎝
2
(
)
⎞
c c Exp[c3 / T ] ⎛ 12c 2 c3 Exp[c3 / T ] 3c3
∂ 2 A2 / RT
ep 2 m 2
⎜⎜
=
[−6 sum1 2 3
+
+ 6 ⎟⎟
3 4
2
c 4T
c 4T
T
∂T
c 4 T tgz
⎝
⎠
+
⎞
⎞
36c 2 c3 Exp[c3 / T ] ⎛⎜ ⎛ c 2 c3 Exp[c3 / T ] ⎛
⎞
6tg 7
⎟ − 6tg 43 ⎟ ]
⎜
⎜
⎟
η
sum
sum
sum
3
2
−
1
−
2
1
⎜
⎟
⎟
⎜⎜
⎟
c 4T
c 4T
tgz
⎝
⎠
⎠
⎝⎝
⎠
where
8η − 2η 2
tg 3
Tgz = 1 + m
+ (1 − m ) 2
4
tg1
tg1 tg 2 2
Tg1 = 1 - η
Tg2 = 2 – η
T12g= (1 − η ) (2 − η ) =tg12tg22
2
2
Tg3 = 20 η - 27 η 2 + 12 η 3 - 2 η 4
Tg4=20-54 η +36 η 2-8 η 3.
Tg5=
m
1− m
(tg 4 + 2tg 3 / tg 2 + 2tg 3 / tg1)
8 − 4η + 4 8η − 2η 2 / tg1 +
4
t12 g
Tg1
(
(
)
)
91
Tg6=
m
1− m
8(8 − 4η ) / tg1 + 4 + 20 8η − 2η 2 / tg12 +
− 54 + 72η − 24η 2 + (4tg 4 / tg 2) + (4tg 4 / tg1)
4
12
t
g
Tg1
(
(
)
)
((
+ (6tg 3 / tg 2 ) + (8tg 3 / tg1 / tg 2 ) + (6tg 3 / tg1 )
2
)
)
2
Tg7=
ym((16η − 4η 2 ) / tg15 + (4 − 2η ) / tg14 ) +
η (1 − m )
t12 g
((tg 3 / tg 2) + (tg 3 / tg1) + (tg 4 / 2))
Tg30=
18c1 2tg 7
18.2.tg 7
12.2.tg 7
Sum2 −
Sum1 −
Sum1
2
2
2
2
T tgz
ρ .c 4T .Tgz
ρ .c 2 c3 Exp[c3 / T ]c 42T .tgz 2
Tg31=
18c1η
12c1c 4
6(tg 27 + tg 28 + tg 29 )
Sum3 +
Sum1 +
Sum2
2
3 2
2
c 2 c3 Exp[c3 / T ].T .tgz
T tgz
ρ .c 4 T .Tgz
Tg32=
36c1tg 7
12(12tg 7 2 )
Sum
1
+
Sum2
T 2 .tgz 2
ρ .c 43T 2 .Tgz 3
Tg28=
(1 − m ) ((60η − 324η 2 + 324η 3 − 96η 4 ) + 18η 2 tg 3 / tg 2 2 + 24η 2 tg 3 / tg 2 / tg1 + 6ηtg 3 / tg 2)
t12 g
Tg29=
(1 − m ) (18η 2 (tg 3) / tg12 + 6η (tg 3) / tg1 + 6η 2 (tg 4) / tg 2 + 6η 2 (tg 4) / tg1)
t12 g
Tg43=6sum1-
4c 2 .c3 .Exp[c3 / T ]
(3ηsum 2 − 6tg 7 sum1 / tgz ) +
c 4T
⎞
c2 .c3 .Exp[c3 / T ] ⎛ 9c 2 .c3 .Exp[c3 / T ]
6c .c .Exp[c3 / T ]
3c sum2
[η ⎜⎜
ηsum3 + 2 3
sum2 − 3
− 6 sum2 ⎟⎟ −
c 4T
c 4T
c 4T
T
⎝
⎠
36c 2 .c3 .Exp[c3 / T ]
tg 42
tg 7ηsum2 +
sum1]
c 4Ttgz
tgz
tg 42 = −(tg 40 + tg 41 +
(1 − m )tg10 + 12(1 − m)c 2 .c3 .Exp[c3 / T ] tg 4(1 / tg1 + 1 / tg 2)] +
c 4Ttg12 tg 2 3
(tg1tg 2)2
36c 2 .c3 .Exp[c3 / T ]
tg 7tg 7
c 4Ttgz
92
c c Exp[c3 / T ] y
⎛c
⎞
tg10 = −3 y⎜ 3 + 2 ⎟tg 4 + 2 3
(
120 − 810 y + 864 y 2 − 264 y 3 )
c 4T
⎝T
⎠
⎞⎛ 1
(1 − m)ηtg 3 ⎛⎜ c2 c3 Exp[c3 / T ]η ⎛⎜ 23 − 30η + 10η 2 ⎞⎟ ⎛ 12c2 c3 Exp[c3 / T ] 6c3
1 ⎞ ⎞⎟
⎜
⎟
⎜
tg 41 =
+
−
−
+
18
12
⎟⎜ tg1 tg 2 ⎟⎟ ⎟
⎜ 2 − 3η + η 2 2 ⎟ ⎜⎝
c 4T
c 4T
T
(tg1.tg 2)2 ⎜⎝
⎠⎠
⎠⎝
⎠
⎝
2
2
2
η 8η − 2η ⎛ 12c3
m c 2 c3 Exp[c3 / T ] 24η 32 + 14η − η
⎞
[
−
+ 24 ⎟ +
tg 40 =
⎜
4
c 4Ttg1
tg1
tg1
tg1
⎝ T
⎠
(
(
)
) (
)
12(c3 + 2T )(η − 2)η
c 2 c3 Exp[c3 / T ]
η (48 − 68η ) +
]
c 4T
T
6.5
Methodology and strategy for development of PC-SAFT equation of state have been
studied. PC-SAFT EOS for pure fluids by modifying SAFT equation of state with an
application of perturbation theory of Barker and Henderson is reviewed. In PC-SAFT,
reference hard chain system is based on Carnahan-Starling equation. First- and secondorder perturbation terms are based on results of Barker and Henderson.
Different derivatives of residual Helmholtz energy are derived for two purposes.
Firstly, these derivatives are used to program a fitting package. Secondly, these derivatives
are used to calculate thermodynamic properties of fluids such as pressure, enthalpy, and
entropy.
93
References
[6.1] A. Müller, J. Winkelmann, J. Fischer, Backone family of equations of state: 1.
Nonpolar and polar pure fluids. AIChE J., 42 (1996) 1116-1126.
[6.2] U. Weingerl, M. Wendland, J. Fischer, A. Müller, J. Winkelmann, Backone family of
equations of state: 2. Nonpolar and polar fluid mixtures. AIChE J., 47 (2001) 705- 717.
Perturbation Theory for Chain Molecules, Ind. Eng. Chem. Res. 40 (2001) 1244-1260.
[6.4] J.A. Barker, D. Henderson, Perturbation theory and equation of state for fluids: The
square-well potential, The Journal of Chemical Physics, 47 (1967) 2856-2861.
[6.5] J.A. Barker, D. Henderson, Perturbation theory and equation of state for fluids. II. A
successful theory of liquids, The Journal of Chemical Physics, 47 (1967) 4714-4721.
[6.6] T. Boublik, I. Nezbeda, K. Hlavaty, statistical thermodynamics of simple liquids and
their mixtures, Elsevier, 1990.
[6.7] D.A. McQuarrie, Statistical Mechanics, University Science Books, 2000
[6.8] S. S. Chen, A. Kreglewski, Applications of the augmented Van der Waals theory for
fluids. I. Pure fluids, Ber. Bunsenges. Phys. Chemie 81 (1977) 1048 – 1052.
[6.9] N. F. Carnahan, K.E. Starling, Equation of state for nonattracting rigid spheres, J.
Chem. Phys., 51 (1969) 635-636
Chain molecules with multiple bonding sites, Molec. Phys. 65 (1988) 105 –1079.
state for associating liquids, Ind. Eng.Chem. Res. 29 (1990) 1709–1721.
thermodynamics, J. Stat. Phys. 35 (1984) 19-34.
perturbation theory and integral equations, J. Stat. Phys. 35 (1984) 35 – 47.
94
attraction sites. J. Stat. Phys. 42 (1986) 459 – 476.
polymeri-zation. J. Stat. Phys. 42 (1986) 477 – 492.
[6.16] Yee C. Chiew, Percus-Yevick integral equation theory for athermal hard-sphere
chains. II. Average intermolecular correlation functions, Molecular Physics, 73 (1991) 359373
[6.17] J. Gross, G. Sadowski, Application of perturbation theory to a hard-chain reference
fluid: An equation of state for square-well chains, Fluid Phase Equilib., 168 (2000) 183199.
[6.18] H. Liu, Y. Hu, Molecular thermodynamic theory for polymer systems. II. Equation
of state for chain fluids, Fluid Phase Equilib., 122 (1996) 75-97.
95
7
Description of linear siloxanes with PC-SAFT equation*
Abstract
The chapter is aimed at a thermodynamic description of linear siloxanes by a
molecular based equation of state. First, experimental data of the five linear siloxanes
hexamethyldisiloxane,
octamethyltrisiloxane,
decamethyltetrasiloxane,
dodecamethylpentasiloxane, and tetradeca-methylhexasiloxane are compiled. As data at
higher temperatures are scarce, it is helpful to extrapolate saturated vapor pressures and
liquid densities by appropriate equations using the critical temperatures and pressures. The
three parameters of the molecular based PC-SAFT equation of state can be fitted either
directly to the experimental vapour pressures and saturated liquid densities or to the
extrapolation equations. Comparisons of thermodynamic data from PC-SAFT for MDM
based on these two fitting routes with experimental data show good agreement for both
modes with a better performance in case of fitting to the extrapolation equations. Hence,
PC-SAFT parameters were determined for all five siloxanes by fitting to the extrapolation
equations. Comparisons of resulting PC-SAFT thermodynamic data with a variety of
experimental data show good agreement. Finally, thermodynamic data are presented in
tables and graphs.
7.1
Introduction
Presently there are strong efforts to develop new environmentally friendly processes
for energy conversion. Interesting processes for that purpose are Clausius-Rankine cycles
for conversion of heat to power which use organic substances as working fluids, simply
called organic Rankine cycles (ORC) [7.1, 7.2]. A crucial question is then the selection of
appropriate working fluids for specified upper and lower temperatures [7.3 –7.9]. A related
problem is the selection of working fluids for heat pump cycles (HPC). For cycles which
work at low temperatures, i.e up to about 100°C, alkanes, fluorinated alkanes, ethers and
fluorinated ethers are potential candidates for subcritical and transcritical cycles. For the
thermodynamic description of these substances the molecular based equation of state
BACKONE is appropriate [7.8, 7.10, 7.11]. This is more problematic for cycles in which
* See also: N. A. Lai, M. Wendland, J. Fischer, Description of linear siloxanes with PC-SAFT equation, Fluid
Phase Equilibria 283 (2009) 22-30
96
the working fluids reach temperatures higher than 200°C. Smaller alkanes might be used in
transcritical cycles. But with increasing chain lengths the self ignition temperature
decreases to about 200°C so that longer alkanes which are environmentally friendly and
yield good thermal efficiencies can not be used any more for safety reasons. Fluorinated
alkanes have a strong globing warming potential and extremely long atmospheric lifetimes
and hence should not be used for environmental reasons. Hence, the use of siloxanes [7.3 –
7.6] for higher temperature ORC and HPC, i.g. heat pumps of the HI splitting section of the
sulfur-iodine process for hydrogen manufacture [7.7], was considered.
A problem in the ORC cycles is the heat transfer from the heat carrier to the
working fluid. A pure fluid as working fluid in a subcritical cycle yields a pinch point [7.8]
which limits the transferable heat. In order to increase the transferred heat, one may use
transcritical cycles with pure fluids, or subcritical and transcritical cycles with mixtures.
Hence, in order to optimize an ORC cycle under given conditions for the heat carrier and
the cooling medium, equations of state (EOS) are needed for potential pure or mixed
working fluids.
First, mixtures were considered [7.3] using the cubic Peng-Robinson-Stryjek-Vera
equation [7.12] with the Wong-Sandler mixing rules [7.13]. The limitations of that
approach have, however, already been pointed out elsewhere [7.14]. Recently, pure fluids
were considered in [7.9] using the DIPPR database [7.15] which, however, contains only
pure substance properties. In [7.14] four pure siloxanes were described by short
fundamental equations of state of the Span-Wagner type [7.16, 7.17]. These can in principle
be extended to mixtures but so far this was only done for natural gas mixtures [7.18] and
requires a wealth of experimental data.
In this situation it is challenging to describe the siloxanes by some molecular based
equations of state which can be extended more easily to mixtures [7.11, 7.19]. First, we
tried BACKONE which was recently shown again to be a very good equation of state for
compact molecules [7.20]. As siloxanes have, however, a chain-like structure, we will use
here the PC-SAFT equation [7.19] which shows better performance for these fluids than
BACKONE.
97
Hence, in this chapter we determine parameters of PC-SAFT equation [7.19] for the
pure siloxanes hexamethyldisiloxane, octamethyltrisiloxane, decamethyltetrasiloxane,
dodecamethylpentasiloxane, and tetradecamethylhexasiloxane and compare PC-SAFT
results with experimental data. As data at higher temperatures are scarce, it is helpful to
extrapolate saturated liquid densities and vapour pressures by appropriate equations. The
three parameters of the molecular based PC-SAFT equation of state can be fitted either
directly to the experimental data or to the extrapolation equations.
In Sec. 2 an overview is given over available experimental data. In Sec. 3
extrapolation procedures are discussed first and then the PC-SAFT equation [7.19] is
outlined and the optimization procedure for the determination of the parameters is
presented. In Sec. 4 we investigate two different fitting modes for PC-SAFT. In the first
mode the input data are taken directly experimental data, in the second the input data are
taken from extrapolation equations. In Sec. 5 the PC-SAFT parameters will be given and
comparisons of the PC-SAFT results with experimental data will be shown. Finally, in Sec.
6 thermodynamic data will be presented in tables and graphs.
7.2
Experimental data
Instead of the full chemical names of the siloxanes frequently the following
abbreviations are used: MM for hexamethyldisiloxane (C6H18OSi2), MDM for
octamethyltrisiloxane (C8H24O2Si3), MD2M for decamethyltetrasiloxane, (C10H30O3Si4),
MD3M
for
dodecamethyl-pentasiloxane
(C12H36O4Si5),
and
MD4M
for
tetradecamethylhexasiloxane (C14H42O5Si6). Drawings of these molecules can be found in
the webbook of NIST [7.21].
In this Section we compile available experimental data for MM, MDM, MD2M,
MD3M, and MD4M. The references for these data were found mainly in the DECHEMA
data series DETHERM [7.22]. The data are taken from the original sources as far as
possible. Also helpful are the webbook of NIST [7.21] and the data compilation of the
Thermodynamics Research Center (TRC) which is now also run by NIST.
In Table 7.1 experimental data for the critical temperature Tc, the critical pressure
pc, and the critical density ρc are compiled. These data have been critically reviewed by
others or by us and the values selected for further evaluations in thic chapter are presented.
98
Table 7.1. Critical data of linear siloxanes: experimental (exp) and selected (sel)
Property
Tc [K]
Value
518.7
518.8
518.6
518.45
518.7
pc [MPa]
ρc [mol/l]
Tc [K]
pc [MPa]
ρc [mol/l]
Tc [K]
pc [MPa]
ρc [mol/l]
Tc [K]
pc [MPa]
1.925
1.910
1.910
1.925
1.637
1.637
564.13
565.4
562.9
564.4
564.13
1.4150
1.46
1.4196
1.4150
1.1341
1.152
1.1341
599.4
599.4
599.15
599.4
1.265
1.19
1.19
0.8643
0.8643
629.0
627.6
629.0
0.945
0.945
Type
Source, Uncertainty
MM (C6H18OSi2)
exp
McLure, Dickinson [7.23]
exp
Dickinson, McLure [7.24]
exp
Young [7.25]
exp
Pollnow [7.26, 7.27]
±0.5 K TRC, NIST[7.22],
sel
DETHERM[7.21]
exp
exp
exp
Young [7.25]
sel
±0.01 TRC, NIST[7.22]
exp
Pollnow [7.26, 7.27]
sel
Pollnow [7.26, 7.27]
MDM (C8H24O2Si3)
exp
Lindley, Hershey [7.28]
exp
exp
Young [7.29]
exp
Pollnow [7.26, 7.27]
sel
exp
exp
exp
Young [7.29]
sel
exp
exp
Pollnow [7.26, 7.27]
sel
MD2M (C10H30O3Si4)
exp
exp
exp
Young [7.29]
Pollnow [7.26, 7.27]
sel
exp
Young [7.29]
exp
sel
exp
Pollnow [7.26, 7.27]
sel
Pollnow [7.26, 7.27]
MD3M (C12H36O4Si5)
exp
Young [7.29]
exp
Pollnow [7.26, 7.27]
sel
Young [7.29]
exp
Young [7.29]
sel
Young [7.29]
99
ρc [mol/l]
Tc [K]
pc [MPa]
ρc [mol/l]
0.7143
0.7143
653.2
653.2
0.804
0.804
0.5970
exp
Pollnow [7.26, 7.27]
sel
Pollnow [7.26, 7.27]
MD4M (C14H42O5Si6)
exp
Young [7.29]
sel
Young [7.29]
exp
Young [7.29]
sel
Young [7.29]
exp
sel
This work, Eq. (2)
Tables 7.2 and 7.3 contain temperature ranges, numbers of data points and sources
of experimental vapour pressures ps and saturated liquid densities ρ’. In these compilations
we have not included experimental data which did not seem to be reliable on the basis of
critical investigations made by previous authors or by us.
Table 7.2. Experimental vapour pressures: temperature ranges, numbers of data points and
sources.
Number
Source
of exp data
MM (C6H18OSi2) Tc = 518.7
McLure, Dickinson
491.60 - 518.70
19
[7.23]
309.36 - 411.57
21
Scott et al. [7.30]
313.15 - 373.61
14
Guzman [7.31]
302.78 - 383.30
15
Flaningam [7.27]
MDM (C8H24O2Si3) Tc = 564.3
346.10 - 436.49
12
Flaningam [7.27]
322.44 - 564.13
74
MD2M (C10H30O3Si4) Tc = 599.4
366.20 - 479.17
15
Flaningam [7.27]
MD3M (C12H36O4Si5) Tc = 629.0
395.61-515.36
15
Flaningam [7.27]
MD4M (C14H42O5Si6) Tc = 653.2
449.17-545.71
11
Flaningam [7.27]
Temperature range [K]
Considering the vapour pressures in Table 7.2 we observe that for MM
experimental data are available from 303 K up to 412 K [7.27, 7.30] and close to the critical
temperature [7.23]. For MDM data are available from 322 K up to the critical temperature
[7.28]. For the higher siloxanes, the data were measured up to 133 kPa [7.27] which
corresponds to temperature ranges up to 0.80 Tc for MD2M and 0.84 for MD4M.
100
Table 7.3. Experimental saturated liquid densities: temperature ranges, numbers of data
points and sources.
273.15 – 313.15
273.15 – 353.15
273.12 - 564.13
273.15 - 353.15
273.15 – 353.15
293.15 - 363.15
273.15 – 353.15
273.15-353.15
Number
Source
of exp data
MM (C6H18OSi2) Tc = 518.7
3
Hurd [7.32]
10
Gubareva [7.33]
MDM (C8H24O2Si3) Tc = 564.3
37
5
Hurd [7.32]
MD2M (C10H30O3Si4) Tc = 599.4
5
Hurd [7.32]
8
Golik, Cholpan [7.34]
MD3M (C12H36O4Si5) Tc = 629.0
5
Hurd [7.32]
MD4M (C14H42O5Si6) Tc = 653.2
5
Hurd [7.32]
The experimental situation for the saturated liquid densities displayed in Table 7.3
is still worse. For MDM, Lindley and Hershey [7.28] measured ρ’ from 273 K up to Tc with
a gap in the temperature range between 361 K and 426 K. For the other siloxanes the
highest temperature is about 353 K [7.32, 7.33, 7.34].
Additional pvT-data sets were given by McLure et al. [7.35] and Marcos et al.
[7.36]. In [7.35] liquid densities at ambient pressure are given for all linear siloxanes
considered here, whilst in [7.36] densities in the vapor phase have been reported for MM
and MDM.
7.3
7.3.1
Equations
Extrapolation equations
A usual method to determine parameters of molecular based equations of state
(EOS) is a fit to experimental vapour pressure and saturated liquid density data. The
particular problem here is that for most siloxanes the experimental data are limited to lower
temperatures. In case that data at higher temperatures are also available (MM, MDM) there
may occur a gap in an intermediate temperature range. At this point we may follow two
routes. The first is simply to fit the parameters of the EOS to the available experimental
data. In the second, one may use extrapolated data for fitting the EOS-parameters to a
101
larger temperature range which requires appropriate equations. This extrapolation may be
only an upward extrapolation from the low temperatures or a matching extrapolation from
low and high temperatures.
For the extrapolation of vapour pressures we considered the Antoine-equation
[7.37], the Wagner-equation [7.38], and the Iglesias-Silva-equation [7.39]. The latter could
not be used as it requires the vapour pressure at the triple point which is not available for
any of the siloxanes. Moreover, we found that for upward extrapolation the Wagnerequation is in general more useful than the Antoine-equation. Hence we use here the
equation of Wagner [7.38] which writes as
ln(pr) = (1/Tr) [A(1-Tr) + B(1-Tr)1.5 + C(1-Tr)3 + D(1-Tr)6],
(7.1)
where Tr = T/Tc and pr = ps/pc and A, B, C, and D are fit parameters. Both Tc and pc
are taken from Table 7.1 and the vapour pressure can be obtained from the sources listed in
Table 7.2. The fit criterion is to minimize Σi [(ps,exp,i - ps,cal,i ) / ps,exp,i]2.
The extrapolation of saturated liquid densities was studied recently [7.40]. It was
found that for MDM the best extrapolation is achieved with the equation of Spencer and
Danner [7.41] which we write here in the form
ln ρ’ = ln ρp – (lnZp)(1-Tr)(2/7).
(7.2)
Eq. (2) requires the experimental critical temperature Tc and the experimental
critical pressure pc and contains the pseudo-critical density ρp as the only fitting parameter.
The pseudo-compression factor Zp is related to Tc, pc and ρp by Zp = pc/ρpRTc. The
objective function for the parameter fitting is the minimization of Σi [(ρ’exp,i - ρ’cal,i /
ρ’exp,i]2. For the siloxanes considered here, experimental saturated liquid densities are
available for all substances in the temperature range from 273.15 K to 353.15 K and in
addition for MDM from 273.12 K to 564.13 K. In order to test the performance of Eq. (2)
for the particular situation of the linear siloxanes, we fitted it to the five data points of Hurd
[7.32] for MDM and compared the extrapolation against the experimental data of Lindley
and Hershey [7.28] over the whole temperature range. In addition, we also tested Eqs. (3),
(4), and (6) from [7.40]. From the results shown in Figure 7.1 we see that the extrapolation
with Eq. (2) is surprisingly good and will be used throughout thic chapter.
102
4.0
100(ρ 'exp -ρ 'cal )/ρ'exp
3.0
2.0
1.0
0.0
-1.0
-2.0
273
353
564
-3.0
-4.0
260
360
460
560
T [K]
Figure 7.1. Deviations of extrapolated saturated liquid densities from experimental data
points [7.28] for MDM. The fitting was made to the experimental data of [7.32] between
273.15 K and 353.15 K. The extrapolation equations used are: - o - Eq.(3) from [7.40], -■Eq.(4) from [7.40], -●- present Eq. (2), -Δ- Eq. (6) from [7.40].
7.3.2
PC-SAFT equation
The working equations of the molecular based PC-SAFT equation of Gross and
Sadowksi [7.19] shall be summarized here for the case of pure fluids. The underlying
model assumes the molecules as chains composed of spherical segments. The pair potential
for the segments of the chain is taken as square well with step potential according to Chen
and Kreglewski [7.42]. The step ranges from 0.88 σ to σ and has a height of 3ε, the well
has the depth ε. Following the perturbation theory of Barker and Henderson [7.43] a
temperature dependent hard sphere diameter d(T) for the segments is introduced as
d = σ [1-0.12exp(-3ε/kT)],
(7.3)
with k being the Boltzmann-constant. Let us now consider a system of N chain
molecules where ρ is the number density of chains and m the number of segments in the
chain. For that system the residual Helmholtz energy Ares is written according to the second
order perturbation theory of Barker and Henderson [7.44] as
Ares/NkT = AHC/NkT + A1/NkT + A2/NkT.
(7.4)
In Eq. (4) AHC is the residual Helmholtz energy of the hard chains and A1 and A2
are the first and second order perturbation terms.
103
The hard chain contribution AHC is obtained by using the SAFT approach of
Wertheim [7.44, 7.47] and its applications by Chapman et al. [7.48, 7.49]. As in [7.19] only
the compression factor ZHC is given we follow here another SAFT application [7.50] and
present AHC explicitly
AHC/NkT = m[y(4-3y)/(1-y)2] - (m-1)ln[(1- ½ y)/(1-y)3],
(7.5)
with y = (π/6)mρd3 being the hard segment packing fraction.
The perturbation terms are expressed by power series in the packing fraction y as
A1/NkT = - 2πρσ3 m2(ε/kT) ∑i ai(m)yi
(i = 0, 1,…6),
(7.6)
A2/NkT = - πρσ3 m3(ε/kT)2 [kT(∂ρ/∂p)HC] ∑i bi(m)yi (i = 0, 1,…6), (7.7)
where the compressibility (∂ρ/∂p)HC of the hard chain reference system can be
derived straightforward from the residual Helmholtz energy AHC given in Eq. (5). The
dependence of the coefficients on the segment number m was shown earlier [7.51] to be
well represented by an ansatz of Liu and Hu [7.52]
ai(m) = a0i + [(m-1)/m] a1i + [(m-1)(m-2)/m2] a2i,
i = 1,2,…6,
(7.8)
bi(m) = b0i + [(m-1)/m] b1i + [(m-1)(m-2)/m2] b2i,
i = 1,2,…6.
(7.9)
The 36 coefficients a0i, a1i, a2i, b0i, b1i, b2i, i = 1, 2,…6, were fitted to experimental
data of alkanes and are given in Table 1 of [7.19]. It seems worth to mention that this
procedure followed that in BACKONE [7.10] where the contribution of the attractive
dispersion forces to the Helmholtz energy was also obtained by a fit to experimental data.
The PC-SAFT equation as described above has three substance-specific parameters,
the energy parameter ε representing the well depth, the size parameter σ representing the
segment diameter and the chain length m. These three parameters shall be determined here
for the linear siloxanes by fitting to experimental or extrapolated vapour pressures and
saturated liquid densities. The data selection will be the subject of the next chapter. Here,
we deal with the fitting procedure for finding ε, σ, and m for given sets of ps,exp,i and ρ’exp,i.
In the procedure for finding optimized parameters ε, σ, and m one has to calculate
for some given parameter vector k = {εk, σk,mk} at some given temperature Ti the vapour
pressure ps,cal,i (k) and the saturated liquid density ρ’cal,i(k). This is done by starting from
104
the expressions for the residual Helmholtz energy, Eqs. (3) to (9), and by calculating with
the parameter vector k at a given temperature Ti the vapour pressure ps,cal,i and the saturated
liquid density ρ’cal,I from the equilibrium conditions of equal pressure p and equal Gibbs
energy G/NKT for liquid and vapour. The objective function D(k) to be minimized is
D(k) = wp Σi [(ps,exp,i - ps,cal,i(k)) / ps,exp,i]2 + wρ Σj [(ρ’exp,j - ρ’cal,j(k)) / ρ’exp,j]2.
(7.10)
In this optimization the vapour pressures and the saturated liquid densities may be
taken at different temperatures which is indicated by the different indices i and j. Moreover,
different weights wp and wρ may be assigned to the vapour pressures and the saturated
liquid densities. If nothing else is said about these weights they are taken to be unity. In
order to find the minimum of D(k) we used the Simplex algorithm of Nelder and Mead
[7.53].
7.4
Fitting modes for PC-SAFT
The molecular based PC-SAFT equation contains the three substance specific
parameters ε, σ, and m. In order to determine these by fitting, at least three experimental
data are required. The item at issue is now, which data should be used and how many. In
the case of the molecular based BACKONE equation two fitting modes were tested [7.10].
In the correlative mode a large number of data was used, whereas in the predictive mode
the parameters were fit to only two vapour pressures and two saturated liquid densities. It
was found that the results from the predictive mode were rather similar to those from the
correlative mode. We should, however, add that the two vapour pressures were taken at low
and high reduced temperatures Tr and the same holds for the saturated liquid densities. The
particular problem here is that for most siloxanes the experimental data are limited to lower
reduced temperatures.
The question is now, how does the accuracy of PC-SAFT depend on different data
sets used for the fitting of the three parameters. We investigate the problem for MDM,
because for that substance the experimental data set extends from low to high temperatures
[7.27], [7.28, 7.32]. The four fitting modes considered are described in Table 7.4, which
also contains the resulting PC-SAFT parameters. The vapour pressures ps and the saturated
liquid densities ρ’ resulting from the four different modes for PC-SAFT are compared with
the experimental data of [7.28] in Figures 7.2 and 7.3.
105
Table 7.4. PC-SAFT parameters for MDM resulting from different fitting modes
Mode
ε/k [K]
σ [nm]
m
1
208.34
0.40965
5.4119
2
208.51
0.41008
5.3972
3
213.38
0.41689
5.1504
4
212.36
0.41575
5.2055
Data used for fit of PC-SAFT parameters
ps at 346.10 K and 436.49 K from [7.27]
ρ’ at 273.15 K and 353.15 K from [7.32]
all ps between 346.10 K and 436.49 K from [7.27]
all ρ’ between 273.15 K and 353.15 K from [7.32]
Eq. (1) is based on all ps in 346.10 K - 436.49 K [7.27].
From Eq. (1) 19 ps-values are generated up to 0.9 Tc.
Eq. (2) is based on all ρ’ in 273.15 K - 353.15 K [7.32].
From Eq. (2) 27 ρ’-values are generated up to 0.9 Tc.
Eq. (1) is based on all ps in 346.10 K - 436.49 K [7.27] and
in 322.44 – 562.13 [7.28]. From Eq. (1) 23 ps-values are
generated up to 0.96 Tc.
From From ρ’ correlation, Eq. (6) from [7.28], 28 ρ’-values
are generated up to 0.96 Tc.
We learn from the parameters in Table 7.4 as well as from the results displayed in
Figure 7.2 and 7.3 that modes 1 and 2, which both use experimental low temperature data
directly, yield rather similar results. Further, we see that modes 3 and 4 yield also results
which are rather similar and are clearly in better agreement with experimental data than
those from mode 1 and 2. It is not surprising that mode 4 yields good results as all available
experimental data, also those at high temperatures, have been used in determining PCSAFT parameters. The crucial point is now, that mode 3 uses as input the same low
temperature experimental data as mode 2. The difference between these two modes is in the
fact that mode 3 uses as intermediate step the extrapolation equations Eq. (1) and (2) which
take into account the experimental values of the critical temperature Tc and the critical
pressure pc.
106
100(ps,exp -ps,cal)/ps,exp
4.0
2.0
0.0
-2.0
-4.0
-6.0
-8.0
300
350
400
450
500
550
600
T [K]
Figure 7.2. Comparison of vapour pressures ps for MDM from four different PC-SAFT
fitting modes with exp data of Lindley and Hershey [7.28]. The fitting modes are described
in Table 7.4. Δ mode 1, X mode 2, O mode 3, + mode 4.
5.0
100(ρ'exp -ρ'cal)/ρ'exp
4.0
3.0
2.0
1.0
0.0
-1.0
-2.0
200
300
400
500
600
T [K]
Figure 7.3. Comparison of saturated liquid densities ρ’ for MDM from four different PCSAFT fitting modes with exp data of Lindley and Hershey [7.28]. The fitting modes are
described in Table 7.4. Δ mode 1, X mode 2, O mode 3, + mode 4
The conclusion of this study is that fitting mode 3 is best suited for determining PCSAFT parameters of all considered linear siloxanes. It requires as experimental input data
low temperature vapour pressures and saturated liquid densities as well as the critical
107
temperature and the critical pressure for a given substance, which all are available for the
considered linear siloxanes.
7.5
Results and discussion
In Sec. 4 we came to the conclusion that in case of linear siloxanes the best fitting
mode for the PC-SAFT equation is mode 3. For that one has to find first extrapolation
equations for the vapour pressure and the saturated liquid densities. Appropriate
extrapolation equations are Eq. (1) for the vapour pressures as suggested by Wagner [7.38]
and Eq. (2) for the saturated liquid densities as suggested by Spencer and Danner [7.41].
The PC-Saft parameters are fitted to vapour pressures and saturated liquid densities
resulting from these extrapolation equations.
The experimental sources and the temperature ranges of the input data for the
Wagner equation, Eq. (1), are listed in Table 7.5 which also contains the fitted parameters
of that equation. The experimental sources and the temperature ranges of the input data for
the Spencer-Danner equation, Eq. (2), are listed in Table 7.6 which also contains the fitted
pseudo-critical density ρp of that equation. The critical data required in Eqs. (1) and (2) are
the selected data given in Table 7.1.
Table 7.5. Coefficients of the Wagner-equation [7.38] for the vapour pressures, Eq. (1),
fitting range in reduced temperatures Tr, and sources of underlying experimental data. All
data are used with the same weight. The critical data are taken from Table 7.1.
A
B
C
D
MM
-7.20937
-0.729325
0.738444
-27.0088
MDM
MD2M
MD3M
MD4M
-9.09304
-9.79160
-9.08403
-9.61691
3.07918
3.23549
1.08936
2.13118
-9.17109
-9.54538
-8.00774
-14.1658
5.77518
0.16326
-4.92236
51.6756
108
Fitting range Sources of exp.
Data
in Tr
0.58-0.79 and [7.23],[7.27],
0.95-1.00
[7.30],[7.31]
0.61-0.77
[7.27]
0.61-0.80
[7.27]
0.63-0.82
[7.27]
0.69-0.84
[7.27]
Table 7.6. Pseudo-critical densities ρp of the Spencer-Danner equation [7.40, 7.41] for the
saturated liquid densities, Eq. (2), fitting range in reduced temperatures Tr, and sources of
underlying experimental data. All data are used with the same weight. The critical data are
taken from Table 7.1.
ρp
MM
MDM
MD2M
MD3M
MD4M
1.6655
1.1608
0.9085
0.7171
0.5970
Fitting range
in Tr
0.53 – 0.68
0.48 – 0.63
0.46 – 0.61
0.43 – 0.56
0.42 – 0.54
Sources of exp.
data
[7.32], [7.33]
[7.32]
[7.32], [7.34]
[7.32]
[7.32]
The resulting PC-SAFT parameters are given in Table 7.7. This table contains also
the ranges of reduced temperatures from which the vapour pressures and the saturated
liquid densities have bee taken from Eqs. (1) and (2) with intervals of 10 K for fitting the
PC-SAFT parameters. Regarding the results in Table 7.7 one observes trends in all three
parameters which, however, are not very regular. The trend in m is satisfying because it had
to be expected according the physical meaning of this parameter.
Table 7.7. PC-SAFT parameters of linear siloxanes and fitting ranges in reduced
temperatures Tr in the fits to Eq. (1) and (2). In the fitting ranges the data were taken with
intervals of 10 K.
MM
MDM
MD2M
MD3M
MD4M
ps fitting range ρ’ fitting range
in Tr
in Tr
0.58 - 0.90
0.52 - 0.90
0.57 - 0.90
0.48 - 0.90
0.61 - 0.90
0.45 - 0.90
0.63 - 0.90
0.43 - 0.90
0.67 - 0.90
0.41 - 0.90
ε/k [K]
σ [nm]
m
209.4933
213.3824
212.6004
215.3387
219.1483
3.97997
4.168933
4.24578
4.36766
4.51132
4.24260
5.150368
6.19610
6.95400
7.48151
The extrapolation equations as well as the thermodynamic properties resulting from
the PC-SAFT equation will be discussed in the following subsections. We remind that
similar studies were made previously for the refrigerants with BACKONE [7.54, 7.55].
7.5.1
Hexamathyldisiloxane (MM)
From the extrapolation equation for the vapour pressure for MM we observed that
the experimental data [7.23, 7.27, 7.30, 7.31] are reproduced by Eq. (1) with deviations less
than 0.7%. Moreover, we found that the deviations between the vapour pressures of
109
Flaningam and of Scott et al. amount up to 0.55%. Regarding the saturated liquid densities,
the pseudo-critical density ρp from Eq. (2) is seen from Table 7.6 to differ from the selected
critical density ρc in Table 7.1 by 1.7%. The experimental saturated liquid densities used
for fitting are reproduced by Eq. (2) with a maximum deviation of 0.26%.
Next we show results from PC-SAFT equation. The comparison of the vapour
pressures from PC-SAFT with the experimental data and with the extrapolated values is
shown in Figure 7.4. The relative differences Δps = (ps,exp - ps,cal)/ ps,exp of the experimental
values are seen to range from –2.5% to +1% except in the critical region. Moreover, PCSAFT results are in good agreement with the extrapolation equation. In Figure 7.5 a
comparison of densities is given which includes the saturated liquid densities of Hurd
[7.32] and Gubareva [7.33] and those from the extrapolation equation as well as the liquid
densities at 1 atm of McLure et al. [7.35]. This figure shows that there is a discrepancy
between the measured densities at the lowest temperatures of about 1% and that the PCSAFT predictions are within the accuracy of the experimental data. Moreover, PC-SAFT
results agree with the extrapolated data within 1% up to Tr = 0.89. Finally, we show in
Figure 7.6 deviations of predicted PC-SAFT densities in the vapour phase from the
experimental values of Marcos et al. [7.36] who measured from 448 K up to 573 K
bracketing the critical temperature at pressures from 0.065 MPa to 0.375 MPa. We see that
most densities predicted with PC-SAFT are within ± 0.5 % of experimental values with
maximum deviations of -2 % to +1%. One might argue that because of the low pressure the
vapour densities should be close to the ideal gas law. A test calculation, however, shows
that at 448 K and 360 kPa the ideal gas volume deviates from the measured value by more
than 12%. Hence, we come to the conclusion that the PC-SAFT predictions are remarkably
good for the vapour phase.
110
2.0
1.0
0.0
-1.0
-2.0
-3.0
-4.0
-5.0
-6.0
280
518.7
330
380
430
480
530
T [K]
Figure 7.4. Deviation of PC-SAFT vapour pressures for MM from experimental data of ■
Flaningam [7.27], ● Guzman [7.31], Scott et al. [7.30], Δ McLure and Dickinson [7.23],
and from extrapolation Eq. (1) --- . The vertical line indicates the critical temperature Tc.
1.2
100(ρexp -ρcal)/ρexp
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
250
300
350
400
450
T [K]
Figure 7.5. Deviations of PC-SAFT saturated liquid densities for MM from experimental
data of ■ Hurd [7.32] and ● Gubareva [7.33], from extrapolation Eq. (2) ---, and of PC
SAFT densities at 1 atm from exp data of ♦ McLure et al. [7.35].
111
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
0
0.1
0.2
0.3
0.4
P [MPa]
Figure 7.6. Deviations of PC-SAFT vapour densities for MM from experimental data of
Marcos et al. [7.36]. ◊ at 448.26K, at 498.28K, Ο at 523.15K, × at 548.04K, and Δ at
573.04K.
7.5.2
Octamethyltrisiloxane (MDM)
In order to check the consistency of the experimental vapour pressures for MDM we
fitted Eq. (1) to all vapour pressures of Flaningam [7.27] and Lindley and Hershey [7.28].
We found that all data are reproduced with deviations less than 1% with some scattering in
each data series. At about 350 K the vapour pressures from [7.28] are on the average about
0.5% higher than those of [7.27], at about 430 K the differences tend to zero.
The experimental saturated liquid densities of Hurd [7.32] and those of Lindley and
Hershey [7.28] have already been compared in [7.28] and agree within 0.01%.
For the extrapolation equations on which the PC-SAFT parameters are based we
used as experimental input data only the vapour pressures of Flaningam [7.27] and the
saturated liquid densities of Hurd [7.32] as can be seen from Tables 7.5 and 7.6. The
pseudo-critical density ρp from Eq. (2) presented in Table 7.6 differs from the selected
critical density ρc in Table 7.1 by 2.4 %.
Next we turn to the results from PC-SAFT equation. We remind that fitting mode 3
is used and that the underlying experimental data are the vapour pressures of Flaningam
[7.27] and the saturated liquid densities of Hurd [7.32], both at low temperatures, as well as
the critical temperature and the critical pressure of Lindley and Hershey [7.28]. For the
112
vapour pressures it is seen from Figure 7.2, mode 3, that the relative differences Δps =
(ps,exp - ps,cal )/ps,exp between the PC-SAFT predictions and the experimental data of [7.28]
range from –2% to +3.0 % except at the lowest temperature. We remind that the
experimental vapour pressures of [7.27] and [7.28] agree within 0.5 %. For the saturated
liquid densities Figure 7.3, mode 3, shows that the relative differences Δρ’ = (ρ’exp - ρ’cal)/
ρ’exp between the PC-SAFT predictions and the experimental data of [7.28] range from -1%
to +2.0 %. In addition, Figure 7.7 shows the deviations of PC-SAFT saturated liquid
densities for MDM from experimental data of Hurd [7.32] and from Lindley and Hershey
[7.28] and of PC SAFT homogeneous densities at 1 atm from experimental data of McLure
et al. [7.35] in the temperature range from 273 to 413 K. It is seen that PC-SAFT represents
all densities within -0.6% to +0.0% with the exception of the density at 362 K from [7.28]
which also drops out from a correlation with Eq. (2). In Figure 7.8 we show deviations of
PC-SAFT densities in the homogeneous vapour phase from the experimental values of
Marcos et al. [7.36] who measured from 448 K up to 573 K (Tc = 564.13 K) at pressures
from 0.035 MPa to 0.377 MPa. We see that there is an increase in the deviations with
increasing temperature and pressure. At 448 K, the deviations range from -1.0% to +0.5%,
whilst at the supercritical temperature 573 K the deviations range from -2% to -4%. Figure
7.9 finally shows deviations of predicted PC-SAFT saturated vapour densities ρ” from the
experimental values of Lindley and Hershey [7.28]. In essence the deviations increase again
with increasing temperature up to -4.5% which matches the results from Figure 7.8. These
deviations together with those shown in Figure 7.7 indicate that the coexistence dome of
PC-SAFT is in comparison with experimental data a little more skewed to higher densities.
A rough estimate on the basis of the calculated saturated vapour and liquid densities shows
that the critical temperature from PC-SAFT is about 1% higher than the experimental value.
113
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
-1.2
-1.4
260
310
360
410
T [K]
Figure 7.7. Deviations of PC-SAFT saturated liquid densities for MDM from experimental
data of ■ Hurd [7.32] and ● Lindley and Hershey [7.28] and of PC SAFT densities at 1 atm
from experimental data of ♦ McLure et al. [7.35].
0.5
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
-3.0
-3.5
-4.0
0.02
0.12
0.22
0.32
0.42
P [MPa]
Figure 7.8. Deviations of PC-SAFT vapour densities for MDM from experimental data of
Marcos et al. [7.36]. ◊ at 448.15K, at 473.15K, + at 498.15, Ο at 523.15K, × at 548.15K,
and Δ at 573.15K.
114
100(ρ"exp -ρ"cal)/ρ"exp
2.0
1.0
0.0
-1.0
-2.0
-3.0
-4.0
-5.0
440
460
480
500
520
540
560
T [K]
Figure 7.9. Deviations of PC-SAFT saturated vapour densities for MDM from experimental
data of Lindley and Hershey [7.28].
7.5.3
Decamethyltetrasiloxane (MD2M), dodecamethylpentasiloxane (MD3M), and
tetradecamethylhexasiloxane (MD4M)
For the substances MD2M, MD3M, and MD4M there is only a limited number of
experimental data [7.27, 7.32, 7.34, 7.35]. Hence we show the PC-SAFT deviations from
experimental values in one figure for each of the properties vapour pressures, saturated
liquid densities, and densities at 1 atm for all three substances together. An advantage of
this presentation is that it allows observing systematic trends in the deviations.
Figure 7.10 shows the deviations of the PC-SAFT vapour pressures from the
experimental data of Flaningam [7.27] and from the extrapolated vapour pressures. We
learn that the deviations range from -3% to +2% and that the deviations are shifted in a
rather systematic way to higher temperatures with increasing chain length. The deviations
found here match with those shown in Figure 7.2 for MDM and in Figure 7.4 for MM.
Moreover, we observe that the PC-SAFT results deviate from the extrapolated vapour
pressures at higher temperatures within -2% up to Tr = 0.92, 0.91, and 0.90 for MD2M,
MD3M, and MD4M, respectively.
115
3.0
2.0
1.0
0.0
-1.0
-2.0
-3.0
-4.0
330
380
430
480
T [K]
530
580
630
Figure 7.10. Deviation of PC-SAFT vapor pressures from experimental data of Flaningam
[7.27] for MD2M, O MD3M, Δ MD4M, and from extrapolation Eq. (1)1 for ⎯ MD2M, −
− − MD3M, ….. MD4M.
Figure 7.11 shows the deviations of the PC-SAFT saturated liquid densities from
the experimental data of Hurd [7.32] for MD2M, MD3M, and MD4M and from the
experimental data of Golik and Cholpan [7.34] for MD2M. We learn that the deviations
from the Hurd-data range from -0.7% to -1.0% for MD2M and MD3M and are nearly
identical. The deviations for MD4M are slightly smaller. The deviations from the data of
[7.34] for MD2M show a steeper descent than those from [7.32]. The deviations found
here match with those shown in Figure 7.3 for MDM, but do not match with those shown in
Figure 7.1 for MM. Moreover, we observe that the PC-SAFT results deviate from the
extrapolated saturated liquid densities at higher temperatures within 2% up to Tr = 0.90 for
all three substances.
116
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
260
360
460
560
T [K]
Figure 7.11. Deviation of PC-SAFT saturated liquid densities from experimental data of
Hurd [7.32] for x MD2M, O MD3M, Δ MD4M, and of Golik and Cholpan [7.34] for
MD2M, and from extrapolation Eq. (2) for ⎯ MD2M, − − − MD3M, ….. MD4M.
Finally, Figure 7.12 shows the deviations of the PC-SAFT liquid densities at 1 atm
from the experimental data of McLure [7.35]. The deviations range from -0.7% to -1.3%
The deviations do not show a systematic trend, as those of MD2M are the smallest whilst
those of MD3M are the largest and MD4M is in between. Anyhow, the deviations are
negative and remain within 1.2% which agrees with nearly all density deviations for all
substances considered.
-0.6
-0.7
-0.8
-0.9
-1.0
-1.1
-1.2
-1.3
290
340
390
440
T [K]
Figure 7.12. Deviation of PC-SAFT liquid densities from experimental data of McLure at 1
atm [7.35] for: x MD2M, O MD3M, Δ MD4M.
117
7.6
Tables of saturation properties and T,s-diagrams
Thermodynamic properties of fluids and their mixtures calculated from equations of
state play important roles for different applications. Examples can be in the designing of
distillation columns, extraction equipment, designing and optimization of different energy
conversion cycles and so on. As a first step for different applications, compilations of
thermodynamic properties of MM, MDM, MD2M, MD3M, and MD4M are given in tables
from 7.8 to 7.12. In these tables, the caloric properties are calculated with PC-SAFT
supplemented by the ideal gas heat capacities from [7.56]. The reference state for the
enthalpy and the entropy is T0 = 298.15 K and p0 = 0.101325 MPa with h0 = 0.0 J/mol and
s0 = 0.0 J/mol K. The thermodynamic properties in tables from 7.8 to 7.12 and some
isobaric data are graphically shown in figures from 7.13 to 7.17.
Table 7.8. Thermodynamic properties of MM from PC-SAFT EOS
T [K]
ps [Mpa]
270
280
290
300
310
320
330
340
350
360
370
380
390
400
410
420
430
440
450
460
470
0.00112
0.00208
0.00366
0.00614
0.00989
0.01537
0.02312
0.03376
0.04804
0.06676
0.09084
0.12128
0.15917
0.20569
0.26211
0.32981
0.41028
0.50512
0.61608
0.74509
0.89427
h’’
ρ’’
ρ’
h' [kJ/mol]
3
[kJ/mol]
[mol/dm ] [mol/dm3]
4.86410
0.00050
-8.308
30.724
4.80100
0.00090
-5.397
32.925
4.73760
0.00152
-2.446
35.175
4.67370
0.00248
0.544
37.469
4.60920
0.00388
3.577
39.807
4.54370
0.00586
6.652
42.185
4.47700
0.00859
9.773
44.601
4.40890
0.01226
12.941
47.052
4.33910
0.01708
16.157
49.534
4.26740
0.02332
19.424
52.044
4.19340
0.03125
22.743
54.578
4.11680
0.04121
26.117
57.133
4.03720
0.05360
29.549
59.705
3.95420
0.06887
33.042
62.288
3.86720
0.08760
36.598
64.878
3.77570
0.11050
40.222
67.469
3.67870
0.13840
43.920
70.054
3.57540
0.17240
47.697
72.624
3.46440
0.21400
51.562
75.170
3.34410
0.26520
55.525
77.678
3.21210
0.32890
59.602
80.130
118
s’
s’’
[kJ/molK] [kJ/molK]
-0.02919
0.11537
-0.01860
0.11826
-0.00825
0.12148
0.00189
0.12497
0.01183
0.12870
0.02159
0.13263
0.03119
0.13673
0.04064
0.14096
0.04995
0.14531
0.05914
0.14975
0.06822
0.15426
0.07720
0.15882
0.08609
0.16341
0.09490
0.16801
0.10364
0.17262
0.11233
0.17721
0.12098
0.18176
0.12961
0.18626
0.13822
0.19068
0.14685
0.19501
0.15552
0.19920
T [K]
ps [Mpa]
480
490
500
510
1.06598
1.26293
1.48818
1.74527
h’’
ρ’’
ρ’
h' [kJ/mol]
3
[kJ/mol]
[mol/dm ] [mol/dm3]
3.06460
0.40920
63.814
82.502
2.89620
0.51280
68.200
84.754
2.69620
0.65170
72.823
86.817
2.44150
0.85110
77.831
88.540
s’
s’’
[kJ/molK] [kJ/molK]
0.16427
0.20320
0.17318
0.20696
0.18235
0.21034
0.19207
0.21307
Table 7.9. Thermodynamic properties of MDM from PC-SAFT EOS
T [K]
270
280
290
300
310
320
330
340
350
360
370
380
390
400
410
420
430
440
450
460
470
480
490
510
520
530
540
550
ps [Mpa] ρ’ [mol/dm3]
0.00007
0.00016
0.00032
0.00062
0.00112
0.00195
0.00324
0.00520
0.00806
0.01211
0.01772
0.02529
0.03528
0.04824
0.06475
0.08547
0.11111
0.14248
0.18041
0.22585
0.27982
0.34342
0.41788
0.60490
0.72063
0.85361
1.00596
1.18007
3.56780
3.52580
3.48390
3.44210
3.40020
3.35800
3.31550
3.27250
3.22890
3.18460
3.13930
3.09310
3.04560
2.99690
2.94660
2.89460
2.84060
2.78440
2.72570
2.66420
2.59920
2.53040
2.45690
2.29210
2.19770
2.09190
1.97030
1.82430
ρ’’
[mol/dm3]
0.00003
0.00007
0.00013
0.00025
0.00044
0.00073
0.00119
0.00185
0.00280
0.00411
0.00588
0.00822
0.01126
0.01516
0.02009
0.02625
0.03387
0.04326
0.05473
0.06873
0.08577
0.10650
0.13180
0.20130
0.24920
0.31010
0.38930
0.49650
119
h'
[kJ/mol]
-11.407
-7.408
-3.358
0.744
4.901
9.113
13.382
17.709
22.096
26.543
31.052
35.626
40.264
44.970
49.744
54.589
59.507
64.502
69.575
74.731
79.975
85.311
90.749
101.966
107.779
113.762
119.962
126.470
h’’
s’
s’’
[kJ/mol] [kJ/molK] [kJ/molK]
37.672 -0.04007
0.14170
40.807 -0.02553
0.14667
44.014 -0.01132
0.15203
47.289
0.00259
0.15774
50.632
0.01622
0.16374
54.041
0.02959
0.16999
57.513
0.04272
0.17645
61.046
0.05564
0.18310
64.638
0.06835
0.18990
68.285
0.08087
0.19682
71.985
0.09322
0.20385
75.734
0.10541
0.21096
79.529
0.11745
0.21813
83.366
0.12935
0.22535
87.241
0.14113
0.23259
91.150
0.15279
0.23984
95.088
0.16434
0.24708
99.050
0.17580
0.25431
103.031
0.18717
0.26151
107.024
0.19846
0.26866
111.022
0.20969
0.27575
115.017
0.22088
0.28276
119.000
0.23203
0.28968
126.870
0.25430
0.30313
130.721
0.26549
0.30961
134.476
0.27677
0.31585
138.086
0.28822
0.32178
141.457
0.29999
0.32724
Table 7.10. Thermodynamic properties of MD2M from PC-SAFT EOS
T [K]
270
280
290
300
310
320
330
340
350
360
370
380
390
400
410
420
430
440
450
460
470
480
490
500
510
520
530
540
550
560
570
580
0.00001
0.00001
0.00003
0.00007
0.00015
0.00029
0.00053
0.00094
0.00159
0.00259
0.00409
0.00624
0.00929
0.01346
0.01908
0.02648
0.03606
0.04827
0.06361
0.08262
0.10592
0.13418
0.16814
0.20862
0.25654
0.31289
0.37879
0.45552
0.54447
0.64723
0.76561
0.90159
2.84130
2.80940
2.77770
2.74630
2.71490
2.68340
2.65190
2.62010
2.58810
2.55570
2.52280
2.48940
2.45540
2.42060
2.38500
2.34850
2.31090
2.27220
2.23210
2.19050
2.14720
2.10210
2.05470
2.00490
1.95220
1.89600
1.83570
1.77050
1.69900
1.61950
1.52870
1.42130
ρ’’
[mol/dm3]
0.00000
0.00001
0.00001
0.00003
0.00006
0.00011
0.00019
0.00033
0.00055
0.00087
0.00134
0.00200
0.00291
0.00413
0.00575
0.00785
0.01054
0.01395
0.01822
0.02353
0.03008
0.03814
0.04801
0.06009
0.07489
0.09304
0.11540
0.14320
0.17810
0.22240
0.27980
0.35690
120
h'
[kJ/mol]
-13.889
-9.021
-4.091
0.905
5.967
11.096
16.295
21.564
26.906
32.320
37.808
43.372
49.013
54.732
60.531
66.410
72.371
78.417
84.549
90.770
97.082
103.488
109.993
116.600
123.317
130.150
137.109
144.208
151.466
158.912
166.593
174.592
h’’
s’
s’’
45.226 -0.04879
0.17015
49.053 -0.03109
0.17632
52.971 -0.01379
0.18298
56.979
0.00315
0.19006
61.075
0.01975
0.19751
65.256
0.03603
0.20528
69.523
0.05203
0.21332
73.872
0.06776
0.22160
78.302
0.08324
0.23009
82.810
0.09849
0.23874
87.394
0.11353
0.24754
92.050
0.12836
0.25646
96.775
0.14301
0.26548
101.567
0.15748
0.27457
106.421
0.17180
0.28372
111.333
0.18595
0.29292
116.299
0.19997
0.30213
121.315
0.21386
0.31135
126.376
0.22762
0.32057
131.476
0.24128
0.32977
136.610
0.25483
0.33893
141.771
0.26829
0.34804
146.952
0.28167
0.35709
152.146
0.29497
0.36607
157.343
0.30823
0.37494
162.531
0.32144
0.38371
167.698
0.33462
0.39234
172.826
0.34782
0.40081
177.893
0.36104
0.40909
182.866
0.37434
0.41712
187.697
0.38780
0.42483
192.298
0.40155
0.43208
T [K] ps [Mpa]
300
310
320
330
340
350
360
370
380
390
400
410
420
430
440
450
460
470
480
490
500
510
520
530
540
550
560
570
580
590
600
610
0.00001
0.00002
0.00005
0.00010
0.00020
0.00036
0.00063
0.00106
0.00172
0.00272
0.00416
0.00619
0.00900
0.01279
0.01782
0.02437
0.03275
0.04334
0.05654
0.07281
0.09265
0.11663
0.14536
0.17954
0.21993
0.26742
0.32295
0.38764
0.46270
0.54953
0.64966
0.76483
ρ’
[mol/dm3]
h'
s’’
ρ’’
h’’ [kJ/mol] s’ [kJ/molK]
[kJ/mol]
[kJ/molK]
[mol/dm3]
2.27540
2.25060
2.22590
2.20130
2.17650
2.15170
2.12660
2.10130
2.07570
2.04970
2.02340
1.99650
1.96900
1.94090
1.91220
1.88260
1.85210
1.82070
1.78820
1.75440
1.71920
1.68250
1.64400
1.60340
1.56050
1.51470
1.46560
1.41240
1.35400
1.28890
1.21440
1.12580
0.00000
0.00001
0.00002
0.00004
0.00007
0.00012
0.00021
0.00035
0.00055
0.00084
0.00126
0.00184
0.00262
0.00366
0.00502
0.00677
0.00899
0.01179
0.01528
0.01961
0.02494
0.03150
0.03953
0.04939
0.06148
0.07637
0.09479
0.11780
0.14670
0.18350
0.23160
0.29640
1.145
7.546
14.035
20.611
27.275
34.030
40.875
47.812
54.841
61.964
69.181
76.492
83.900
91.404
99.006
106.707
114.508
122.411
130.417
138.530
146.750
155.082
163.530
172.098
180.793
189.622
198.598
207.734
217.053
226.587
236.389
246.562
121
65.560
70.869
76.289
81.818
87.454
93.196
99.041
104.986
111.029
117.167
123.397
129.715
136.117
142.600
149.159
155.790
162.487
169.246
176.061
182.926
189.834
196.778
203.750
210.740
217.739
224.733
231.707
238.642
245.512
252.282
258.896
265.252
0.00398
0.02497
0.04556
0.06580
0.08569
0.10527
0.12456
0.14356
0.16231
0.18081
0.19907
0.21713
0.23497
0.25263
0.27010
0.28739
0.30453
0.32151
0.33835
0.35506
0.37164
0.38812
0.40449
0.42077
0.43697
0.45311
0.46922
0.48531
0.50142
0.51761
0.53395
0.55060
0.21869
0.22923
0.24011
0.25128
0.26269
0.27432
0.28613
0.29809
0.31017
0.32235
0.33462
0.34694
0.35930
0.37169
0.38408
0.39647
0.40883
0.42116
0.43344
0.44567
0.45781
0.46987
0.48183
0.49368
0.50539
0.51695
0.52834
0.53953
0.55049
0.56116
0.57146
0.58124
T [K]
320
330
340
350
360
370
380
390
400
410
420
430
440
450
460
470
480
490
500
510
520
530
540
550
560
570
580
590
600
610
620
630
0.00001
0.00002
0.00005
0.00010
0.00018
0.00033
0.00056
0.00093
0.00149
0.00231
0.00350
0.00517
0.00745
0.01052
0.01457
0.01982
0.02653
0.03499
0.04553
0.05851
0.07432
0.09343
0.11634
0.14360
0.17584
0.21377
0.25818
0.30996
0.37012
0.43979
0.52024
0.61286
1.89820
1.87810
1.85800
1.83780
1.81760
1.79720
1.77660
1.75590
1.73480
1.71350
1.69170
1.66960
1.64700
1.62390
1.60020
1.57580
1.55080
1.52500
1.49820
1.47050
1.44170
1.41170
1.38030
1.34720
1.31230
1.27520
1.23550
1.19280
1.14610
1.09450
1.03630
0.96840
ρ’’
[mol/dm3]
0.00000
0.00001
0.00002
0.00003
0.00006
0.00011
0.00018
0.00029
0.00045
0.00068
0.00101
0.00146
0.00207
0.00287
0.00392
0.00527
0.00697
0.00912
0.01179
0.01510
0.01918
0.02418
0.03032
0.03784
0.04706
0.05840
0.07241
0.08983
0.11170
0.13930
0.17500
0.22230
122
h'
[kJ/mol]
16.438
24.134
31.930
39.828
47.828
55.931
64.137
72.448
80.864
89.386
98.014
106.749
115.592
124.544
133.606
142.778
152.062
161.459
170.971
180.600
190.349
200.219
210.216
220.343
230.607
241.016
251.579
262.310
273.228
284.363
295.758
307.495
h’’
s’
s’’
85.406
0.05337
0.26889
91.965
0.07705
0.28260
98.646
0.10032
0.29655
105.450
0.12322
0.31071
112.372
0.14575
0.32504
119.411
0.16795
0.33952
126.565
0.18984
0.35412
133.831
0.21142
0.36881
141.206
0.23273
0.38358
148.688
0.25377
0.39841
156.272
0.27456
0.41327
163.956
0.29511
0.42815
171.736
0.31544
0.44304
179.607
0.33555
0.45791
187.566
0.35546
0.47276
195.607
0.37518
0.48758
203.725
0.39471
0.50235
211.916
0.41408
0.51705
220.173
0.43328
0.53169
228.490
0.45233
0.54623
236.861
0.47124
0.56069
245.278
0.49002
0.57503
253.733
0.50867
0.58926
262.216
0.52722
0.60335
270.718
0.54567
0.61729
279.225
0.56404
0.63107
287.723
0.58235
0.64466
296.195
0.60062
0.65805
304.616
0.61888
0.67119
312.954
0.63718
0.68405
321.163
0.65559
0.69656
329.162
0.67422
0.70861
550
5
M
Pa
MM
1.5 MPa
500
1 MPa
T [K]
450
0.5 MPa
400
0.1 MPa
M
Pa
350
0.
00
1
0.01 MPa
300
-50
0
50
100
150
200
250
s [kJ/molK]
Figure 7.13. T, s-diagram of MM showing the saturated liquid and the saturated vapour
curve and several isobars
600
MDM
1 MPa
5
M
Pa
550
T [K]
500
450
0.5 MPa
0.1 MPa
400
0.01 MPa
350
0.001 MPa
300
-100
0
100
200
300
400
s [J/molK]
Figure 7.14. T, s-diagram of MDM showing the saturated liquid and the saturated vapour
123
600
MD2M
M
Pa
0.75 MPa
0.5 MPa
5
550
500
T [K]
0.1 MPa
450
0.01 MPa
400
0.001 MPa
350
300
-100
0
100
200
300
400
500
s [J/molK]
Figure 7.15. T, s-diagram of MD2M showing the saturated liquid and the saturated vapour
MD3M
5
T [K]
600
M
Pa
0.5 MPa
0.1 MPa
500
0.01 MPa
400
0.001 MPa
300
0
100
200
300
400
500
600
s [J/molK]
124
700
MD4M
5
M
Pa
0.5 MPa
600
T [K]
0.1 MPa
500
0.01 MPa
0.001 MPa
400
300
0
200
400
600
800
s [J/molK]
We learn from these figures that the coexistence curves of MM, MDM, MD2M,
MD3M, and MD4M in the T,s-diagrams are skewed. The steep of coexistence curves
increase with chain length of the molecules.
7.7
In this chapter we have shown an effective way to determine PC-SAFT parameters
for the siloxanes. The procedure is to firstly construct extrapolation equations for the
vapour pressures and the saturated liquid densities and to fit thereafter PC-SAFT
parameters to the extrapolation equations. Comparisons of resulting PC-SAFT
thermodynamic data with a variety of experimental data show good agreement.
The present results pave the way for optimizing ORC cycles with pure siloxanes as
working fluids either in subcritical or in supercritical cycles. As a first step for designing
these cycles we have given Tables of the saturation properties of the siloxanes and T,sdiagrams.
Moreover and perhaps more important, PC-SAFT [7.19] allows also the
thermodynamic description of mixtures in a rather simple way. This has some practical
importance for optimizing subcritical ORC cycles for the production of electricity from
125
biomass [7.2], which was claimed to be one of the most efficient methods for climate
protection by biomass [7.57].
126
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131
8
Description of aromates with BACKONE equation of state *
Abstract
The paper aims at a thermodynamic description of aromates by the molecular based
equation of state, BACKONE. First, experimental data of the seven aromates benzene,
toluene, ethylbenzene, butylbenzene, o-xylene, m-xylene, and p-xylene are compiled. Then
four parameters of BACKONE for these substances are found by fitting to the vapour
pressures and saturated liquid densities. The predictive power of BACKONE is investigated
in detail for benzene and toluene. The study shows that BACKONE predicts saturated
vapour density very accurately. The deviations of saturated vapour densities of benzene and
toluene from BACKONE and experiments are mostly within 1%. The average absolute
deviations of BACKONE results and 432 tested data points of benzene in homogeneous
region of vapour, liquid, and supercritical gas is 1.27%. The deviations of experimental
data of toluene and predicted data from BACKONE EOS are in the same magnitude with
those of benzene, except in the region with temperature slightly higher than critical
temperature. Finally, in order to demonstrate the usefulness of BACKONE for aromates in
technical applications, a hypothetic Organic Rankine Cycle with benzene as working fluid
is considered. The enthalpies and entropies at different state points as well as the thermal
efficiencies from BACKONE and Bender-type equation are agreed within 2%.
8.1
Introduction
The use of fossil fuels and the increase of population [8.1] did lead to environmental
and energy problems. In order to solve the problems scientists, industrial organizations and
governments have paid a lot of attention to energy saving solutions as well as the use of
renewable energy sources like geothermal energy, solar energy, biomass energy. One very
promising idea for exploiting geothermal heat, solar heat, heat from biomass burning, waste
heat and other heat sources is to use the Clausius-Rankine cycle to generate electricity.
Examples can be found in [8.2 - 8.4].
One of the most potential and promising concept projects for exploiting renewable
energy sources is DESERTEC project [8.5]. Intended technologies of this concept project
*
N. A. LAI, J. Fischer, M. Wendland, Description of aromates with BACKONE equations of state, to be
submitted in refereed journal
132
are to use mirrors to concentrate sunlight to create heat, to use steam turbines to generate
electricity, and to use heat storage tanks (e.g. molten salt tanks or concrete blocks) to store
heat during the day to power the turbines during the night. For the solar energy sources,
temperature can reach as high as about 3000C to 3500C if parabolic collectors are used.
To our knowledge, working fluid is a key of the Clausius-Rankine cycles. It is obvious that
there is no working fluid for all temperature ranges. In the low-temperature, i.e up to about
100°C, we found out that the supercritical Rankine cycle using refrigerant R143a as
working fluid yields about 20% more power than subcritical working fluids if the pinch
problem in the evaporator is also taken into consideration [8.6]. In the high-temperature
range, i.e above 400°C, water has been used as working fluid. A crucial question now is
which working fluids are suitable for medium-temperature range, i.e above 200 °C.
In order to give an answer for the question, accurate equations of state for potential
working fluids of medium temperature Rankine cycles must be known. There are different
types of equations of state (EOS) such as multi-parameter equations of state, molecular
based equations of state and cubic equations of state. Cubic equations of state are not
sufficiently accurate [8.7]. Multi-parameter equations of state are very accurate provided
that sufficient accurate experimental data are used in fitting. Thus molecular based
equations of state should be used when available experimental data are limited.
Various siloxanes and aromates are potential candidates for medium temperature
range. Because experimental data of almost all our interested substances are limited we
decide to use BACKONE [8.8, 8.9] and PC-SAFT [8.10] molecular based equations of
state. Parameters of equations of state for siloxanes have recently been found [8.11, 8.12].
For aromates, we first tried both BACKONE equation and PC-SAFT equation for benzene.
We find out that BACKONE EOS outperforms PC-SAFT EOS.
Hence, in this chapter we use BACKONE EOS to describe seven aromates
including benzene, toluene, ethylbenzene, butylbenzene, o-xylene, m-xylene, and p-xylene.
In section 8.2, a compilation of experimental data as well as equations for correlation and
extrapolation of vapour pressure and saturated liquid density are given. In section 8.3, an
overview of BACKONE EOS and parameters of BACKONE EOS for seven aromates are
presented. In section 8.4, we make a discussion of BACKONE results. Thermodynamic
133
data from BACKONE EOS are given in tables and graphs, section 8.5. Finally, in order to
demonstrate the usefulness of BACKONE for aromates in technical applications, a
hypothetic Organic Rankine Cycle with benzene as working fluid is considered in section
8.6.
8.2
Experimental data and auxiliary equations
In this Section, we compile available experimental data for benzene, toluene,
ethylbenzene, butylbenzene, o-xylene, m-xylene, and p-xylene. The critical temperature Tc,
critical pressure pc and critical density ρc of these substances have been reviewed by
Tsonopoulos and Ambrose [8.13]. They recommended average values of all available data
in their review. In our review, the critical temperatures and critical pressures of all
substances, except butylbenzene, are selected to be consistent with Ambrose’s experimental
vapour pressures, [8.14]. For butylbenzene, experimental vapour pressures are not available
up to critical points. Fortunately, Ambrose and co-workers have also measured and
published critical pressure of butylbenzene [8.15] so we select their published data. The
experimental critical densities of our selected substances have not been extensively
investigated, except those of benzene and toluene. There is only one source having critical
densities of all our interested substances. These data are based on Simson’s 1938 thesis,
which was published by Timmermans in 1957, [8.13, 8.16]. The data of Simson are
consistent with other experimental data, if available, so we select these data in our study.
The critical temperatures, critical pressures and critical densities which were selected in this
study are given in table 8.1.
Table 8.1. Critical data of aromates.
Substance
Tc [K]
pc [MPa]
ρc [mol/l]
benzene
toluene
ethylbenzene
butylbenzene
o-xylene
m-xylene
p-xylene
562.16
591.80
617.20
660.05
630.33
617.05
616.23
4.8980
4.1087
3.6088
2.8870
3.7318
3.5412
3.5107
3.8970
3.1615
2.6704
2.0094
2.7099
2.6581
2.6440
134
Tc and pc
sources
[8.14]
[8.14]
[8.14]
[8.15]
[8.14]
[8.14]
[8.14]
ρc source
[8.13, 8.16]
[8.13, 8.16]
[8.13, 8.16]
[8.13, 8.16]
[8.13, 8.16]
[8.13, 8.16]
[8.13, 8.16]
Tables 8.2 and 8.3 contain temperature ranges, numbers of data points and sources
of experimental vapour pressures ps and experimental saturated liquid densities ρ’. In this
compilation, we have not included experimental data sources which did not seem to be
reliable on the basis of our critical investigation or previous investigations. We observe in
Table 8.2 that experimental data are available up to critical point for all selected substances
except butylbenzene.
Table 8.2. Experimental vapour pressures: temperature ranges, numbers of data points and
sources.
Temperature range
[K]
Reduced
temperature
308.332 - 388.847
285.957 - 383.175
403.98 - 562.16
0.55 – 0.69
0.51 – 0.68
0.72 – 1.00
273.15 - 323.15
273.15 - 322.41
398.37 - 591.8
0.46 – 0.55
0.46 – 0.54
0.67 – 1.00
306.244 - 450.056
424.02 - 617.2
0.50 – 0.73
0.69 – 1.00
343.359 - 500.97
0.52 - 0.76
273.15 - 333.15
312.66 - 458.822
432.17 - 630.33
0.43 – 0.53
0.50 – 0.73
0.69 – 1.00
273.15 - 333.15
308.64 - 452.947
428.06 - 617.05
0.44 – 0.54
0.50 – 0.73
0.69 – 1.00
298.15 - 333.15
286.435 - 365.954
286.432 - 452.341
428.16 - 616.23
0.48 – 0.54
0.46 – 0.59
0.46 – 0.73
0.69 – 1.00
Number
of exp data
benzene
15
80
19
toluene
5
7
49
ethylbenzene
23
43
butylbenzene
23
o-xylene
6
23
40
m-xylene
6
24
48
p-xylene
4
12
25
48
135
Source
Scott and Osborn [8.17]
Ambrose [8.18]
Ambrose [8.14]
Pitzer and Scott [8.19]
Munday et al. [8.20]
Ambrose [8.14]
Chirico et al. [8.21]
Ambrose [8.14]
Ambrose [8.14]
Pitzer, and Scott [8.19]
Ambrose [8.14]
Osborn and Douslin [8.25]
Ambrose [8.14]
Table 8.3. Experimental saturated liquid density: temperature ranges, numbers of data
points and sources.
Temperature range
[K]
Reduced
temperature
293.15 - 490
0.52 – 0.87
293.15 - 490
0.50 – 0.83
293.15 – 490.0
0.47 – 0.79
323.136 - 523.11
0.49 – 0.79
293.15 - 490
323.14 - 523.11
0.51 – 0.83
0.47 – 0.78
323.14 - 523.11
293.15 – 490.0
0.52 – 0.85
0.48 – 0.79
293.15 – 490.0
423.15 - 613.15
0.48 – 0.80
0.69 – 1.00
Number
of exp data
benzene
17
toluene
14
ethylbenzene
14
butylbenzene
9
o-xylene
14
9
m-xylene
9
14
p-xylene
14
21
Source
Hales and Townsend [8.27]
Francis [8.28]
In this study we use Wagner equation, [8.29], for correlation and extrapolation of
vapour pressure:
ln pr = (1/Tr) [A(1-Tr) + B(1-Tr)1.5 + C(1-Tr)3 + D(1-Tr)6],
(8.1)
where Tr = T/Tc and pr = ps/pc, and with the fitted parameters A, B, C, and D. Both Tc and
pc are taken from Table 8.1 and the vapour pressures are taken from the sources in Table
8.2. The objective function is Σi [(ps,exp,i - ps,cal,i ) / ps,exp,i]2. Results are given in table 8.4 and
are discussed as follow.
Vapour pressures of benzene from 290.076 K to critical point were published by
Ambrose [8.14, 8.18]. Scott and Osborn [8.17] published vapour pressures of benzene from
308.332 K to 388.847 K. The data of Scott and Osborn are in very good agreement with
data of Ambrose. We fit Wagner equation to both data set of Ambrose and of Scott and
Osborn. We find out that the average absolute deviation between calculated data and
experimental data (AAD) for all 114 data points is 0.0165% and the maximum absolute
deviation is 0.11% at 494.89 K.
136
Table 8.4. Coefficients of the Wagner equation, Eq. (8.1), for the vapour pressures, ranges
of reduced temperatures Tr in the fits, and sources of underlying experimental data. All
data are taken with the same weight. The critical data are taken from Table 8.1.
A
B
C
D
Fit range
in Tr
benzene
-6.97547
1.31019
-2.57520
-3.51443
0.51-1.00
toluene
-7.28102
1.35632
-2.74464
-3.12870
0.46-1.00
ethylbenzene
butylbenzene
-7.52339
-8.11868
1.51973
1.85276
-3.37718
-4.32328
-2.48294
-2.30124
0.50-1.00
0.52-0.76
o-xylene
-7.53784
1.42385
-3.10565
-3.02946
0.43-1.00
m-xylene
-7.70907
1.71173
-3.87727
-0.46621
0.44-1.00
p-xylene
-7.62288
1.49156
-3.20109
-2.53592
0.46-1.00
Sources of exp.
data
[8.14], [8.18],
[8.17]
[8.14], [8.19],
[8.20]
[8.14], [8.21]
[8.22]
[8.14], [8.19],
[8.23]
[8.14], [8.19],
[8.24]
[8.14], [8.19],
[8.25], [8.26]
Vapour pressures of toluene are taken from published data of Pitzer and Scott
[8.19], Ambrose [8.14], and Munday et al. [8.20]. Experimental data of Pitzer et al. [8.19]
and Munday et al. [8.20] are in the same temperature range, from 273.15 K to about 323 K.
Experimental data of toluene from Ambrose [8.14] range from 398.37 K to Tc. We fit
Wagner equation to all 61 selected data points of toluene and receive an AAD of 0.0619%.
Vapour pressures of o-xylene and m-xylene are available from 273.15 K to critical
point. For o-xylene, vapour pressures of Pitzer and Scott [8.19], Ambrose [8.14], and
Chirico et al. [8.23] are used in this study. For m-xylene, we also use the same data sources
for o-xylene, but using reference [8.24] instead of using reference [8.23]. We fit Wagner
equation to the data of o-xylene and m-xylene and have AADs of 0.15% and 0.20%,
respectively. Vapour pressures of p-xylene from Pitzer and Scott [8.19], Ambrose [8.14],
Osborn and Douslin [8.25], and Chirico et al. [8.26] are in a temperature range from
286.432 K to 616.23 K or from 0.46 Tc to Tc. The experimental data of p-xylene can be
described well with Wagner equation. The AAD and maximum absolute deviation of
Wagner equation and 89 data points are 0.088% and 0.57% respectively.
Experimental data of ethylbenzene from Ambrose [8.14] and Chirico et al. [8.21]
range from 306.244 K to 617.2 K or from 0.5 Tc to Tc. We fit Wagner equation to all 66
137
data points of ethylbenzene and have AAD of 0.086% and maximum absolute deviation of
0.42%.
Vapour pressures of butylbenzene from Chirico et al. [8.22] range from 0.52 Tc to
0.76 Tc. In this case, the use of extrapolation equation for vapour pressure is of vital
importance for improvement of the description of BACKONE EOS. Similar to other
substances in Table 8.4, we use Wagner equation to correlate and extrapolate vapour
pressure. Input data of vapour pressure for construction of EOS are generated with an
interval of 10 K.
Experimental saturated liquid densities have not been fully investigated for all of
our studied substances. In this study, we use saturated liquid densities from Hales and
Townsend [8.27], Chirico et al. [8.22], and Francis [8.28], Table 8.3. Hales and Townsend
published saturated liquid densities of almost all our studied substances from 293.15 K to
490 K except butylbenzene. Chirico et al. and Steele et al. published saturated liquid
densities of o-xylene, m-xylene, and butylbenzene in different papers [8.23], [8.24], [8.22]
respectively. We observe that saturated liquid densities of Chirico and Steele have large
uncertainty, especially in critical region. Thus we do not use the data of Chirico and Steele
in critical region.
Accurate experimental saturated liquid densities are available up to about 0.8 Tc,
except p-xylene which has experimental data up to critical point [8.28]. Thus equation for
extrapolation of saturated liquid density is needed for improving quality of description of
BACKONE EOS. Our recent study on the possibility of upward prediction of saturated
liquid densities has pointed out that if experimental data are available in low temperature
range Rackett equation of Spencer and Danner [8.30] is the best equation, [8.12]. If
experimental data from 0.5 Tc to 0.75 Tc are used to predict saturated liquid densities at
higher temperature, the best prediction up to 0.95 Tc can be obtained with two-parameter
equation, [8.31]:
lnρ’ = B – A(1-Tr)(2/7).
(8.2)
where Tr = T/Tc. Critical temperature Tc is taken from Table 8.1. A and B are fitted
parameters. We remind that in [8.31] we write B as lnρp. The objective function for fitting
Eq. (8.2) is Σi [(ρ’exp,i – ρ’cal,i)/ρ’exp,i]2.
138
Parameters of Eq. (8.2) in Table 8.5 for all studied substances are found by fitting to
the experimental data set in Table 8.3. These parameters are used to generate input data of
saturated liquid densities for construction of BACKONE EOS with an interval of 10 K.
Table 8.5. Parameters of Eq. (8.2) for the saturated liquid density, AADs, ranges of reduced
temperatures Tr in the fits, and sources of underlying experimental data. All data are taken
with the same weight. The critical data are taken from Table 8.1.
benzene
toluene
ethylbenzene
butylbenzene
o-xylene
m-xylene
p-xylene
8.3
A
B
AAD [%]
Fit range in Tr
-1.31815
-1.29826
-1.31578
-1.33535
-1.29488
-1.29956
-1.33724
1.35077
1.17264
1.00434
0.725718
1.03120
1.01494
0.983161
0.098
0.055
0.059
0.034
0.055
0.073
0.327
0.52 – 0.87
0.50 – 0.83
0.47 – 0.79
0.49 – 0.79
0.47 – 0.83
0.48 – 0.85
0.48 – 1.00
Sources of exp.
data
[8.27]
[8.27]
[8.27]
[8.22]
[8.23], [8.27]
[8.24], [8.27]
[8.27], [8.28]
Equation of state
BACKONE EOS has been successfully applied to many pure fluids and their
mixtures, [8.8, 8.9, 8.32, 8.33, 8.34]. Strictly speaking, BACKONE is a family of
physically based EOS [8.8] developed for nonpolar, dipolar and quadrupolar fluids.
Recently, independent authors [8.35] have made comparison among BACKONE, PC-SAFT
[8.10], soft-SAFT [8.36], and SAFT-VR [8.37] for representing virial coefficients of some
fluids. The study pointed out that BACKONE EoS outperformes SAFT-type EoSs in
describing second virial coefficients.
Before discussion of BACKONE results for aromates, we give a brief introduction
of BACKONE EOS. In BACKONE, the Helmholtz energy is written as a sum of molecular
hard-body contribution FH, attractive dispersion force contribution FA, dipolar contribution
FD and quadrupolar contribution FQ:
F = FH + FA + FD + FQ
The hard-body contribution FH is determined by Boublik [8.38] as following:
FH/RT = (α2 - 1).ln(1 - ξ)+{(α2 + 3α).ξ - 3.α.ξ2}/(1-ξ)2
139
where ξ is the packing fraction and α is the anisotropy parameter [8.8]. In
BACKONE, α is assumed to be state independence but ξ is state dependent. The packing
fraction is a function of characteristic critical density ρ0, characteristic critical temperature
T0, density ρ and temperature T.
Similar to the hard-body contribution, the attractive dispersion force in BACKONE
depends on the same three parameters α, ρ0, T0. The equation for FA is:
FA/ RT = Σi[ci(ρ/ρ0)mi(T/T0)ni/2αjiexp{-oi(ρ/ρ0)li}]
All exponents and coefficients of FA were found by fitting to experimental data of
ethane, methane and oxygen. Their values are given in [8.8].
For polar fluids, the dipolar contribution FD and the quadrupolar contribution FQ to
the Helmholtz energy are given as:
FD/ RT = Σi[ci(ρ/ρ0)mi/2(T/1.13T0)ni/2(μ*2)ki/4exp{-oi(ρ/ρ0)2}]
and
FQ/ RT = Σi[ci(ρ/ρ0)mi/2(T/1.13T0)ni/2(Q*2)ki/4exp{-oi(ρ/ρ0)2}]
where μ* is a reduced dipole moment and Q* is a reduced quadrupole moment.
Exponents and coefficients of FD and FQ were determined by fitting to simulation data.
Depending on type of fluids, BACKONE EOS has 3 parameters T0, ρ0, and α for
nonpolar fluids and one more parameter μ* or Q* for dipolar or quadrupolar fluids
respectively. If fluids have both dipolar and quadrupolar contributions, BACKONE EOS
has 5 parameters. The parameters of BACKONE EOS are found by fitting to vapour
pressures and saturated liquid densities. For all considered aromates, we assumed that the
dipole moment is zero and hence μ* = 0. Thus BACKONE EOS for aromates has four
parameters α, ρ0, T0 and Q*.
8.4
Results and discussion
As given in Table 8.2 and Table 8.3, vapour pressures and saturated liquid densities
are taken from different sources. The data are not distributed over the same temperature
interval. Saturated liquid densities are normally not available up to critical points like
vapour pressures. Thus, we first fit Eqs. (8.1) and (8.2) to the selected experimental data
including the critical data. Then input data for BACKONE are generated from Eq. (8.1) and
140
Eq. (8.2) in the temperature range from the lowest vapour pressure temperatures up to 0.95
Tc with a temperature interval of 10 K.
BACKONE parameters as well as quadrupole moment in Table 8.6 are found by
fitting to input data generated from Eq. (8.1) and Eq. (8.2). This Table also contains the
ranges of reduced temperatures from which the vapour pressures and the saturated liquid
densities have been taken for fitting. About 36 data points of vapour pressure from Eq. (8.1)
and saturated liquid density from Eq. (8.2) have been used in the fit for each substance.
Table 8.6. BACKONE parameters as well as quadrupole moment of aromates and ranges of
reduced temperature Tr in the fits to data calculated from Eq. (8.1) and Eq. (8.2).
Temperature interval of fitting range is 10K.
T0 [K] ρ0 [mol/l]
benzene
toluene
ethylbenzene
butylbenzene
o-xylene
m-xylene
p-xylene
538.93
567.41
585.32
598.68
594.91
587.69
588.78
3.80457
3.10164
2.62272
1.94725
2.64340
2.60975
2.59856
α
μ*2
Q*2
Q [D A]
fit range
in Tr
1.37977
1.40682
1.42411
1.44871
1.42209
1.43046
1.42988
0
0
0
0
0
0
0
2.08642
2.47697
2.91600
4.05099
2.99888
2.98906
2.92245
- 9.3261
- 12.2909
- 15.5217
- 23.5998
- 15.7720
- 15.5217
- 15.6873
0.51-0.95
0.46-0.95
0.50-0.95
0.52-0.95
0.43-0.95
0.44-0.95
0.46-0.96
o
The quadrupole moment of benzene from BACKONE, Table 8.6, is in a very good
agreement with both experimental values resulting from electric gradient-field-induced
o
birefringence method by Ritchie and Watson [8.39] of Qzz = - 9.11 ± 0.36 D A and ab
initio calculation with MP2 type correlation method by Meijer and Sprik [8.40] of Q = o
9.63 D A .
8.4.1
Benzene and toluene
Input data for fitting BACKONE parameters of benzene and toluene range from
0.51 Tc to 0.95 Tc and from 0.46 Tc to 0.95 Tc, respectively. We observe that the relative
deviations Δps = (ps,exp - ps,cal)/ ps,exp between BACKONE results and experimental data of
Ambrose [8.14] and [8.18] range from -0.98% to +0.57%. For the saturated liquid densities
of Hales and Townsend [8.27], the relative deviations range from -0.23% to 0.32%. For
toluene, deviations of vapour pressures and saturated liquid densities are from -0.4% to
141
1.4% and from -0.54% to 0.23% respectively. Besides comparison with fitted data, we also
make comparison of BACKONE results and data from reference equations for benzene and
toluene from Goodwin [8.41, 8.42]. Figure 8.1 shows that the description of BACKONE
100(ps,ref-ps,B1)/ps,ref
for vapour pressures, saturated vapour and liquid densities is quite good.
2
1
0
-1
-2
250
300
350
400
450
500
550
600
500
550
600
500
550
600
T [K]
100(ρ'ref-ρ'B1)/ρ'ref
2
1
0
-1
-2
250
300
350
400
450
100(ρ"ref-ρ"B1)/ρ"ref
T [K]
2
1
0
-1
-2
250
300
350
400
450
T [K]
Figure 8.1. Deviations of BACKONE results for the vapour pressure ps, saturated liquid
density ρ’, and saturated vapour density ρ’’ of benzene (solid lines) and toluene (open
lines) from reference (ref) equations (Goodwin [8.41, 8.42]). × Ambrose [8.14, 8.18] for
benzene, Ο Pitzer and Scott [8.19] for toluene, + Ambrose [8.14, 8.18] for toluene. Hales
and Townsend [8.27]: Δ for benzene, for toluene
142
In addition to the comparisons of vapour pressures and saturated liquid densities
which have been used for determining the parameters of BACKONE, we also compare, in
the following, predictions of BACKONE for other data with experimental results.
We use pvT data of Straty et al. [8.43] for benzene. These data range from 423.155
K to 723.187 K for temperature, from 0.886 MPa to 35.506 MPa for pressure and from
1.241 mol/l to 9.363 mol/l for both vapour and liquid densities. We remind that the fitted
data for BACKONE are saturated liquid densities from 6.76 mol/l to 11.32 mol/l, vapour
pressures from 0.007MPa to 3.44 MPa, corresponding to temperatures from 285.96 K to
534.05 K.
First we use BACKONE, input temperatures and densities from Straty et al. [8.43]
to calculate pressures of all 438 data points. Then comparison between BACKONE results
and experimental data of pressures shows that the average absolute deviation is 1.64%. We
observe that there are 6 experimental data points having strange absolute deviations, from
9.5% to 63.4%. One may think that there are typing mistakes or unreliabilities of the
published data points because around the strange points, other experimental data are still
consistent with BACKONE results. Thus we exclude the 6 data points.
The AAD between BACKONE results and experimental data of pressures for 432
points is 1.27%. Further comparison between BACKONE results and data of Straty et al. is
made for number of data points having absolute deviation larger than 4%. With this
criterion, BACKONE provides only 5 points with absolute deviation up to 4.5%. The
deviations of BACKONE results and pressures of Straty for benzene are shown in figure
8.2.
We also compare pvT data of Straty and co-workers [8.44] with BACKONE results
for toluene. The data range from 348.07 K to 673.184 K for temperature, from 0.2102 MPa
to 36.2038 MPa for pressure and from 1.45 mol/l to 8.34 mol/l for vapour and liquid
densities. Deviations of experimental data of Straty et al. [8.44] and BACKONE results at
some typical temperatures are given in figure 8.3.
143
4
100(pexp-pB1)/pexp
2
0
-2
-4
-6
450
500
550
600
650
700
750
T [K]
Figure 8.2. Deviations of BACKONE pressures (B1) of benzene in homogeneous regions
and experimental data (exp) of Straty et al. [8.43].
4
100(ρexp-ρB1)/ρexp
2
0
-2
-4
-6
-8
-10
520
540
560
580
600
620
640
660
680
700
T [K]
Figure 8.3. Deviations of BACKONE densities of toluene in homogeneous regions and
experimental data of Straty et al. [8.44].
8.4.2
Ethylbenzene, butylbenzene
The description of BACKONE for vapour pressures, saturated liquid densities of
ethylbenzene and butylbenzene is quite good, figure 8.4 and figure 8.5. The deviations of
144
experimental data of ethylbenzene and BACKONE vapour pressures and saturated liquid
densities are from -0.9% to 1.8% and from -0.5% to 0.2% respectively.
The deviations of experimental data and BACKONE results of butylbenzene are
higher than those of ethylbenzene. In detail, the deviations of vapour pressures and
saturated liquid densities are from -1.2% to 1.8% and from -0.9% to 1.6% respectively. The
large deviations of butylbenzene can be explained by three reasons. Firstly, experimental
vapour pressures range only from 0.52Tc to 0.76Tc. Secondly, saturated liquid densities of
Chirico et al. [8.22] are not very accurate, figure 2 in [8.22]. Finally, molecular structure of
butylbenzene is larger than that of ethylbenzene.
2
1.5
1
0.5
0
-0.5
-1
-1.5
250
350
450
550
650
T [K]
Figure 8.4. Comparison of vapour pressures ps from BACKONE with: experimental data of
ethylbenzene from × Ambrose [8.14] and ▪ Chirico et al. [8.21]; --- calculated values of
butylbenzen from Eq. (8.1); experimental data of butylbenzene from Chirico et al. [8.22].
145
2.0
100(ρ'cal-ρ'exp )/ρ'exp
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
240
340
440
540
640
T [K]
Figure 8.5. Comparison of saturated liquid density ρ’ from BACKONE with experimental
results: X experimental data of ethylbenzene from Hales and Townsend [8.27] and …...
calculated data of ethylbenzen from Eq. (8.2); experimental data of butylbenzene from
Chirico et al. [8.22] and − − − calculated data of butylbenzene from Eq. (8.2).
8.4.3
o-xylene, m-xylene, p-xylene
The other considered alkylbenzenes are o-xylene, m-xylene, and p-xylene.
Available vapour pressures of the xylenes range from about 0.45 Tc to critical point. For
saturated liquid densities of m-xylene and o-xylene, available data range from about 0.5 Tc
to about 0.85 Tc. Saturated liquid densities of p-xylene are available up to critical point. We
observe that experimental data of p-xylene from Francis [8.28] range from 0.69Tc to 1.00Tc
have larger uncertainties than those of Hales and Townsend [8.27]. The deviations of
experimental data and BACKONE vapour pressures and saturated liquid densities of
xylenes are shown in Figures 8.6 and 8.7, respectively.
146
4
3
2
1
0
-1
-2
260
360
460
560
660
T [K]
Figure 8.6. Comparison of vapour pressures ps from BACKONE results with experimental:
O experimental data of o-xylene from Pitzer and Scott [8.19], Ambrose [8.14], and Chirico
et al. [8.23]; X experimental data of m-xylene from Pitzer and Scott [8.19], Ambrose
[8.14], and Chirico et al. [8.24]; and experimental data of p-xylene from Pitzer and Scott
[8.19], Ambrose [8.14], Osborn and Douslin [8.25], and Chirico et al. [8.26].
0.8
0.6
100(ρ'cal-ρ'exp )/ρ'exp
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
-1.2
260
360
460
560
T [K]
Figure 8.7. Comparison of saturated liquid densities ρ’ from BACKONE results and
experimental data: O experimental data of o-xylene from Hales and Townsend [8.27],
Chirico et al. [8.23] and ⎯ calculated data of o-xylene from Eq. (8.2); X experimental data
of m-xylene from Hales and Townsend [8.27], Chirico et al. [8.24] and …... calculated data
of m-xylene from Eq. (8.2); experimental data of p-xylene from Hales and Townsend
[8.27], and Francis [8.28].
147
In figure 8.6, two data points of Pitzer and Scott [8.19] for m-xylene and o-xylene in
very low temperature have strange large deviations. Because they are not consistent with
other data points so one may obviously think of the unreliability of the two data points. We
observe from figure 8.6 that most experimental data points have deviations within ± 1%.
The deviations of vapour pressure of xylenes increase to about 3.4% in the temperature
range from 0.9 Tc up to around critical points.
Deviations of saturated liquid density are smaller than those of vapour pressure. In
detail, the deviations of o-xylene and m-xylene are from -0.77% to 0.52% and from -0.75%
to 0.43% respectively, figure 8.7. The deviations of saturated liquid density of p-xylene are
from -0.60% to 0.32% for experimental data of Hales and Townsend [8.27] and from 1.05% to 0.43% for the data of Francis [8.28].
8.5
Thermodynamic properties from BACKONE for ethylbenzene, butylbenzene, oxylene, m-xylene, and p-xylene
8.5.1
Ideal gas heat capacity
In order to calculate caloric properties, BACKONE needs additional information of
the ideal gas heat capacity. The isobaric ideal gas heat capacities of almost all the
investigated substances can be found in NIST homepage [8.45], in the book of Prausnitz et
al. [8.46], and in paper of Pitzer and Scot [8.19]. In these sources, NIST give values of
isobaric heat capacity at different temperatures and the original sources but we can not
access their original sources. However, NIST have made comparisons of its values with
statistical calculated values from different sources and pointed out that it’s values and data
of Pitzer and Scot [8.19] and some other sources for xylenes are agreed within 1.5 J/mol*K.
Prausnitz et al. give parameters for calculation of isobaric ideal gas heat capacity with Eq.
(8.3):
c0p/R = A + B.T + C.T2 + D.T3 + E.T4
(8.3)
For further study, it is now interesting to investigate the difference of data from the
mentioned sources. Firstly we fit Eq. (8.3) to data of benzene from NIST homepage. Our
fitted parameters and parameters from [8.46] are given in Table 8.7. Secondly, we make
comparison of data from NIST homepage, c0p,1, and data from Eq. (8.3) with parameters
148
from [8.46] , c0p,2, and with our fit parameters, c0p,3. The comparison is given in Table 8.8
and Table 8.9.
Table 8.7. Parameters for calculation of isobaric heat capacity of benzene
A
3.55100E+00
4.02743E+00
B
-6.18400E-03
-1.13354E-02
C
1.43650E-04
1.60495E-04
D
-1.98070E-07
-2.20137E-07
E
8.23400E-11
9.24976E-11
Data source
[8.46]
[8.45]
Table 8.8. Comparison of isobaric ideal gas heat capacity between NIST data and data from
Eq. (8.3)
T [K]
50
100
150
200
273.15
298.15
300
400
500
600
700
800
900
1000
c0p,1
33.27
35.11
41.94
53.17
74.55
82.44
83.02
113.52
139.35
160.09
176.78
190.45
201.82
211.41
c0p,2
29.738
34.748
43.474
54.937
74.842
82.129
82.674
112.186
139.342
161.656
178.283
190.022
199.316
210.248
(c0p,1- c0p,2)/ c0p,1
-10.6%
-1.0%
3.7%
3.3%
0.4%
-0.4%
-0.4%
-1.2%
0.0%
1.0%
0.9%
-0.2%
-1.2%
-0.5%
c0p,3
31.886
35.652
43.585
54.601
74.284
81.575
82.121
111.842
139.245
161.652
178.234
190.005
199.828
212.409
(c0p,1- c0p,3)/ c0p,1
-4.2%
1.5%
3.9%
2.7%
-0.4%
-1.1%
-1.1%
-1.5%
-0.1%
1.0%
0.8%
-0.2%
-1.0%
0.5%
We learn from table 8.8 and Table 8.9 that differences of isobaric gas heat
capacities from different approaches are large in low temperature range, but only within 1%
for the temperature higher than 298.15K. Furthermore, we also make comparison between
data from Pitzer and Scot [8.19] and data from Prausnitz et al. [8.46] and find out that the
differences are also mostly with 1%. Because data from NIST and Prausnitz et al. are
compiled from different sources which we can not access so we will use data from other
original sources if we have. In case we do not have data from original sources we will use
data from either Prausnitz et al. [8.46] or NIST homepage [8.45]. Table 8.10 shows
parameters, temperature ranges and data sources of our studied aromates.
149
Table 8.9. Comparison of isobaric ideal gas heat capacity from Eq. (8.3) with parameters
from this study and from Prausnitz et al. [8.46].
T [K]
50
75
100
125
150
175
200
225
250
275
300
325
350
375
400
425
450
475
500
525
550
575
600
625
650
675
700
725
750
775
800
825
850
875
900
925
950
975
1000
c0p,2
[8.46]
29.738
31.714
34.748
38.710
43.474
48.920
54.937
61.417
68.261
75.376
82.674
90.075
97.505
104.895
112.186
119.321
126.252
132.938
139.342
145.436
151.196
156.606
161.656
166.343
170.669
174.644
178.283
181.609
184.650
187.440
190.022
192.444
194.758
197.026
199.316
201.699
204.258
207.076
210.248
c0p,3
This study
31.886
33.176
35.652
39.168
43.585
48.771
54.601
60.958
67.731
74.817
82.121
89.553
97.032
104.484
111.842
119.047
126.044
132.790
139.245
145.379
151.168
156.596
161.652
166.336
170.651
174.610
178.234
181.547
184.585
187.388
190.005
192.491
194.909
197.329
199.828
202.490
205.408
208.679
212.409
150
(c0p,2- c0p,3)/ c0p,3
-6.7%
-4.4%
-2.5%
-1.2%
-0.3%
0.3%
0.6%
0.8%
0.8%
0.7%
0.7%
0.6%
0.5%
0.4%
0.3%
0.2%
0.2%
0.1%
0.1%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
-0.1%
-0.2%
-0.3%
-0.4%
-0.6%
-0.8%
-1.0%
Table 8.10. Parameters of Eq. (8.3) for calculations of isobaric gas heat capacity
Substance
A
B
C
D
E
Tmin
Tmax
benzene
toluene
ethylbenzene
butylbenzene
o-xylene
m-xylene
p-xylene
- 5.7060
-4.7793
2.8611
6.4900
-1.3865
-3.4749
-2.7508
6.6468E-02
7.0821E-02
2.4422E-02
1.9080E-02
6.8366E-02
7.4572E-02
6.9888E-02
- 5.4042E-05
-4.7711E-05
9.8673E-05
1.5665E-04
-3.4018E-05
-4.1203E-05
-3.2768E-05
2.2087E-08
1.4068E-08
-1.5176E-07
-2.2059E-07
2.2944E-09
6.0963E-09
-1.1154E-10
-3.5307E-12
-1.0756E-12
6.3489E-11
8.8870E-11
2.1532E-12
1.3830E-12
3.0077E-12
298
298
0
200
298
298
298
1500
1500
1000
1000
1500
1500
1500
8.5.2
c0 p
sources
[8.19]
[8.19]
[8.47]
[8.46]
[8.19]
[8.19]
[8.19]
Tables of saturation properties and T,s-diagrams
For the investigated substances, reference equations of state are available for only
benzene and toluene. Because thermodynamic properties of benzene and toluene can be
calculated from NIST homepage easily, we present only data from equations of state for
ethylbenzene, butylbenzene, o-xylene, m-xylene, and p-xylene in tables 8.11 to 8.15 and
figures 8.8 to 8.12. The reference state is selected at T0 = 298.15 K and p0 = 0.101325 MPa,
2
M
Pa
h0 = 0.0 J/mol and s0 = 0.0 J/mol K.
600
1 MPa
T [K]
500
0.1 MPa
400
0.01 MPa
0.001 MPa
300
-50
0
50
100
150
200
250
s [J/mol.K]
Figure 8.8. T, s-diagram of ethylbenzene showing the saturated liquid curve, the saturated
vapour curve and several isobars
151
Table 8.11. Thermodynamic properties of ethylbenzene from BACKONE EOS
T [K]
ps [Mpa]
270
280
290
300
310
320
330
340
350
360
370
380
390
400
410
420
430
440
450
460
470
480
490
500
510
520
530
540
550
560
570
580
590
600
0.0002
0.0004
0.0008
0.0014
0.0024
0.0040
0.0064
0.0099
0.0149
0.0217
0.0309
0.0430
0.0587
0.0787
0.1037
0.1345
0.1720
0.2170
0.2706
0.3338
0.4074
0.4928
0.5908
0.7027
0.8297
0.9730
1.1338
1.3135
1.5135
1.7352
1.9802
2.2500
2.5462
2.8700
h'
h’’
s’
ρ’’
ρ’
3
[mol/dm ] [mol/dm3] [kJ/mol] [kJ/mol] [kJ/molK]
8.3847
0.0001
-4.60
38.70
-0.0161
8.3103
0.0002
-3.04
39.88
-0.0105
8.2329
0.0003
-1.40
41.10
-0.0047
8.1528
0.0006
0.31
42.37
0.0011
8.0702
0.0009
2.10
43.66
0.0069
7.9853
0.0015
3.94
45.00
0.0128
7.8985
0.0024
5.84
46.38
0.0186
7.8097
0.0035
7.79
47.79
0.0245
7.7193
0.0052
9.79
49.23
0.0303
7.6274
0.0073
11.84
50.71
0.0360
7.5341
0.0102
13.94
52.22
0.0418
7.4394
0.0139
16.08
53.76
0.0475
7.3435
0.0186
18.26
55.33
0.0531
7.2463
0.0244
20.48
56.92
0.0587
7.1479
0.0316
22.74
58.54
0.0643
7.0481
0.0404
25.05
60.18
0.0699
6.9469
0.0509
27.39
61.85
0.0754
6.8442
0.0635
29.78
63.53
0.0808
6.7397
0.0784
32.20
65.23
0.0863
6.6332
0.0958
34.67
66.95
0.0917
6.5244
0.1163
37.17
68.67
0.0970
6.4129
0.1400
39.72
70.41
0.1024
6.2982
0.1676
42.32
72.15
0.1077
6.1796
0.1994
44.96
73.89
0.1130
6.0564
0.2363
47.65
75.63
0.1183
5.9277
0.2789
50.39
77.36
0.1236
5.7919
0.3284
53.19
79.08
0.1288
5.6475
0.3859
56.05
80.77
0.1341
5.4918
0.4532
58.97
82.42
0.1394
5.3215
0.5327
61.98
84.03
0.1448
5.1310
0.6280
65.08
85.56
0.1502
4.9117
0.7442
68.30
87.00
0.1557
4.6487
0.8903
71.69
88.29
0.1614
4.3149
1.0820
75.31
89.35
0.1673
152
s’’
[kJ/molK]
0.1442
0.1428
0.1418
0.1413
0.1410
0.1411
0.1415
0.1421
0.1429
0.1440
0.1452
0.1466
0.1482
0.1498
0.1516
0.1535
0.1555
0.1576
0.1597
0.1618
0.1641
0.1663
0.1686
0.1709
0.1731
0.1754
0.1777
0.1799
0.1820
0.1841
0.1861
0.1879
0.1895
0.1907
Table 8.12. Thermodynamic properties of butylbenzene from BACKONE EOS
T [K]
ps [Mpa]
300
310
320
330
340
350
360
370
380
390
400
410
420
430
440
450
460
470
480
490
500
510
520
530
540
550
560
570
580
590
600
610
620
630
640
0.0002
0.0003
0.0006
0.0010
0.0017
0.0027
0.0043
0.0065
0.0097
0.0141
0.0200
0.0278
0.0379
0.0508
0.0670
0.0871
0.1116
0.1413
0.1766
0.2185
0.2675
0.3244
0.3901
0.4653
0.5508
0.6476
0.7565
0.8786
1.0146
1.1658
1.3332
1.5179
1.7212
1.9443
2.1885
h'
h’’
s’
ρ’’
ρ’
[mol/dm3] [mol/dm3] [kJ/mol] [kJ/mol] [kJ/molK]
6.3751
0.0001
0.27
48.09
0.0010
6.3317
0.0001
1.92
49.87
0.0064
6.2837
0.0002
3.78
51.69
0.0123
6.2318
0.0004
5.81
53.57
0.0185
6.1762
0.0006
7.99
55.50
0.0250
6.1173
0.0009
10.31
57.48
0.0318
6.0555
0.0014
12.76
59.51
0.0386
5.9910
0.0021
15.31
61.58
0.0456
5.9242
0.0031
17.96
63.70
0.0527
5.8553
0.0044
20.70
65.87
0.0598
5.7846
0.0061
23.52
68.07
0.0669
5.7124
0.0083
26.42
70.32
0.0741
5.6388
0.0111
29.39
72.62
0.0812
5.5641
0.0146
32.42
74.94
0.0884
5.4883
0.0189
35.52
77.31
0.0955
5.4115
0.0242
38.67
79.71
0.1026
5.3338
0.0306
41.88
82.14
0.1096
5.2552
0.0383
45.14
84.61
0.1166
5.1757
0.0474
48.46
87.10
0.1236
5.0951
0.0581
51.83
89.61
0.1305
5.0133
0.0706
55.25
92.15
0.1374
4.9300
0.0852
58.72
94.71
0.1443
4.8452
0.1021
62.24
97.29
0.1511
4.7584
0.1216
65.81
99.88
0.1578
4.6692
0.1442
69.44
102.49
0.1646
4.5772
0.1701
73.12
105.09
0.1713
4.4817
0.2001
76.87
107.70
0.1780
4.3819
0.2347
80.67
110.30
0.1847
4.2767
0.2747
84.54
112.88
0.1914
4.1647
0.3213
88.48
115.44
0.1980
4.0440
0.3760
92.51
117.96
0.2047
3.9117
0.4409
96.63
120.42
0.2115
3.7638
0.5190
100.87 122.80
0.2183
3.5938
0.6150
105.24 125.07
0.2252
3.3919
0.7367
109.81 127.16
0.2323
153
s’’
[kJ/molK]
0.1604
0.1610
0.1620
0.1632
0.1648
0.1665
0.1685
0.1707
0.1731
0.1756
0.1783
0.1812
0.1842
0.1873
0.1905
0.1938
0.1971
0.2006
0.2041
0.2076
0.2112
0.2148
0.2185
0.2221
0.2258
0.2294
0.2331
0.2367
0.2402
0.2437
0.2472
0.2505
0.2537
0.2567
0.2594
Table 8.13. Thermodynamic properties of o-xylene from BACKONE EOS
T [K]
ps [Mpa]
270
280
290
300
310
320
330
340
350
360
370
380
390
400
410
420
430
440
450
460
470
480
490
500
510
520
530
540
550
560
570
580
590
600
610
0.0001
0.0003
0.0005
0.0010
0.0017
0.0029
0.0047
0.0073
0.0111
0.0165
0.0238
0.0335
0.0463
0.0626
0.0833
0.1090
0.1405
0.1787
0.2244
0.2787
0.3423
0.4164
0.5020
0.6001
0.7119
0.8384
0.9809
1.1406
1.3188
1.5168
1.7360
1.9779
2.2439
2.5357
2.8547
h'
h’’
s’
ρ’’
ρ’
8.5182
0.0001
-4.63
39.93
-0.0162
8.4473
0.0001
-3.07
41.17
-0.0106
8.3732
0.0002
-1.42
42.45
-0.0048
8.2961
0.0004
0.32
43.77
0.0011
8.2163
0.0007
2.13
45.12
0.0070
8.1340
0.0011
4.01
46.52
0.0130
8.0495
0.0017
5.95
47.95
0.0190
7.9630
0.0026
7.95
49.41
0.0249
7.8747
0.0039
9.99
50.91
0.0309
7.7848
0.0056
12.08
52.44
0.0368
7.6933
0.0078
14.22
54.00
0.0426
7.6005
0.0108
16.40
55.58
0.0484
7.5064
0.0146
18.63
57.20
0.0542
7.4110
0.0193
20.89
58.84
0.0599
7.3144
0.0252
23.18
60.50
0.0656
7.2166
0.0325
25.52
62.19
0.0712
7.1176
0.0412
27.89
63.89
0.0768
7.0172
0.0517
30.30
65.61
0.0823
6.9153
0.0642
32.75
67.35
0.0878
6.8118
0.0789
35.23
69.10
0.0932
6.7064
0.0961
37.74
70.86
0.0986
6.5988
0.1162
40.30
72.63
0.1040
6.4887
0.1394
42.89
74.41
0.1093
6.3756
0.1663
45.53
76.19
0.1146
6.2590
0.1974
48.20
77.97
0.1198
6.1383
0.2331
50.92
79.74
0.1251
6.0124
0.2744
53.68
81.50
0.1303
5.8802
0.3219
56.50
83.25
0.1355
5.7404
0.3770
59.37
84.97
0.1407
5.5908
0.4412
62.30
86.66
0.1459
5.4285
0.5163
65.30
88.30
0.1512
5.2496
0.6055
68.39
89.89
0.1565
5.0476
0.7129
71.58
91.38
0.1618
4.8124
0.8451
74.90
92.76
0.1673
4.5271
1.0135
78.40
93.95
0.1730
154
s’’
[kJ/molK]
0.1488
0.1474
0.1465
0.1459
0.1457
0.1458
0.1462
0.1469
0.1478
0.1488
0.1501
0.1515
0.1531
0.1548
0.1566
0.1585
0.1605
0.1626
0.1647
0.1669
0.1691
0.1713
0.1736
0.1759
0.1782
0.1805
0.1828
0.1850
0.1873
0.1894
0.1915
0.1935
0.1954
0.1971
0.1985
Table 8.14. Thermodynamic properties of m-xylene from BACKONE EOS
T [K]
ps [Mpa]
270
280
290
300
310
320
330
340
350
360
370
380
390
400
410
420
430
440
450
460
470
480
490
500
510
520
530
540
550
560
570
580
590
600
0.0002
0.0004
0.0007
0.0012
0.0021
0.0036
0.0057
0.0089
0.0135
0.0198
0.0283
0.0396
0.0543
0.0730
0.0966
0.1257
0.1613
0.2042
0.2553
0.3157
0.3863
0.4683
0.5627
0.6706
0.7933
0.9318
1.0876
1.2619
1.4561
1.6717
1.9101
2.1729
2.4616
2.7775
h'
h’’
s’
ρ’’
ρ’
3
[mol/dm ] [mol/dm3] [kJ/mol] [kJ/mol] [kJ/molK]
8.3674
0.0001
-4.50
39.32
-0.0158
8.2946
0.0002
-2.99
40.49
-0.0103
8.2185
0.0003
-1.38
41.70
-0.0047
8.1394
0.0005
0.31
42.95
0.0011
8.0577
0.0008
2.07
44.24
0.0069
7.9736
0.0013
3.90
45.57
0.0127
7.8873
0.0021
5.79
46.94
0.0185
7.7991
0.0032
7.74
48.34
0.0243
7.7091
0.0047
9.74
49.78
0.0301
7.6174
0.0067
11.79
51.25
0.0359
7.5244
0.0093
13.88
52.76
0.0416
7.4299
0.0128
16.02
54.29
0.0473
7.3341
0.0172
18.20
55.85
0.0529
7.2371
0.0226
20.42
57.44
0.0586
7.1388
0.0294
22.68
59.05
0.0641
7.0393
0.0377
24.98
60.68
0.0697
6.9383
0.0477
27.32
62.33
0.0752
6.8358
0.0596
29.70
64.00
0.0806
6.7316
0.0737
32.11
65.68
0.0860
6.6255
0.0904
34.56
67.38
0.0914
6.5171
0.1098
37.06
69.08
0.0967
6.4061
0.1325
39.59
70.80
0.1020
6.2920
0.1589
42.17
72.51
0.1073
6.1741
0.1894
44.78
74.23
0.1126
6.0517
0.2248
47.45
75.94
0.1178
5.9239
0.2657
50.16
77.64
0.1230
5.7892
0.3131
52.93
79.33
0.1283
5.6460
0.3682
55.76
80.99
0.1335
5.4918
0.4328
58.65
82.61
0.1387
5.3230
0.5090
61.63
84.18
0.1440
5.1344
0.6003
64.69
85.69
0.1494
4.9170
0.7115
67.88
87.09
0.1548
4.6555
0.8510
71.23
88.35
0.1604
4.3203
1.0339
74.82
89.38
0.1664
155
s’’
[kJ/molK]
0.1465
0.1450
0.1439
0.1432
0.1429
0.1429
0.1432
0.1437
0.1445
0.1455
0.1467
0.1480
0.1495
0.1511
0.1528
0.1547
0.1566
0.1586
0.1606
0.1627
0.1649
0.1670
0.1692
0.1715
0.1737
0.1759
0.1781
0.1802
0.1823
0.1843
0.1862
0.1879
0.1894
0.1906
Table 8.15. Thermodynamic properties of p-xylene from BACKONE EOS
T [K]
ps [Mpa]
270
280
290
300
310
320
330
340
350
360
370
380
390
400
410
420
430
440
450
460
470
480
490
500
510
520
530
540
550
560
570
580
590
600
0.0002
0.0004
0.0007
0.0013
0.0022
0.0037
0.0060
0.0092
0.0139
0.0203
0.0290
0.0405
0.0555
0.0745
0.0984
0.1278
0.1637
0.2070
0.2586
0.3194
0.3905
0.4729
0.5677
0.6760
0.7991
0.9381
1.0943
1.2689
1.4634
1.6793
1.9179
2.1809
2.4696
2.7852
h'
h’’
s’
ρ’’
ρ’
8.3304
0.0001
-4.53
39.06
-0.0159
8.2570
0.0002
-3.00
40.23
-0.0103
8.1806
0.0003
-1.38
41.43
-0.0047
8.1014
0.0005
0.31
42.67
0.0011
8.0197
0.0009
2.07
43.96
0.0069
7.9357
0.0014
3.90
45.28
0.0127
7.8497
0.0022
5.78
46.64
0.0185
7.7619
0.0033
7.72
48.04
0.0242
7.6724
0.0048
9.71
49.46
0.0300
7.5813
0.0069
11.74
50.92
0.0357
7.4888
0.0096
13.82
52.42
0.0414
7.3950
0.0131
15.94
53.94
0.0470
7.3000
0.0175
18.10
55.48
0.0527
7.2036
0.0231
20.30
57.06
0.0582
7.1061
0.0300
22.54
58.65
0.0637
7.0072
0.0384
24.81
60.27
0.0692
6.9069
0.0484
27.13
61.90
0.0747
6.8050
0.0605
29.48
63.56
0.0801
6.7014
0.0747
31.88
65.22
0.0854
6.5958
0.0915
34.31
66.90
0.0907
6.4878
0.1112
36.78
68.59
0.0960
6.3772
0.1341
39.29
70.29
0.1013
6.2633
0.1606
41.85
71.99
0.1065
6.1456
0.1914
44.44
73.69
0.1117
6.0233
0.2269
47.09
75.38
0.1169
5.8953
0.2681
49.78
77.06
0.1221
5.7602
0.3159
52.53
78.73
0.1273
5.6162
0.3715
55.34
80.37
0.1325
5.4608
0.4367
58.22
81.97
0.1377
5.2900
0.5138
61.18
83.52
0.1430
5.0979
0.6062
64.23
84.99
0.1483
4.8746
0.7192
67.42
86.36
0.1537
4.6019
0.8619
70.78
87.58
0.1594
4.2430
1.0507
74.42
88.55
0.1654
156
s’’
[kJ/molK]
0.1456
0.1440
0.1430
0.1423
0.1420
0.1420
0.1423
0.1428
0.1436
0.1446
0.1457
0.1470
0.1485
0.1501
0.1518
0.1536
0.1555
0.1575
0.1595
0.1616
0.1637
0.1659
0.1680
0.1702
0.1724
0.1746
0.1767
0.1788
0.1809
0.1829
0.1847
0.1864
0.1879
0.1889
1.
5
M
Pa
700
T [K]
600
1 MPa
500
0.1 MPa
400
0.01 MPa
0.001 MPa
300
0
50
100
150
200
250
300
s [J/mol.K]
2
M
Pa
Figure 8.9. T, s-diagram of butylbenzene showing the saturated liquid curve, the saturated
600
T [K]
1 MPa
500
0.1 MPa
400
0.01 MPa
0.001 MPa
300
-50
0
50
100
150
200
250
s [J/mol.K]
Figure 8.10. T, s-diagram of o-xylene showing the saturated liquid curve, the saturated
157
Pa
M
2
600
T [K]
1 MPa
500
0.1 MPa
400
0.01 MPa
0.001 MPa
300
-50
0
50
100
150
200
250
s [J/mol.K]
2
M
Pa
Figure 8.11. T, s-diagram of m-xylene showing the saturated liquid curve, the saturated
600
T [K]
1 MPa
500
0.1 MPa
400
0.01 MPa
0.001 MPa
300
-50
0
50
100
150
200
250
s [J/mol.K]
Figure 8.12. T, s-diagram of p-xylene showing the saturated liquid curve, the saturated
158
8.6
ORC cycle with benzene
Benzene is a carcinogen thus it should not be used as working fluid for ORC.
However, in order to demonstrate the usefulness of BACKONE for aromates in technical
applications, a hypothetic Organic Rankine Cycle with benzene as working fluid is
considered for following two reasons. Firstly, critical temperature of benzene (562.16 K) is
close to that of MDM (564.13 K) which allows a good comparison of the cycles using
fluids from either aromates or siloxanes. Secondly, reference equations of state for benzene
have been developed [8.48], [8.41]. The Bender-type equation [8.48] can be accessed easily
via NIST’s homepage [8.45] so we can make comparison of cycle efficiencies resulting
from the Bender-type equation and BACKONE EOS.
The intention is not an optimization of the process which requires also consideration
of the heat transfer from the heat carrier to the working fluid [8.6] and from the working
fluid to the cooling medium and a variation of the cycle parameters [8.49]. The idea is
simply to compare cycle efficiencies ηth of ORC from BACKONE equation and from
Bender-type equation.
The cycle with the state points is shown for benzene in a T,s-diagram, Figure 8.13.
The cycle starts at state point 1, which is the saturated liquid at the minimum temperature
Tmin = 311.15 K and the corresponding pressure pmin. Then the working fluid is compressed
to state point 2 with the isentropic pump efficiency ηs,P = 0.65. Next, it is heated up
isobarically and vaporized till it reaches just the dew point (state point 3) at Tmax = 523.15
K with the corresponding pressure pmax. This saturated vapour enters the turbine where it
expands to pmin (state point 4) with an isentropic turbine efficiency ηs,T = 0.85. Then it is
cooled and condensed isobarically to reach state point 1. For this cycle the thermal
efficiency ηth is calculated as:
ηth = - [(h4 - h3) + (h2 - h1)] / (h3 - h2).
(8.4)
where h1, h2, h3, and h4 are the specific enthalpies at the respective state points.
159
600
550
2.9993 MPa
3
T [K]
500
450
400
4
2a
350
0.0224 MPa
2
300
4a
1
250
-60
-40
-20
0
20
40
60
80
100
120
s [J/mol.K]
Figure 8.13. T,s-diagram for benzene with BACKONE EOS. The state points of the cycle
are specified in the text and given explicitly in Table 8.16.
We also consider internal heat exchanger (IHE) for heat recovery. The state point at
the outlet of the hot stream is named as 4a, the state point at the outlet of the cold stream is
named as 2a. We assume that the temperature of point 4a is just 10 K higher than that of
point 2. The state points 4a and 2a are also shown in the T,s-diagram of benzene in Figure
8.13. For the cycle with IHE the thermal efficiency ηth is calculated as
ηth = - [(h4 - h3) + (h2 - h1)] / (h3 - h2a).
(8.5)
In order to make comparisons of state points and cycle efficiencies resulting from
BACKONE equation and Bender-type equation, reference state point is selected to have h=
0 J/mol and s = 0 J/mol.K. Reference temperature is normal boiling point temperature.
Reference density is saturated liquid density at reference temperature. Typical properties
from BACKONE equation and Bender-type equation for all state points of ORC with above
boundary conditions are given in Table 8.16 and Table 8.17, respectively.
Based on values in Table 8.16 and Table 8.17, thermal efficiencies of the cycle
without and with IHE can be readily calculated from Eqs. (8.4) and (8.5). The results are
shown in the last lines of Table 8.16 and Table 8.17. We observe from Table 8.16 and
Table 8.17 that deviations of pvT data from the two equations are smaller than 0.6%. Both
160
deviations of enthalpy and entropy from Table 8.16 and Table 8.17 are smaller than 2%.
Cycle efficiencies which are calculated from BACKONE EOS for the case with and
without internal heat exchanger are 27.6% and 25.1%, respectively. With Bender-type
equation of state, cycle efficiencies of ORC are 27.2% and 24.9% for the case with and
without internal heat exchanger, respectively. The cycle efficiencies of ORC resulting from
BACKONE equation and Bender-type equation are in very good agreement. The small
difference, within 0.4%, of the cycle efficiencies shows the uncertainty due to equations of
state.
Table 8.16. Thermodynamic properties of benzene with BACKONE at the state points of
the cycle specified in the text. The last line shows the thermal efficiencies for cycles
without internal heat exchanger (-IHE) and with internal heat exchanger (+IHE).
State
point
1
2
3
4
4a
T [K]
s
h
[J/(mol.K)] [J/mol]
0.0224
-17.58
-5849
2.9993
-17.12
-5432
2.9993
98.62
46210
0.0224
105.18
32853
0.0224
91.66
28163
ηth (+IHE) = 27.6%
ρ [mol/l] p [MPa]
311.1500 10.97727
313.0358 10.99384
523.1500 1.08859
370.9125 0.00731
323.0358 0.00842
ηth (-IHE) = 25.1%
Table 8.17. Thermodynamic properties of benzene with Bender-type equation [8.48] at the
state points of the cycle specified in the text. The last line shows the thermal efficiencies for
cycles without internal heat exchanger (-IHE) and with internal heat exchanger (+IHE).
State
point
1
2
3
4
4a
8.7
T [K]
s
h
[J/(mol.K)] [J/mol]
0.0224
-17.92
-5958
2.9842
-17.46
-5545
2.9842
98.67
46163
0.0224
105.21
32852
0.0224
91.84
28214
ηth (+IHE) = 27.2%
ρ [mol/l] p [MPa]
311.15 11.00800
312.98 11.01800
523.15 1.09510
370.64 0.00732
322.98 0.00843
ηth (-IHE) = 24.9%
Parameters of BACKONE equation of state for 7 aromates have been determined.
The predictive power of BACKONE EOS has been tested for benzene and toluene. The
study shows that prediction for saturated vapour density from BACKONE is very accurate.
The deviations of saturated vapour densities of benzene and toluene from BACKONE and
experiments and are mostly within 1%. It is shown that the average absolute deviations of
161
BACKONE results and 432 tested data points of benzene in homogeneous region of
vapour, liquid, and supercritical gas is 1.27%. The deviations of predicted data from
BACKONE EOS for toluene are in the same magnitude with those of benzene, except in
the region with temperature slightly higher than critical temperature.
All tested properties from BACKONE equation and Bender-type equation are in
good agreement. Furthermore, thermal efficiencies of Organic Rankine Cycle with benzene
as working fluid resulting from BACKONE EOS and reference equation of state are also in
good agreement. The present results pave the way for optimizing ORC cycles using pure
aromates as working fluids. We will use BACKONE EOS for aromates in Chapter 9 for
calculation of ORC.
162
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Chem. Eng. Data, 19 (1974) 114-117
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Equilibria in Xylene Isomerization. 1. The Thermodynamic Properties of p-xylene, J.
Chem. Eng. Data, 42 (1997) 248-261
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hydrocarbons, J. Chem. Thermodynamics, 4 (1972) 763-772.
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Ind. Eng. Chem., 49 (1957) 1779–1786
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density, J. Chem. Eng. Data, 17 (1972) 236-241
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Statistical Associating Fluid Theory for Chain Molecules with Attractive Potentials of
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and magnetic properties of C6H6 and C6F6, Chem. Phys. Lett. 322 (2000) 143-148.
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CH4 , J. Chem. Phys. 109 (1998) 7176-7184.
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to 1000 Bar, J. Phys. Chem. Ref. Data 17 (1988) 1541-1636.
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to 1000 Bar, J. Phys. Chem. Ref. Data 18 (1989) 1565-1636.
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temperatures to 723 K, J. Chem. Eng. Data, 32 (1987) 163-166
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167
9
Comparison between BACKONE and PC-SAFT *
Abstract
Siloxanes are considered as working fluids for medium-temperature ORC processes.
Here, a description of the thermodynamic properties of compact linear and cyclic siloxanes
with BACKONE equation is given. Then, BACKONE and PC-SAFT are compared in
views of correlation and prediction for benzene and hexamethyldisiloxane. Results from
this study for benzene show that BACKONE outperforms PC-SAFT in both correlation and
prediction. For MM, PC-SAFT outperforms BACKONE.
9.1
9.1.1
Development of BACKONE Equations of State for Siloxanes
Introduction
Presently there are strong efforts to develop processes for energy conversion from
renewable heat sources. One very promising process for that purpose is the ClausiusRankine cycle which uses organic substances as working fluids, simply called organic
Rankine cycle (ORC) [9.1]. One important matter is the choice of working fluids suitable
for the different temperature ranges of available heat sources. To answer this question, we
are currently developing equations of state (EOS) to describe thermodynamic properties of
different working fluids. We have already applied the BACKONE EOS [9.2] to study
working fluids for low-temperature organic Rankine cycles [9.3], e.g., for the utilization of
geothermal heat. In the temperature range, we found that the supercritical refrigerant R143a
yields about 20% more power than subcritical working fluids if the pinch problem in the
evaporator is also taken into consideration [9.3].
The problem arises for cycles in which the working fluids reach temperatures higher
than 200°C. Smaller alkanes might be used in supercritical cycles but with increasing chain
lengths the self ignition temperature decreases to about 200°C so that longer alkanes which
are environmentally friendly and yield good thermal efficiencies can not be used any more
for safety reasons. Fluorinated alkanes have a strong global warming potential and
*
Mostly available in: N. A. LAI, M. Wendland, J. Fischer, Development of equations of
state for siloxanes as working fluids for ORC Processes, Proceeding of 24th European
symposium on applied thermodynamics, Santiago de Compostela, Spain, (2009) 200-205
168
extremely long atmospheric lifetimes and hence should not be used for environmental
reasons. Hence, siloxanes and their mixtures [9.4] are considered for this medium
temperature range.
Various siloxanes are potential candidates as working fluids for mediumtemperature ORC processes, but experimental data is mostly not sufficient to set up an
empirical multi-parameter EOS. Thus, it is necessary to use physically based EOS which
have few physically meaningful parameters, need only a small experimental database for
fitting and still give reliable results for thermal and caloric data over a wide range.
BACKONE, for example, needs only vapour pressures and saturated liquid densities for
fitting between 3 and 5 substance specific parameters. But these data must be available over
the whole temperature range which is not the case for many siloxanes, especially at higher
temperatures.
Here, BACKONE EOS was applied for various siloxanes as hexamethyldisiloxane
(MM),
octamethyltrisiloxane
(MDM),
hexamethylcyclotrisiloxane
(D3),
and
octamethylcyclotetrasiloxane (D4). The experimental vapour pressures and saturated liquid
densities of selected siloxanes are generally not covered in full fluid region. Thus, we use
the version of the Rackett equation of Spencer and Danner [9.5] to predict saturated liquid
densities of siloxanes at temperature ranges where experimental data are not available. We
also use Wagner equation [9.6] to correlate and upward extrapolate vapour pressure data of
the studied siloxanes. BACKONE EOS parameters were fitted to these correlated and
extrapolated data. With the addition of ideal gas heat capacities [9.7], we are able to
calculate cycle efficiencies of ORC processes with these working fluids.
9.1.2
Availability of experimental data
In Table 9.1 the critical temperature Tc, pressure pc, and density ρc are given which
have been selected after a critical evaluation of experimental data. Tables 9.2 and 9.3
contain temperature ranges and sources of experimental vapour pressures ps and
experimental saturated liquid densities ρ’ which did seem to be reliable.
169
Table 9.1. Critical data of selected siloxanes
MM (C6H18OSi2)
MDM (C8H24O2Si3)
D3 (C6H18O3Si3)
D4 (C8H24O4Si4)
Tc [K]
518.7 [9.8]
564.13 [9.11]
554.15 [9.9], [9.10]
586.5 [9.12]
Pc [MPa]
1.925 [9.8]
1.4150 [9.11]
1.663 [9.9]
1.3841 [9.12]
ρc [mol/L]
1.637 [9.9], [9.10]
1.1340 [9.11]
1.4144 [9.9], [9.10]
1.0309 [9.9], [9.10]
Vapour pressures are only available for MDM over the whole temperature, range
from 322 K up to the critical temperature [9.11]. For MM experimental data are available
from 303 K up to 412 K [9.9, 9.13] and close to the critical temperature [9.8], for D3 up to
0.76 Tc [9.9], and for D4 from 361 K to the critical point with a gap from 460 K to 505 K
[9.12]. For the correlation and upward extrapolation of vapour pressures we used the
Wagner-equation [9.6]:
ln pr = (1/Tr) [A(1-Tr) + B(1-Tr)1.5 + C(1-Tr)3 + D(1-Tr)6],
(9.1)
where Tr = T/Tc and pr = ps/pc and A, B, C, and D are fit parameters. Both Tc and pc are
taken from Table 9.1. Fitted parameters for selected substances are given in Table 9.2.
Table 9.2. Experimental vapour pressures and parameters of Wanger equation.
Data points
Source
MM (A = -7.2094, B = -0.729325, C = 0.738444, D = -27.0088)
491.60 - 518.70
19
309.36 - 411.57
21
Scott et al. [9.13]
313.15 - 373.61
14
Guzman [9.14]
302.78 - 383.30
15
Flaningam [9.9]
MDM (A = -8.5550, B = 1.65229, C = -6.02152, D = -8.15538)
346.10 - 436.49
12
Flaningam [9.9]
322.44 - 564.13
74
D3 (A = -7.5527, B = 0.00640184, C = -3.91874, D = -6.9479)
342.62 - 419.67
15
Flaningam [9.9]
D4 (A = -8.7381, B = 1.55154, C = -7.29876, D = -4.13027)
361.71- 459.65
13
Flaningam [9.9]
505.40- 586.50
16
Young [9.12]
170
Table 9.3. Experimental saturated liquid densities and parameters of Eq. (9.2).
Data Points
Source
MM (ρp =1.6655 mol/L)
273.15 – 313.15
3
Hurd [9.15]
273.15 – 353.15
10
Gubareva [9.16]
MDM (parameters from [9.11])
273.12 - 564.13
37
273.15 - 353.15
5
Hurd [9.15]
D4 (ρp=1.07303 mol/L)
273.15-353.15
5
Hurd [9.15]
The situation is worse for saturated liquid densities (Table 9.3). For MDM, Lindley
and Hershey [9.11] measured ρ’ from 273 K up to Tc with a gap between 361 K and 426 K
and give a correlation equation used within the present paper. For D3, no experimental
saturated liquid densities are available. However, at low pressures, the differences between
liquid density and saturated liquid density can be neglected and the liquid density of
Palczewska-Tulinska [9.17] at 343.15 K to 387.85 K and 1 atm where used for fitting with
a Rackett equation. For D4, experimental data are available up to critical point with a gap
from 408 K to 503 K. For extrapolation of saturated liquid densities, we found in a previous
study [9.18, 9.19] that the best results are achieved with the Rackett equation of Spencer
and Danner [9.5] which we write here in the form
ln ρ’ = ln ρp - (lnZp)(1-Tr) 2/7.
(9.2)
Eq. (9.2) requires the experimental data for Tc and pc and contains the pseudocritical density ρp as the only fit parameter. The pseudo-compression factor Zp is related to
Tc, pc and ρp by Zp = pc/ρpRTc. Fitted parameter ρp is given in table 9.3. For D3, ρp =
1.40018 mol/L was estimated from homogeneous liquid densities at 1 atm.
9.1.3
BACKONE equation of state
BACKONE is a family of physically based equation of state. The Helmholtz energy
is written in term of a sum of contributions from characteristic intermolecular interactions
[9.20] as F = FH + FA+ FD +FQ, where FH is the hard-body contribution, FA the attractive
dispersion energy contribution, FD is the dipolar contribution, and FQ is the quadrupolar
contribution [9.20, 9.21]. Thus, five substance specific parameters are used: a characteristic
temperature and density T0 and ρ0 and the anisotropy factor α for the first two unpolar
171
contributions and a reduced dipolar moment μ*2 and quadrupolar moment Q*2 for polar
fluids. These parameters are fitted to vapour pressures and saturated liquid densities.
Homogeneous phase pvT data, if available, were used for testing the accuracy of
predictions. Parameters of BACKONE EOS for selected siloxanes are given in table 9.4.
Table 9.4. Parameters of BACKONE EOS for selected siloxanes.
Q*2
Parameters
T0 [K]
μ*2
ρ0 [mol/L]
α
MM
391.2140
1.4540
1.3848
0
6.2260
MDM
381.54963
0.996950
1.431399 6.25986 6.98579
D3
409.38689 1.2340714 1.434726
0
6.98109
D4
311.62587
0.875144
1.350704 11.97262 8.79145
9.1.3.1 MM
The comparison of the vapour pressures from BACKONE with the experimental
data and with the extrapolation equation is shown in Figure 9.1. The relative differences
Δps = (ps,exp - ps,cal)/ ps,exp of the experimental values are seen to range from -2.0% to +1.6%
except in the critical region. For the saturated liquid densities of Hurd [9.15] and Gubareva
[9.16], the relative differences between experimental data and BACKONE results are from
-1.5% to -0.5%. The predictive strength of BACKONE is examined by comparison with
experimental data of McLure et al. [9.22] in the liquid phase and of Marcos et al. [9.23] in
the vapour phase. Almost all relative deviations are in range from -0.5% to 1.0%. This
proves that BACKONE is very good for the prediction in both vapour and liquid phases.
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
-3.0
280
518.7
330
380
430
480
530
T [K]
Figure 9.1. Deviations of BACKONE results for the vapour pressure of MM from
experimental data of ■ Flaningam [9.9], ● Guzman [9.14], Scott et al. [9.13], Δ McLure,
Dickinson [9.8], and from Eq. (9.1) --- .
172
9.1.3.2 MDM
Relative deviations between experimental data of Flaningam [9.9] and Lindley et al.
[9.11] and BACKONE results for MDM are in range from 0% to 2%. For the saturated
liquid density, the deviations are in range from -2.5% to 4%. For predicted liquid phase
densities, the deviations with experimental data of McLure et al. [9.22] are from -2.3% to
0.5% and for vapour phase densities, mostly in the range from -2.5% to 0%.
9.1.3.3 D3
The relative differences between experimental vapour pressures of Flaningam [9.9]
and BACKONE results are in the range from -3.8% to 1.2%. No saturated liquid densities
are available. For homogeneous liquid densities [9.17], the deviations are of from -1.6% to
-0.9%.
9.1.3.4 D4
Compared to the other substances, deviations between experimental data and
BACKONE results are larger for D4. In detail, the differences of the vapour pressure of
Flaningam to BACKONE range from -3.5% to 3.5% and of the saturated liquid densities of
Hurd from -4.9% to 2.0%. Deviations of saturated liquid densities of Young are in between
-0.6% and 2.4%. Experimental liquid densities of McLure and Palczewska-Tulinska give
quite similar results with differences from BACKONE of -5.5% to -0.4%.
BACKONE gives good results for compact molecules as MM, MDM and D3 which
are still relatively close to the underlying physical model of a sphere or dumbbell. With
increasing size, results get more and more unsatisfying, as for D4, and MD2M, MD3M or
MD4M, which are not given here. We were able to get better results for the larger linear
siloxanes with PC-SAFT equation starting from MD2M elsewhere [9.19].
9.1.4
ORC cycle with MDM
To give an example for the application of the new siloxane equation, thermal
efficiencies ηth of ORC with MDM as working fluid were calculated. The idea is simply to
compare cycle efficiencies of working fluids with results from other equations of state
[9.19, 9.24] for the various cases given there.
173
The cycle with the state points is shown for MDM in a T,s-diagram in Figure 9.2.
The cycle starts at state point 1, which is the saturated liquid at the minimum temperature
Tmin = 298.15 K and the corresponding pressure pmin. Then the working fluid is compressed
to state point 2 with the isentropic pump efficiency ηs,P = 0.65. Next, it is heated up
isobarically and vaporized till it reaches just the dew point (state point 3) at Tmax = 533.15
K with the corresponding pressure pmax. This saturated vapour enters the turbine where it
expands to pmin (state point 4) with an isentropic turbine efficiency ηs,T = 0.85. Then it is
cooled and condensed isobarically to reach state point 1. For this cycle the thermal
efficiency ηth is given as
ηth = - [(h4 - h3) + (h2 - h1)] / (h3 - h2).
(9.3)
where h1, h2, h3, and h4 are the specific enthalpies at the respective state points.
Because of the large overheating of the expanded vapour in point 4, an internal heat
exchanger (IHE) can be used for heat recovery. In that case, the state point at the outlet of
the hot stream is 4a, the state point at the outlet of the cold stream is 2a, and a pinch occurs
between state points 4a and 2. We assume that temperature difference at the pinch point is
just 10 K. The state points 4a and 2 are also shown in the T,s-diagram of MDM in Figure
9.2. For the cycle with IHE the thermal efficiency ηth is given as
ηth = - [(h4 - h3) + (h2 - h1)] / (h3 - h2a).
(9.4)
The thermal efficiency of the cycle with IHE is 33.9% and without IHE 18.1%. PCSAFT gives 32.2% and 17.6% at the same conditions which compares fairly well [9.19].
The results point out that for fluids with strongly re-entrant saturated vapour lines as MDM,
IHE can improve cycle efficiency up to 87%.
174
600
550
0.875 MPa
3
T [K]
500
450
4
2a
400
350
300
2
0.60E-3 MPa
4a
1
250
-100
0
100
200
300
400
s [J/molK]
Figure 9.2. T,s-diagram for ORC using MDM as working fluid (reference point: T0 =
298.15 K, p0 = 0.101325 MPa, h0 = 0.0 J/mol, s0 = 0.0 J/mol K)
Angelino and Colonna [9.24] have calculated thermal efficiencies of a
MM/MDM/MD2M mixture with the cubic PRSV equation of state under the same
conditions with IHE and got a lower value with 29.9%. The difference of 4% in ηth
between BACKONE for pure MDM and PRSV for the mixture might be attributed to an
effect of the mixed towards a pure fluid or to an uncertainty of the equations of state.
9.2
Comparison between BACKONE and PC-SAFT
The experimental data of benzene is mostly sufficient for both single and two phase
regions. Thus we use this substance to examine correlation and extrapolation properties of
BACKONE and PC-SAFT equations. For siloxanes, we select MM as a tested substance
because experimental data for both single and two phase regions are available and the
square reduced dipolar moment of MM is reasonable with its structure.
9.2.1
Benzene
In construction of BACKONE and PC-SAFT we use vapour pressures from [9.25 9.27] and saturated liquid densities from [9.28]. Critical temperature and pressure are taken
from [9.25]. Wagner equation for vapour pressure [9.6] and two-parameter equation for
correlation and extrapolation of saturated liquid density [9.18] are used to generate input
175
data from 0.51 Tc to 0.95 Tc for the construction of BACKONE and PC-SAFT.
Comparisons of the vapour pressures and saturated liquid densities from BACKONE and
PC-SAFT with the experimental data are shown in Figures 9.3 and 9.4. The relative
differences Δps = (ps,exp - ps,cal)/ ps,exp of the experimental values of Ambrose [9.25, 9.26]
range from -0.98% to +0.57% for BACKONE and from -1.24% to 0.76% for PC-SAFT.
Relative differences between saturated liquid densities of Hales and Townsend [9.28] and
BACKONE results range from -0.23% to 0.32%, figure 9.4. These deviations are smaller
than those of PC-SAFT, from -0.40% to 0.98%.
1
0.5
0
-0.5
-1
-1.5
280
330
380
430
480
530
580
T [K]
Figure 9.3. Deviations of PC-SAFT vapour pressure for benzene from experimental data of
Δ Ambrose [9.25] and × Ambrose [9.26]. Deviations of BACKONE vapour pressure for
benzene from experimental data of Ambrose [9.25] and Ο Ambrose [9.26].
176
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
260
310
360
410
460
510
T [K]
Figure 9.4. Deviations of BACKONE and Δ PC-SAFT saturated liquid density for
benzene from experimental data of Hales and Townsend [9.28].
We remind that the strength of physically based equations of state is predictive.
Thus, in case that accurate experimental data are not sufficient enough to set up an
empirical multi-parameter EOS, physically based equations of state should be employed.
Now we turn to test the possibilities of extrapolation with the two physically based
equations of state, BACKONE and PC-SAFT.
Experimental data for examination of predictabilities of BACKONE and PC-SAFT
are taken from Straty et al. [9.29]. The published pvT data by Straty et al. range from
423.155 K to 723.187 K for temperature, from 0.886 MPa to 35.506 MPa for pressure and
from 1.241 mol/l to 9.363 mol/l for both vapour and liquid densities. In fitting BACKONE
and PC-SAFT we use saturated liquid densities from 6.76 mol/l to 11.32 mol/l, vapour
pressures from 0.007MPa to 3.44 MPa, corresponding to temperatures from 285.96 K to
534.05 K.
In previous chapter we point out that the average absolute deviation (AAD) between
BACKONE and experimental pressures for 432 points of [9.29] is 1.27%. In this study, the
AAD between PC-SAFT and experimental pressures for the same 432 points is 4.79%.
Further comparison between BACKONE and PC-SAFT with data by Straty et al. is made
177
by comparing total number data points which have absolute deviation larger than 4%. With
this criterion, BACKONE provides only 5 points with absolute deviation up to 4.5%. PCSAFT provides 247 points with absolute deviation up to 13.8%. We also check PC-SAFT
for criterion of 8% and find out that there are 74 points having the absolute deviation larger
than 8%. The results confirm the predictive strength of BACKONE. It surely outperforms
the PC-SAFT for compact molecules.
9.2.2
MM
BACKONE and PC-SAFT equations of state have been developed for MM by using
the same input data set as given in part 9.1.3. Comparison of experimental vapour pressures
and vapour pressures from BACKONE and PC-SAFT is given in figure 9.5. Figure 9.6
shows deviations between experimental saturated liquid densities with data from
BACKONE and PC-SAFT.
2.0
1.0
0.0
-1.0
-2.0
-3.0
-4.0
-5.0
-6.0
280
518.7
330
380
430
480
530
T [K]
Figure 9.5. Deviations of BACKONE (solid line for predicted data) and Δ PC-SAFT
(open line for predicted data) vapour pressure for MM from experimental data of
Flaningam [9.9], Scott et al. [9.13], McLure et al. [9.8], Guzman [9.14].
178
1.2
0.8
0.4
0.0
-0.4
-0.8
-1.2
-1.6
250
300
350
400
450
T [K]
Figure 9.6. Deviations of BACKONE (solid line for predicted data) and Δ PC-SAFT
(open line for predicted data) saturated liquid density for MM from experimental data of
Hurd [9.15] and Gubareva [9.16].
Figures 9.5 and 9.6 show that PC-SAFT EOS can correlate both vapour pressures
and saturated liquid densities better than BACKONE EOS can. The extrapolation of
BACKONE and PC-SAFT equations are investigated with experimental liquid densities
from [9.30] and vapour densities from [9.31]. Results for comparison of extrapolation of
BACKONE and PC-SAFT are given in table 9.5. The results from table 9.5 prove that PCSAFT outperforms BACKONE for MM.
Table 9.5. Extrapolation possibilities of BACKONE and PC-SAFT equations
Equation
BACKONE
PC-SAFT
9.3
Liquid density of Vapour density of
[9.31]
[9.30]
AAD
0.46
0.26
Max. absolute deviations
1.38
0.86
AAD
0.32
0.10
Max. absolute deviations
1.94
0.19
Quantity
We have studied BACKONE equation of state for selected pure siloxanes. It is
shown that BACKONE can describe thermodynamic properties of selected siloxanes in a
good agreement with available experimental data. For compacted molecules as MM and D3,
179
BACKONE performs better than for larger molecules as MDM and D4. The results pave
the way for optimization of ORC processes.
Our investigation shows that both BACKONE and PC-SAFT equations of state are
accurate for both correlation and prediction of thermodynamic properties of working fluids.
Each equation has pros and cons for certain group of substances. In this investigation,
BACKONE outperforms PC-SAFT for benzene but not for MM.
180
References
[9.1]
G. Rogers, Y. Mayhew, Engineering Thermodynamics, Work and Heat Transfer.
4th ed. Harlow: Longman, 1992.
[9.2]
U. Weingerl, M. Wendland, J. Fischer, A. Müller, J. Winkelmann, The BACKONE
family of equations of state: 2. Nonpolar and polar fluid mixtures, AIChE-J., 47 (2001) 705
-717.
[9.3]
B. Saleh, G. Koglbauer, M. Wendland, J. Fischer, Working fluids for low
temperature Organic Rankine Cycles, Energy, 32 (2007) 1210-1221.
[9.4]
G. Angelino, P. Colonna di Paliano, Multicomponent working fluids for organic
Rankine cycles (ORCs), Energy, 23 (1998) 449-463.
[9.5]
C. F. Spencer, R. P. Danner, Improved equation for prediction of saturated liquid
density, J. Chem. Eng. Data, 17 (1972) 236-241.
[9.6]
W. Wagner, New vapour pressure measurements for argon and nitrogen and a new
method for establishing rational vapour pressure equations, Cryogenics, 13 (1973) 470-482.
[9.7]
N.R. Nannan, P. Colonna, C.M. Tracy, R.L. Rowley, J.J. Hurly, Ideal-gas heat
calculations, Fluid Phase Equilib., 257 (2007) 102-113.
[9.8]
I.A. McLure, E. Dickinson, Vapour pressure of hexamethyldisiloxane near its
critical point: corresponding-states principle for dimethylsiloxanes, J. Chem. Thermodyn., 8
(1976) 93-95.
[9.9]
O.L. Flaningam, Vapour pressures of poly(dimethylsiloxane) oligomers, J. Chem.
Eng. Data, 31 (1986) 266-272.
[9.10] G. P. Pollnow, Dow Corning, Midland, MI. 1957.
[9.11] D.L. Lindley, H.C. Hershey, The orthobaric region of octamethyltrisiloxane , Fluid
Phase Equilib., 55 (1990) 109-124.
[9.12] C. L. Young, Equilibrium properties of octamethylcyclotetrasiloxane near its critical
point and applicability of the principle of corresponding states, J. Chem. Thermodynamics,
4 (1972) 65-75.
181
[9.13] D.W. Scott, J.F. Messerly, S.S. Todd, G.B. Guthrie, I.A. Hossenlopp, R.T. Moore,
A. Osborn, W.T. Berg, J.P. McCullough, Hexamethyldisiloxane: chemical thermodynamic
properties and internal rotation about the siloxane linkage, J. Phys. Chem., 65 (1961) 13201326.
[9.14] J.A. Guzman, Vapour-liquid equilibrium relations in non-ideal systems. The binary
systems: hexamethylsiloxane-toluene, hexamethylsiloxane-ethylalcohol, and ethyl alcoholtoluene at 40, 50, 60, and 70°C, Thesis, (1973) 1-253.
[9.15] C.B. Hurd, Studies on siloxanes. I. The specific volume and viscosity in relation to
temperature and constitution, J. Am. Chem. Soc., 68 (1946) 364-370.
[9.16] A.I. Gubareva, Temperature dependence of physicochemical characteristics of
hexamethyldisiloxane and diphenylmethylsilane, Deposited Doc. Oniitekhim (1983) 874883.
[9.17] M. Palczewska-Tulinska, P.J. Oracz, Selected physicochemical properties of
hexamethylcyclotrisiloxane,
octamethylcyclotetrasiloxane,
and
decamethylcyclopentasiloxane , J. Chem. Eng. Data, 50 (2005) 1711-1719.
densities, Fluid Phase Equilib., 280 (2009) 30-34.
[9.20] A. Müller, J. Winkelmann, Backone Family of Equations of State: 1. Nonpolar and
Polar Pure Fluids , J. Fischer, AIChE J, 42 (1996) 1116-1126.
BACKONE equations, Fluid Phase Equilib., 152 (1998) 1-22.
[9.22] I. A. McLure, A. J. Pretty, P. A. Sadler, Specific volumes, thermal pressure
coefficients, and derived quantities of five dimethylsiloxane oligomers from 25 to 140°C, J.
Chem. Eng. Data, 22 (1977) 372 – 376.
[9.23] D. H. Marcos, D. D. Lindley, K. S. Wilson, W. B. Kay, H. C. Hershey, A (p, V, T)
study of tetramethylsilane, hexamethyldisiloxane, octamethyltrisiloxane, and toluene from
423 to 573 K in the vapour phase, J. Chem. Thermodyn. 15 (1983) 1003-1014.
182
fuel cells, 2000 Fuel Cell Seminar, Portland, 114 (2000) 667-670.
[9.25] Ambrose, D., Vapour pressures of some aromatic hydrocarbons, J. Chem.
Thermodyn., 19 (1987) 1007-1008
[9.26] D. Ambrose, Reference values of the vapour pressures of benzene and
hexafluorobenzene, J. Chem. Thermodynamics, 13 (1981) 1161-1167
[9.27] D. W. Scott, and A. G. Osborn, Representation of vapour-pressure data, J. Phys.
Chem., 83 (1979) 2714-2723
[9.28] J. L. Hales and R. Townsend, Liquid densities from 293 to 490 K of nine aromatic
hydrocarbons, J. Chem. Thermodynamics, 4 (1972) 763-772.
[9.29] G. C. Straty, M. J. Ball, and T. J. Bruno, PVT measurements on benzene at
temperatures to 723 K, J. Chem. Eng. Data, 32 (1987) 163-166
[9.30] I. A. McLure, A. J. Pretty, P. A. Sadler, Specific Volumes, Thermal Pressure
Coefficients, and Derived Quantities of Five Dimethylsiloxane Oligomers from 25 to 140
O
C, J. Chem. Eng. Data., 22 (1977) 372 – 376.
[9.31] D. H. Marcos, D. D. Lindley, K. S. Wilson, W. B. Kay, H. C. Hershey, A (p,V,T)
study of tetramethylsilane, hexamethyldisiloxane, octamethyltrislioxane, and toluene from
423 to 573 K in the vapour phase, J. Chem. Thermodyn., 15 (1983) 1003-1014.
183
10 Working fluids for medium-temperature Organic Rankine cycles *
Abstract
Thermodynamic properties of different working fluids from BACKONE and PCSAFT EOS have been used to investigate medium temperature Organic Rankine cycles
(ORC) with maximum temperature from 523.15 K to 623.15 K and minimum temperatures
from 311.15 K to 358.15 K. The temperature rangea are suitable for electricity generation
from different renewable energy sources such as biomass, solar, geothermal and waste
heats. Medium temperature ORC are investigated with 2 small alkanes as n-pentane, isopentane, with 6 aromates as toluene, ethylbenzene, butylbenzene, o-xylene, m-xylene, pxylene,
and
with
4
siloxanes
as
hexamethyldisiloxane,
octamethyltrisiloxane,
decamethytetrasiloxane, dodecamethylpentasiloxane, and with cyclopentan. The study
shows that the cycle efficiency for the case with internal heat exchanger (IHE) increases
with the increase of the critical temperature. Furthermore, the cycle efficiency does not
only depend on the temperature boundary but it also depends on the maximum pressure of
the cycle. The study also shows that if heat carrier is heated up and circulated in a loop,
selection of working fluids should be based on cycle efficiency and size of equipment.
Aromates prove to be the most potential working fluids for this case. If heat carrier isn’t
circulated in a loop, the criteria for selection of working fluid are the size of equipment and
the total efficiency, not the cycle efficiency. In this case, the investigated alkanes,
cyclopentane, MM, MDM, toluene and o-xylene are the most potential working fluids,
depending on the working temperature range.
10.1 Introduction
There have been great deals of renewable energy sources such as solar energy,
geothermal heat, biomass, and waste heat from different industrial plants. In general,
temperature of the energy sources varies in large range, from decades degree to hundreds
degree. For example, temperature of solar energy can reach as high as about 3000C in
southern areas if parabolic collectors are provided. Temperature of geothermal heat source
can be over 2000C, depending on location and the deep of the thermal sources [10.1]. Waste
*
See also in N. A. LAI, M. Wendland, J. Fischer, Working fluids for medium-temperature
Organic Rankine cycles, to be submitted in refereed journal.
184
heat from different industrial plants can have temperature up to 3000C, for example in
cement plants [10.2, 10.3]. Temperature of burning biomass is higher and more stable than
the above mentioned heat sources.
One approach to utilize the renewable energy sources is to use Organic Rankine
cycles (ORC). In the ORC, organic fluids are used as working fluids instead of water in
steam power plant. The reason for using organic substances instead of conventional water is
that in low and medium temperature range, ORC gives higher efficiency than that of water
cycle. In economic point of view, investment in steam power plants to utilize renewable
energy sources directly is not as feasible as investment in ORC plants.
We want to emphasize that organic power plants already exist. Example of existing
ORC is a wood fired combined heat-and-power plant using Octamethyltrisiloxane (MDM)
as working fluid in Oerlinghausen, Germany [10.4, 10.5]. Other combined heat-and-power
plants using silicon oil as working fluid were built in Admont [10.6] and Lienz [10.7],
Austria. Last example here is an ORC using n-pentane as working fluid for recovering
waste heat from a cement plant in Heidelberger, Germany [10.3]. More information about
existing ORC plants and current state of the art of ORC can be found in a survey of
Windmann [10.8].
A crucial question for medium temperature ORC processes with maximum
temperature higher than 200°C is to select appropriate working fluids for given temperature
ranges of the available heat sources. Smaller alkanes might be used in supercritical cycles
but with increasing chain lengths the auto-ignition temperature decreases to about 200°C.
Thus longer alkanes which are environmentally friendly and yield good thermal efficiencies
can not be used for safety reasons. Fluorinated alkanes have a strong global warming
potential and extremely long atmospheric lifetimes and hence should not be used for
environmental reasons. Aromates, important industrial compounds, have high critical
temperature and high auto-ignition temperature may be used. Another group of substances
can be considered as working fluids for the medium temperature ORC processes is
siloxanes.
A problem in the ORC is the heat transfer from the heat carrier to the working fluid.
A pure fluid as working fluid in a subcritical cycle causes a pinch point [10.9] which limits
185
the transferable heat. In order to increase the transferable heat, one may use transcritical
cycles with pure fluids, subcritical cycles with mixtures or transcritical cycles with
mixtures. Hence, in order to optimize an ORC cycle under given conditions for the heat
carrier and the cooling, accurate equations of state (EOS) are needed for potential pure
fluids and mixtures.
The cubic Peng-Robinson [10.10] and Peng-Robinson-Stryjek-Vera (PRSV)
equation [10.11] has been applied for study on working fluids for ORC processes [10.1210.15]. The limitations of the cubic equations of state have, however, already been pointed
out elsewhere, example in [10.16]. Currently two physically based equations of state,
BACKONE and PC-SAFT, have been proved to be accurate for various working fluids
[10.9, 10.17-10.23]. The advantage of the physically based equations of state is that they
have possibilities of prediction and use small numbers of accurate experimental vapour
pressures and saturated densities. Further importance of these equations is that they can
easily extend to mixture accurately with only one parameter for each binary. Thus we have
currently applied BACKONE and PC-SAFT equations of state for different working fluids
[10.18, 10.19, 10.24, 10.25] which are considered as working fluids for medium
temperature ORC processes.
In this paper, to continue our project [10.26], we present some results for medium
temperature ORC processes. Part 2 describes typical ORC diagrams. Part 3 presents our
selected 13 potential fluids from different molecular classes together with accurate
available equations of state currently developed by us. Analyses of cycle efficiencies and
volume flow rates at the inlet and outlet of the turbine for subcritical and transcritical cycles
are given in part 4. Heat transfer from heat carrier to working fluids is considered in part 5.
10.2 Cycle description
10.2.1 Organic Rankine cycles
In investigation of cycle efficiencies of different working fluids we consider simple
cycles, figure 10.1. In figure 10.1, four main components of the cycle are pump, evaporator,
turbine and condenser for case ‘A’ without internal heat exchanger (IHE) and one more
component for the case ‘B’ with IHE.
186
mc
3
mc
m
3
5
m
5
Qin
Qin
Wout
6
Wout
4
2a
6
2
4
2
Qout
Win
4a
Qout
Win
1
1
A
B
Figure 10.1. Flow diagram of organic Rankine cycle: ‘A’ without IHE, ‘B’ with IHE
Pump process (states 1–2)
We assume that the state of working fluids leaving the condenser and entering the
pump is saturated liquid. Temperature and pressure of fluid at this state point 1 are T1 and
p1. Pressure of point 1 is the lowest one, called as pmin (pmin = p1). Due to driving
mechanism on the circulation pump, working fluid regains high pressure pmax (or p2 at state
point 2) after the pump. With an isentropic efficiency ηsP, work for pumping liquid from
state point 1 to state point 2 is calculated by the following equation:
w12 = h2 – h1 = (h2’-h1)/ηsP
where h2’ is enthalpy of an ideal isentropic compression from state point 1 (T1, pmin) to
pressure pmax.
Pre-heated process (states 2-2a)
If internal heat exchanger for recovering heat of fluid at the turbine outlet is used,
state point of fluid changes from 2 to 2a. In this study, we assume no heat and pressure loss
in this process, if not stated otherwise (e.g. in 10.3.3).
187
Evaporation process (states 2a-3 or 2–3)
Working fluid receives energy from heat carrier in evaporator at constant pressure.
After leaving the evaporator and before entering the turbine, the state of fluid (state point 3)
is either saturated vapour or superheated vapour. We assume that there is no pressure loss
in this process, heat receiving from this process is calculated, for the case without IHE, as:
q23 = h3 – h2
and for the case with IHE as:
q2a3 = h3 – h2a
Expansion process (states 3–4)
The superheated or saturated vapour, state point 3, passes through the turbine to
generate mechanical power. After an expansion process of vapour in the turbine, pressure
of fluid drop from pmax to pmin. The state of fluid (state point 4) at the turbine outlet is
superheated vapour. The work of the cycle is generated in this process. With an isentropic
efficiency ηsT, work generates in this process is calculated as:
w34 = (h4-h3) = (h4’-h3).ηsT
where h4’ is enthalpy of an ideal isentropic expansion from state point 3 (Tm, pmax) to
pressure pmin.
Pre-cooled process (states 4-4a)
If internal heat exchanger is used, state point of fluid at the turbine outlet changes
from 4 to 4a. In this study, we assume no pressure loss in the IHE. If efficiency of internal
heat exchanger is ηHE we have a following relation:
h2a – h2 = (h4 – h4a)ηHE
or
h2a = h2 + (h4 – h4a)ηHE
Condensation process (states 4a-1 or 4–1)
After leaving the turbine or the IHE, the fluid passes through the condenser where
heat is removed at constant pressure pmin. We assume that the state of fluid after condenser
188
is saturated liquid, state point 1. Heat of the fluid is discharged to cooling media and can be
calculated for the case with IHE as:
q4a1 = h4a – h1
and for the case without IHE as:
q41 = h4 – h1
Cycle types
Depending on boundary conditions and coexistence curves, ORC can be devided
into different types such as b1, b2, b3, s1, s2, o1, o2, and o3 [10.26]. In this study there
exist only three types as o2, o3, and s2. Thus we give out short description of the three
cycle types.
Let us present coexistence curves and different processes in T-s diagram. The o2
cycle is shown in fig. 10.2. In the o2-cycle state of working fluid at the turbine inlet is
saturated vapour. The maximum temperature and pressure of the cycle are smaller than
those at critical point. The state of working fluid at the turbine outlet is superheated vapour.
600
550
3
T [K]
500
450
4
2a
400
2
4a
1
350
300
250
-20
0
20
40
60
80
100
120
140
s [J/mol.K]
Figure 10.2. ORC cycle o2 in the T,s-diagram for a fluid with overhanging coexistence
curve and saturated vapour at the turbine inlet.
189
The typical difference between the o2-cycle and the o3-cycle is the state of fluid at
the turbine inlet. In the o3-cycle, the state of working fluid at the turbine inlet is
superheated vapour, fig. 10.3. In this cycle, maximum pressure of the cycle is always
smaller than critical pressure. The maximum temperature of the cycle may be equal to or
higher or lower than critical temperature. The state of working fluid at the turbine outlet is
also superheated vapour.
650
3
600
550
T [K]
500
4
2a
450
400
2
4a
1
350
300
250
-20
0
20
40
60
80
100
120
140
160
s [J/mol.K]
Figure 10.3. ORC cycle o3 in the T,s-diagram for a fluid with overhanging coexistence
curve and superheated vapour at the turbine inlet.
Difference from the o2 and o3 cycles, the maximum pressure of the s2-cycle is
always higher than critical pressure. Futhermore, maximum temperature of the s2-cycle is
also higher than critical temperature, fig. 10.4.
190
650
3
600
550
T [K]
500
2a
4
450
400
2
350
1
4a
300
250
-20
0
20
40
60
80
100
120
140
160
s [J/mol.K]
Figure 10.4. ORC cycle s2 in the T,s-diagram for a fluid with overhanging coexistence
curve and superheated vapour at the turbine inlet.
Cycle efficiency
Cycle efficiency for the case without IHE is calculated as:
ηth,-IHE = - (w34 + w12)/q23 = -[(h4-h3) + (h2 – h1)]/(h3 – h2)
(10.1)
For the case with internal heat exchanger, cycle efficiency is calculated as:
ηth,+IHE = - (w34 + w12)/q2a3 = -[(h4-h3) + (h2 – h1)]/(h3 – h2a)
(10.2)
10.2.2 Water cycle with extraction
Typical differences between Rankine cycles using water and our studied organic
substances as working fluids are the shape of coexistence curves and the state of fluid at the
turbine outlet. The shape of coexistence curves of siloxanes and aromates is skew or
overhanging. Thus, if fluid at the turbine inlet is saturated vapour or superheated vapour,
fluid at the turbine outlet is superheated vapour. In order to recover heat of fluid after the
turbine or increase cycle efficiency, internal heat exchanger is used, figure 10.1. In water
cycle, if water at the turbine inlet is saturated vapour, water at the turbine outlet is in two
phase region. To avoid corrosion of the turbine’s blades, water at the turbine inlet is heated
to ensure that water at the turbine outlet is superheated vapour or wet vapour with vapour
191
content of about 95% or more. Thus, internal heat exchanger like that of organic cycles can
not be used. In order to improve cycle efficiencies, extraction cycle is used, figure 10.5.
3
4
5
1
2
9
8
6
7
Figure 10.5. Water cycle with extraction
The extraction cycle is used to compare efficiencies of water Rankine cycle with
those of organic Rankine cycles. Comparison of efficiencies is based on the same
maximum and minimum temperature. In this section, we give a brief introduction for
assumptions and steps for calculation of the extraction cycle.
We assume that there is no heat loss in the heat exchangers. We assume p2 = p3 = p7
= p8 = p9, p5 = p6 and p1 = p4. Other assumptions are that state point 1 is saturated liquid
and state of point 4 is saturated vapour. We firstly calculate properties of points 1 and 4
from minimum temperature of the cycle. We secondly calculate properties of point 3 from
maximum temperature of the cycle, efficiency of the turbine and properties of point 4.
Pressure of points 2, 7, 8, and 9 is identical to that of point 3. Properties of point 8 are
calculated from pressure of point 8, pump efficiency and properties of point 1. Extraction
flow rate and extraction pressure of point 5 are determined by an optimization process for
the cycle. We vary values of extraction flow rate and pressure of point 5 and calculate cycle
efficiency. Optimized cycle has the highest efficiency corresponding to certain values of
pressure and extraction flow rate. Properties of point 5 are determined from pressure of
point 5 and properties of points 3 and 4.
192
In order to determine properties of points 6 and 9 from properties of points 8 and 5,
we assume that temperature of point 6 is 10K higher than that of point 9 and the heat
exchanger has parallel flows. Properties of points 6 and 9 are determined from properties of
points 8 and 5, ratios of flow rates, pressures of points 6 and 9, and energy balance.
Properties of point 2 are determined from enthalpies and ratios of flow rates of points 6 and
9.
If ratio of extraction flow is mex, the ratio of main flow is (1- mex). Efficiency of
extraction cycle is calculated as:
ηth,+IHE = [(h3-h5) + (1- mex) (h5 – h4)- (1- mex)( h8 – h1)- mex(h7 – h6)]/(h3 – h2)
(10.3)
10.3 Screening of fluids and thermodynamic data
10.3.1 Selection of fluids
In general, criteria for consideration of working fluids are thermal efficiency,
stability, compatibility with contacted materials of cycle, safety, health and environmental
aspects, and costs. In this study, we pay attention on thermal efficiency and size of
equipment.
Our interest for this research is medium temperature ORC processes with
temperature above 2000C for utilize waste heat from industrial plants, waste heat in flue gas
from biogas combustion, heat from biomass combustion, waste heat from high temperature
fuel cell, and other sources. For the mentioned temperature range, alkanes, aromates, and
siloxanes, cycloalkanes are potential substances and will be investigated in following
section.
Alkanes are environmental fluids and yield good thermal efficiencies. Auto ignition
temperature of alkanes normally decreases with the chain length. In this study we mainly
consider only iso-pentane, n-pentane. For cycloalkanes, we consider only cyclo-pentane.
another group of substances which is considered as working fluids for medium temperature
ORC is aromates toluene (C7H8), ethylbenzene (C8H10), butylbenzene (C10H14), and xylenes
(C8H10).
The last potential group of substances which is considered in this study is siloxanes
or
silicon
oils
including
MM
(Hexamethyldisiloxane,
193
C6H18OSi2),
MDM
(Octamethyltrisiloxane, C8H24O2Si3), MD2M (Decamethytetrasiloxane, C10H30O3Si4), and
MD3M
(Dodecamethylpentasiloxane,
C12H36O4Si5).
Siloxanes
have
high
critical
temperature and suitable in our interested temperature range. Siloxanes have been used in
different ORC [10.4, 10.5, 10.7]. For this type of substances, there is lack of intensive
experimental data in full fluid region so physically based equations of state with small
number of fitted parameters are suitable for describe thermodynamic properties of pure
fluids and their mixtures.
10.3.2 Equation of state and caloric properties of selected fluids
In order to evaluate the thermal efficiency of a cycle, its thermodynamic properties
must be known and described accurately by fundamental equations of state. There are
different types of equations of state such as multi-parameters, cubic, BACKONE, PCSAFT. Empirical multi-parameter equations are the most accurate equations provided that
there are sufficient accurate experimental data in large space for construction of the
equation. Some accurate equations have been developed by Wagner group [10.28 - 10.30].
For fluids with limited experimental data, physically based equations of state with few
parameters should be employed.
Among different types of physically based equations of state, BACKONE and PCSAFT have been proved to be accurate and reliable. We have recently developed and
applied BACKONE EOS for different fluids and applications [10.9, 10.17, 10.20 - 10.22].
BACKONE is a family of physically based EOS. BACKONE EOS is able to describe
thermodynamic properties of nonpolar, dipolar and quadrupolar fluids with good to
excellent accuracy.
In BACKONE, the Helmholtz energy is written in term of a sum of contributions
from characteristic intermolecular interactions [10.17] as F = FH + FA+ FD +FQ, where FH is
the hard-body contribution, FA the attractive dispersion energy contribution, FD is the
dipolar contribution, and FQ is the quadrupolar contribution [10.17]. Thus, five substance
specific parameters are used: a characteristic temperature and density T0 and ρ0 and the
anisotropy factor α for nonpolar fluids and a reduced dipole moment μ*2 and/or quadrupole
moment Q*2 for polar fluids. BACKONE for pure fluids have only from three to five
parameters which need to be found by fitting to accurate experimental vapour pressures and
194
saturated densities. For mixture, BACKONE need one more parameter for each binary
[10.20].
PC-SAFT, a physically based EOS [10.23], was developed for chain molecules. The
development of PC-SAFT was based on a perturbation theory and statistical associating
fluid theory. Thus, for chain-like molecules such as linear siloxanes, PC-SAFT has better
performances than BACKONE [10.18]. For this reason, PC-SAFT EOS is used for linear
siloxanes. For compact molecules and aromates, BACKONE EOS is employed.
In construction of BACKONE and PC-SAFT EOS for aromates and siloxanes, we
fitted the equations to vapour pressures and saturated liquid densities. Because
experimental saturated liquid densities are not available up to critical point so we use
upward extrapolation equations which have recently been studied [10.18, 10.31, 10.32] to
generate data. Details of procedures for construction as well as parameters of the
BACKONE and PC-SAFT EOS can be found in [10.18, 10.19, 10.26]. Table 10.1 presents
critical data, types and sources of EOS as well as auto-ignition temperatures, if available,
for our studied working fluids.
Table 10.1. Critical data, type and source of equation of state, and auto-ignition
temperature.
Name
n-butane
iso-pentane
n-pentane
cyclo-pentane
toluene
ethylbenzene
butylbenzene
o-xylene
m-xylene
p-xylene
MM
MDM
MD2M
MD3M
a
: From
CAS No
Tc[K]
Auto-ignition
Types of Sources of
temperature,
EOS
EOS
[°C]
3.80
BACK1
[10.26]
365 a
3.27
BACK1
[10.26]
420 a
3.29
BACK1
[10.26]
309 a
3.85
BACK1
[10.26]
361b
3.16
BACK1
[10.19]
480 a
2.67
BACK1
[10.19]
432 a
2.01
BACK1
[10.19]
510 a
2.71
BACK1
[10.19]
463 a
2.66
BACK1
[10.19]
527 a
2.64
BACK1
[10.19]
528 a
1.64
PC-SAFT [10.18]
341 c
1.13
PC-SAFT [10.18]
350 c
0.86
PC-SAFT [10.18]
N/A
0.71
PC-SAFT [10.18]
430 d
From [10.34]; d :From [10.35]; N/A: Not
ρc
pc [MPa]
[mol/l]
106-97-8
425.20
3.922
78-78-4
460.90
3.386
109-66-0
469.65
3.370
287-92-3
511.7
4.510
108-88-3
591.80
4.109
100-41-4
617.20
3.609
104-51-8
660.05
2.887
95-47-6
630.33
3.732
108-38-3
617.05
3.541
106-42-3
616.23
3.511
107-46-0
518.70
1.925
107-51-7
564.13
1.415
141-62-8
599.40
1.190
141-63-9
629.00
0.945
[10.27]; b :From [10.33]; c :
available
195
In the calculation of caloric properties such as the enthalpy or/and the entropy, the
residual contributions of BACKONE and PC-SAFT have to be supplemented by ideal gas
heat capacity. The ideal gas heat capacity can be represented with following equation in
[10.36]:
Cp0/R = A+BT+CT2+DT3+ET4
(10.4)
Where R is ideal gas constant, R = 8.314472 J/(mol.K) [10.37]. A, B, C, D, and E
are coefficients. The fit coefficients of ideal gas heat capacity together with original data
sources and temperature ranges are given in table 10.2.
Table 10.2. Coefficients, temperature ranges and sources of ideal gas heat capacity
Name
A
n-butane
iso-pentane
n-pentane
cyclo-pentane
toluene
ethylbenzene
butylbenzene
o-xylene
m-xylene
p-xylene
MM
MDM
MD2M
MD3M
2.5706
-2.2928
-0.2584
5.0190
-4.7793
2.8611
6.4900
-1.3865
-3.4749
-2.7508
6.3472
9.4056
10.0356
12.8945
B
C
D
E
3.1702E-02 4.2294E-06 -1.6889E-08 5.8222E-12
6.9473E-02 -4.9760E-05 1.9555E-08 -3.2983E-12
5.9954E-02 -3.5457E-05 1.0485E-08 -1.2117E-12
-1.9734E-02 1.7917E-04 -2.1696E-07 8.2150E-11
7.0821E-02 -4.7711E-05 1.4068E-08 -1.0756E-12
2.4422E-02 9.8673E-05 -1.5176E-07 6.3489E-11
1.9080E-02 1.5665E-04 -2.2059E-07 8.8870E-11
6.8366E-02 -3.4018E-05 2.2944E-09 2.1532E-12
7.4572E-02 -4.1203E-05 6.0963E-09 1.3830E-12
6.9888E-02 -3.2768E-05 -1.1154E-10 3.0077E-12
8.5604E-02 -4.6759E-05 1.0523E-08 0.0000E+00
1.2051E-01 -6.8020E-05 1.5776E-08 0.0000E+00
1.5327E-01 -8.6846E-05 2.0222E-08 0.0000E+00
2.0199E-01 -1.2386E-04 3.0840E-08 0.0000E+00
Tmin Tmax
200
298
298
50
298
0
200
298
298
298
298
298
298
298
1500
1500
1500
1000
1500
1000
1000
1500
1500
1500
1400
1400
1400
1400
cp 0
sources
[10.38]
[10.39]
[10.39]
[10.36]
[10.40]
[10.41]
[10.36]
[10.40]
[10.40]
[10.40]
[10.42]
[10.42]
[10.42]
[10.42]
10.3.3 Comparison of BACKONE data with those from reference EOS
Accuracy of cycle efficiency is strongly depended on thermodynamic data of the
state points. Thus it is interesting to investigate the differences of thermodynamic data from
different equations of state and the differences of cycle efficiencies. In this subsection we
compare thermodynamic data of toluene from BACKONE EOS [10.19] and those from
Bender-type reference equation of state [10.43] which can be easily accessed via NIST
homepage [10.44].
196
In order to compare caloric properties, the two equations must have the same
reference state point. In this comparison, temperature of the reference state point is selected
to be the normal boiling temperature. Density of the reference state point is selected to be
saturated liquid density at the normal boiling point temperature. Enthalpy and entropy of
the reference state point are set to be zero.
For convenience in comparisons of cycle efficiencies from different equations of
state and in investigation of the accuracies of the equations, we compare thermodynamic
properties from BACKONE equation and reference equation for all typical state points of
the cycle given by Drescher and Brueggemann [10.12]. In the paper of Drescher and
Brueggemann [10.12], saturated liquid toluene at 363 K is pumped into internal heat
exchanger. The efficiencies of the pump and the internal heat exchanger are 0.8 and 0.95,
respectively. Toluene is heated up to saturated vapour at 2MPa in evaporator and enters the
turbine. After the expansion of toluene in the turbine with efficiency of 0.8 to generate
work, toluene discharges heat in the internal heat exchanger and the condenser. The
minimum temperature difference between state point 4a and state point 1 is 10K. When
toluene is cooled to saturated liquid state, it is pumped to internal heat exchanger and
continues its cycle.
Thermodynamic properties of toluene from BACKONE equation [10.19] and from
reference equation [10.43] at typical state points of the cycle are given in table 10.3. Table
10.4 shows deviations of thermodynamic properties from BACKONE equation [10.19] and
from reference equation [10.43]. We observe from table 10.4 that results from BACKONE
equation and from reference equation are in very good agreement. Thus cycle efficiencies
from BACKONE equation of state in following section are expected to be reliable.
Table 10.3. Thermodynamic properties of toluene at all state points of ORC from
BACKONE and from reference equations of state.
State
point
1
2
3
4
4a
BACKONE equation [10.19]
p
s [J/
h
T [K] ρ [mol/l]
[MPa] (mol.K)] [J/mol]
363.0000 8.6909 0.0540 -10.02 -3745
363.9046 8.7114 2.0000 -9.86 -3465
536.1920 0.6436 2.0000 108.74 53159
446.0676 0.0147 0.0540 114.69 42733
373.0000 0.0178 0.0540 89.05 32236
197
T [K]
363.00
363.88
535.76
444.10
373.00
Reference equation [10.43]
s [J/
h [J/mol]
ρ [mol/l] p [MPa]
(mol.K)]
8.6835 0.0540 -10.05
-3757
8.6995 2.0000
-9.89
-3477
0.6738 2.0000 107.60 52507
0.0148 0.0540 113.49 42242
0.0178 0.0540
88.44 32011
Table 10.4. Deviations of thermodynamic properties of toluene from BACKONE (B1)
equation [10.19] and reference (Ref) equation [10.43].
State
point
1
2
3
4
4a
(TRef-TB1)/TRef (ρRef-ρB1)/ ρRef (pRef-pB1)/pRef (sRef-sB1)/sRef (hRef-hB1)/hRef
0.0000
-0.0001
-0.0008
-0.0044
0.0000
-0.0008
-0.0014
0.0448
0.0062
0.0034
0.0001
0.0000
0.0000
0.0001
0.0001
0.0031
0.0032
-0.0106
-0.0106
-0.0070
0.0032
0.0033
-0.0124
-0.0116
-0.0070
10.4 Thermal efficiencies
10.4.1 Cycle efficiencies of all considered substances
In this subsection we assume that the isentropic efficiency of turbine and pump are
0.85 and 0.65, respectively. In case that the internal heat exchanger (IHE) is used,
efficiency of the IHE is 100% and the temperature difference between fluid at the
condenser inlet and fluid at the pump outlet is just 10 K. This is the minimum temperature
difference between hot and cold fluids in the IHE. The calculation of flow rates of working
fluids is based on 1MW net power output.
The intention of this section is not a systematic optimization for a real practical
problem with consideration for both cost and technical points of view. The idea is simply to
calculate and compare cycle efficiencies of all working fluids with some typical boundary
conditions, which are similar to those of some recent studies [10.12, 10.13, and 10.45]. In
this study, we consider for four typical temperature boundary conditions. The first
temperature boundary condition has maximum temperature of 250°C and minimum
temperature of 85°C. This is similar to boundary condition of Gaderer [10.45] who studied
on combined-heat-and-power (CHP) plants. The heat from the condenser of the cycle can
be the energy source for central heating system or other consumers. The ORC can use
energy from biomass or/and solar sources.
The second temperature boundary condition has maximum temperature of 250°C
and minimum temperature of 38°C. The intention of this case is only for electricity
generation. The third and the last temperature boundary conditions are for combined-heatand-power (CHP) plants with maximum temperature higher than that of the first
temperature boundary condition. In detail, maximum temperatures of the third case and the
198
last case are 300°C and 350°C, respectively. The minimum temperatures of the third case
and the last case are 38°C
One remaining question is the selection of maximum pressure for the o3 and s2
cycles. Before calculation for all substances, we try with calculations of cycle efficiencies
of n-pentane and toluene with various maximum pressures, from 0.5 pc to 1.4 pc. The step
of the increase of maximum pressure is 0.1pc. The minnimum temperature of the
investigated cycles is 38°C. The maximum temperature of the cycle with n-pentane is
250°C and that of the cycle with toluene is 350°C. Results for investigated cycles with npentane and toluene are given in table 10.5. In this table, we show only results for the case
with T2a smaller than saturation temperature at the maximum pressure of the investigated
cycles.
Table 10.5. Cycle efficienies with various maximum pressure
Fluid
T4 [K]
pmax /
pc
pmax
[MPa]
vْ 3
[l/s]
ْv4[l/s]
ْ
m
[kg/s]
ηth,
[%]
-IHE
ηth,
[%]
+IHE
n-pentane
Tc = 469.65 K, Tmax = 523.15 K, Tmax/ Tc = 1.11, Tmin = 358.15 K, ηcarnot = 0.315
n-pentane
476.24
0.8
2.70
238
1684
13.4 11.3 18.0
n-pentane
471.82
0.9
3.03
197
1609
13.0 11.9 18.5
n-pentane
463.04
1.1
3.71
142
1513
12.5 12.7 19.0
n-pentane
458.54
1.2
4.04
122
1484
12.4 13.0 19.1
n-pentane
453.87
1.3
4.38
107
1464
12.3 13.3 19.1
n-pentane
448.98
1.4
4.72
93
1452
12.4 13.5 19.0
toluene
Tc = 591.80, Tmax = 623.15 K, Tmax/ Tc = 1.05, Tmin = 358.15 K, ηcarnot = 0.425
toluene
532.07
0.5
2.05
151 6750
6.5
18.7 28.0
toluene
525.63
0.6
2.47
117 6470
6.3
19.4 28.6
toluene
519.47
0.7
2.88
94
6298
6.2
19.9 28.9
toluene
513.32
0.8
3.29
77
6160
6.2
20.4 29.0
toluene
506.96
0.9
3.70
64
6074
6.1
20.7 29.1
toluene
492.25
1.1
4.52
44
5922
6.2
21.3 28.8
toluene
482.61
1.2
4.93
36
5936
6.4
21.4 28.4
toluene
469.10
1.3
5.34
28
6008
6.6
21.5 27.5
toluene
454.30
1.4
5.75
22
6208
7.0
21.4 26.5
We observe from table 10.5 that cycle with n-pentane reachs the highest value of
cycle efficiencies of 19.1% for the case with internal heat exchanger at pmax = 1.2pc and
199
pmax = 1.3pc. If one increases more number of digits after the comma, cycle efficiency has
maximum value at pmax = 1.2pc. In this case, supercritical cycle yields the highest cycle
efficiency. Whist cycle with toluene reachs the highest cycle efficiency at 0.9pc.
In this study, we do not investigate cycle efficiencies with various maximum
pressures for all substances. We simply choose maximum pressure of 0.90pc for o3 cycle
and 1.2 pc for s2 cycle for all substances. All results for calculations of cycle efficiencies
for the four temperature boundary conditions are given in tables 10.6, 10.7, 10.8, and 10.9.
Table 10.6. Volume flow rate, volume ratio, efficiencies and other properties of cycle with
Tmax = 523.15 K, Tmin = 358.15 K, ηsP = 0.65, ηsT = 0.85.
Fluid
Tmax/Tc T4 [K]
n-butane
iso-pentane
n-pentane
cyclo-pentane
1.23
1.14
1.11
1.02
n-butane
iso-pentane
n-pentane
1.23
1.14
1.11
MDM
toluene
MD2M
p-xylene
m-xylene
ethylbenzene
MD3M
o-xylene
butylbenzene
0.93
0.88
0.87
0.85
0.85
0.85
0.83
0.83
0.79
pmin [MPa]
pmax
[MPa]
vْ 3
[l/s]
ْv4[l/s]
ْ
m
[kg/s]
Tmax > Tc,
pmax/pc = 0.9, Type o3
487.21 1.13E+00
3.53
326
1043
18.2
477.56 5.16E-01
3.05
223
1466
14.4
471.82 4.20E-01
3.03
197
1609
13.0
431.90 2.89E-01
4.06
99
1764
10.3
Tmax < Tc,
pmax/pc = 1.2, Type s2
475.38 1.13E+00
4.71
197
886
15.7
465.52 5.16E-01
4.06
139
1330
13.5
458.54 4.20E-01
4.04
122
1484
12.4
Tmax < Tc,
pmax = ps(Tmax)
Type o2
482.18 1.13E-02
0.76
283
26772
17.9
432.82 4.60E-02
1.67
182
7289
8.7
487.80 2.37E-03
0.33
610
103544 18.9
444.21 1.90E-02
0.99
287
15700
8.6
444.54 1.84E-02
0.98
288
16057
8.5
446.24 2.03E-02
1.02
278
14861
8.7
493.46 5.69E-04
0.16 1219 372914 19.9
445.71 1.54E-02
0.88
318
18763
8.3
459.52 3.94E-03
0.41
603
62960
8.7
ηth,
ηth,
pmax /
[%]
[%]
pc
-IHE +IHE
8.4
10.9
11.9
16.6
14.8
17.9
18.5
20.2
0.90
0.90
0.90
0.90
10.1
12.1
13.0
16.6
18.8
19.1
1.20
1.20
1.20
12.5
19.0
12.4
18.7
18.7
18.5
11.6
18.7
17.9
22.4
22.6
22.9
23.2
23.2
23.2
23.1
23.3
23.9
0.54
0.41
0.28
0.28
0.28
0.28
0.16
0.24
0.14
Table 10.6 shows typical results for almost all substances except MM and
cyclopentane with s2 cycle. We do not use results of MM with subcritical cycle and MM
with supercritical cycle because the critical temperature of MM is very close to the
maximum temperature of the cycle. The critical temperature of cyclopentane is close to the
maximum temperature of the cycle so in order to avoid the affection of the uncertainty of
200
thermodynamic data in critical region on the results we do not use results of supercritical
cycle with cyclopentane.
Observation from cycle efficiency and temperature at the turbine outlet of o3 cycle
and s2 cycle for the same fluid and boundary condition, we find out that s2 cycle has higher
cycle efficiency than that of o3 cycle. Temperature of working fluid at the turbine outlet of
s2 cycle is smaller than that of o3 cycle. Furthermore, volume flow rate of s2 cycle is
smaller than that of o3 cycle. For these reasons, supercritical cycle seems better than
subcritical cycle.
Cycle efficiencies of alkanes and cyclopentane are much smaller than those of
aromates and siloxanes. We observe from results for siloxanes and aromates that volume
flows ْv3 and ْv4 of the siloxanes are dramatically higher than those of aromates.
Furthermore, temperatures of siloxanes at the turbine outlet are higher than those of
aromates. The cycle efficiencies of aromates and siloxanes are similar for the case with
IHE, thus aromates seem better than siloxanes for this boundary condition in view of size of
equipment.
ηth,
ηth,
Tmax/
ْv3
pmax
ْm
Fluid
T4 [K] pmin [MPa]
ْv4[l/s]
[%]
[%]
Tc
[MPa] [l/s]
[kg/s]
-IHE +IHE
Tmax > Tc,
iso-pentane
1.14 452.12 1.43E-01
3.05
124
2873 8.0
16.5 25.9
n-pentane
1.11 445.36 1.09E-01
3.03
113
3440 7.4
17.5 26.2
cyclo-pentane 1.02 396.71 6.89E-02
4.06
62
4403 6.5
22.6 26.9
Tmax > Tc,
iso-pentane
1.14 440.13 1.43E-01
4.06
81
2720 7.8
17.6 26.3
n-pentane
1.11 432.17 1.09E-01
4.04
72
3283 7.3
18.4 26.4
Tmax < Tc,
pmax = ps(Tmax)
Type o2
MDM
0.93 465.29 1.20E-03
0.76
182
156431 11.5 16.3 29.5
toluene
0.88 396.10 7.16E-03
1.67
122
29021 5.8
24.8 29.0
MD2M
0.87 471.35 1.61E-04
0.33
395
954471 12.2 16.2 30.0
p-xylene
0.85 409.80 2.38E-03
0.99
193
77850 5.8
24.4 29.8
m-xylene
0.85 410.17 2.28E-03
0.98
193
80746 5.7
24.4 29.8
ethylbenzene
0.85 412.62 2.58E-03
1.02
186
72588 5.8
24.2 29.8
MD3M
0.83 478.69 2.59E-05
0.16
790 5154084 12.9 15.1 30.3
o-xylene
0.83 411.45 1.83E-03
0.88
214
98556 5.6
24.4 30.0
butylbenzene
0.79 429.96 3.51E-04
0.41
408
447981 5.9
23.7 31.3
201
pmax
/ pc
0.90
0.90
0.90
1.20
1.20
0.54
0.41
0.28
0.28
0.28
0.28
0.16
0.24
0.14
In table Table 10.7, we use the similar boundary condition to that in table 10.6
except minimum temperature. The table 10.7 shows the advantage of s2 cycle in view of
cycle efficiency and size of the turbine. This table also shows the advangtage of aromates
over siloxanes in view of size of the turbine. We observe from tables 10.6 and 10.7 that
cycle efficiencies of toluene and ethylbezene are higher than those of MDM and MD2M for
the case with Tmin = 358.15K, respectively. When Tmin = 311.15K, cycle efficiencies of
toluene and ethylbezene are lower than those of MDM and MD2M, respectively.
We now consider for the case with higher maximum tempertature than those of
previous two cases. Table 10.8 shows results for the case with Tmax = 573.15 K. In this
table we show only results for fluids having T2a smaller than saturation temperature at
maximum pressure of o3 cycle. With this criterion, all considered alkanes are dropped out.
The results for supercritical cycle with MDM also is dropped out from table 10.8 because
the maximum temperature of the cycle is close to critical temperature of MDM.
We observe that cycle efficiency of cylopentane for the case with IHE is higher than
that of MM for o3 cycle but not for s2 cycle. S2 cycle yields higher cycle efficiency than
that of o3 cycle for the same working fluid. The temperture of cyclopentane at the turbine
outlet is lower than that of MM for both o3 and s2 cycles. Furthermore flow rates of
cyclopentane are much smaller than that of MM. Thus cyclopentane should be used instead
of MM.
Butylbenzene yields the highest cycle efficiencies. Comparison with siloxanes in o2
cycle, butylbenzene yields smaller size of equipment than siloxanes. Furthermore,
evaporation pressure of butylbenzene is higher than those of siloxanes. Thus, if criteria are
cycle efficiency and the size of the turbine, butylbenzene should be used instead of
siloxanes. It should be noticed that volume flow rate of butylbenzene at the turbine outlet is
about 4 times higher than those of xylenes and ethylbenzene and about 8.5 times higher
than that of toluene. Thus this characteristic should be considered in economical analyses.
202
ηth,
ηth,
Tmax/
ْv3
pmax
ْm
Fluid
T4 [K] pmin [MPa]
ْv4[l/s]
[%]
[%]
Tc
[MPa] [l/s]
[kg/s]
-IHE +IHE
Tmax > Tc,
cyclopentane
1.12 494.21
2.89E-01
4.06
109
1678
8.5 16.2
23.6
MM
1.10 525.40
6.29E-02
1.73
183
6225
14.7 11.7
23.5
MDM
1.02 523.36
1.13E-02
1.27
140
24329
15.0 12.4
25.6
Tmax < Tc,
cyclopentane
1.12 478.26
2.89E-01
5.41
70
1579
8.3 17.3
23.9
MM
1.10 517.66
6.29E-02
2.31
117
5944
14.3 12.4
24.0
Tmax < Tc,
pmax = ps(Tmax)
Type o2
toluene
0.97 450.31
4.60E-02
3.22
64
6430
7.4 20.7
25.6
MD2M
0.96 525.08
2.37E-03
0.81
165
91262
15.4 12.8
26.6
p-xylene
0.93 470.68
1.90E-02
2.00
104
13672
7.1 20.4
26.8
m-xylene
0.93 471.26
1.84E-02
1.99
104
13981
7.0 20.4
26.8
ethylbenzene
0.93 473.50
2.03E-02
2.06
101
12947
7.1 20.1
26.8
MD3M
0.91 534.47
5.69E-04
0.41
331 326054
16.1 11.9
27.1
o-xylene
0.91 470.68
1.90E-02
2.00
104
13672
7.1 20.4
26.8
butylbenzene
0.87 493.96
3.94E-03 0.92
212
54445
7.0 19.4
28.0
If Tmax = 623.15K, only aromates are suitable for working fluids because either the
maximum temperature is higher than auto-ignition temperatures of siloxanes or T2a of
alkanes and cyclopentane is higher than saturated temperature corresponding to maximum
pressure of o3 cycle. In table 10.9, we drop out results for supercritical cycle with mxylene, p-xylene, and ethylbenzne because the maximum temperature of cycle is close to
their critical temperatures.
Observation from table 10.9 we find out that supercritical cycle with toluene yields
the lowest cycle efficiency. Butylbenzene yields the highest cycle efficiency. However,
volume flow rate of butylbenzene at the turbine outlet is about 3.3 times higher than those
of m-xylene, p-xylene and ethylbenzene and about 8.3 times higher than that of toluene in
o3 cycle. For this reason, o3 cycle with toluene, m-xylene, p-xylene and ethylbenzene
should be considered in application.
For convenience in the comparison of cycle efficiencies among different groups of
substances, we show graphically relation between cycle efficiencies and critical
temperatures of alkanes, cylopentane, aromates, and siloxanes from table 10.6 for the case
with and without IHE in figure 10.6. For the case without IHE, when critical temperatures
203
pmax
/ pc
0.9
0.9
0.9
1.2
1.2
0.8
0.7
0.6
0.6
0.6
0.4
0.6
0.3
increase, cycle efficiencies of alkanes increase and cycle efficiencies of alkylbenzenes and
siloxanes decrease. When critical temperatures increase, cycle efficiencies of all fluids for
the case with IHE increase. This figure also shows the large improvement in cycle
efficiency of siloxanes with IHE.
ηth,
ηth,
Tmax/
ْv3
pmax
ْm
Fluid
T4 [K] pmin [MPa]
ْv4[l/s]
[%]
[%]
Tc
[MPa] [l/s]
[kg/s]
-IHE +IHE
Tmax > Tc,
toluene
1.05 506.96 4.60E-02
3.70
64
6070
6.1
20.7
29.1
p-xylene
1.01 503.29 1.90E-02
3.16
54
12679
6.1
21.0
29.6
m-xylene
1.01 503.03 1.84E-02
3.19
53
12959
6.1
21.0
29.6
ethylbenzene 1.01 506.99 2.03E-02
3.25
53
12007
6.1
20.7
29.7
Tmax > Tc,
toluene
1.05 482.61 4.60E-02
4.93
36
5968
6.4
21.4
28.4
Tmax < Tc,
pmax = ps(Tmax)
Type o2
o-xylene
0.99 493.46 1.54E-02
3.32
42
14996
6.0
21.5
29.5
butylbenzene 0.94 525.94 3.94E-03
1.79
82
49600
6.0
20.2
31.3
26
24
22
ηth [%]
20
18
16
14
12
10
8
6
400
450
500
550
600
650
700
Tc [K]
Figure 10.6. Relation between cycle efficiency and critical temperature for boundary
conditions with maximum temperature of 523.15K and minimum temperature of 358.15K:
Δ alkanes with IHE, ◊ alkanes without IHE, siloxanes with IHE, + siloxanes without IHE,
Ο aromates with IHE, × aromates without IHE, ∇ cyclopentane without IHE, and ⊕
cyclopentane with IHE.
204
pmax
/ pc
0.90
0.90
0.90
0.90
1.20
0.89
0.62
In the design of the turbine, the volume flow rate ْv3 at the inlet and at the the outlet
ْv4 of the turbine are important, which are given in Tables 10.6, 10.7, 10.8, 10.9. For
convenience of the reader, some data in Table 10.6 are also presented graphically in Figures
10.7 and 10.8. Figures 10.7 presents relation between critical temperatures of working
fluids and volume flow rates ْv3. It is shown that when critical temperature of alkanes
increases volume flow rate at the turbine inlet decreases. The situation for alkylbenzenes
and siloxanes is different from that of alkanes. In detail, if critical temperature increases
volume flow rate increases.
700
600
V3 [l/s]
500
400
300
200
100
0
400
450
500
550
600
650
700
Tc [K]
Figure 10.7. Relation between volume flow rate ْv3 and critical temperature for the case
with maximum temperature of 523.15K and minimum temperature of 358.15K: Δ alkanes,
siloxanes, Ο aromates, and • cyclopentane.
Figures 10.8 shows the thermal efficiency ηth via the volume flow rate vْ 3 at the inlet
of the turbine for cycle with IHE. Consideration for both thermodynamic point of view and
size of equipment for the given boundary conditions, normal alkylbenzenes and xylenes are
better than siloxanes and alkanes. Butylbenzene has the highest cycle efficiency. However
the volume flow rate and volume ratio are much higher than those of other substances so
butylbenzene should not be used. Xylenes and ethylbenzene have similar characteristics.
The last considered normal alkylbenzene is toluene. Comparison between toluene and oxylene we find out that cycle efficiency of toluene is 22.6% and that of o-xylene is 23.3%.
205
Observations of critical temperature of siloxanes and aromates we find out that
MD2M and toluene have similar critical temperature and cycle efficiency. MD3M and oxylene have also similar critical temperature and cycle efficiency. Volume flow rate ْv3 of
the two siloxanes are much higher than those of the two aromates. Figure 10.8 shows that
ethylbenzene, m-xylene, and p-xylene yield similar volume flow rate ْv3 and higher cycle
efficiency than those of MDM.
26
m-xylene
p-xylene
o-xylene
Ethylbenzene
th
η [%]
24
22
Toluene
20
cyclopentane
Butylbenzene
MD2M
MDM
n-pentane (sup)
n-pentane
iso-pentane
iso-pentane (sup)
n-butane (sup)
18
16
n-butane
14
0
200
400
600
800
v3 [l/s]
Figure 10.8. Thermal efficiency ηth vs. volume flow rate ْv3 at the inlet of the turbine of
ORC with IHE for the case with maximum temperature of 523.15K and minimum
temperature of 358.15K, “sup” stands for supercritical.
Relation of the thermal efficiency ηth via the volume flow rate ْv3 in table 10.7 for
cycle with IHE is shown in figures 10.9. Observations from this figure we find out that
alkylbenzenes are better than siloxanes in views of cycle efficiency and size of the turbine.
206
32
31
Butylbenzene
p-xylene
m-xylene
o-xylene
Ethylbenzene
th
η [%]
30
MD2M
MDM
29
Toluene
28
cyclopentane
27
iso-pentane (sup)
iso-pentane
n-pentane
n-pentane (sup)
26
25
0
100
200
300
400
500
v3 [l/s]
Figure 10.9. Relation between ηth and ْv3 for the case with IHE, maximum temperature of
523.15K and minimum temperature of 311.15K, “sup” stands for supercritical.
Relation between thermal efficiency for the case with IHE and volume flow rate at
the turbine outlet for the case with Tmax = 573.15K is given in figure 10.10. This figure
again shows the advantage of aromates over siloxanes and cyclopentane in view of cycle
efficiency and size of the turbine.
Figure 10.11 shows relation between thermal efficiency for the case without IHE
and outlet volume flow rate. With the maximum temperature of 623.15K, ORC using
toluene as working fluid can be either subcritical cycle or supercritical cycle. In this case,
subcritical cycle have lower cycle efficiency than that of supercritical cycle. If cycle do not
have IHE, supercritical cycle with toluene is the best choice in both views of cycle
efficiency and the turbine size.
207
29
28
Butylbenzene
Ethylbenzene
xylenes
MD2M
th
η [%]
27
26
Toluene
MDM
25
24
23
cyclopentane (sup)
MM (sup)
cyclopentan
MM
0
20
40
60
80
100
120
v4 [m3/s]
Figure 10.10. Relation between ηth and ْv4 for the case with IHE, maximum temperature of
573.15K and minimum temperature of 358.15K, “sup” stands for supercritical.
21.6
o-xylene
21.4
Toluene (sup)
21.2
p-xylene
th
η [%]
21.0
m-xylene
20.8
Toluene Ethylbenzene
20.6
20.4
Butylbenzene
20.2
20.0
0
10
20
30
40
50
60
v4 [m3/s]
Figure 10.11. Relation between ηth and ْv3 for the case without IHE, maximum temperature
of 623.15 K, minimum temperature of 358.15 K, “sup” stands for supercritical.
In order to investigate the affection of boundary temperatures on cycle efficiency,
we select MDM from siloxanes and toluene from aromates for our study. The reason for the
selection is that MDM is already used in existing power plants. Toluene yields good cycle
208
efficiency and yields small size of the turbine. We plot cycle efficiencies of the two
substances with maximum temperature of 523.15 K and variation minimum temperature
from 303.15 K to 383.15 K in figure 10.12. This figure shows that at temperature of 339 K,
the two fluids yield the same cycle efficiency of 25 %. If the minimum temperature smaller
than 339 K, MDM yield higher cycle efficiency than that of toluene and vice versa.
32
Cycle efficiency [%]
30
28
26
24
22
20
18
300
320
340
360
380
T [K]
Figure 10.12. Relation between cycle efficiency and minimum temperature of - - -MDM
and ⎯ toluene for the case with maximum temperature of 523.15 K.
To exam the similar effect of maximum temperature on cycle efficiency of the two
fluids, we fix minimum temperature of 363.15 K. The maximum temperature is varied from
480 K to 540 K. Figure 10.13 shows a parallel between the two cycle efficiency lines whilst
figure 10.12 shows an intersection between the two cycle efficiency lines. We also observe
that volume flow rates ْv3 and ْv4 of MDM are higher than those of toluene. These prove that
MDM is not an optimal working fluid for combined heat and power plants in view of cycle
efficiency and size of the turbine.
209
24
Cycle efficiency [%]
23
22
21
20
19
18
17
470
490
510
T [K]
530
550
Figure 10.13. Relation between cycle efficiency and maximum temperature of - - -MDM
and ⎯ toluene for the case with minimum temperature of 363.15 K.
10.4.2 Comparison of cycle efficiencies from different equations of state
Cycle efficiency of ORC using toluene as working fluid for the same case by
Drescher and Brueggemann [10.12] can be easily calculated from table 10.3. Cycle
efficiencies of ORC with internal heat exchanger from reference equation of state [10.43]
and BACKONE equation of state [10.19] are 21.6% and 21.7%, respectively. The results
from these two equations are in very good agreement thus we believe in our other results.
Whilst, cycle efficiency of ORC with internal heat exchanger using toluene as working
fluid for the same conditions from Drescher and Brueggemann [10.12] is 23.2%. The
relative deviation between result from Drescher and Brueggemann [10.12] and result from
reference equation of state is -7.4%. The reason for the difference is that Drescher and
Brueggemann use Peng–Robinson EOS which is well-known for insufficient accurate.
Other comparisons of cycle efficiencies for the same conditions from Drescher and
Brueggemann [10.12] and from our study are given for ethylbenzene, butylbenzene and
octamethyltrisiloxane (MDM), Table 10.10. We observe from the table that the relative
differences of cycle efficiencies of the two investigated working fluids as toluene and
MDM are -6.9% and -1.8%, respectively.
210
Table 10.10. Comparisons of cycle efficiencies from different equations of state
ηth,+IHE from
BACKONE or PCSAFT EOSs (1)
0.217
0.246
0.255
0.221
Substance
toluene
ethylbenzene
butylbenzene
MDM
ηth,+IHE from Peng–
Robinson EOS (2)
(ηth,+IHE,1ηth,+IHE,2)/ηth,+IHE,1
0.232
0.243
0.253
0.225
-6.9%
1.2%
0.8%
-1.8%
10.4.3 Efficiency of medium-temperature Rankine cycle using water as working fluid
In order to compare efficiencies of extraction cycle using water as working fluid
with those from ORC, we also assume pump efficiency of 0.65 and turbine efficiency of
0.85. Optimized efficiencies of extraction cycle for different boundary conditions are given
in table 10.11.
We observe from table 10.11 that the increase of maximum temperature of the cycle
leads to the increase of extraction flow ratio, extraction pressure, cycle efficiency, and leads
to the decrease of turbine size and mass flow rate of water. The difference of cycle
efficiency for the case with and without extraction increases with the increase of maximum
temperature. We also observe from tables 10.6 to 10.9 and table 10.11 that ORC yields
much higher cycle efficiency than water cycle with extraction for our study temperature
range.
Table 10.11. Optimized efficiencies of water extraction cycle (ηsP = 0.65, ηsT = 0.85).
Minimum temperature of cycle is 358.15K
T3 [K] mext [%] p5 [MPa] ْv3 [l/s]
ْv4/ْv3
ْ
m
[kg/s]
ηth,[%]
-IHE
ηth,[%]
+IHE
480.15
523.15
580.15
630.15
2.53
3.38
4.80
6.39
6.53
4.77
3.55
3.01
7.70
10.27
13.51
16.19
7.98
10.86
14.55
17.76
5
6
8
11
0.21
0.31
0.45
0.69
6083
3332
1743
1110
10.5 Heat transfer from the heat carrier to the working fluids
The highest cycle efficiency of ORC process may not lead to highest power output
from available heat sources. In order to have the highest power output from available heat
211
sources, one needs to consider both cycle efficiency as well as the supply and processing of
heat carrier fluid.
Let us assume that mass flow rate of the heat carrier is ْmc. The heat carrier enters
external heat exchanger (EHE) at state point 5 with temperature T5 > T3, figure 10.1. It
leaves EHE with state point 6 having temperature of T6. The minimum temperature T6 and
the mass flow rate of the heat carrier are determined by a pinch point analysis for
generation of 1MW net power output.
The pinch point or minimum temperature difference of heat carrier and working
fluid in EHE is strongly depended on type of cycles, heat capacity of heat carrier cp,c,
temperature T5 and the temperature difference at pinch point ΔTp. In o2 cycle or o3 cycle,
the minimum temperature difference of the heat carrier and working fluid can have relation
with either point 2a or saturated liquid point corresponding to the maximum pressure of the
cycles. In s2 cycle, the pinch point can be at any point from point 2a to point 3. The
determination of pinch point of s2 cycle is rather complicated than o2 cycle or o3 cycle. In
our determination, we analyze T-ΔْH diagram, figure 10.14.
T [K]
T5
T3
Tp
ΔT p
T6
T 2a
o
ΔH (MW)
Figure 10.14. T-Δ ْH diagram for analysis of a pinch point in s2 cycle.
212
In this study we consider three cases for heat carrier corresponding to three
maximum temperatures of the investigated cycles. We firstly assume that heat carrier for
cycle with Tmax = 523.15 K is thermal oil with heat capacity cp,c = 2.5 kJ/kgK, temperature
T5 = 553.15 K and the temperature difference at pinch point ΔTp = 10 K. Thermal oil cycle
is circulated in a loop. For the last two cases, we assume that heat carrier is flue gas with
heat capacity cp,c = 1.18 kJ/kgK and the temperature difference at pinch point ΔTp = 30 K.
Temperatures T5 for cycle with Tmax = 573.15 K and Tmax = 623.15 K are 623.15 K and
723.15 K, respectively. Flue gas leaving the EHE is discharged into environment.
Results for the mass flow rate of heat carrier, the heat supply for the cycle Q56 from
heat carrier, temperature T5 and Tp corresponding to the data from tables 10.6 to 10.9 are
shown in tables 10.12 to 10.15.
Table 10.12. The mass flow rate of heat carrier, the heat supply for the cycle Q56 from heat
carrier, temperature T5 and Tp corresponding to cycle with Tmax = 523.15 K, Tmin = 358.15
K, ηsP = 0.65, ηsT = 0.85 for generation of 1MW net power output.
Fluid
iso-pentane
n-pentane
cyclo-pentane
iso-pentane
n-pentane
MDM
toluene
MD2M
p-xylene
m-xylene
ethylbenzene
MD3M
o-xylene
butylbenzene
Tmax/Tc
ηth,[%]
+IHE
T5 [K]
ْmc
ْQ56 pmax/
cp,c
ΔTp
T [K] Tp [K]
[kJ/kgK] [kg/s] [MW] pc
[K] 6
Tmax > Tc,
pmax/pc = 0.9,
17.9 553.15 10 450.1
18.5 553.15 10 447.7
20.2 553.15 10 455.2
Tmax > Tc,
pmax/pc = 1.2,
1.14
18.8 553.15 10 446.6
1.11
19.1 553.15 10 443.1
Tmax < Tc,
pmax = ps(Tmax)
0.93
22.4 553.15 10 500.1
0.88
22.6 553.15 10 510.9
0.87
22.9 553.15 10 508.0
0.85
23.2 553.15 10 513.6
0.85
23.2 553.15 10 513.6
0.85
23.2 553.15 10 513.4
0.83
23.1 553.15 10 511.0
0.83
23.3 553.15 10 514.6
0.79
23.9 553.15 10 516.1
1.14
1.11
1.02
Type o3
464.6
2.5
472.2
2.5
513.4
2.5
Type s2
460.5
2.5
469.4
2.5
Type o2
533.2
2.5
533.2
2.5
533.2
2.5
533.2
2.5
533.2
2.5
533.2
2.5
533.2
2.5
533.2
2.5
533.2
2.5
21.7
20.5
20.2
5.59 0.90
5.41 0.90
4.95 0.90
20.0
19.0
5.33 1.20
5.23 1.20
33.7
41.9
38.7
43.7
43.7
43.5
41.1
44.6
45.2
4.47
4.43
4.37
4.32
4.32
4.32
4.33
4.29
4.19
0.54
0.41
0.28
0.28
0.28
0.28
0.16
0.24
0.14
Tables 10.12 and 10.13 show that butylbenzene requires the lowest energy from
heat carrier, Q56. Iso-pentane requires the highest energy. In this case, alkanes and
cyclopentane require more energy from heat carrier than aromates and siloxanes do.
213
Siloxanes require more energy than aromates do. Thus, for a close loop of heat carrier,
butylbenzene is the best substance for this case in view of energy efficiency because it
requires the lowest external energy. Consideration for group of substances, aromates are the
best choice.
We also observe that butylbenzene requires the highest heat carrier flow rate.
Siloxanes require less heat carrier flow rate than aromates. With the same working fluid, s2
cycle requires less heat carrier flow rate than o3 cycle. In o3 cycle, cyclopentane requires
the lowest heat carrier flow rate. Among all considered substances in table 10.12,
supercritical cycle with n-pentane requires the lowest heat carrier flow rate. If the heat
carrier isn’t circulated in a loop, butylbenzene is the worst substance and n-pentane is the
best substance. Butylbenzene requires 4.4 times more heat carrier flow rate than n-pentane
in supercritical cycle does. As a consequency, n-pentane yields 4.4 times more power
output than butylbenzene does for a given heat sources.
Fluid
iso-pentane
n-pentane
cyclo-pentane
iso-pentane
n-pentane
MDM
toluene
MD2M
p-xylene
m-xylene
ethylbenzene
MD3M
o-xylene
butylbenzene
Tmax/Tc
ηth,[%]
+IHE
T5 [K]
ْmc
ْQ56 pmax/
cp,c
ΔTp
T6 [K] Tp [K]
[K]
Tmax > Tc,
pmax/pc = 0.9,
25.9 553.15 10 425.0
26.2 553.15 11 423.2
26.9 553.15 12 436.9
Tmax > Tc,
pmax/pc = 1.2,
1.14
26.3 553.15 14 345.3
1.11
26.4 553.15 15 408.8
Tmax < Tc,
pmax = ps(Tmax)
0.93
29.5 553.15 18 490.4
0.88
29.0 553.15 19 504.1
0.87
30.0 553.15 20 499.9
0.85
29.8 553.15 21 507.4
0.85
29.8 553.15 22 507.5
0.85
29.8 553.15 23 507.2
0.83
30.3 553.15 24 503.6
0.83
30.0 553.15 25 508.7
0.79
31.3 553.15 26 511.4
1.14
1.11
1.02
214
Type o3
464.6
2.5
472.2
2.5
513.4
2.5
Type s2
416.3
2.5
408.8
2.5
Type o2
533.2
2.5
533.2
2.5
533.2
2.5
533.2
2.5
533.2
2.5
533.2
2.5
533.2
2.5
533.2
2.5
533.2
2.5
12.1
11.7
12.8
3.86 0.90
3.82 0.90
3.72 0.90
7.3
7.0
3.80 1.20
3.79 1.20
21.6
28.1
25.0
29.4
29.4
29.2
26.7
30.0
30.6
3.39
3.45
3.33
3.36
3.36
3.36
3.30
3.33
3.19
0.54
0.41
0.28
0.28
0.28
0.28
0.16
0.24
0.14
Fluid
cyclopentane
MM
MDM
cyclopentane
MM
toluene
MD2M
p-xylene
m-xylene
ethylbenzene
MD3M
o-xylene
butylbenzene
Tmax/Tc
ηth,[%]
+IHE
T5 [K]
ΔTp
T6 [K] Tp [K]
[K]
Tmax > Tc,
pmax/pc = 0.9,
23.6 623.15 30.0 490.7
23.5 623.15 30.0 515.5
25.6 623.15 30.0 523.4
Tmax > Tc,
pmax/pc = 1.2,
1.12
23.9 623.15 30.0 485.9
1.10
24.0 623.15 30.0 511.8
Tmax < Tc,
pmax = ps(Tmax)
0.97
25.6 623.15 30.0 550.9
0.96
26.6 623.15 30.0 547.9
0.93
26.8 623.15 30.0 567.7
0.93
26.8 623.15 30.0 568.0
0.93
26.8 623.15 30.0 567.3
0.91
27.1 623.15 30.0 564.5
0.91
26.8 623.15 30.0 567.7
0.87
28.0 623.15 30.0 576.5
1.12
1.10
1.02
cp,c
[kJ/kgK]
Type o3
533.4
1.18
534.1
1.18
584.9
1.18
Type s2
548.3
1.18
519.2
1.18
Type o2
603.2
1.18
603.2
1.18
603.2
1.18
603.2
1.18
603.2
1.18
603.2
1.18
603.2
1.18
603.2
1.18
ْmc
ْQ56 pmax/
[kg/s] [MW] pc
27.2
33.5
33.2
4.24
4.26
3.91
0.9
0.9
0.9
25.8
31.7
4.18
4.17
1.2
1.2
45.9
42.4
57.0
57.3
56.6
53.3
57.0
64.8
3.91
3.76
3.73
3.73
3.73
3.69
3.73
3.57
0.8
0.7
0.6
0.6
0.6
0.4
0.6
0.3
In table 10.14 we show our results for cycle with Tmax = 573.15K and flue gas is
heat carrier. We find out that cyclopentane requires the lowest heat carrier flow rate. The
required heat carrier flow rate of cyclopentane is only 0.78 times of MDM, 0.56 times of
toluene, and 0.40 times of butylbenzene. In other words, cyclopentane is the best working
fluid, the second one should be MDM. In this situation, siloxanes prove to be better than
aromates in view of energy efficiency.
When Tmax=623.15K, only aromates are suitable fluids, table 10.15. In this table,
toluene and o-xylene prove to be the best fluids because they require the lowest heat carrier
flow rates.
215
Fluid
toluene
p-xylene
m-xylene
ethylbenzene
toluene
o-xylene
butylbenzene
Tmax/Tc
ηth,[%]
+IHE
T5 [K]
ْmc
ْQ56 pmax/
cp,c
ΔTp
T [K] Tp [K]
[K] 6
Tmax > Tc,
pmax/pc = 0.9,
1.05
29.1 723.15 30 497.7
1.01
29.6 723.15 30 496.3
1.01
29.6 723.15 30 495.9
1.01
29.7 723.15 30 500.0
Tmax > Tc,
pmax/pc = 1.2,
1.05
28.4 723.15 30 478.5
Tmax < Tc,
pmax = ps(Tmax)
0.99
29.5 723.15 30 488.5
0.94
31.3 723.15 30 520.4
Type o3
497.7
1.18
496.3
1.18
495.9
1.18
500.0
1.18
Type s2
491.5
1.18
Type o2
488.5
1.18
520.4
1.18
12.9
12.6
12.6
12.8
3.44
3.38
3.38
3.37
0.90
0.90
0.90
0.90
12.2
3.52 1.20
12.2
13.4
3.39 0.92
3.19 0.62
10.6 Summary and conclusions
This chapter investigates potential working fluids for medium-temperature ORC
processes by using thermodynamic data from BACKONE and PC-SAFT equations of state.
The study of alkanes, cyclopentane, aromates and siloxanes show that large siloxanes are
not the potential working fluids because they cause dramatically large size of the turbine.
The study show that the cycle efficiency for the case with IHE increases with the
increase of the critical temperature. Furthermore, the cycle efficiency depends not only on
the temperature boundary but also on the maximum pressure of the cycle. Thus in design
and operation of ORC, one should consider also the optimum pressure.
Our investigation for four typical boundary conditions shows that there is no
optimal working fluid for the maximum temperature from 250oC to 350oC. The selection of
working fluid should be based not only on cycle efficiency but also the size of equipment
and further on supply and processing of heat carrier fluid.
temperature after the EHE. In this case, selection of working fluid should be based on cycle
efficiency and size of equipment. For this case, aromates prove to be the most potential
working fluids.
216
that investigated alkanes, cyclopentane, MM, MDM, toluene and o-xylene are the most
potential working fluids, depending on the working temperature range.
The finding in this study paves the way for optimization of practical problems
where both investment and operation costs via size of ORC, cycle efficiency, total
efficiency and other criteria should be considered for renewable sources. Other extensions
for mixtures of potential fluids and optimization of the cycle with pure fluids and mixtures
will be presented for other individual projects.
217
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11 Summary and conclusions
Thermodynamic data of working fluids for energy engineering application have been
studied. Investigation of extrapolation of vapour pressure is carried out for different
equations such as Antoine equation, Wagner equation, Van der Waals equation, Korsten
equation, and new one-parameter equation with six typical substances representing for
different molecular classes such as argon, ethylene, ethane, sulfur hexafluoride, benzene,
and water. The input reduced temperature ranges for the investigation are from 0.5 to 0.6,
from 0.7 to 0.8, and from 0.9 to around critical point. With different offsets of reduced
temperature ranges, it is shown that new one-parameter equation is the most stable and
reliable one for downward extrapolation. Wagner equation is the most reliable equation for
upward extrapolation with input reduced temperature from 0.7 to 0.8.
A new set-up apparatus is constructed for measurement of vapour pressures. This
apparatus can be used to measure vapour pressures up to 2.67 MPa for temperature higher
than 107°C and up to 6.9 MPa for temperature lower than 107°C. The working temperature
range of this apparatus is from - 54°C to +200 °C. This apparatus has total uncertainty of
0.17 kPa for pressure up to 100 kPa.
Because available saturated liquid densities of our interested substances are mostly
limited in low reduced temperature range so we investigate the correlation and
extrapolation of saturated liquid densities with different equations. We use 18 substances
from different molecular classes in our investigation. Extrapolation methods with different
inputs from the critical point are used: (a) no critical point data, (b) critical temperature, (c)
critical temperature and pressure, and (d) critical temperature and compression factor. It is
found that upward extrapolations of the saturated liquid densities without using critical
point data can be done with some care and that the additional use of the critical temperature
improves the quality of the predictions substantially. With input data in range of reduced
temperature from 0.50 to 0.75, two-parameter equation give the best prediction up to
0.95Tc. The AADs of predicted data with two-parameter equation for all studied substances
at 0.90Tc, 0.95Tc, and Tc are 0.42%, 0.90%, and 3.48%, respectively. One-parameter
equation is the best for prediction of critical density with AADs less than 1.82%. Further
222
study with input data in lower reduced temperature shows that one parameter equation of
Spencer and Danner is the best one for prediction from low reduced temperature range.
The study on correlation and extrapolation of vapour pressures and saturated liquid
densities allows us to select the best available equation for each practical case. Saturated
liquid densities of siloxanes are available in low reduced temperature; hence Spencer and
Danner equation is used to predict data up to 0.9Tc. Saturated liquid densities of aromates
are available in higher reduced temperature range than that of siloxanes and it is suitable to
use two-parameter equation. Vapour pressures of siloxanes and butylbenzene are not
available in high reduced temperature range so we use Wagner equation for upward
prediction of vapour pressures.
In this study, we review theory studies and make some minor own contributions on
equations of state and on Helmholtz energies of hard convex bodies and hard chain systems
which are the core for developments of physically based equations of state such as
BACKONE and PC-SAFT. Results from this study show that the application of hard
convex bodies to hard chain systems is good for compact molecules, but not for large
chain-like molecules. Comparison between simulation data and results from SAFT of
Wertheim using hard sphere description of CS and that of KBN shows good agreement.
Methodology and strategy for development of PC-SAFT equation of state are
studied. PC-SAFT EOS for pure fluids by modifying SAFT equation of state with an
application of perturbation theory of Barker and Henderson is reviewed. Then, different
derivatives of residual Helmholtz energy are derived for two purposes. Firstly, these
derivatives are used to program a fitting package. Secondly, these derivatives are used to
calculate thermodynamic properties of fluids such as pressure, enthalpy, and entropy.
Program from the derivatives allows us to study on potential working fluids. Primary
study of MDM with BACKONE and PC-SAFT equations of state shows that PC-SAFT
outperforms BACKONE. Thus parameters of PC-SAFT equation of state for all five
studied linear siloxanes are determined. As experimental data at higher temperatures are
scarce, it is helpful to extrapolate vapour pressures and saturated liquid densities by
appropriate equations using the critical temperatures and pressures. With these strategies
performance of PC-SAFT is better than the case of fitting directly to experimental data or
223
to correlation data. Comparisons of PC-SAFT results with a variety of experimental data
show good agreement.
Besides siloxanes, we also develop equation of state for another potential group of
substances, aromates. First, parameters of BACKONE and PC-SAFT for benzene are
determined. Comparison of BACKONE and PC-SAFT results with experimental data of
benzene shows that BACKONE outperforms PC-SAFT. Thus parameters of BACKONE
for seven aromates are determined. Comparisons of resulting BACKONE thermodynamic
data with experimental data using in fitting show good to excellent agreement. The study
also shows that prediction of saturated vapour densities from BACKONE is very accurate.
The deviations of saturated vapour densities of benzene and toluene from BACKONE and
experiments and are mostly within 1%.
The governing principles of this work are to develop equations of state for
calculation of thermodynamic data and to apply them to energy engineering field. For
energy engineering application, we use data of different substances from the developed
equations of state to calculate cycle efficiency of medium-temperature organic Rankine
cycles. Our study shows that ORC plants have higher thermal efficiency than conventional
steam power plants. Furthermore, working fluid yields the highest thermal efficiency may
not yields highest power output and normally yields large size of the turbine, consequently
large ORC or high investment cost. Thus, the selection of working fluid should be based
not only on cycle efficiency but also the size of equipment and further on supply and
processing of heat carrier fluid.
temperature after the EHE. In this case, selection of working fluid should be based on cycle
efficiency and size of equipment. For this case, aromates prove to be the most potential
working fluids.
that investigated alkanes, cyclopentane, MM, MDM, toluene and o-xylene are the most
potential working fluids, depending on the working temperature range.
224
The finding in this study paves the way for extension to mixtures and for
optimization of practical problems where power output, investment and operation costs via
size of ORC, and other criteria should be considered for renewable sources with variation
of temperature and flow rate. Extensions and optimization of the cycle with pure fluids and
mixtures will be presented for other individual projects.
225

THERMODYNAMIC DATA OF WORKING FLUIDS FOR

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