
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/AA00041171/00001
Material Information
 Title:
 Thermodynamic properties of compressed liquids and liquid mixtures from fluctuation solution theory
 Creator:
 Huang, YungHui, 1953
 Publisher:
 [s.n.]
 Publication Date:
 1986
 Language:
 English
 Physical Description:
 xiii, 159 leaves : ill. ; 28 cm.
Subjects
 Subjects / Keywords:
 Bulk modulus ( jstor )
Correlations ( jstor ) Crossovers ( jstor ) High pressure ( jstor ) Isotherms ( jstor ) Liquids ( jstor ) Mathematical independent variables ( jstor ) Pressure ( jstor ) Thermodynamic properties ( jstor ) Volume ( jstor ) Chemical Engineering thesis Ph. D Dissertations, Academic  Chemical Engineering  UF Liquids ( lcsh ) Liquids  Thermodynamics ( lcsh ) Molecular dynamics ( lcsh )
 Genre:
 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph. D.)University of Florida, 1986.
 Bibliography:
 Bibliography: leaves 149158.
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by YungHui Huang.
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 University of Florida
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 University of Florida
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 The University of Florida George A. Smathers Libraries respect the intellectual property rights of others and do not claim any copyright interest in this item. This item may be protected by copyright but is made available here under a claim of fair use (17 U.S.C. Â§107) for nonprofit research and educational purposes. Users of this work have responsibility for determining copyright status prior to reusing, publishing or reproducing this item for purposes other than what is allowed by fair use or other copyright exemptions. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder. The Smathers Libraries would like to learn more about this item and invite individuals or organizations to contact the RDS coordinator (ufdissertations@uflib.ufl.edu) with any additional information they can provide.
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Full Text 
THERMODYNAMIC PROPERTIES OF COMPRESSED LIQUIDS AND LIQUID MIXTURES FROM FLUCTUATION SOLUTION THEORY
BY
YUNGHUI HUANG
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1986
To my parents and beloved sisters
ii
ACKNOWLEDGEMENTS
I would like to express my sincere gratitude to
Dr. J.P O'Connell for his interest and guidance throughout the course of this study. He provides me with not only the kind of intellectual atmosphere that I have never experienced before but also many valuable opportunities to interact with some leading researchers in the field of fluid phase equilibria.
I also wish to thank Drs. G.B. Hoflund,
W. Weltner, Jr. , S.O. Colgate , and G.K. Lyberatos for serving on the supervisory committee and Drs. E.A. Brignole and M.L. Michelsen for their helpful discussions.
Special thanks are due to professors A. Fredenslund and P. Rasmussen for their hospitality during my stay in the Instituttet for Kemiteknik, Danmarks Tekniske
Hdjskole and Mr. R.P. Currier and Mr. T. Schmidt for their assistance during my stay in the Kbenhavns Amts
Sygehus I Gentofte.
It is a pleasure to thank Mrs. Sun for her excellent typing and patience with my manuscripts and Mr. B.A. Klein and Mr. C.T. Skowlund for their help in using VAX and UNIX computing systems.
iii
Finally, I am grateful to the Northeast Regional Computing Center, Florida, for use of their facilities and Department of Energy, National Science Foundation
and University of Florida for providing the financial support that made this work possible.
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . iii
KEY TO SYMBOLS . . . . . . . . . . . . . . . . . . vii
ABSTRACT . . . .. . . . . . . . . . . . . . . .xi
CHAPTERS
1 INTRODUCTION . . . . . . . . . . . . . . 1
2 FLUCTUATION SOLUTION THEORY FOR
COMPRESSSED LIQUIDS AND LIQUID
MIXTURES . . . . . . . . . . . . . . . 6
Introduction . . . . . . . . . *. .... 6
Fluctuation Solution Theory and Direct
Correlation Function Integrals . . . . 7
Thermodynamic Consequences . . . . . . . 12
3 VOLUMETRIC BEHAVIOR OF COMPRESSED
LIQUIDS AND LIQUID MIXTURES . . . . . .15
Introduction . . . . . . . . . . . . . . 15
Pure Liquids . . . . . . . . . . . . . . 23
Liquid Mixtures . . . . . . . . . . . . 32
4 CORRESPONDING STATES CORRELATION FOR THE
DIRECT CORRELATION FUNCTION INTEGRALS . 36
Introduction . . . . . . . . . . . . . . 36
Model Development . . . . . . . . . . . 37
Model Parameterization . . . . . . . . . 38 Results and Discussion . . . . . . . . . 40
5 APPLICATION OF THE MODEL TO THE PURE
LIQUIDS . . . . . . . . . . . . . . . . 51
Introduction . . . . . . . . . . . . . . 51
Derived Thermodynamic Properties . . . . 51 Pure Liquid Compression Data Bank . . . 57 Results and Discussion . . . . . . . . . 59 Group Contribution Method . . . . . . . 83
6 APPLICATION OF THE MODEL TO LIQUID
MIXTURES . . . . . . . . . . . . . . . . 89
Introduction . . . . . . . . . . . . . . 89
Derived Thermodynamic Properties . . . . 90 Liquid Mixture Compression Data Bank . . 94 Results and Discussion . . . . . . . . . 94
7 CONCLUSIONS AND RECOMMENDATIONS . . . . 102
APPENDICES
A PERCUSYEVICK HARD SPHERE DIRECT
CORRELATION FUNCTION INTEGRALS FROM
VERLETWEIS ALGORITHM . . . . . . . . . 106
B THERMODYNAMIC PROPERTIES OF COMPRESSED
LIQUID MIXTURES FROM DCFI MODEL . . . . 108
C COMPUTER PROGRAMS . . . . . . . . . . . 117
REFERENCES . . . . . . . . . . . . . . . . . . . . . 149
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . 159
vi
KEY TO SYMBOLS
A = Helmholtz free energy ai = ith order reduced density coefficient of DCFI model ai coefficient matrix of DCFI model b = constant
C = direct correlation function integral c = molecular direct correlation function C = heat capacity at constant pressure d = effective hard sphere diameter f = state dependent function g = pair distribution function, state dependent function H = enthalpy, spatial integrals of total correlation function h = total correlation function I = identity matrix k = Boltzmann's constant ki = binary parameter M = general property m = integer
N = number of moles, number of components n = integer, refractive index P = pressure
R = gas constant
vii
r = position vector S = entropy
s = position vector T = temperature U = internal energy u = velocity of sound, intermolecular potential V = volume
v = V/N, molar volume vi = partial molar volume of component i X = matrix of mole fraction x = mole fraction, position Y = density dependent function of hard sphere diameter a = isobaric expansivity yi = activity coefficient of component i Yv = thermal pressure coefficient
6 = Kronecker delta
6 = parameter set ei = parameter of molecule i
6 = c/kT
c = energy parameter, dielectric constant q = packing fraction )T = isothermal compressibility y = chemical potential V = Stochiometric coefficient p = density
a = collision diameter
viii
T = inverse of reduced temperature cpi = volume fraction of component i Q = orientation normalization factor w = angular orientation coordinate
Subscripts
c = critical property i,j,k = component ij,jk,ik = pairs of components m = mixture
o = reference property, saturation property hs = hard sphere
Superscripts
E = excess property r = reference state
* = characteristic parameter o = standard state
= first composition derivative
= second composition derivative
Special Symbols
(as in X) Matrix quantity
(as in v) Partial molar property
(as in p) reduced property
o (as in v?) pure component property
ix
< > ensemble average
< > angular orientation average
x
Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
THERMODYNAMIC PROPERTIES OF COMPRESSED LIQUIDS AND
LIQUID MIXTURES FROM FLUCTUATION SOLUTION THEORY By
YungHui Huang
August 1986
Chairman: John P. O'Connell Major Department: Chemical Engineering
For every pure liquid there exists a characteristic volume for which the reduced bulk modulus, v/XTRT= (@(P/RT)/3p)T, is independent of temperature. In addition to the possible physical significance of this effect, the crossover point serves as the basis for a correlation of
the spatial integrals of the statistical mechanical direct correlation function (DCFI) which can be related to the density derivatives of thermodynamic properties in the fluctuation solution theory.
The pressure variation of the volume and reduced bulk modulus of essentially all pure and mixed liquids are correlated with a threeparameter corresponding states correlation over temperatures from the triple to nearly the critical point and densities from the saturation to
xi
The parameterization is based on the
detailed analysis of the most accurate experimental data. The three parameters each serve a particular mathematical
purpose, ensuring flexible and tractable numerical calculations.
The correlation method has been applied to over 200 pure and mixed liquids with errors probably within those
of the experiments, making it significantly more general and accurate than previous models. The systems investigated cover all orders of complexity of molecular structure and intermolecular interaction. These include liquefied noble gases, short and longchain hydrocarbons,
small and large globular chain or ring compounds, polar and highly associating species, metallic liquids, molten salts, and polymer melts. Mixtures of simple and complex systems are carefully examined.
The results are relatively insensitive to the characteristic temperature and all parameters may be estimated by a group contribution method. Simple mixing rules work
successfully in a onefluid form for mixtures with small excess volume. In other cases, a single binary parameter is adequate for good agreement. These findings indicate that the direct correlation function integrals are less sensitive to the exact details of the intermolecular forces for dense liquids than other statistical mechanical quantities.
xii
the freezing point.
This correlation, along with accurate saturation values, will provide all thermodynamic properties of compressed liquids and liquid mixtures.
xiii
CHAPTER 1
INTRODUCTION
High pressure liquids and liquid mixtures are present in many natural and manmade situations.1 They vary from geological systems to high pressure processes and techniques. Thermodynamic properties of compressed liquids and their mixtures are important both in basic scientific research and practical engineering applications. Thus the theoretical study of the volumetric behavior of compressed liquids is motivated by the concept that the structure of dense liquids is dominated by the
shortrange repulsive forces and that their physical properties can be treated in terms of a hard sphere model.2 From the practical point of view, liquid volumes are used exclusively in equipment sizing and descriptions of occupancy in natural reservoirs and formations.
Since the pioneering work of Bridgman,3 there have
been many volumetric measurements of compressed pure liquids, whereas much less effort has been dedicated to compressed liquid mixtures. A recent survey of PVT properties of liquid and liquid mixtures listed about 350 pure components and 170 binary mixtures. However, half of the numbers cited are restricted to very limited pressure
1
2
or temperature ranges, including many of atmospheric or single isotherm measurements. Compression data of pure liquids have been correlated with equations such as those
of Tait,5 Hudleston,6 Hayward,7 Chaudhuri,8 and others. Despite their substantial value for smoothing, interpolation, and computation of compressional thermodynamic functions, these equations share common disadvantages. First, they contain no explicit temperature dependence, so extensions by using temperaturedependent parameters are still uncertain. Second, two parameters are insufficient for wide pressure range isotherms, so extrapolations to either low or extreme high pressures are inaccurate. Third, their empirical nature renders molecular interpretation difficult. Fourth, these equations do not lead to mixing rules for compressed liquid mixtures.
An acceptable equation of state, apart from quantitative success, should provide some insight into the molecular characteristics in terms of a minimum possible set of parameters. Agreement between theory and experiment not only enhances one's faith in the molecular theory but also leads to the ability to generate the volumetric
properties with confidence from very limited experimental efforts. Theoretical equations of state which are based on the modification of Prigogine's cell theory such as Flory et al.,10 Simha and Somcynsky,11 and others have been applied to compressed liquids and liquid mixtures using
scaling parameters evaluated from lowpressure density data.
3
However, as noted by Bridgman,3 experimental behavior at low pressures is a poor guide to what is likely to happen at high pressure. As a result, these equations of state can not correctly represent the PVT behavior of compressed
liquids at high pressure. Empirical modifications by Zoller,12 and Simha and his coworkersl316 using elevated pressure data make only marginal improvement. Apparently, the theory cannot furnish an ideal representation of experimental PVT data at elevated pressure.
Recently, the Tait equation has been extended and
generalized by Thomson et at.17 This correlation, along with the correlation developed by Hankinson and Thomson18 for saturated liquid densities, comprises the COSTALD method. These authors claim that COSTALD is the most general and accurate compressed liquid density correlation yet published. This is true when compared with those
methods commonly used in chemical process calculations such as YenWoods,19 ChuehPrausnitz,20 LeeKesler,21 and others. However, the correlation method is purely empirical; application is limited to nonpolar and slightly polar liquids and pressures no higher than 700 bar.
A judicious alternative is to combine the simplicity
of an empirical equation with the advantage of statistical mechanical theory in developing accurate and general relations for thermodynamic computations. Fluctuation solution theory is explored here for this purpose. The
4
basis of the method is from the statistical mechanical grand canonical ensemble which relates composition derivatives of the pressure and the chemical potential, thus the activity coefficient, to integrals of the molecular direct correlation function. As recognized by
O'Connell,2223 and Gubbins and O'Connell,24 the density dependence of the reduced bulk modulus behaves differently
and quite simply in liquid phase compared to the gas phase, regardless of the complexity of the intermolecular forces. Hence, this work focuses on describing the liquid state alone.
The most surprising feature of the reduced bulk modulus versus reduced density plot is that there exists a
unique point of crossover of the isotherms for all of the compressed liquids studied. This finding is obviously a most important contribution to any liquid state theory. In view of the wellknown fact that it is density, not pressure, that is the key variable for compressed liquids, a corresponding states correlation, based on the density and temperature expansion of reduced bulk modulus, has been constructed to predict accurately the pressure effect on the compressional thermodynamic properties.
In the chapters that follow, detailed descriptions
of the theoretical background, experimental observations, model development, and successful applications, including pure and mixed liquids are presented. Chapter 2 describes the fluctuation solution theory and how useful
5
thermodynamic properties are related to the statistical mechanical direct correlation function integrals. Chapter 3 displays the reduced bulk modulus for many compressed liquids, providing significant features that are further exploited. For compressed liquid mixtures, volumetric data are fully analyzed and deeper insights concerned with the excess properties are carefully evaluated.
Chapter 4 is devoted to the process of model construction. It is demonstrated how the theoretical concept can be transformed into an actual working equation that simulates, to within experimental accuracy, the compressed liquid behavior. Chapter 5 applies the model to pure liquids. Also
basic thermodynamic properties such as entropy, internal energy, and others are derived through the Maxwell relations. Characteristic parameters and comparison with a variety of pure liquids compression data are tabulated. An initial attempt at a group contribution correlation for the parameters is also given. Chapter 6 extends the model to liquid mixtures. Mixture properties such as partial molar volume, activity coefficient, and others are expressed in a onefluid form. Binary parameters and comparison with liquid mixture compression data are tabulated. Chapter 7 closes this work with some remarks on the numerical techniques and our present concept of compressed liquid properties.
CHAPTER 2
FLUCTUATION SOLUTION THEORY FOR COMPRESSED LIQUIDS AND LIQUID MIXTURES
Introduction
Over the years there have been a growing number of uses of statistical mechanics in physical property research. There are two general approaches in the development of liquid state theory. One may be called the formal approach, based on distribution functions, while the other may be called the model approach, based on partition functions. The former was pioneered by Mayer and Mayer,25 Kirwood,26 and others. Thermodynamic properties can be calculated through the energy equation and the pressure equation with known pairwise additive potentials and pair distribution functions.27 The latter approach visualizes a liquid with a certain physical picture, such as made up of cells,28 holes,29 or molecular arrangements. Thermodynamic properties then can be obtained by transforming these physical models into a partition function.
All suffer the disadvantage of assuming pairwise additivity of intermolecular potentials.
Another alternative may be called the fluctuation solution theory.24 The compressibility equation which relates concentration derivatives of pressure and
6
7
chemical potential to the spatial integrals of the total
correlation function, and hence to direct correlation function integrals (DCFI) in the grand canonical ensemble is free from this assumption. Furthermore, the DCFI appear to be insensitive to the exact details of intermolecular forces. Therefore, from a numerical point of
view, the theory involves integration of a simple model, instead of taking the derivative of a complicated expression as in the others. Experience shows that this should be less sensitive to errors of modeling. It is the number density and temperature, not the usual mole fraction, pressure, and temperatures that are the most appropriate set of independent variables for theoretical work. However, the equations allow for consistent relationships to be made among the variables.
Fluctuation Solution Theory and
Direct Correlation Function Integrals
In the statistical mechanical grand canonical
ensemble, a relation exists between composition fluctuations and derivatives of the mean composition with respect to chemical potentials.
2
iT T,  (N > (21)
8
where the brackets denote a grand canonical ensemble average. These averages are also related to the spatial integrals of the radial distribution function by
KNi >KN .>
 6 (N > = g (r)d;
(22)
where 65 is the Kronecker delta. As recognized by Kirkwood and Buff,30 g1 (r) is not necessarily limited to be the radial distribution functions for spherical molecules; it can also be interpreted as a more general pair distribution function with all possible orientations. Using the definition of total correlation function,
1
(23)
A combination of equations (21), (22), and (23) yields
3
@yi/kT T,V'Pk/i
= 6. 
+ V j ,, dr
(24)
9
where
W =  hi(rwiWj)dwidwj
= the angle averaged total correlation function.
2 = dw = the nomalization constant of the
orientation dependence.
r = ri  r = the translational change of the position vector i and j. Equation (24) can be written as
pi/
Tpi/RT IT,VnPk~ei
6ijt i + pip dr
(25)
This is the basic equation of the KirkwoodBuff solution theory.30 From Orstein and Zernike,31 the direct correlation function is defined as
ci(i ij = h .(r riwiW.)  Z P
j JCik(rilkwiWk) hkj(rkrwkw.j)drkdwk
(26)
The second term on the righthand side represents the indirect correlations due to all possible third
10
molecules. Similarly, the angle averaged version of Equation (26) is
w = w  Z Pk
k
JCik(s ds (27)
where W = C ( o )dwidw
= the angle averaged direct correlation
function.
s = ri  rk = the translational change of position
vector i and k.
rs = rk  rj = the translational change of position
vector k and j.
The angle averaging operation in the last term of Equation (27) may not be appropriate for longrange potentials.32 Integration of Equation (27) yields
P J w dr = p j W dr  p2 Z Xk
k
dr d; (28)
11
In matrix notation, Equation (28) can be expressed as
C = H  C X H (29)
where X is a diagonal matrix whose nonzero elements are mole fractions. After basic matrix manipulations, Equation (29) can be written as
I + H X (I  C X)l (210)
or
6.. + pi dr = (6j  Pi d )~<
(211)
Comparing Equation (211) with Equation (25) and recognizing that the elements of the lefthand side of Equation (25) are the matrix inverse of a more desirable partial derivative,
/ =  (212)
api TPkj 1
where Ci = p 5 W d; = the direct correlation function integrals. This is the basic connection between
thermodynamic variables and direct correlation function integrals.
12
Thermodynamic Consequences
From the isothermal GibbsDuhem equation, the composition derivative of the pressure and the chemical potentials are related to each other by
3P/RT
3P TPk/j
P ii /RT
i jpI TPk/j
(213)
Substitution of Equation (212) into Equation (213) yields
kP/RT
@ j IT Pk j
(214a)
= xi(lC. *)
1 1
VI.
= 3
xyRT
(214b)
where the subscript m indicates mixture properties and )T is the isothermal compressibility. Summing all equations (214) yields
3P RT T ,x= xjx(lCi )
ap I T , i ij(215a )
1
(215b)
13
The partial molar volume V then can be expressed in terms of density and the DCFI
z xi(1Ci )
pm 1 ij i)(216)
V PM Z Z xixi(lCii. 2)6
i j
where pm is mixture molar density. An isothermal change of pressure can be obtained by integration of Equation (214)
ppr (PPr) P P dp.
pi
(217)
while an isothermal change of chemical potential is
Pi11r n i 4j Ci p
RT n r  r p Typ j
p. j p. /
1 J
(218)
where the superscript r denotes a reference property. This can be any fluid state, including ideal gas, saturation, or other welldetermined conditions. Naturally, the model for the C.. must be appropriate over the entire range of the integration. In phase equilibrium calculations,.it is usually more convenient to work with activity coefficient than chemical potential.
14
pI(T,p) = po(T,pr) + RTlnxiyi(T,p)
1 i1 
(219)
where the superscript o denotes the standard state properties. Substitution of Equation (219) into Equation (212) yields
3lny
3p T ,Pkij
1Cij Ic
P
(220)
Composition integration of equation (220) yields
(221)
. p 1C..
in r = Z J (1J) dpYji j P r
i p.
For a pure component,
9P/RT
P I T
(222a)
(222b)
PXTRT
This is the reduced bulk modulus of pure liquids, the quantity of primary interest in the present work.
CHAPTER 3
VOLUMETRIC BEHAVIOR OF COMPRESSED LIQUIDS AND LIQUID MIXTURES
Introduction
Theoretical investigations of the physical properties of dense liquids indicate that the structure of liquids,
static or dynamic, is similar to that of a rigidbody system. In order to examine the effects of the differences between real and rigidbody systems, the temperature and density effects should be separated requiring experimental studies at high pressures. In describing the thermodynamic state of a liquid, the applied pressure is considered high if it is comparable to or greater than the kinetic pressure.33 The thermodynamic equation of state is
P T( )  ( (31)
where the thermal pressure coefficient,T( the
kinetic pressure resulting from the molecular motion and the energyvolume coefficient,(aV)T, is a measure of the internal pressure arising from the intermolecular forces. When the energyvolume coefficient ( 3 )T is zero, the external pressure P is equal to the kinetic pressure
TP
T (5) , and
15
16
P)  P (32)
7 V T
This is the case for a rigidbody fluid. When the applied pressure P is close to the kinetic pressure, a perturbation approach is suggested using a rigid body reference with a small perturbation attractive potential term. Under these circumstances, the volume plays the dominant role in affecting the molecular behavior in liquids, because the molecules are so closely packed together that the forces between them arise almost exclusively from the shortrange repulsive forces.
There has been much theoretical speculation concerning this physical picture of liquids. The first controversy comes from the experimental evidence that the internal energy passes through a minimum which is characteristic of the liquid and a function of temperature.34 This brings in a temperature dependent effective hardcore diameter which would be necessary to make Equation (32) behave like Equation (31). Further experimental evidences for highly compressed liquids indicate that theoretical values of the hard sphere liquid deviate systematically from the experimental values.35 This suggests that the effective hardcore diameter becomes density dependent at high packing fractions (pa3>0.93) due to the softness of the repulsive potential and/or orientational ordering in compressed liquids. Figure 31 shows the reduced bulk modulus of a hard sphere liquid calculated
17
10.0
kT/c=. 376 .542
3.0 .687
* 794 1.0
6.0
In(2C)
2.0
.6932
0
0 0.5 1.0 1.5
pc.3 Figure 31. Theoretical DCFI of Effective Hard Sphere Liquid
18
from the PercusYevick compressibility equation. The VerletWeis algorithm is adopted to simulate the temperature and density dependent effective hardcore diameters of WeeksChandlerAndersen perturbation theory.2 Detailed
calculations are summarized in Appendix A. Figure 32 shows the experimental reduced bulk modulus of liquid argon calculated from Twu's algorithm.36 Comparison of Figures 31 and 32 shows that the differences are significant, especially at high pressure where the crossover is observed for argon and the temperature dependence is very small.
Since theoretical calculations of compressed liquids are improbable, the alternative is to analyze fully the experimental data available and develop an empirical relationship. There are several interesting findings concerning the volumetric behavior of compressed liquids and liquid mixtures. The earliest one is made by Bridgman on the pressure effect of isobaric expansivity.3 At low pressure, the thermal expansion coefficient of liquids increases with rising temperature. At high pressure, this relation is reversed. The reversal in the sign of the temperature derivative of isobaric expansivity is due to the nonlinear effect of the intermolecular forces. It is interesting to relate this phenomenon to the behavior of constant pressure heat capacity.3
19
4.0
.910, . 866
3.5 .811+0T/Tc=.6823.0
In(2C)
2.5
2.0
1.5
1.0
1.5 2.0 2.5 3.0 3.5
PVc Figure 32. Experimental DCFI of Liquid Argon
20
) = T( ) (33a)
VT(a2 P (33b)
where ap is the isobaric expansivity. The effect found
by Bridgman raised the possibility of a minimum in Cp as a function of pressure.
Similar observations on the change of internal energy with pressure were also made by Bridgman.3
( ) = T( )  P( ) (34)
1 7 T T( PT T T
The pressure at which the lefthand side of (34) is zero is given by the same form as Equation (32).
a )
= T( DP) (35)
These variations were further explored by Jenner and his coworker3740 on the compression measurement of 1bromoalkanes from C2 to C7 in the liquid state. The applied pressure is up to 6 Kbar in the temperature range between 2030K and 4480K. For all of the bromides
21
investigated, there exists a welldetermined pressure for which the isobaric expansivity is independent of temperature. This inversion pressure is further confirmed by experimental data from the piezothermal method of Ter Minassian and his coworkers41,42 on liquid water, carbon dioxide, and nbutane. The curves of the pressure derivatives of internal energy and of heat capacity show similar behavior as a function of pressure. In addition to the observed minimum at different temperatures, all of the isotherms pass through a single point at the inversion pressure noted above. Further, the isothermal variation of the heat capacity difference CPCv exhibits the same crossover at the inversion pressure.
In all of these cases, there must exist a thermodynamic relationship at the inversion pressure pi. First,
daPi i( 2PV) 2
d V =piT api
=0 (36)
leading to
a = a 2 ) 1/2 (37)
3T
Second,
22
3C 2
(gE) = T()
@P T @T2 Pi
= ( 38)
where bi is a constant. Combination of Equation (37) and (38) yields
b
TV
ap
Third,
CP  Cv = T( 3) ( )V
= TVa* y
CI
Pi
=b2
(39)
(310)
where b2 is another constant. From Equation (310), the thermal pressure coefficient yV can be written as
Y*=  b2 (311)
V F1 Pi
From the triple product rule, the isothermal compressibility XT can then be given as
23
a*.
P i
T y*
b
(312)
2
This is true only when inversion pressures are exactly the same for all of the properties investigated above. Unfortunately, data analysis on the isothermal compressibility is unavailable in Jenner's work to confirm this completely. Careful analysis of this property will be described in next section.
Pure Liquids
Volumetric properties of large globular and longchain hydrocarbons have been measured by Grindley43 for temperatures from 2980K to 4531K and pressures from zero to
8 Kbar. The data are adequate to determine the first and second volume derivatives of the internal energy and
entropy. The first isothermal volume derivatives yield the thermodynamic equation of state.
P = T(ia ) ( TU (313)
Both terms on the righthand side of (313) are temperature dependent. However, scaling can be achieved by a twoparameter corresponding states relation. The second
24
isothermal volume derivatives are related to the bulk
modulus
1 V U2
RThT = RT 7 T R %V2T (314)
In contrast to the external pressure expression (313), where the entropy term dominates throughout the whole density region, the bulk modulus has a large contribution from both terms. The entropy term is larger at low densities while the energy term is larger at high densities. This can be seen clearly from Figure 33 for the second isothermal volume derivative properties of nnonane. The crossing of the bulk modulus isotherms is due to the additive effect of both entropy and energy term, though crossing of the energy term also appears at higher densities.
The thermodynamic implications of this behavior have been analyzed further in this work. Our examination of the experimental reduced bulk modulus data available reveals a similar feature: A density at which the reduced
bulk modulus, which is related to the integral of the direct correlation function of Chapter 2, is independent of temperature. Figure 34 shows the experimental behavior of the DCFI for methane from Mollerup's computer programs.44 At the critical point, the DCFI is unity because the reduced bulk modulus is zero. At the ideal gas limit, the DCFI is zero, and hence the reduced bulk
25
2.
1.5
I.0
70.5~ 30*
.(V/R) (6?2s/aV2)T
0 BAR
150*
0
(RT) ~150*
300 150*
(V/RT)(a2U/a V2T
0.5
1.0 1.2 1.4 1.6
SPECIFIC VOLUME, cm3/gm
Figure 33. Entropy and Energy Contributions to
the Bulk Modulus of nnonane
26
4.0k
3.0h
In(2C)
2.0
1.0 
155K 0
165K
1750K
0
1100K
1300K
155*K
1650K
1750K 190.530K
1.0
2.0
3.0
PVC
Figure 34. Experimental DCFI of Liquid Methane
5.0
.6932
27
modulus is unity. Generally, the DCFI is small and positive (less than unity) in the vapor phase; it is large and negative in the liquid phase. The high density limits of
the isotherms are the freezing line, while the low density limits of the liquid isotherms are the saturated liquid.
Figures 35 and 36 show the DCFI of two sets of isomers: nnonane, 3,3diethylpentane, and nheptadecane, 5,5dibutylnonane in the liquid state alone. All the experimental curves behave similarly with relative insensitivity to the temperature in the higher density range, and the crossover is apparent. By shifting the curves to superimpose the crossover DCFI and density, complete overlap can be obtained by rotating the graphs slightly to bring the isotherms into coincidence. This provides the basis for a threeparameter corresponding state correlation of the reduced bulk modulus. Similar behavior has been found for essentially every other substance studied except water. Figure 37 shows the DCFI of water generated from the computer programs of the National Bureau of Standards.45 The temperatures are from boiling to the critical point and pressures up to 10 Kbars. Except for the low temperature isotherms (below 500C), water exhibits the same volumetric behavior as the other substances over a very broad range of state conditions. The differences at low temperature can be attributed to hydrogen bonding which decreases rapidly with temperature, a conclusion which has been reached on evidence from diverse sources.46
28
6.0
nheptadecane
5.5
5.0 nnonane
In(2C) 0
600C
4.5 90 C0
130 0C 300
4.0~
500C
180Oc 700C 3.5  100 0C
1300C
3.0 150 C
2.0 2.5 3.0 3.5 4.0
PVc
Figure 35. Experimental DCFI of Liquid nnonane and nheptadecane
29
5,5dibutylno
2.5
6.0r~
3.0 PVc
3.5
4.0
Figure 36. Experimental DCFI of 3,3diethylpentane and
5,5dibutylnonane
nane
30 C
60 0c
100c 3,3diethylpentane
300C
1600c 70CO
110 0C 150 C
5.0
In(2C)
3.5k
2.0
4. 0 r
3.0
30
4.0
3.0
In(2C) 1000 C
168. 690c
2.0
236.700C
270.650c
1.0
338.640c 374.520c
1.0 2.0 3.0 4.0
Figure 37. Experimental
I
DCFI of Liquid Water
31
Volume and temperature have been chosen as the
independent variables since they are the most suitable for theoretical treatment. In general, volume scaling is the result of an increase in molecular size, while temperature scaling is due to the ratio of potential to kinetic energy effects and variation in rotational degrees of freedom. The relative insensitivity of the DCFI with temperature in the compressed state indicates that internal degrees of freedom are not appreciably altered when the density changes above twice the critical density. The crossover reduced bulk modulus variation which leads to DCFI scaling arises probably from the combined effects of size and intermolecular forces because of the additive effects of entropy and energy contributions to the crossing of the liquid bulk modulus. To better understand this conjecture, two separate studies from Gibson and Loeffler47 and Grindley and Lind48 can be considered. For the same sized
molecules, if a polar group is introduced as in the former case, or upon charging a set of neutral molecules to form a molten salt as in the latter case, the major contribution to the decreased pressure and isothermal compressibility (or increased DCFI) comes from an increased (u ) aV T
a2U
which makes (@2U2 T more positive. Only a small contribu@S a 2d
tion comes from a decrease of (s)T and  2) T ~VT (V
32
Liquid Mixtures
Insights into the thermodynamic properties of liquid mixtures can be obtained from the excess volume as a function of pressure, temperature and composition. An equation of state based on the excess volume VE can be written as
Vm(T,P,x) = Z xiV (T,P) + VE(T,P,x) (315)
where Vm is the mixture volume and V is the volume of pure component i. The absolute value of excess volume is small in comparison to the mixture volume itself, whereas the variations of the excess volume with pressure, temperature and composition can be important and comparable to those of the mixture volume.
3VE E
 3P T,x m (316)
3VE E 17)
T Px Vm (317
and for binary mixtures,
T,P ~ ( UVm) + Vo  V0 (318a)
2 2) 2 1
I / V1 (318b)
33
E
@V T,P I  (V2 m) + V  V (319a)
(V m l) + V1  V (319b)
Since very few measurements have been made to determine the equation of state for liquid mixtures, significant progress in their high pressure thermodynamics has not occurred. However, it is known that the experimental behavior of the excess volume as a function of temperature and composition is exceedingly complicated due to a variety of physical effects and intermolecular interactions. One behavior of interest is the pressure dependence of the excess volume of liquid mixtures. Figure 38 shows the excess volumes of the carbon monoxidemethane system from Calado et al.49 and the nheptaneethanol system from Ozawa et al.50 The absolute value of 3VE
(7) rapidly decreases with pressure, implying that
high pressure mixtures for a wide variety of liquids show more ideal volumes of mixing than do low pressure systems. Generally, the pressure influence is very significant. For simple liquid mixtures, the excess volume is small and negative at low pressure, becoming almost zero at high pressures. For nalkane mixtures, the excess volume is negative at low pressure, but it eventually
34
nheptaneethanol (Ozawa, et al.)
Excess Vol ume CM3MOL
1.01 0.5 0.0
n 4i
0
0
Excess Vo I ume CM3MOL
 1
0
XETOH.32
T. K
.298
348
.r
100
200
11.0
T=348.20 K
x ETOH=0.82
~0.5
=0.30 .
=0.57
004
0 100 200
Pressure, MPA
CO/CH at xCO =0.5 (Calado, et al.)
4 c oT i
40
Pressure, MPA
Figure 38. State Dependence of Excess Volume
116.3K
 120
125

8o
35
becomes small and positive at high pressures. For nalkane and 1alkanol mixtures, the excess volume is positive at low pressures becoming negative at high pressures. For aqueous systems, the excess volume is large and negative at low pressures, becoming small and positive at high pressures. These are all closely related to the packing effect at highly compressed states.
CHAPTER 4
CORRESPONDING STATES CORRELATION FOR THE
DIRECT CORRELATION FUNCTION INTEGRALS
Introduction
The practical value of the rigorous formulas derived in Chapter 2 and experimental observations discussed in Chapter 3 depends on a mathematical function to simulate the DCFI behavior. The ideal model for DCFI would yield a robust mathematical form, giving all of the variations nature shows with a minimum of experimentally accessible parameters that depend only weakly on thermodynamic states. Treatments using the corresponding states principle (CSP) have this desirable feature. However, most corresponding states correlations are based on the critical properties and other macroscopic fluid measurements such as saturation properties, so they are limited to classes of substances that have simple intermolecular forces. Furthermore, these properties are not always measurable as in the case of heavy hydrocarbons and coal liquids. However, a corresponding states correlation for the DCFI would be very useful in chemical process calculations if suitable parameters are found. In this chapter, a threeparameter corresponding states correlation has been developed to describe the DCFI of all liquids. The
36
37
three characteristic parameters of the correlation are the volume, V*, and DCFI, C*, at the crossover of the isotherms along with a temperature, T*, which is in the range of the critical temperature.
Model Development
The first apparent utility of the DCFI to describe the relationship of pressure and density for liquids was
described by Brelvi51 and Brelvi and O'Connell.5253 They noted that density, not temperature, was the dominant variable and the DCFI of many substances could be correlated in a oneparameter corresponding states form. In principle, their characteristic volume parameter could be determined from a single compression measurement. This correlation holds within 510% for a variety of substances over the dense liquid region. However, the correlation is uncertain at lower densities where the experimental data are sparse and temperature effect becomes significant. Further examination by Mathias and O'Connell5455 showed that the temperature dependence could be taken into account with a characteristic temperature in a twoparameter corresponding states form. Since temperature dependence was included, the results at low densities were improved. Unfortunately, the correlation used hard sphere expressions corrected by a linear density term so applications to high pressure were not of high accuracy.
38
In the present work, liquid compression data have been analyzed even more carefully and the most general form of the corresponding states relation is shown in Figure 41. The behavior of the liquid DCFI for methane is plotted in reduced coordinates along with some extensions using data for nheptadecane. At the crossover point, the reduced density and DCFI are defined to be unity. While variations away from this characteristic density and DCFI are somewhat dependent on the temperature, they can be correlated in reduced form.
Model Parameterization
The present correlation takes advantage of the
similarity of the crossover behavior that all dense liquid DCFI data show. The success of the model relies on the construction of an equation of state that can describe the reference substance accurately. Since this work has been concerned with formulating the thermodynamic properties of petrochemicals and coalderived liquids, methane is used as the reference substance. The basic form of the correlation is chosen to express the reduced DCFI, =C/C*, as the polynomial expansions of the reduced density, p=pV*, and the inverse reduced temperature, T=T*/T.
m n
C = ai (7)J (P) (41)
i=l j=l
39
CH
4.0 _  nc H .0
.80
.74 .66
3.0
.59
C/C* T/T*=.54
2.0
1.0 ~
0.0
0.6 0.8 1.0 1.2 1.4
p/p* Figure 41. Corresponding States Correlation of Liquid DCFI
40
where at is the coefficient matrix of the reduced variables.
After a thorough test using the liquid DCFI data of
methane generated from Mollerup's computer program, the optimal expression has been determined as the lowest order polynomial that can achieve the same accuracy as his equation. In this case, m=4 and n=3. The coefficient matrix ai of equation (41) is given in Table 41. The agreement with experimental data overall is better than 0.2% except for low density points near the critical isotherms. The valid range of the reduced variables is
0.7% pVts 1.3 and 0.5;T/T*50.99. This range could be expanded by use of a modified form of the polynomial and data for substances at lower reduced temperatures and higher reduced densities. These coefficients give the crossover value of E=l to within 1% over the specified temperature range. The reduced bulk modulus then can be written as
@P/RT 1
p T p'TRT
1 C*t (42)
Results and Discussions
Equation (41) with the coefficient matrix of Table 41 has been applied to all of the substances for which liquid DCFI data have been found. The compressibilities
41
TABLE 41
Coefficient Matrix m\n 1
1
9.8642
2
28.465
3
27.542
for Equation (41)
2
10.191
30.864
32.898
3
1.5356
6.0294
8.7130
8.2606 12.737
4
4.0170
42
are generally from fitting volume changes with pressure and sonic velocities. The present reduced bulk modulus data bank includes 33 substances ranging from noble gases to petrochemicals to water and methanol. (Compressions of
many more substances are discussed in Chapter 5.) Table 42 lists all of the characteristic parameters and the data ranges from which the parameters are evaluated. The values of V* are about 3040% of the critical volume while the values of T* are normally within 20% of the critical temperature. The values of C* vary from 7 for very small molecules to 167 for large polyatomics. The results are in excellent agreement with the experimental data. For some substances with wide data ranges, the fitted pair parameters coincide with the crossover point observed experimentally. For substances with lower pressure ranges, the fitted pair parameters are close to the crossover point obtained from graphical extrapolation of the experimental results. This is also true when water is considered individually at high and low temperatures. There are two sets of characteristic parameters for water at different temperature ranges. They are listed separately because water behaves unusually at low temperatures56 (below 751C).
Discussion of the results can be developed according
to the complexity of the substance, the range of the state conditions, and the accuracy of the experimental results.
Table 42
Characteristic Parameters and Comparisons with Liquid DCFI
Data
Substance
Argon Krypton Xenon Nitrogen Oxygen Methane Ethylene Ethane Propane nButane Isobutane nHexane nNonane nDecane nDodecane nHexadecane nHeptadecane
19.0696 16.2128 16.4803 16.9860 20.3201 14.7575
24.2217 27.7190 30.0627 32.9989 32.1510 59.6000 79.5211 98.2235 119.925 167.000 129.582
C*
T*,Ok
V*, cc/mol
28.2294 35.2617
45.3441 34.6485 27.6618 38.7154 47.8650 53.0997 71.7628 90.3628 91.7398
122.802 174.750
190.584 223.532
288.290 315.920
No. of AAE,% Points
139.854 201.966 277.367 119.965
146.000 190.573 261.171 295.785
340.923 391.759
379.441 454.693 508.562
541.824 540.761
522.111 603.156
0.5530 1.0809 0.7612
0.4628 0.2665 0.1835 0.8795 0.9608 0.9285 0.9298 0.8390 0.9387 1.3717
2.4417 2.3157
1.4783 0.9444
Data Range
T,0k P,bar
43 28 31
20 17 77
43 45 56 32 32
24 34 32 32 26 29
Ref.
90140 120200 170260 77101 90118 95185 160250 160260 200320
240360 240360 273333
303423 298358
298358 298358 333453
25.232497 25.333040 25.332533 21.37519.4
11.29504.3
0.654785 51100
6.34743.8 0.2711.6
0.24717.4 0.4711.1
05018 06000
1.015171
1.014137 1.012757 08000
36 56 57 58 58
44 59 60 61
62 63
64 43 65 65 65
43
Table 42 (Continued)
Substance
3,3Diethylpentane
4,4Dipropylheptane
5,5Dibutyl
nonane Benzene Chloro
benzene
Bromobenzene Nitrobenzene Aniline Carbon tetra
chloride
Ethylene
glycol
Methanol Water
76.0363
99.1199
134.342 40.5966
42.7971
44.9976 51.8347 51.6100
44.1436
33.9809 15.9110 15.7897 7.19912
V*, cc/mol
169.647
241.133 308.186 88.5489 105.199 109.707
108.264 96.0762
T*, 0k
594.818
611.932
624.183
492.013
546.936 564.097 582.678 573.472
95.0715 486.377
59.6111 39.7147
17.9400 20.1522
603.269
481.700 298.093
445.452
No. of AAE,% Points
2.0520
0.9592
1.5203
0.4674
0.2961 0.2836 0.3171 0.2393
0.4886
0.2354 1.1346 0.8414 1.8048
Data T, 0k
31 303423 30 303373
31
24
24 24 24 24
303433 298358
298358 298358 298358 298358
24 298358
24 68
24 42
298358 298473 298358 348573
Range
P,bar
08000 05000 08000 01000 01000 01000 01000 01000 01000
01000 0.171000
01000 0.391000
Ref.
43 43 43 47
47 447 47 47
66
66 67 66
45
Table 42 (Continued)
Substance
Tetraethylsilane
Tetraethylammonium tetrapropylborate
Tetrapropylammonium tetraethylborate
Tetrabutylammonium tetrabutylborate
V*.cc/mol
94.0941 96.8865 69.1909
183.536
T*. 0k
510.458
198.677 695.258
204.873
836.801
AAE .
2.5058
0.5314
0.8200
No. of Points
32
40
Data Range
T,0k P,bar
303423 363403
16 393433
299.692 716.369 0.7023 20
Ref.
43 43
08000
04200
.1::.
01400
43
146.013
393433 05000 43
,
46
In the case of monatomic substances from Streett et
al.56,57 and diatomic species from Bender,58 the temperature ranges from just above the triple point to near the critical point, and pressures from saturation to freezing point. The agreement with the present correlation is within 1%. The ratios of the characteristic parameters to the critical properties are slightly lower than the reference methane values, while the characteristic DCFI values,
C*, are slightly higher. In the case of shortchain hydrocarbons from the National Bureau of Standards,6063
the reduced temperature range is between 0.5 and 0.97, and the pressures are from saturation to 700 bars. The agreement with the present correlation is within 1%. The ratio of the characteristic volume to critical volume is about
0.35 and the ratio of the characteristic temperature to critical temperature is about 0.92. In the case of longchain hydrocarbons from Snyder and Winnick,65 the temperatures are 250 to 850C and pressures from about 1 bar to over 5 Kbars or the freezing pressure, whichever is lower. The agreement with the present correlation is within 2.5%. This is the worst for all of the substances tested. However, examination of the deviation patterns indicate that the major source of the disagreement comes from the atmospheric measurements of each isotherm. There are several possibilities related to this observation. First, the extrapolation of high pressure data to atmospheric pressure either by a graphic technique or by a
47
specific equation may cause errors, though it is a common practice in high pressure work. Second, the adoption of the atmospheric measurements from the other sources may cause inconsistency in the complete data set. Third, the initial compressibility of the longchain hydrocarbons may behave differently from that of methane. If the deviations do result from the experimental uncertainty of the atmospheric DCFI, then the agreement with the present correlation is definitely within the experimental errors. Similarly, for long chain and large globular hydrocarbons
of Grindly,43 the temperatures are from 250 to 1900C and pressures from zero to 8 Kbars. The agreement with the present correlation is within 2%. The experimental precision of the lowest pressure compressibilities is only 1 or 2%, while above 400 bars, the precision is about .5%. Comparisons of the experimental data with the calculated results indicate that the deviation patterns are consistent with the previous assertions that the disagreement between the experimental DCFI and the present correlation at zero pressure is due to the experimental uncertainty at this state. For these larger hydrocarbons, the ratio of
the characteristic to critical volume is about 0.31 and the ratio of the characteristic to critical temperature is between 0.8 and 0.9. The only exception is 3,3diethylpentane which has an unusually high ratio of the characteristic to critical temperature; this is associated with the abnormally large disagreement of DCFI at zero pressure
48
where the temperature effect is significant. For benzene and some of its derivatives, the work of Gibson and Loeffler,47,66 is ranked as one of the most accurate among all high pressure measurements. The temperatures are from 250 to 850C and pressure from zero to 1 Kbars. The agreement
with the present correlation is within 0.5% which is significantly better than for the rest of the substances. This is partly due to the insensitivity of the experimental DCFI with temperature at low pressures for these substances. Figure 42 shows the experimental behavior of DCFI for benzene. The value of C* increases from liquid to liquid in the order of benzene, chlorobenzene, bromobenzene, nitrobenzene and aniline. This is consistent with the argument in the previous chapter that the introduction of a polar group acts on the liquid DCFI of benzene in the same way as the increase of energy volume coefficient. It changes the attractive potential and
2U
makes ()T more positive. The ratio of the characteris3V
tic to critical volume is about 0.34 and characteristic to critical temperature is about 0.85. In the case of the molten salts from Grindly,43 the temperatures are from 900 to 1600C and pressures from zero to 5 Kbars. The agreement with the present correlation is within 0.8%. The ratio of the characteristic to critical volume is about
0.32 and characteristic to the estimated critical temperature is about 1. Comparison of C* values for
49
5.0
4.5
4.0 In(2C) 20c 25 0C 3.5  400c
55 C 70 0c
3.0  85 0c
2.5
2.0 2.5 3.0 3.5 4.0
PVc
Figure 42. Experimental DCFI of Liquid Benzene
50
tetrabutylammonium tetrabutylborate and 5,5dibutylnonane which have nearly the same V* confirms that the charges have the same effect on liquid DCFI as a polar group does. For highly associating species such as water45,66 and methanol67 the agreement with the present correlation over
a broad range of state conditions is as good as the other substances; only lower temperature water is anomalous.
CHAPTER 5
APPLICATION OF THE MODEL TO OTHER PURE LIQUIDS
Introduction
Thermodynamic properties of a pure liquid can be calculated from the DCFI model using the parameter developed in the last chapter. In addition to isothermal compressibilities, there are compression data at high pressures, so model parameters based on the pressure equation and compression data can be obtained for these substances as well. These parameter tables include almost all of the significant liquid compression data available. The crossover parameters appear to be group additive and related to
critical properties. Some of the group parameters for characteristic volume V* and characteristic DCFI C* are tabulated.
Derived Thermodynamic Properties
Isothermal integration of the DCFI model from a reference density, po to final density, p, yields the pressure, P
51
0
52
P  P0 + p*RT{(3. )C*[(9.8642 .'9.1.5356
P P + P * T I  oT T 2
(14.232515.432 3.0147 )( 2_ 2)+(9. 18067 10.966_ 2.904 33
T T T 2
( 3_3 )(2.06515_3.18425 1.00425 (4_ 4)
0 T T
(51)
where Po is the pressure at P. and T. Substitution of the DCFI model into Equation (222b) yields the liquid bulk modulus XT'
X = pRT (1C*[(9.864210.191 1.5356 )(2B.46530.864 T Y[( T T
6.0294 )p+(27.542 32.898 8.7130 ) 2(8.260612.737
T 7 T T
4.0170 3]
72 (52)
Isothermal integration of P from the reference to the
2
p
final density yields the change in the Hemholtz free energy.
A(p,T)A (p ,T) Po P P0 1 1
1)  In  ) C*{(9.8642RT p p p*RT p 0
10.191 1.5356  9 10.191_1.5356
T )In 0T[(9.8642 T
53
15.432_3.0147 +(9.1806710.966_2.90433 )2(2.06515T T2 0 T
3.18425 1.00425) ] (o )+(14.2325 15.432_3.0147
T T p T T
(4.590335.483 1.45217) 2_2 )(.688381.06142_0.33475
T T
(3O3)j } (53) Isobaric temperature differentiation of the pressure yields the thermal pressure coefficient, yv.
Y =(dP ) + (PP 0)  RT(dp2)  p*C*R{![(0.191+3.0712)
dT T dT 7 T
)(15.432+6.0294 22 )+(10.966 5.80867 3 3
(3.18425+2.0085 )(~4_4)]  [(9.864210.191_1.5356
5 0 2 T
2
(.)(14.2325 15.432 3.0147 dp dT 7 T dT
~3
10.966 2.90433 d 3.18425
7 T d! 7
d4
1.00425 X
7 dT (54)
where the derivatives of the reference properties Po and o are along the path chosen for their variation, usually
along saturation line.
54
The isobaric expansivity ap then can be calculated through the relation
a ="Tyv (55)
From the Maxwell relation
a T = (T3T) (56)
The entropy derivative is
T = ~Y (57)
Substitution of Equation (54) into Equation (57) and isothermal integration from the reference density to the final density yields the entropy change. S(p,T)S (p ,T) dP 0 d0
R R ) T()() +
o p P dT
1  C*{(9.864220382 4.6068 )ln (14.232530.864 9.0441)
T T 0
(00 )+(4.5903310.966_4.3565)( 20 )(0.688382.12283
0Y T2 0
1.00425 3 3 1011 .35
 25)( 3P0 )T[(9.864210.191_1 5356d )(14.2325T TT dT
~2 ~3 15.432 3.0147 d ( 8 10.9662.90433 dp
2 )( )+(..9067)(o) Y 2 dY T T2 dY
4
(2.06515 18425 1.00425) ~1( ~ )}+{1C*[(9.86422
T T dT p p0
20.382 4.6068)(14.2325 30.864_9.0441) +(9.1806721.932 S2 T2 0 T
8.7130) 26.3685 3.01275 3 o T T 0 (58)
55
Internal energy U is given by
dU = TdS  Pdv (59)
Isothermal integration and substitution of Equations (51) and (58) into Equation (59) yield the internal energy change.
U(p,T)U (poT) P0 dP dp
R( 0)[ 0 )]  1) +
o T dT pp0 dT
C*{1(10.19113.071 )lnL(15.432A6.0294 ))+(5.483+
T T p T
2.90433 )2 )(1.06142+0.6695 33 )]+[(10.191+3.0712)
T T T Y
(15.432+ 6.0294) +(10.966+ 5.80867)2(3.18425+2.0085) 3
( )+ [( .8 42 0 I9 .5356 dp 2 )(14.2325 15.432dT T
 2 3
3.0147 p0 2do
3.2 )( )+(9.180670.9662 .90433)(2.06515T dT T dT
d4
3.18425 1.00425 d(1)]( _ (510)
T T dT ppo
The enthalpy H is found from the definition
H = U + PV
(511)
56
Substitution of Equations (51) and (510) into Equation (511) gives the enthalpy,
21 11 dP P
H Uj(p,T) + Po(; ) + T(( ) +
0~ p p p 0 T
(dP ) + C*RT{!(10 .191+3.2712 )1n +(14.2325 30.864_9.0441 dT T T T
(33 )(9.180716.449 5.80863 23 )+(2.60515 4.24567
0 T T T
1.67375) 3_ 3)+[ (9.8642+1.5356 )(14.2325 3.0147 + F2 o Y2 0
(9.1807 . 3 (2.60515+1. ]( 1)+Y[(9.8642A 3
10.191 1.5356 dp 15.432 3.0147 dp
~ ~2 ~ d )(14.2325 y  )2 ( )+
T T dT T T d
~3
10.966 2.90433 0 . 3.18425 1.00425
T T dT T T
4
(d 1)](  ) (512)
dT p p
In the above equations, the property value at the state p,T can be obtained completely in terms of the property at the state p,,T and, where necessary, derivatives such as
( ) of that state.
57
Pure Liquid Compression Data Bank
Critical compilations of the experimental data of compressed liquids are scarce. A recent survey of PVT properties of liquids, as referenced in Chemical Abstracts before the end of 1983, was conducted by Tekic et al.4 and published in the Journal of Fluid Phase Equilibria in 1985. This survey may serve as a source of references to some original papers on PVT properties of compressed liquids. Our present pure liquid compression data bank contains 133 organic and inorganic liquids (including liquefied gases and metallic liquids), 13 molten salts, 17 polymer melts, and 7 deuteriated liquids. They are listed
separately in the first column from Table 51 to Table 54. The substance order is based on the Chemical Abstracts system. In the case where the substance has been studied by several authors or by the same author with several investigations, the reference numbers are ordered chronologically. Only those data sets with the pressure ranges greater than 300 bar and temperatures more than two isotherms have been selected. There are 234 total data sets. Several misprints have been found in the printed values of the experimental data in the original articles. They have been corrected in the present data bank.
Experimental determination of the volumetric properties of compressed liquids may be divided into two methods: direct measurements and indirect measurements. The major sources of the experimental data in the present
58
data bank come from the direct measurements of volume change or volume itself as a function of pressure at constant temperature such as from piezometric68 or pycnometric methods.69 Isochoric data70 are generally not included in the data bank because they are not available in isothermal form. Only a small portion comes from the indirect methods where liquid density can be related to the other physical properties such as ultrasonic velocity, dielectric constant, refractive index, and heat capacity, etc. In the case of the ultrasonic velocity measurement,
1 2
%T I + (513)
pu P
where u is the velocity of sound. Isothermal integration from the reference pressure P0 to the final pressure P
yields the density expression.71
P P a2
P = dP + T P dP (514)
u P
Po TT
In the case of the dielectric constant measurements, the theoretical expression of the ClausiusMossotti Equation29 can be written as
1 = (515)
59
where CM is the ClausiusMossotti function7274 and E is the dielectric constant. Similarly, for infractive index measurements the LorentzLorenz Equation25 can be written as
2
1 n 1 (516)
n +2
where LL is the LorentzLorenz analogue74 of the ClausiusMossotti function CM which replaces E in Equation (515) with n2, the square of the refractive index.
The direct measurements tend to be more accurate than the indirect measurements. The accuracy of the experimental data varies from 0.01% to 1% in density. Generally, a precision of 0.1% in density corresponds to la in pressure or 110% in isothermal compressibility.
Results and Discussions
The pressure Equation (51) can be used as an equation of state for pure liquids. If the characteristic parameters are known for a specific substance, then the change in pressure due to an isothermal change in density can be calculated. On the other hand, Equation (51) can also be used as a correlation scheme to fit the liquid compression data. Tables 51 to 54 list the characteristic parameters for all of the substances in the present data bank. The results are in excellent agreement with the experimental data for all of the substances over all data ranges. In
Table 51
Characteristic Parameters and Comparisons
Substance C*
Argon 17.7363
16.5475
Bismuth 18.2663
Dichlorodifluoromethane 37.3342
Trichlorofluoromethane 30.7966
Carbon tetrachloride 41.9381
42.0350 44.3621
Carbon
monoxide 20.7868 Carbon
disulfide 29.0507 Chlorodifluoromethane 24.8969
V*,cc/mol
28.6692 28.8229 27.7070 75.9700
89.4542 95.6553 95.5192
95.0146 34.1649 59.6196 59.9348
T*,"k
145.647 150.113 751.293 373.366 455.306
489.902 488.580 507.552 125.638
424.798 356.229
AAE,%
0.9052 0.5093 1.1289
1.4629 0.4633 0.2141 1.5006
0.9514 0.9061
1.1426 1.5669
with Liquid Compression Data
No. of Data Range
Points T,"k P,bar
48 90140 22.42497
154 110147 12.51422
9 550950 08000
38 253313 1.521608 16 341460 1071594
24 298358 01000
25 273323 11977
44 313373 1.012622 72 82125 19.81397
44 223298 12028
38 253313 2.431596
Ref.
36
75,76 77 78 79 66 80 81 82 80 78
Table 51 (Continued)
Substance
J*. cc/mol
T*. Ok
No. of D
AAE % Pointf T Ol
ata Range
r h ' .
Tr ch oromethane Dichloro
methane
Methyl
bromide
Chloromethane Methyl iodide Methane
Methanol
Acetonitrile
Ethylene
54.7671 36.9601
250.711
26.3340
21.6065 25.5235 26.6306 29.2767 14.7376 20.2269 17.2607 15.9908 23.8523 16.4759
22.6415 20.2608 23.9617
75.3199
79.1835
44.2208 64.0808
54.5703
48.1691 62.4797 61.0318 38.7225 37.1619 37.9177 39.5268 37.6472 39.5920 37.8518
51.8140 47.9370
414.111 468.398
281.633
462.224
443.891 366.157
433.610 465.803 190.711 182.236
180.204 506.620
420.711 492.130 420.349 438.583 265.921
0.9860 0.6586
3.6907
0.8314
1.0955 0.7966 1.7119 0.6581
0.1452 1.3459 0.7636 1.5166 1.9036 1.1207 0.8369 1.0996 0.7727
14 32
20 24
36 36
20 36 80 19
109 32 61 80 35
21
303363
273348
303323
298348
253313 253313 273363 253313
95185 105160
110147 323473 283348 298473 278323 303393
43 160250
304000
0.1987
1.015189 0.61013
0.41595 1.221595
12500 0.11595 0.654785 0.572094
101281
1341392 12075
0.171000
12750
304500
51300 59
83
84
85
84
ON
78 78 86 78
44 87
88,89 90 91 67 92 93
R f
Table 51 (Continued)
Substance
1,1Dichloroethane
1,2Dichloroethane
Acetic acid Ethyl bromide Ethane Ethanol
Dimethyl
sulfoxide Ethylene
glycol
Ethylene
carbonate Propylene Acetone Methyl acetate
28.2808
30.4401 32.8191 27.7857 25.1652 19.3553
38.1417 22.3241
61.5869
33.9809
68.3229 23.6708 35.0617
63.4474
V*, cc/mol
T*, 0k
84.8104 488.322
82.0062 55.7849 73.3477 53.6389 59.2663
54.5444 58.0450
70.0619
517.867 493.835
474.360 293.737 538.568 415.781
486.624
442.503
59.6111 603.269
65.2805 67.5402 72.3087 72.7310
646.152 355.272 566.224
428.005
AAE %
0.9888
0.8907
0.3249 1.3446 1.0062
2.3471 1.3305
1.4371
2.1925
0.3862
1.4247 1.5945 1.3519 1.3335
No. of Points
Data Range T,0k P,bar
40 298398
40 18 36
64 15
46 43
298398 298328 253313
140280 293353 298323 298348
31 294323
24
14 65
49 36
298358
353393 273353 298398 253313
0.31013
0.11013
1.012533 0.11568 0.6745
04903
13104 11962
Ref.
94
94 95 78 60 96 97 50
01500 98 01000 66
11400 19.61029
10.14119
01568
99
100 101 78
0'~
Table 51 (Continued)
Substance
Propionic acid Propane Isopropanol Glycerol Perfluorocyclobutane Furan
Vinyl acetate nButyraldehyde Methyl ethyl
ketone
Tetrahydrofuran
Ethyl acetate
Isobutyric
acid
nButane Isobutane Diethyl ether
39.1760 29.2700 26.8345 71.1156
39.4863
49.1557 35.4288 41.7832
V*. cc/mol
73.5143
72.0274 78.6194 76.3347
114.173 67.7580 91.0150 90.3996
67.0723 82.3974
197.721 146.386
*54.8846
53.3627 32.3974 30.7976 51.5640
63.5865 79.7130 93.1047
90.2104 90.6076 92.2552 95.1122
T*, uk
515.340
345.560 750.459 736.593
0
0
1
1
389.817
428.410 421.669 552.808
420.479
299.328 305.153 460.697
520.614 396.630 383.286 377.696
No. of AAE,% Points
.2189 18
.5484 56
.3412 60
.3414 38
1.2250
1.2001 0.5092 2.4365
5.2120
3.3331 2.7803 1.2965
0.5695 0.5258 0.5606
1.8304
20
169 55 36
Data Range
1, "k
298328
200320 298398 323523
323373
225257
298398 303333
18 303343
14 14 36
18 32 32 70
303323 303323 253313
298328
240360 240360 293353
P,bar
1.012533
0.2712
10.14119 1351477
501740 18830 10.13923 1.012765
01720
1.015170 1.015127 01568
1.012533
0.24714 0.4711 1.0112159
Ref.
95 61
101 90
102 103 101
104
105
85 85 78
95 62 63 96
Substan,
Table 51 (Continued)
Substance
tButanol
Tetramethylsilane
Tetramethyltin
Pyridine Cyclopentane iPentene Neopentane Isopentane nPentane Perfluorobenzene
Bromobenzene
Chlorobenzene
Nitrobenzene
V*,cc/mol
60.8345
52.4182
52.8151
54.9820 36.0670 26.9889 24.3908 35.9999 47.7900 7.48978 77.7432 75.3355
42.2245
*40.3277
40.2752
*48.2536 68.0672
88.4142
122.908 122.752
132.278 83.7184 95.5073
108.467 112.318 106.289 135.279 106.765 108.077 110.530
105.940 109.467 109.179 105.079
T*. 1
464.922
405.826 402.484
527.717 571.697 530.684 453.659 415.085
432.174 628.365
440.043 465.700 568.512 551.255 399.994 586.729 491.797
No. of Dat
AAE,% Points T,"k
0.5197 39 323423
1.0866 1.0729
1.1879
1.4617 0.1859 0.9398 0.7855 0.7293
4.3400 2.9766 0.5286
0.2348 0.2155 1.5115 0.3169
0.4617
40
41
22
43 24 38
16
41 25
42 22
24 24 11
24 28
298373 298373
258398
303423
298353
353448 348373 223298 311361 288423 298373 298358
298358 303323 298358
293313
a Range
P,bar
08000
454500 464500
14000 14000 0.981961
5.7316 6.75316
12028 1.08690 12500
13104 01000 01000 1.015096
01000 11000
Ref.
90
O\
35
106
107 108 109 110 111 80
112 113
114 47 47 85
47 115
L I
V* cc/mol T* OK
Table 51 (Continued)
Substance
Benzene
Aniline
Cyclohexane
Methylcyclopentane 2,2Dimethylbutane
V*, cc/mol
89.2371 87.6248 87.6609 89.8287 85.1985 85.5564 85.5176 96.8373
C*
38.0877
47.4353
47.3037 35.6040
*57.4441 54.2541 54.9147 48.2882
*34. 9788 34.2241
*44.5619 52.4305
*43.0528 54.0977 48.5859
46.4833 24.0891
No. of Points
24 33 32 18
43 30
24 24
Data Range T,_k P
298358 0
303433 1
303433 1
293313 1
298348 1
298373 1
288313 298358 0
T*, k
497.400 487.163 487.028 502.920 473.577 453.078 457.325 576.030
536.969 513.297 539.727
493.310 558.646 501.965 514.633 507.509 480.776
,bar
1000
4544
4000
1289
1051
3914 1350
1000
AAE, %
0.3839 0.9113
0.9445 0.1804 0.5561 0.9287
0.1244 0.2326
1.0726 0.1873 0.2195
0.9742 1.6164 2.7124 0.3733 1.7919 2.3827
Ref.
47 35
106 116 117
114 118
47
119 109
120 117
121 122 118 50
123
ON
U,
49
9
8
35
42 33
24 41
69
344511 313353 298348 298348 313383
287338 288313 298348 373473
53.5677 0.981177
11000 11053
12140 11100 1350
11962
10.1304
110.347 111.266 107.881
105.450 108.490 105.252 106.335 110.127 135.192
Table 51 (Continued)
Substance
V*, cc/mol
T*. k
AAE %
No. of
Points T k Pbr Ref.
Data Range
nHexane
2Methylpentane 3Methylpentane Isopropyl ether Toluene
Anisole
Cycloheptane Methycyclohexane 3Ethylpentane
*23.1691
71.9438 60.8798 53.0649
74.4525
*67.0976 28.6649 32.2747 73.4113
59.2518 45.5918 53.9271
49.6542 73.9784 58.9151
136.583
119.539 122.219 123.896
119.114 120.896 132.906
130.184 128.570
104.145 107.629 112.126 123.283
122.249 139.668
498.619 424.813 453.699
423.590 423.637
449.005 480.616 488.963 367.933
479.754 574.807 535.928 533.605 478.370
489.463
1.7140 0.5782
0.4880 0.2909
0.8048 1.7835 1.2660 2.3598
2.1644 1.1031
0.4805 0.5358 0.6745 1.5786 0.3305
43 44 24 24 31 18 37
47 14 43 30
24 16 30
373498 273333 273333 298353 298373 298348 373473
348473 303323 223298 293313 298353
298353 203298
5.67316 1.015066
05018 0.981961
15640 15000 5.66316 5.67316 1.015113
12028
12000 0.981961 0.981961
15000
24 298353 0.981961
124 125
64
109 126
120 124 110 85 80
116 109 109 127 109
Ref
Table 51 (Continued)
Substance
V* cc/mnol
T*.0
No. of Da AAE % Points T 0Lk
ta Range
P bar
_P hlr
nHeptane 84.1373
49.8654
64.3441
49.8654 50.8954 52.4577
2,2,3Trimethyl
butane
oXylene nOctane
Hexamethylethane 2,2,4Trimethylpentane
45.5561
33.4430 43.4517 87.9996 57.2025 82.9251 50.6789
32.7867 91.9651
135.399
144.694 140.665
144.694 144.512 143.682
140.797 128.981 163.870
154.689 162.127 155.088 158.191
171.242 153.133
433.293 521.251
462.152 521.251
499.512 493.186
524.835 656.224 518.991 462.599 508.535 462.530 532.879
528.894 473.690
0.5074 1.8289 1.5689 0.9351
0.2749 1.8294
0.4893 0.4682 3.9254 0.5436
0.6614 1.5882 2.5329
0.6635 0.8582
44 72
34
40 24 45
15 19 56 39
28 28 38
46 47
273333 311511 273393 303473 298353
298348
298353 293313 373523
273333 303393
298348 393543
373523 298373
1.015066
60691
01177 05000 0.981961
11962
0.981569
12000
5.07304 1.015066
01177 14795
10.1304
5.07304
15389
125
128 129 130 109 50
109 116 123 125 129 131 132
123 133
I
R f
Table 51 (Continued)
Substance
Octamethylcyclo
V* ,cc/mol
T*. Ok
No. of AAE.% Points
Data Range
T 0k
P *bar
tetrasiloxane
3,3Diethylpentane nNonane
transDecalin nDecane
nDecanol Isodecanol nUndecane Crown ether nDodecane
143.728
68.3054 70.7412 56.6153
70.7588
64.4222 95.9055
102.460 42.0686 51.9110 66.5577 113.901
231.541 95.0230 101.328 91.4995 106.101
295.684 171.636 178.153 181.817 177.199 165.611
190.643 189.488 216.366 208.538 217.978
242.730 207.772 230.069 227.886 230.699
226.146
489.515 625.689
547.242 587.459 536.600 588.268
498.819 532.570 680.358 786.533 615.740 767.519
488.844 583.176 575.898 607.689
542.992
0.8358 1.1587 0.3318 0.8950 0.6852 0.7836 1.0601
1.9948 3.0555 1.8039
1.1974 1.0210 1.9995
0.3470 1.5527 3.0083 1.0108
28 31
28
40 34 24 19 32 26
44 48 32 78
24 32 96
34
313373
303423 303393 303523
303423 298353
298373 298358 298353
298353 303523 373523
311408
303393 298358
298358 298373
1.011662 81
08000 43 01177 129 05000 130 06000 43 0.981961 109
14202 131 1.015171 65
01373 134 01961 134 05000 130 1421547 90 1.016891 135
01177 129 1.014137 65 1.014169 65
14419 131
Ref.
a'
Table 51 (Continued)
Substance
V*.cc/mol
T*. Ok
No. of AAE.% Points
Data Range T"k Pbr
4, 4Dipropylheptane nTridecane 1,1Dephenylethane
nTetradecane 1Methoxypheny11Phenylethane
1Methoxycyclohexyl1Cyclohexylethane nPentadecane nHexadecane
90.2694
73.7403 100.605 119.758
92.2505
99.1289 179.006
135.480 134.131 105.988 120.028
243.422 255.137 189.535
261.484
217.670
255.878 269.053 297.194 295.995 304.393 299.559
637.160 654.379
627.060 563.229
0.6208 1.3647
0.3984 2.3681
669.302 0.7435
633.814 554.691 595.105 567.607 617.277 602.368
0.6137 1.4937
0.4203 1.2604 3.1369 0.5354
30 303373 48 303523
30 311372 117 298358
24 298353
24 69
24 26 73 23
298353
311408 303393 298358 318358 298373
05000 05000
1.013400 1.013668
0.981961
0.981961 1.016546
01177 1.012757 1.012902
14505
Ref
43
130 136 65
109
ON I'D
109 135 129 65 65
126
AAE % Points T Ok P bar Pef
Table 51 (Continued)
V*,cc/mol
Substance
5,5Dibutyl
T*,0k
No. of Data Range
AAE,% Points T,Ok P,bar
nonane
nHeptadecane
1,2,3,4,5,6,7,8,
13,14,15,16Dodecahydrochrysene Perhydrochrysene nOctadecane 1,1Diphenylheptane
7,nHexyltridecane nEicosane Bixylyltoluene
123.093 88.1203
120.280
310.847 329.801 318.556
657.741 706.134
627.446
125.191 244.067 751.127
109.013 187.659
270.042 323.130
783.426 588.922
123.163 278.667 683.275
136.211 98.0865
144.750
359.715 386.335 296.618
625.887 737.725 612.170
1.1984 1.2868 0.6331
1.4455 1.4653 0.9911
31
48 29
303433 323573 333453
4 311408
54 53
311408 333408
1.3626 41 311408
0.8747 1.3859
0.1874
40 40 24
311408 373573 298353
08000 05000 08000
1.013400 1.013400 1.015513
1.013400 1.013400
05000
0.981961
Ref.
43
130
43
4
136 136 135 136 136 130 109
V* cc/mol T* Ok
Table 51 (Continued)
Substance
Bicyclohexyltoluene 16
2,4Bis(aphenylethyl)phenylmethylether 11 2,4Bisatcyclohexylethyl)cyclohexylmethylether 11
1phenyl3(2phenylethyl)
hendecane 11
1aNaphtaylpentadecane 26
9(2phenylethyl) heptadecane 23
1,7Dicyclopentyl4
(3cyclopentylpropyl) heptane 20
V*,cc/mol
0.457 352.058
2.682
326.812
6.082 378.016 7.564 398.589 6.794 372.792 3.172 408.388
T*, k
No. of AAE,% Points
Data Range T,Ok P,bar
606.071 0.3060 24 298353 0.981961 728.355 0.3975 24 298353 0.981961 683.115 0.5837 24 298353 0.981961
739.649 1.0506 135
311408 1.018958
615.362 1.0329 48 333408 1.015513
618.091 1.1503 132
311408 1.019647
0.999 403.698 660.418 1.1609 119
Ref.
109 109
109 135
135 135
311408 1.0110337 135
Table 51 (Continued)
Substance
1cyclohexyl3(2cyclohexylethyl) hendecane
I*. ce/mol
179.644 422.795
T*.O
No. of Data Range
AAE,% Points T,Ok P,bar
662.385 1.0291 109 311408
Ref.
1.018958 135
1cyclopentyl4(3cyclopentylpropyl)
dodecane 233.201
1aDecaly1pentadecane 221.510
9(2Cyclohexylethyl)
heptadecane 227.263
9(3Cyclopentylpropyl)
heptadecane 222.792
9nOctylheptadecane 228.384
413.155
414.182 428.837
430.734 444.701
596.681 1.6144 155 311408 1.0110337 135
rN)
674.410 638.810 619.719 599.505
0.7107 0.9820
1.0514 1.2509
54
133
145 146
333408 311408 311408 311408
1.015857 135 1.0110337 135 1.0110337 135 1.0110337 135
_ I
,* ,cm l *
Table 51 (Continued)
Substance
1*. cc/mnl
T*. I
No. of Da
AAE % Pninft T OL,
ta Range
P b
_ hou. T* F
1,1Diphenyltetradecane Squalene Squalane nTriacontane 1, 1Di (decalyl)hendecane
11nDecylheneicosane
13nDodecylhexa
cosane
183.555
258.312
223.451 140.659
398.592
491.140 541.486 570.480
276.663 474.895 275.270 539584 285.514 664.015
688.199 559.716 594.137 783.517
2.0528
0.6349 0.6172 1.6095
676.267 1.5014 613.412 0.9488
637798
0.8561
31
24 24 40
41
333408 298353 298353 373573
311408
33 311408
1.013400 0.981961 0.981961
05000
1.013000
1.013400
35 311408 1.013400
nTetracontane 203.694 745.195
136 109 109 130
A4
136 136
136
R f
780.353 2.1338 32 423573
05000 130
Table 51 (Continued)
Substance
C* V*.cc/mol
T*. k
No. of Data Range
AAE.% Points T."k P.bar
Water
Ammonia Mercury Krypton
Nitrogen
Oxgen Sulphur Dioxide Xenon
15.6558
6.80856 19.7417 15.8098 7.49609 12.2033 27.0455 15.6019 16.3006 16.7950 17.3745
20.3573
9.04891 16.3046
17.9402 32.0515
17.3482 17.8588 20.1175 26.0062 17.5290
35.4823 35.1179
34.7248 34.4744 27.6646
48.4053 45.4322
326.221
604.809 427.493 318.117
424.182 380.914 848.180 207.290 199.391
124.803 124.388 147.217
444.752 279.633
0.8070 7.5498
2.4946 1.6205
0.9470 1.7643
0.4784
0.4752 0.4546 0.8033 0.8252 0.4277 2.0502 0.3132
24 18
40 58
42 41 56
34 60 29 32 17 37 36
298358 303343 273373 283348 348573 253313
303423 120200 129147 77122
110120 90118
323423 170260
01000 01720
01000 12077
0.3910000 1.971797 08000
5.073040
10.41147 8.64519.4
191377 7.92504.3 8.6318.5 5.072533
Ref.
66
105 137 91
45
138 139 56 76 58 88 58
140 57
p.
, ,
Table 52
Characteristic Parameters and Comparisons with Molten Salt Compression Data
Substance
Potassium
chloride Potassium
nitrate RUbidium
nitrate
Silver nitrate Tetrapropylphosphonium tetrafluoroborate Tetrapropylarsonium tetrafluoroborate
Tetrabutylammonium
tetrafluoroborate
C* V*.cc/mol T*.Ok
No. of AAE.% Points
16.9363 20.5829 1526.36 1.0932
14.1174 63.7239 1208.28
76.0519 63. 3071
57.0353
44.5619
666.713
942.390
2.0503 1.0963
0.5432
154.376 285.130 789.574 0.7171 545.175 243.825 567.151 1.8690
Data Range T. 0k
43 1073.151323.15 20 673.15773.15
15
20
673.15773.15 573.15773.15
16 520.94530.80 24 503.55512.75
AA ont kP bahr RP f
16000 09119.25 09119.25
09119.25
141 142 142 142
1.011725.80 143
1.012 503.55
143
165.853 346.826 947.901 0.3511 39 420.15480.15
4
P bar
Ref
101000 144
Table 52 (Continued)
Substance
Tetrapentylammonium tetrafluoroborate
Tetrahexylammonium
tetrafluoroborate
Tetraheptylammonium tetrafluoroborate
Tetraethylammonium tetrapropylborate
C* V* cc/mol T*.Ok
No. of AAE.% Points
161.310 423.485 924.715 0.5654 231.879 484.140 808.913 0.1484 239.349 561.309 886.090 0.3640
T.k
56 402.88494.51 42 394.15473.15 54 407.15503.15
Data Range
P bahr
102000 102000
102000
86.3565 201.036 733.253 0.1562 39 363.15403.15
,R , , , , f
144
144
144
Ref
04200 48
Table 52 (Continued)
Substance
Tetrapropylammonium tetraethylborate
Tetrabutylammonium tetrabutyl
borate
C* V* cc/mol T* Ok
No. of AAE 9. Pnints
66.3607 205.720 849.931 0.4424
Data Range
T i
16 393.15433.15
P bar
01400
48
128.281 303.900 757.227 0.1661 20 393.15433.15
,A % P i T 0L, ,_ _
R f
05000 48
Table 53
Characteristic Parameters and Comparisons with Polymeric Liquid
Compression Data
Substance C*
Poly(tetrafluoethylene) 150.116 Poly(vinyl
chloride) 64.6474 Low Density
Polyethylene 106.473 High Density
Polyethylene 394.030 Branched
Polyethylene 356.322 Linear
Polyethylene 550.584 High Molecular Weight Linear Polyethylene 773.266
V*, cc/mol
5.09707 0.85999 1.43576 1.27585 1.28835 1.21975
T*,v k
669.489 1896.22 1289.71
744.323 798.313 685.795
AAE, %
1.9100
0.4133 0.9439 1.0049 0.9673
0.8014
No. of
Points
19 17 38 32 78 67
Data Range
T, 0k
615.95645.55 355.15370.15 423.55476.15
402.95447.65 398.25471.15 415.25472.85
P,bar
0392.27
0.981373.91 0.981962.31 0.981962.31
12000 12000
617.496 0.7953 60 409.85472.65
Ref.
145 146 146 146 147 147
j
1.18514
12000 147
Table 53 (Continued)
Substance
Silicone Oil
Bayer M 100 Silicone Oil
Bayer M 1000 cis1,4Polybutadiene Poly(methyl
methacrylate) Poly(4methylpentene1) Polystyrene
Poly(nbutyl methacrylate) Poly(orthomethyl styrene)
C* V*,cc/mol
312.579 396.912 628.687 232.062 379.695 222.252 273.239 288.719 215.627
Poly(cyclohexyl methacrylate) 131.047
1.05217
1.02254 1.15168 0.94655 1.27596 1.07308 1.05128 1.05218
1.09430
T*,ok AAE, %
555.488 0.3991 522.659 0.3299 538.466 0.9608 962.326 0.3804 787.879 1.3781 1054.65 1.7755 955.103 1.5730 817.028 0.8120 1207.95 1.2974
No. of
Points , O P .h
Data Range
24 298.15353.15
24
128 41
126 73 69
167 50
298.15353.15 277.15317.65 386.65432.15 513.65592.05 388.55522.05 388.55468.75 307.05472.65 412.55470.85
0.981961.33 0.981961.33
1.012786.44 12000 01961.33
0.981766.18
12000 12000 11800
1.06464 1173.20 0.8374 89 395.85472.05
109 109
148 147 145 146 149 147 149
P bar
R f
12000 147
Table 54
Characteristic Parameters and Comparisons with Deuteriated Liquid Compression Data
Substance
V* cc/mol
T*, 0k
No. of AAE,% Points
Data Range Tak P.bar
Chloroformd Methanold Acetonitriled3
Fluorobenzened5
Benzened6
Cyclohexaned12 Toluned8
61.0950 18.6288 19.6590 50.8996
36.5382
44.5191
53.2186 63.7417 37.8479
70.6727
74.8436 38.8181
52.0451 97.3255
101.294 88.5889 85.5476 84.0675 109.170 102.623
Ref.
406.254 414.399 449.167 472.832 569.539
472.057
466.009 448.796 574.821 486.850
0.7269 0.9990
0.8564 0.9066
1.0932 1.3727 1.2636 0.2365 0.7602
1.5641
17
38
15 11
33
18
34
24 24 38
303363
243323
303363
303373
303423 303393
324374
288313 288313 238473
304500
154905 304000 12000
603500
504000 14032 1350 1350
14000
83
150 93
151 152 153
114 118 118
154
81
fact, they all appear to be within experimental error. This proves the generality and flexibility of the present correlation. The characteristic parameters for each substance evaluated from the liquid compression data with
those from the liquid DCFI data are very close to each other except for the data of Grindley.43 This discrepancy is probably due to the experimental uncertainty of liquid
DCFI at low pressure where Grindley's mathematical treatment of the compression data is questionable. The general similarity of the characteristic parameters for each substance indicates the thermodynamic consistency of the present approach. Both the compressibility Equation (41) and the pressure Equation (51) with the same parameters reproduce the crossover behavior of the volumetric properties of compressed liquids discussed in Chapter 3.
The characteristic parameters are optimized only in the tabulated data ranges. Thus computations performed outside the data ranges for some substances may be less accurate. This is especially true for those highly polar and associating species where the characteristic parameters are evaluated from a very narrow data ranges or from questionable data sets such as the case of Schornack and Eckert.85 The characteristic parameters are evaluated from two isotherms with a temperature span of 201K and the experimental accuracy in density of 0.3% only. Though the fitted results are within the experimental errors (less
than 3% in pressure), the characteristic parameters are
82
probably not as reliable as those from the more accurate experimental results. Even so, extrapolations from the present correlation should be more reliable than from other correlations. This is particularly important in physical properties correlation since experimental data covering the whole state conditions are not always available.
There are a number of substances where several sets of characteristic parameters are tabulated together because several sets of measurements exist. For a substance with similar parameter sets, in most cases, any one set can be used as the characteristic parameter set. This is the case for most of the normal liquids where the characteristic parameters are found over a wide range of state conditions. For substances with different parameter sets, the recommended values are characterized with an asterisk. Variations of the characteristic parameters are due mainly to the precision of the experimental results and the insensitivity of the DCFI and pressure changes to parameter variations. It may be that some sets are not equivalent because of limited ranges of measurement as for some polar and associating substances. The recommended characteristic parameters are based on the data range, experimental precision, and accuracy of the measurements. There is a preference in this work for the data of some authors. Generally, they are liquefied noble gases and others from Streett and his coworkers,56,57 liquefied natural and petroleum gases from the National
83
Bureau of Standards (NBS),6063 hydrocarbons from Dymond et al.,126,131,133 long chain hydrocarbons from Doolittle,130 coal liquids (PSU compounds) from Cutler, et al.135 and Lewitz et al.,136 benzene and its derivatives from Gibson and Loeffler,47,66 cyclic compounds from Kuss and Taslimi,109 polar liquids from Kumagai and his coworkers,78,84,138 alcoholic liquids from Makita and his coworkers,91'97 liquid acids from Karpela,95 molten salts from Barton and Speedy,144 polymer melts from Simha and his coworkers,147'149 and deuteriated liquids from Jonas and his coworkers83,93,150154
Group Contribution Method
In addition to the molecular corresponding states
correlations of the above type, group contribution methods
are very useful for physical property estimation. The concept assumes that each molecule is made up of different types of atomic or structural groups and that the molecular properties can be evaluated from contributions of these atomic or structural groups. Unlike the corresponding states correlation, the fundamental assumption is that various groups in a molecule contribute a definite value to the total properties independent of the presence of the other groups on the molecule.155
There are two general approaches in group contribution method. The first one may be called the propertyadditive approach. It assumes that the groups actually
84
posses thermodynamic properties or make directly additive contributions to the molecular properties.156 The molecular properties then can be calculated from the sum of these group properties.
M = i M (517)
where via is the stoichiometric coefficient, represents the number of groups or type a in molecule i. This approach has been applied to the liquid molar volume at the normal boiling pointl57 and to the liquid heat capacity,158 as well as to the solution of group methods for activity coefficient.159,160 This group contribution method has also been widely used for ideal gas properties such as heat capacity, entropy, and enthalpy of formations155 etc., where intra or intermolecular forces play no role. For the
DCFI, Equation (517) would yield
C = Z v C (518)
1 a
The other general group contribution method may be called the parameteradditive approach. It assumes that the molecular property can be written as a function of the state variables, x, and a set of parameters, G.
M = f(x;e)
(519)
85
where
EI = Z via Ga (520)
This approach has been applied to the estimation of the critical properties by Lydersen.155
Evaluation of the molecular parameters from Tables
51 to 54 shows that these parameters appear to be group additive, according to Equation (520). Thus, the DCFI
can be written as
C. = Ctf(pV*,T*/T) (521)
1 1 1
and
V* = E v. V* (522)
1 la a
a
Ct = Z C* (523)
1 la a
Table 55 lists the group parameters that can be added for each group to estimate the crossover parameters for some of the substances. By adopting T*=0.96Tc, comparisons of the group contribution correlation with liquid nalkane compression data are listed in Table 56. The results are satisfactory.
86
Table 55
Group Contributions to Characteristic Parameters C* and V*
Group (Attachment) C*(+0.5) V*, cm3/mol(+0.05)
CH3 (Linear paraffin) 12.6 26.83
CH3 (nonlinear paraffin) 9.7 26.02
CH2 (linear paraffin) 4.3 18.35
CH 2 (nonlinear paraffin) 7.9 16.54
CH2 (ring) 5.8 18.39
CH< (nonring) 2.3 12.20
CH< (aromatic ring) 6.8 14.77
>C< 3.4 9.95
>Si< (silicon) 11.9 17.72
>Sn< (tin) 16.4 28.20
F (on nonring) 6.7 12.40
Cl (on nonring) 12.5 21.29
Br (on nonring) 9.1 28.23
COOH (on nonring) 21.2 28.64
COO (ester on nonring) 25.0 21.10
Table 56
Group Contribution Correlation and Comparisons with nAlkane Liquid Compression Data
Substance
Ethane Propane nButane nHexane
nHeptane
nOctane
T*,0k
140280
200320
240360 373498 273333 273333 298353 298373 298348 273333 311511 273393 303473 298353 298348 273333 303393 298348
Data Range
P,bar
0.6745 0.2712
0.2714 5.7316
1.015066
05018
0.981961
15640 15000
1.015066
60691
01177 05000
0.981961
11962 1.015066
01177 14795
No. of Points
64 56 32
43 44 24 24 31 18
44 72
34 40 24 45 39
28 28
AAE%
1.1396 2.8171 2.7298 8.3906 4.5357 4.7963 5.1995 7.9198 6.1217 2.2857
1.4267 7.1420 1.8006 3.6664
3.4231 6.1787 4.8369 3.5022
Ref.
60* 61* 62*
124 125
64
109 126
120 125
128 129 130* 109 50
125 129 131
co
j

Full Text 
PAGE 1
THERMODYNAMIC PROPERTIES OF COMPRESSED LIQUIDS AND LIQUID MIXTURES FROM FLUCTUATION SOLUTION THEORY BY YUNGHU I HUANG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1986
PAGE 2
To my parents and beloved sisters 11
PAGE 3
ACKNOWLEDGEMENTS I would like to express my sincere gratitude to Dr. J.P O'Connell for his interest and guidance throughout the course of this study. He provides me with not only the kind of intellectual atmosphere that I have never experienced before but also many valuable opportunities to interact with some leading researchers in the field of fluid phase equilibria. I also wish to thank Drs. G.B. Hoflund, W. Weltner , Jr. , S.O. Colgate, and G.K. Lyberatos for serving on the supervisory committee and Drs. E.A. Brignole and M.L. Michelsen for their helpful discussions. Special thanks are due to professors A. Fredenslund and P. Rasmussen for their hospitality during my stay in the Instituttet for Kemiteknik, Danmarks Tekniske H^jskole and Mr. R.P. Currier and Mr. T. Schmidt for their assistance during my stay in the K^benhavns Amts Sygehus I Gentofte. It is a pleasure to thank Mrs. Sun for her excellent typing and patience with my manuscripts and Mr. B.A. Klein and Mr. C.T. Skowlund for their help in using VAX and UNIX computing systems. iii
PAGE 4
Finally, I am grateful to the Northeast Regional Computing Center, and Department of and University of support that made Florida, for use of their facilities Energy, National Science Foundation Florida for providing the financial this work possible. IV
PAGE 5
TABLE OF CONTENTS Page ACKNOWLEDGEMENTS iii KEY TO SYMBOLS vii ABSTRACT xi CHAPTERS 1 INTRODUCTION 1 2 FLUCTUATION SOLUTION THEORY FOR COMPRESSSED LIQUIDS AND LIQUID MIXTURES 6 Introduction 6 Fluctuation Solution Theory and Direct Correlation Function Integrals .... 7 Thermodynamic Consequences 12 3 VOLUMETRIC BEHAVIOR OF COMPRESSED LIQUIDS AND LIQUID MIXTURES 13 Introduction 13 Pure Liquids 23 Liquid Mixtures 32 4 CORRESPONDING STATES CORRELATION FOR THE DIRECT CORRELATION FUNCTION INTEGRALS . 36 Introduction 36 Model Development 37 Model Parameterization 38 Results and Discussion 40 5 APPLICATION OF THE MODEL TO THE PURE LIQUIDS 51 Introduction 51 Derived Thermodynamic Properties .... 51 Pure Liquid Compression Data Bank ... 57 Results and Discussion 59 Group Contribution Method 83 v
PAGE 6
6 APPLICATION OF THE MODEL TO LIQUID MIXTURES 89 Introduction 89 Derived Thermodynamic Properties .... 90 Liquid Mixture Compression Data Bank . . 94 Results and Discussion 94 7 CONCLUSIONS AND RECOMMENDATIONS .... 102 APPENDICES A PERCUSYEVICK HARD SPHERE DIRECT CORRELATION FUNCTION INTEGRALS FROM VERLETWEIS ALGORITHM 106 B THERMODYNAMIC PROPERTIES OF COMPRESSED LIQUID MIXTURES FROM DCFI MODEL .... 108 C COMPUTER PROGRAMS 117 REFERENCES 149 BIOGRAPHICAL SKETCH 159 v 1
PAGE 7
KEY TO SYMBOLS A = Helmholtz free energy = ith order reduced density coefficient of DCFI model a^j = coefficient matrix of DCEI model b = constant C = direct correlation function integral c = molecular direct correlation function Cp = heat capacity at constant pressure d = effective hard sphere diameter f = state dependent function g = pair distribution function, state dependent function H = enthalpy, spatial integrals of total correlation function h = total correlation function _I = identity matrix k = Boltzmann's constant kÂ• = binary parameter M = general property m = integer N = number of moles, number of components n = integer, refractive index P = pressure R = gas constant Vll
PAGE 8
r = position vector 5 = entropy s = position vector T = temperature U = internal energy u = velocity of sound, intermolecular potential V = volume v = V/N, molar volume v^ = partial molar volume of component i X = matrix of mole fraction x = mole fraction, position Y = density dependent function of hard sphere diameter Oip = isobaric expansivity = activity coefficient of component i y v = thermal pressure coefficient 6 = Kronecker delta Â£ = parameter set 0^ = parameter of molecule i 0 = e/kT e = energy parameter, dielectric constant r = packing fraction Hy = isothermal compressibility p = chemical potential v = Stochiometric coefficient p = density a = collision diameter viii
PAGE 9
T = inverse of reduced temperature cp^ = volume fraction of component i ft = orientation normalization factor oo = angular orientation coordinate Subscripts c = critical property i,j,k = component ijÂ»jkÂ»ik = pairs of components m = mixture o = reference property, saturation property hs = hard sphere Superscripts E = excess property r = reference state * = characteristic parameter o = standard state ' = first composition derivative " = second composition derivative (as in X) (as in v ) ~ (as in p) o (as in v ? ) 1 Special Symbols Matrix quantity Partial molar property reduced property pure component property ix
PAGE 10
ensemble average angular orientation average
PAGE 11
Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THERMODYNAMIC PROPERTIES OF COMPRESSED LIQUIDS AND LIQUID MIXTURES FROM FLUCTUATION SOLUTION THEORY By YungHui Huang August 1986 Chairman: John P. O'Connell Major Department: Chemical Engineering For every pure liquid there exists a characteristic volume for which the reduced bulk modulus, v/hjRT= (3 (P/RT )/9 p ) j , is independent of temperature. In addition to the possible physical significance of this effect, the crossover point serves as the basis for a correlation of the spatial integrals of the statistical mechanical direct correlation function (DCFI) which can be related to the density derivatives of thermodynamic properties in the fluctuation solution theory. The pressure variation of the volume and reduced bulk modulus of essentially all pure and mixed liquids are correlated with a threeparameter corresponding states correlation over temperatures from the triple to nearly the critical point and densities from the saturation to xi
PAGE 12
the freezing point. The parameterization is based on the detailed analysis of the most accurate experimental data. The three parameters each serve a particular mathematical purpose, ensuring flexible and tractable numerical calculations. The correlation method has been applied to over 200 pure and mixed liquids with errors probably within those of the experiments, making it significantly more general and accurate than previous models. The systems investigated cover all orders of complexity of molecular structure and intermolecular interaction. These include liquefied noble gases, shortand longchain hydrocarbons, small and large globular chain or ring compounds, polar and highly associating species, metallic liquids, molten salts, and polymer melts. Mixtures of simple and complex systems are carefully examined. The results are relatively insensitive to the characteristic temperature and all parameters may be estimated by a group contribution method. Simple mixing rules work successfully in a onefluid form for mixtures with small excess volume. In other cases, a single binary parameter is adequate for good agreement. These findings indicate that the direct correlation function integrals are less sensitive to the exact details of the intermolecular forces for dense liquids than other statistical mechanical quantities. xu
PAGE 13
This correlation, along with accurate saturation values, will provide all thermodynamic properties of compressed liquids and liquid mixtures.
PAGE 14
CHAPTER 1 INTRODUCTION High pressure liquids and liquid mixtures are present in many natural and manmade situations. ^ They vary from geological systems to high pressure processes and techniques. Thermodynamic properties of compressed liquids and their mixtures are important both in basic scientific research and practical engineering applications. Thus the theoretical study of the volumetric behavior of compressed liquids is motivated by the concept that the structure of dense liquids is dominated by the shortrange repulsive forces and that their physical pro2 perties can be treated in terms of a hard sphere model. From the practical point of view, liquid volumes are used exclusively in equipment sizing and descriptions of occupancy in natural reservoirs and formations. Since the pioneering work of Bridgman,*^ there have been many volumetric measurements of compressed pure liquids, whereas much less effort has been dedicated to compressed liquid mixtures. A recent survey^ of PVT properties of liquid and liquid mixtures listed about 350 pure components and 170 binary mixtures. However, half of the numbers cited are restricted to very limited pressure 1
PAGE 15
2 or temperature ranges, including many of atmospheric or single isotherm measurements. Compression data of pure liquids have been correlated with equations such as those 5 7 fl of Tait, Hudleston, Hayward, Chaudhuri, and others. Despite their substantial value for smoothing, interpolation, and computation of compressional thermodynamic functions, these equations share common disadvantages. First, they contain no explicit temperature dependence, so extensions by using temperaturedependent parameters are still uncertain. Second, two parameters are insufficient for wide pressure range isotherms, so extrapolations to either low or extreme high pressures are inaccurate. Third, their empirical nature renders molecular interpretation difficult. Fourth, these equations do not lead to mixing rules for compressed liquid mixtures. An acceptable equation of state, apart from quantitative success, should provide some insight into the molecular characteristics in terms of a minimum possible set of parameters. Agreement between theory and experiment not only enhances one's faith in the molecular theory but also leads to the ability to generate the volumetric properties with confidence from very limited experimental efforts. Theoretical equations of state which are based on the modification of Prigogine's cell theory^ such as Flory et al.,^ Simha and Somcynsky, and others have been applied to compressed liquids and liquid mixtures using scaling parameters evaluated from lowpressure density data.
PAGE 16
3 However, as noted by Bridgman,^ experimental behavior at low pressures is a poor guide to what is likely to happen at high pressure. As a result, these equations of state can not correctly represent the PVT behavior of compressed liquids at high pressure. Empirical modifications by Zoller,^ and Simha and his coworkers^"^ using elevated pressure data make only marginal improvement. Apparently, the theory cannot furnish an ideal representation of experimental PVT data at elevated pressure. Recently, the Tait equation has been extended and generalized by Thomson et at.^ This correlation, along with the correlation developed by Hankinson and Thomson^ for saturated liquid densities, comprises the COSTALD method. These authors claim that COSTALD is the most general and accurate compressed liquid density correlation yet published. This is true when compared with those methods commonly used in chemical process calculations such as YenWoods, ChuehPrausnitz , LeeKesler, and others. However, the correlation method is purely empirical; application is limited to nonpolar and slightly polar liquids and pressures no higher than 700 bar. A judicious alternative is to combine the simplicity of an empirical equation with the advantage of statistical mechanical theory in developing accurate and general relations for thermodynamic computations. Fluctuation solution theory is explored here for this purpose. The
PAGE 17
4 basis of the method is from the statistical mechanical grand canonical ensemble which relates composition derivatives of the pressure and the chemical potential, thus the activity coefficient, to integrals of the molecular direct correlation function. As recognized by O'Connell , 2223 anc j Gubbins and O'Connell, ^ the density dependence of the reduced bulk modulus behaves differently and quite simply in liquid phase compared to the gas phase, regardless of the complexity of the intermolecular forces. Hence, this work focuses on describing the liquid state alone . The most surprising feature of the reduced bulk modulus versus reduced density plot is that there exists a unique point of crossover of the isotherms for all of the compressed liquids studied. This finding is obviously a most important contribution to any liquid state theory. In view of the wellknown fact that it is density, not pressure, that is the key variable for compressed liquids, a corresponding states correlation, based on the density and temperature expansion of reduced bulk modulus, has been constructed to predict accurately the pressure effect on the compressional thermodynamic properties. In the chapters that follow, detailed descriptions of the theoretical background, experimental observations, model development, and successful applications, including pure and mixed liquids are presented. Chapter 2 describes the fluctuation solution theory and how useful
PAGE 18
5 thermodynamic properties are related to the statistical mechanical direct correlation function integrals. Chapter 3 displays the reduced bulk modulus for many compressed liquids, providing significant features that are further exploited. For compressed liquid mixtures, volumetric data are fully analyzed and deeper insights concerned with the excess properties are carefully evaluated. Chapter 4 is devoted to the process of model construction. It is demonstrated how the theoretical concept can be transformed into an actual working equation that simulates, to within experimental accuracy, the compressed liquid behavior. Chapter 5 applies the model to pure liquids. Also basic thermodynamic properties such as entropy, internal energy, and others are derived through the Maxwell relations. Characteristic parameters and comparison with a variety of pure liquids compression data are tabulated. An initial attempt at a group contribution correlation for the parameters is also given. Chapter 6 extends the model to liquid mixtures. Mixture properties such as partial molar volume, activity coefficient, and others are expressed in a onefluid form. Binary parameters and comparison with liquid mixture compression data are tabulated. Chapter 7 closes this work with some remarks on the numerical techniques and our present concept of compressed liquid properties .
PAGE 19
CHAPTER 2 FLUCTUATION SOLUTION THEORY FOR COMPRESSED LIQUIDS AND LIQUID MIXTURES Introduction Over the years there have been a growing number of uses of statistical mechanics in physical property research. There are two general approaches in the development of liquid state theory. One may be called the formal approach, based on distribution functions, while the other may be called the model approach, based on partition functions. The former was pioneered by Mayer and Mayer, J Kirwood, Â° and others. Thermodynamic properties can be calculated through the energy equation and the pressure equation with known pairwise additive potentials and pair distribution functions. The latter approach visualizes a liquid with a certain physical picture, such as made up of cells, holes, or molecular arrangements. Thermodynamic properties then can be obtained by transforming these physical models into a partition function. All suffer the disadvantage of assuming pairwise additivity of i nt er m ol ecul ar potentials. Another alternative may be called the fluctuation solution theory. ^ The compressibility equation which relates concentration derivatives of pressure and 6
PAGE 20
7 chemical potential to the spatial integrals of the total correlation function, and hence to direct correlation function integrals (DCFI) in the grand canonical ensemble is free from this assumption. Furthermore, the DCFI appear to be insensitive to the exact details of intermolecular forces. Therefore, from a numerical point of view, the theory involves integration of a simple model, instead of taking the derivative of a complicated expression as in the others. Experience shows that this should be less sensitive to errors of modeling. It is the number density and temperature, not the usual mole fraction, pressure, and temperatures that are the most appropriate set of independent variables for theoretical work. However, the equations allow for consistent relationships to be made among the variables. Fluctuation Solution Theory and Direct Correlation Function Integrals In the statistical mechanical grand canonical ensemble, a relation exists between composition fluctuations and derivatives of the mean composition with respect to chemical potentials. 9 'SUi/kT T, V, p k;I ( 2 1 )
PAGE 21
8 where the brackets denote a grand canonical ensemble average. These averages are also related to the spatial integrals of the radial distribution function by f 6 ij = J gij (r)dp ( 2 2 ) where is the Kronecker delta. As recognized by Kirkwood and Buff,^ g^j (r) is not necessarily limited to be the radial distribution functions for spherical molecules; it can also be interpreted as a more general pair distribution function with all possible orientations. Using the definition of total correlation function, (23) A combination of equations (21), (22), and (23) yields 9 J Sbi/kT T > V,p k? dr V (24)
PAGE 22
9 where 0) = h Â± j ( rw^co j )dco i dwj = the angle averaged total correlation function. ft = J dco = the nomalization constant of the orientation dependence. r = r j = the translational change of the position vector i and j. Equation (24) can be written as 3 Pj SUj/RT T Â’ V,y k*i = 5 i jPi P iPj 0) dr (25) This is the basic equation of the KirkwoodBuff solution theory. From Orstein and Zernike,^*the direct correlation function is defined as C i j(?i? j a3 i a) j ) = h ij (r i p j 0 ) i a) j ) Z p k JJ C ik ( ^i r k a3 i a) i < ) h k j ^ r k r j ^ dr k daj k ( 2 6 ) The second term on the righthand side represents the indirect correlations due to all possible third
PAGE 23
10 molecules. Similarly, the angle averaged version of Equation (26) is w = w Z p k w a) ds (27) where u = J J c L j ( ru^to j ) da^dio j the angle averaged direct correlation function . s = r^ r k = the translational change of position vector i and k. rs = r^ Tj = the translational change of position vector k and j . The angle averaging operation in the last term of Equation (27) may not be appropriate for longrange potentials.^ Integration of Equation (27) yields u) dr pÂ‘ Z k 0 ) w dr ds ( 2 8 )
PAGE 24
11 In matrix notation, Equation (28) can be expressed as C = H C X H ' (29) where X is a diagonal matrix whose nonzero elements are mole fractions. After basic matrix manipulations, Equation (29) can be written as I + H X = (I C X)" 1 (210) or 6 ij + Pi J d? = < 5 ij Pi < c ij (?) a , > d ?)1 ( 2 11 ) Comparing Equation (211) with Equation (25) and recognizing that the elements of the lefthand side of Equation (25) are the matrix inverse of a more desirable partial derivative, SPi/RT T,p k*j ( 2 12 ) where C^j = p j w dr = the direct correlation function integrals. This is the basic connection between thermodynamic variables and direct correlation function integrals.
PAGE 25
12 Thermodynamic Consequences From the isothermal GibbsDuhem equation, the composition derivative of the pressure and the chemical potentials are related to each other by 9P/RT T,p k*j = Z i 3y i /RT Tp~ T,p k*j (213) Substitution of Equation (212) into Equation (213) yields 9P/RT TpT,P k*j = Z i : i ( 1C i j ) ( 214a ) FT (214b) where the subscript m indicates mixture properties and xp is the isothermal compressibility. Summing all equations (214) yields 9P/RT Â“TpÂ— T, x = Z Z X^dCy.) i J J J (215a) 1 p m w Tm RT ( 215b )
PAGE 26
13 The partial molar volume V j then can be expressed in terms of density and the DCFI V . J Z x i(l c j. j ) m l Z x i x(lC i jJ i J (216) where p m is mixture molar density. An isothermal change of pressure can be obtained by integration of Equation (214) PP r RT (pp r ) Z p.C. . i ij T,P dp i*j (217) while an isothermal change of chemical potential is T,P k*j (218) where the superscript r denotes a reference property. This can be any fluid state, including ideal gas, saturation, or other welldetermined conditions. Naturally, the model for the must be appropriate over the entire range of the integration. In phase equilibrium calculations, it is usually more convenient to work with activity coefficient than chemical potential.
PAGE 27
14 Ui(TÂ»p) = UÂ°(T,p r ) + RTlnx^iCT, p) (219) where the superscript o denotes the standard state properties. Substitution of Equation (219) into Equation (212) yields T,p k*j 1C i j P ( 2 20 ) Composition integration of equation (220) yields In Z j 1C. . (Â— rM) dp For a pure component, ( 2 21 ) 3P/RT Â“5p~ T 1C (222a) 1 ( 222b ) PHjRT This is the reduced bulk modulus of pure liquids, the quantity of primary interest in the present work.
PAGE 28
CHAPTER 3 VOLUMETRIC BEHAVIOR OF COMPRESSED LIQUIDS AND LIQUID MIXTURES Introduction Theoretical investigations of the physical properties of dense liquids indicate that the structure of liquids, static or dynamic, is similar to that of a rigidbody system. In order to examine the effects of the differences between real and rigidbody systems, the temperature and density effects should be separated requiring experimental studies at high pressures. In describing the thermodynamic state of a liquid, the applied pressure is considered high if it is comparable to or greater than the kinetic pressure.^ The thermodynamic equation of state is p t ( ^ ^ ^ ) P " 1 ^TTV [ JV J T (31) 3 P where the thermal pressure coefficient , T(^y)y , is the kinetic pressure resulting from the molecular motion and the energyvolume coef f icient , (yy) y , is a measure of the internal pressure arising from the intermolecular forces. When the energyvolume coefficient (yy)j is zero, the external pressure P is equal to the kinetic pressure T( It ) V Â’ and 15
PAGE 29
16 (t)v = T (3 2 ^ This is the case for a rigidbody fluid. When the applied pressure P is close to the kinetic pressure, a perturbation approach is suggested using a rigid body reference with a small perturbation attractive potential term. Under these circumstances, the volume plays the dominant role in affecting the molecular behavior in liquids, because the molecules are so closely packed together that the forces between them arise almost exclusively from the shortrange repulsive forces. There has been much theoretical speculation concerning this physical picture of liquids. The first controversy comes from the experimental evidence that the internal energy passes through a minimum which is characteristic of the liquid and a function of temperature. 3 ^ This brings in a temperature dependent effective hardcore diameter which would be necessary to make Equation (32) behave like Equation (31). Further experimental evidences for highly compressed liquids indicate that theoretical values of the hard sphere liquid deviate systematically from the experimental values.^ This suggests that the effective hardcore diameter becomes density dependent at high packing fractions (pa^>0.93) due to the softness of the repulsive potential and/or orientational ordering in compressed liquids. Figure 31 shows the reduced bulk modulus of a hard sphere liquid calculated
PAGE 30
17 Figure 31. Theoretical DCFI of Effective Hard Sphere Liquid
PAGE 31
18 from the PercusYevick compressibility equation. The VerletWeis algorithm is adopted to simulate the temperature and density dependent effective hardcore diameters of WeeksChandlerAndersen perturbation theory. 2 Detailed calculations are summarized in Appendix A. Figure 32 shows the experimental reduced bulk modulus of liquid argon calculated from Twu's algorithm.56 Comparison of Figures 31 and 32 shows that the differences are significant, especially at high pressure where the crossover is observed for argon and the temperature dependence is very small . Since theoretical calculations of compressed liquids are improbable, the alternative is to analyze fully the experimental data available and develop an empirical relationship. There are several interesting findings concerning the volumetric behavior of compressed liquids and liquid mixtures. The earliest one is made by Bridgman on the pressure effect of isobaric expansivity.5 At low pressure, the thermal expansion coefficient of liquids increases with rising temperature. At high pressure, this relation is reversed. The reversal in the sign of the temperature derivative of isobaric expansivity is due to the nonlinear effect of the intermolecular forces. It is interesting to relate this phenomenon to the behavior of constant pressure heat capacity.5
PAGE 32
19 pv c Figure 32. Experimental DCFI of Liquid Argon
PAGE 33
20 ( 3C P, yp; T ( 33a ) = VT (a 2 da P, p + ir ) ( 3 3b ) where ap is the isobaric expansivity. The effect found by Bridgman raised the possibility of a minimum in Cp as a function of pressure. Similar observations on the change of internal energy with pressure were also made by Bridgman.^ _ y(3V\ (34) T Â’ U TT J P ^TF't u The pressure at which the lefthand side of (34) is zero is given by the same form as Equation (32). P T ( 3 ^ ) U 7T ; P TVÂ— ^TF J T = T ( 3 P FT' V ), (35) These variations were further explored by Jenner and his coworker^"^ on the compression measurement of 1bromoalkanes from CÂ£ to C j in the liquid state. The applied pressure is up to 6 Kbar in the temperature range between 203Â°K and 448Â°K. For all of the bromides
PAGE 34
21 investigated, there exists a welldetermined pressure for which the isobaric expansivity is independent of temperature. This inversion pressure is further confirmed by experimental data from the piezothermal method of Ter dioxide, and nbutane. The curves of the pressure derivatives of internal energy and of heat capacity show similar behavior as a function of pressure. In addition to the observed minimum at different temperatures, all of the isotherms pass through a single point at the inversion pressure noted above. Further, the isothermal variation of the heat capacity difference C p Cy exhibits the same crossover at the inversion pressure. In all of these cases, there must exist a thermodynamic relationship at the inversion pressure p^. First, Minassian and his coworkers^ 1 Â’ 42 on liquid water, carbon da Pi 1 / 3 2 V 2 0 (36) leading to Pi (37) Second ,
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22 T ( _afV) 3T 2 Pi (38) where by is a constant. Combination of Equation (37) and (38) yields TV (39) Third , C D ~ C T ( ^ P 'i C V T( TT ) Pi ( FT ) V = JVa Px t\i c. a pi YV (310) where b 2 is another constant. From Equation (310), the thermal pressure coefficient yy can be written as Y * V FJ Â“Pi (311) From the triple product rule, the isothermal compressibility Hy can then be given as
PAGE 36
23 h? = a Pi *1 (312) This is true only when inversion pressures are exactly the same for all of the properties investigated above. Unfortunately, data analysis on the isothermal compressibility is unavailable in Jenner's work to confirm this completely. Careful analysis of this property will be described in next section. Pure Liquids Volumetric properties of large globular and longchain hydrocarbons have been measured by Grindley^ for temperatures from 298Â°K to 453Â°K and pressures from zero to 8 Kbar. The data are adequate to determine the first and second volume derivatives of the internal energy and entropy. The first isothermal volume derivatives yield the thermodynamic equation of state. dt /3Ss /8U> p = Wi < 3 i3) Both terms on the righthand side of (313) are temperature dependent. However, scaling can be achieved by a twoparameter corresponding states relation. The second
PAGE 37
24 isothermal volume derivatives are related to the bulk modulus In contrast to the external pressure expression (313), where the entropy term dominates throughout the whole density region, the bulk modulus has a large contribution from both terms. The entropy term is larger at low densities while the energy term is larger at high densities. This can be seen clearly from Figure 33 for the second isothermal volume derivative properties of nnonane. The crossing of the bulk modulus isotherms is due to the additive effect of both entropy and energy term, though crossing of the energy term also appears at higher densities . The thermodynamic implications of this behavior have been analyzed further in this work. Our examination of the experimental reduced bulk modulus data available reveals a similar feature: A density at which the reduced bulk modulus, which is related to the integral of the direct correlation function of Chapter 2, is independent of temperature. Figure 34 shows the experimental behavior of the DCFI for methane from Mollerup's computer programs.^ At the critical point, the DCFI is unity because the reduced bulk modulus is zero. At the ideal gas limit, the DCFI is zero, and hence the reduced bulk (314)
PAGE 38
MOLES /cm 25 Figure 33. Entropy and Energy Contributions to the Bulk Modulus of nnonane
PAGE 39
26 PV C Figure 34. Experimental DCFI of Liquid Methane
PAGE 40
27 modulus is unity. Generally, the DCFI is small and positive (less than unity) in the vapor phase; it is large and negative in the liquid phase. The high density limits of the isotherms are the freezing line, while the low density limits of the liquid isotherms are the saturated liquid. Figures 35 and 36 show the DCFI of two sets of isomers: nnonane, 3 , 3diethylpentane, and nheptadecane, 5,5dibutylnonane in the liquid state alone. All the experimental curves behave similarly with relative insensitivity to the temperature in the higher density range, and the crossover is apparent. By shifting the curves to superimpose the crossover DCFI and density, complete overlap can be obtained by rotating the graphs slightly to bring the isotherms into coincidence. This provides the basis for a threeparameter corresponding state correlation of the reduced bulk modulus. Similar behavior has been found for essentially every other substance studied except water. Figure 37 shows the DCFI of water generated from the computer programs of the National Bureau of Standards. ^ The temperatures are from boiling to the critical point and pressures up to 10 Kbars. Except for the low temperature isotherms (below 50Â°C), water exhibits the same volumetric behavior as the other substances over a very broad range of state conditions. The differences at low temperature can be attributed to hydrogen bonding which decreases rapidly with temperature, a conclusion which has been reached on evidence from diverse sources.^
PAGE 41
28 Figure 35. pv c Experimental DCFI of Liquid nnonane and nheptadecane
PAGE 42
29 pv c Figure 36. Experimental DCFI of 3 ,3diethylpentane and 5,5dibutylnonane
PAGE 43
30
PAGE 44
31 Volume and temperature have been chosen as the independent variables since they are the most suitable for theoretical treatment. In general, volume scaling is the result of an increase in molecular size, while temperature scaling is due to the ratio of potential to kinetic energy effects and variation in rotational degrees of freedom. The relative insensitivity of the DCFI with temperature in the compressed state indicates that internal degrees of freedom are not appreciably altered when the density changes above twice the critical density. The crossover reduced bulk modulus variation which leads to DCFI scaling arises probably from the combined effects of size and inter molecular forces because of the additive effects of entropy and energy contributions to the crossing of the liquid bulk modulus. To better understand this conjecture, two separate studies from Gibson and Loeffler 47 and Grindley and Lind 4 Â® can be considered. For the same sized molecules, if a polar group is introduced as in the former case, or upon charging a set of neutral molecules to form a molten salt as in the latter case, the major contribution to the decreased pressure and isothermal compressibility (or increased DCFI) comes from an increased (Â£iL) 9 V T g 2 u which makes ( Â— Â„ Â— more positive. Only a small contribu3 \l 2 tion comes from a decrease of (4)_ and () 3V T T
PAGE 45
32 Liquid Mixtures Insights into the thermodynamic properties of liquid mixtures can be obtained from the excess volume as a function of pressure, temperature and composition. An where V m is the mixture volume and VÂ° is the volume of pure component i. The absolute value of excess volume is small in comparison to the mixture volume itself, whereas the variations of the excess volume with pressure, temperature and composition can be important and comparable to those of the mixture volume. equation of state based on the excess volume can be written as (315) TP~ T , x_ V mx*r E (316) m P ,x = (317) and for binary mixtures, ( 318a ) ( 3 Â— 1 8 b )
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33 ( 319a ) ( 31 9b ) Since very few measurements have been made to determine the equation of state for liquid mixtures, significant progress in their high pressure thermodynamics has not occurred. However, it is known that the experimental behavior of the excess volume as a function of temperature and composition is exceedingly complicated due to a variety of physical effects and intermolecular interactions. One behavior of interest is the pressure dependence of the excess volume of liquid mixtures. Figure 38 shows the excess volumes of the carbon monoxidemethane system from Calado et al.^ and the nheptaneethanol system from Ozawa et al.^ The absolute value of high pressure mixtures for a wide variety of liquids show more ideal volumes of mixing than do low pressure systems. Generally, the pressure influence is very significant. For simple liquid mixtures, the excess volume is small and negative at low pressure, becoming almost zero at high pressures. For nalkane mixtures, the excess volume is negative at low pressure, but it eventually g y u (^p) t x rapidly decreases with pressure, implying that
PAGE 47
34 nheptaneethanol (Ozawa, et al.) Pressure, MPA CO/CH^ at x co =0.5 (Calado, et al.) Figure 38. State Dependence of Excess Volume
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35 becomes small and positive at high pressures. For nalkane and 1alkanol mixtures, the excess volume is positive at low pressures becoming negative at high pressures. For aqueous systems, the excess volume is large and negative at low pressures, becoming small and positive at high pressures. These are all closely related to the packing effect at highly compressed states.
PAGE 49
CHAPTER 4 CORRESPONDING STATES CORRELATION FOR THE DIRECT CORRELATION FUNCTION INTEGRALS Introduction The practical value of the rigorous formulas derived in Chapter 2 and experimental observations discussed in Chapter 3 depends on a mathematical function to simulate the DCF I behavior. The ideal model for DCFI would yield a robust mathematical form, giving all of the variations nature shows with a minimum of experimentally accessible parameters that depend only weakly on thermodynamic states. Treatments using the corresponding states principle (CSP) have this desirable feature. However, most corresponding states correlations are based on the critical properties and other macroscopic fluid measurements such as saturation properties, so they are limited to classes of substances that have simple intermolecular forces. Furthermore, these properties are not always measurable as in the case of heavy hydrocarbons and coal liquids. However, a corresponding states correlation for the DCFI would be very useful in chemical process calculations if suitable parameters are found. In this chapter, a threeparameter corresponding states correlation has been developed to describe the DCFI of all liquids. The 36
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37 three characteristic parameters of the correlation are the volume, V*, and DCFI, C*, at the crossover of the isotherms along with a temperature, T*, which is in the range of the critical temperature. Model Development The first apparent utility of the DCFI to describe the relationship of pressure and density for liquids was described by Brelvi 51 and Brelvi and O'Connell. 5253 They noted that density, not temperature, was the dominant variable and the DCFI of many substances could be correlated in a oneparameter corresponding states form. In principle, their characteristic volume parameter could be determined from a single compression measurement. This correlation holds within 510% for a variety of substances over the dense liquid region. However, the correlation is uncertain at lower densities where the experimental data are sparse and temperature effect becomes significant. Further examination by Mathias and O'Connell 5455 showed that the temperature dependence could be taken into account with a characteristic temperature in a twoparameter corresponding states form. Since temperature dependence was included, the results at low densities were improved. Unfortunately, the correlation used hard sphere expressions corrected by a linear density term so applications to high pressure were not of high accuracy.
PAGE 51
38 In the present work, liquid compression data have been analyzed even more carefully and the most general form of the corresponding states relation is shown in Figure 41. The behavior of the liquid DCFI for methane is plotted in reduced coordinates along with some extensions using data for nheptadecane. At the crossover point, the reduced density and DCFI are defined to be unity. While variations away from this characteristic density and DCFI are somewhat dependent on the temperature, they can be correlated in reduced form. Model Parameterization The present correlation takes advantage of the similarity of the crossover behavior that all dense liquid DCFI data show. The success of the model relies on the construction of an equation of state that can describe the reference substance accurately. Since this work has been concerned with formulating the thermodynamic properties of petrochemicals and coalderived liquids, methane is used as the reference substance. The basic form of the correlation is chosen to express the reduced DCFI, Â£=C/C*, as the polynomial expansions of the reduced density, p=pV*, and the inverse reduced temperature, x=T*/T. m n C = Z Z a A , (T)J _1 (p) i_1 i=l j=l J (41)
PAGE 52
39 p/p* Figure 41. Corresponding States Correlation of Liquid DCFI
PAGE 53
40 where a^j is the coefficient matrix of the reduced variables . After a thorough test using the liquid DCFI data of methane generated from Mollerup's computer program, the optimal expression has been determined as the lowest order polynomial that can achieve the same accuracy as his equation. In this case, m=4 and n=3. The coefficient matrix a^j of equation (41) is given in Table 41. The agreement with experimental data overall is better than 0.2% except for low density points near the critical isotherms. The valid range of the reduced variables is 0.7Â£pV*S1.3 and 0 . 5Â£ T / T 0 . 9 9 . This range could be expanded by use of a modified form of the polynomial and data for substances at lower reduced temperatures and higher reduced densities. These coefficients give the crossover value of C = 1 to within 1% over the specified temperature range. The reduced bulk modulus then can be written as 9P/RT _ 1 5 p T ~ = 1 C*Â£ (42) Results and Discussions Equation (41) with the coefficient matrix of Table 41 has been applied to all of the substances for which liquid DCFI data have been found. The compressibilities
PAGE 54
41 TABLE 41 Coefficient Matrix aÂ— for Equation (41) \ 1 m\ 1 2 3 1 9.8642 10.191 1.5356 2 28.465 30.864 6.0294 3 27.542 32.898 8.7130 4 8.2606 12.737 4.0170
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42 are generally from fitting volume changes with pressure and sonic velocities. The present reduced bulk modulus data bank includes 33 substances ranging from noble gases to petrochemicals to water and methanol. (Compressions of many more substances are discussed in Chapter 5.) Table 42 lists all of the characteristic parameters and the data ranges from which the parameters are evaluated. The values of V* are about 3040% of the critical volume while the values of T * are normally within 20% of the critical temperature. The values of C* vary from 7 for very small molecules to 167 for large polyatomics. The results are in excellent agreement with the experimental data. For some substances with wide data ranges, the fitted pair parameters coincide with the crossover point observed experimentally. For substances with lower pressure ranges, the fitted pair parameters are close to the crossover point obtained from graphical extrapolation of the experimental results. This is also true when water is considered individually at high and low temperatures. There are two sets of characteristic parameters for water at different temperature ranges. They are listed separately because water behaves unusually at low temperatures 56 (below 75 0 C ) . Discussion of the results can be developed according to the complexity of the substance, the range of the state conditions, and the accuracy of the experimental results.
PAGE 56
Characteristic Parameters and Comparisons with Liquid DCFI Data 43 03 VO NO co CO ON o CN rA
PAGE 57
Table 42 (Continued) 44 4rA 1^ rA rr** V0 VO vO LA 0 rA LA LA LA LA LA LA LA LA Y rA rA rA rA rA rA rA rA <3rA LA a I 1 l 1 1 1 1 1 l 1 i I 1 rA rA rA CO CO CO CO ao CO ao CO CO CO lÂ— a a a ON ON ON ON ON ON ON ON ON SO o Os LA Â• Â• Â• Â• OS o VO <3rA CN r^0 * rH CN Â• Â• Â• Â• Â• Â• Â• Â• Â• 1Â— On rÂ— 1 CN ON SO CN rA v0 rA rH CO LA LA VO vO Â— f O rrA vO ON OS a<3CN LA rH 0 CN E * CN so Os vO r' s 0 Q. 0 >S 0 0 0 4> Â“O o ji c O c p 0 C N N 0 Â•H Â•H c P 0 U 0 D c 0 C C 4) 0 O rH CO 0 44 Cl p JD 0 0 i N 0 0 0 O c CJ O 4) Â•H c Â•H Ql Â•H C c o C _Q X 3 C c rH 0 >N c cn Q 0 a 0 o o 0 p 0 O O Â•H a JC rH rH 0 P X 1 Q. i x 1 c N o JO E u rH XI o >N cn _c 0 3 rA O' LA C 1 Â— 1 O 4> Â•H u JI p P t/j Â•s Â•* 0 x u Â•H C 0 P 0 0 rA lA cn u CQ < c_> LJ 2 : 3
PAGE 58
45 (*0 oo rA rA rA rA rA rA rA rA CO CN o rA rA Q ^ co On o ON * r Â— 1 CN CM rH LA ON NO o rA * ON CO On rH CJ a CO rH a 1 Â• Â• Â• Â• N >s 3 rH rH 3 rH CJ >N 0 >Â» Â•H Q. 0 a. H >4 0 >N Â•H >N 0 C JI c C C O 4> o C JO P 4> c 44 u 0 4> 0 H> o P 0 p o J> 0 D o D 0 4> CD Â»H 0 E Q. u CL E 0 P Q E JO p 0 CO Â•H 0 E CO o 0 E 0 O CO E 0 o Q P 0 p 0 P JO p 0 P JD P 0 P 40 3 P JJ P u 41 44 44 CO 0 0 0 0 0 0 0 h1Â— 4) 1Â— JJ rÂ— 44
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46 In the case of monatomic substances from Streett et ai.56,57 gnc j diatomic species from Bender,50 the temperature ranges from just above the triple point to near the critical point, and pressures from saturation to freezing point. The agreement with the present correlation is within 1 %. The ratios of the characteristic parameters to the critical properties are slightly lower than the reference methane values, while the characteristic DCFI values, C*, are slightly higher. In the case of shortchain hydrocarbons from the National Bureau of Standards, 606 5 the reduced temperature range is between 0.5 and 0.97, and the pressures are from saturation to 700 bars. The agreement with the present correlation is within 1 %. The ratio of the characteristic volume to critical volume is about 0.35 and the ratio of the characteristic temperature to critical temperature is about 0.92. In the case of longchain hydrocarbons from Snyder and Winnick, 65 the temperatures are 25Â° to 85Â°C and pressures from about 1 bar to over 5 Kbars or the freezing pressure, whichever is lower. The agreement with the present correlation is within 2.5%. This is the worst for all of the substances tested. However, examination of the deviation patterns indicate that the major source of the disagreement comes from the atmospheric measurements of each isotherm. There are several possibilities related to this observation. First, the extrapolation of high pressure data to atmospheric pressure either by a graphic technique or by a
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47 specific equation may cause errors, though it is a common practice in high pressure work. Second, the adoption of the atmospheric measurements from the other sources may cause inconsistency in the complete data set. Third, the initial compressibility of the longchain hydrocarbons may behave differently from that of methane. If the deviations do result from the experimental uncertainty of the atmospheric DCFI, then the agreement with the present correlation is definitely within the experimental errors. Similarly, for long chain and large globular hydrocarbons of Grindly, 43 the temperatures are from 25Â° to 190Â°C and pressures from zero to 8 Kbars. The agreement with the present correlation is within 2 %. The experimental precision of the lowest pressure compressibilities is only 1 or 2?o, while above 400 bars, the precision is about . 5 %. Comparisons of the experimental data with the calculated results indicate that the deviation patterns are consistent with the previous assertions that the disagreement between the experimental DCFI and the present correlation at zero pressure is due to the experimental uncertainty at this state. For these larger hydrocarbons, the ratio of the characteristic to critical volume is about 0.31 and the ratio of the characteristic to critical temperature is between 0.8 and 0.9. The only exception is 3,3diethylpentane which has an unusually high ratio of the characteristic to critical temperature; this is associated with the abnormally large disagreement of DCFI at zero pressure
PAGE 61
48 where the temperature effect is significant. For benzene and some of its derivatives, the work of Gibson and Loefis ranked as one of the most accurate among all high pressure measurements. The temperatures are from 25Â° to 85 0 C and pressure from zero to 1 Kbars. The agreement with the present correlation is within 0.5?o which is significantly better than for the rest of the substances. This is partly due to the insensitivity of the experimental DCF I with temperature at low pressures for these substances. Figure 42 shows the experimental behavior of DCFI for benzene. The value of C* increases from liquid to liquid in the order of benzene, chlorobenzene, bromobenzene, nitrobenzene and aniline. This is consistent with the argument in the previous chapter that the introduction of a polar group acts on the liquid DCFI of benzene in the same way as the increase of energy volume coefficient. It changes the attractive potential and 2 9 U makes ( more positive. The ratio of the characteris3 \I L 1 tic to critical volume is about 0.34 and characteristic to critical temperature is about 0.85. In the case of the molten salts from Grindly, 1 ^^ the temperatures are from 90Â° to 160Â°C and pressures from zero to 5 Kbars. The agreement with the present correlation is within 0.8?o. The ratio of the characteristic to critical volume is about 0.32 and characteristic to the estimated critical temperature is about 1. Comparison of C* values for
PAGE 62
49 pv c Figure 42. Experimental DCFI of Liquid Benzene
PAGE 63
50 tetrabutylammonium tetrabuty lborate and 5 , 5dibuty lnonane which have nearly the same V* confirms that the charges have the same effect on liquid DCFI as a polar group does. For highly associating species such as water^Â’^ 6 and methanol^ the agreement with the present correlation over a broad range of state conditions is as good as the other substances; only lower temperature water is anomalous.
PAGE 64
CHAPTER 5 APPLICATION OF THE MODEL TO OTHER PURE LIQUIDS Introduction Thermodynamic properties of a pure liquid can be calculated from the DCFI model using the parameter developed in the last chapter. In addition to isothermal compressibilities, there are compression data at high pressures, so model parameters based on the pressure equation and compression data can be obtained for these substances as well. These parameter tables include almost all of the significant liquid compression data available. The crossover parameters appear to be group additive and related to critical properties. Some of the group parameters for characteristic volume V* and characteristic DCFI C* are tabulated . Derived Thermodynamic Properties Isothermal integration of the DCFI model from a reference density, p Q to final density, p, yields the pressure, P 51
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52 P = P 0 + P*RT{ (pp 0 )CÂ»[ (9.8642Â— J .1 . 91 . 1 ;? 3 56 ) (pp Q )(14.23251M21M^)(p 2 ^) + (9.18067J^ 9 i 6 ^Â°^2) T T 2 T T 2 (p 3 p^)(2.06515Â— 1 8425 1 . 00 ^ 25 ) (p 4 _p 4 )]} T T Z 0 (51) where P Q is the pressure at p Q and T. Substitution of the DCF I model into Equation (222b) yields the liquid bulk modulus h y ^ , k" 1 = pRT {lC*[ (9.864210.191 1.5356 ' , 23 /65 30.864 T ^^)p + (2 7.54 221^j 8 . 7 ^ 30 ) p 2_( 8 .2 60 6^ilZlZT Z T T 2 T 4 . 0170 v ~3 )P 3 ]} (52) Isothermal integration of Â— from the reference to the P final density yields the change in the Hemholtz free energy A ( p , T ) A ( p , T ) p p P . . = (^1) InÂ— 2. Â— (Â— Â— ) + C*{ (9.8642RT p*RT p p [ ( 9 . 8 6 4 2 .10^. K5356 ) . ( : 4 . 2 , 2 5 . T T 2 p T T 2
PAGE 66
53 )p o + (^*18067 2 j_ 9^43_3 )p^_(2. 065153.18425 1.00425 ^3, , p o , x , 1 Z2 )P 0 ](Â— 1 ) + ( 14.232515.432 3.0147 w . Â— Z2 ' (PP 0 ) + (4.59033^iLg2l : . 45 217 )( ~2_~2 ) _ (0<68838 _l I 06142 _0_._33475 ) T ? 2 Â° r T 2
PAGE 67
54 The isobaric expansivity a p then can be calculated through the relation a P = h T^v (55) From the Maxwell relation / 3 S v _ / 3 P s The entropy derivative is ( 9S ) j T T. (56) (57) Substitution of Equation (54) into Equation (57) and isothermal integration from the reference density to the final density yields the entropy change. S ( p , T ) S ( p , T ) i i . d P i i dp R R ( pÂ— )( HTÂ“ ) Â“ T( Â“Â— )( Â— ) + o P P Q dT 1 C*{ (9.8642Â° Â‘ 382 4,6 Â° 68 )ln^ (14 . 2?2?30 ' 864 9 Â‘ 0441 ( P P n ) + ( 4 . 5 9 0 3 3 ) ( P 2 P 2 ) ( 0 . 6 8 8 3 8 1 . 00425 v ,~3 ~3 n dp. L ^ Â± ^) (p^p^TE (9.8642Â— 191 1 . b l 56 ) (Â—5.)(14.2325T Â° T T^ dT dp 2 dT dT ^ i 3.18425 1 . 00425 w dp o ,,, 1 1 , (2.06515 ) ( )]( )} + {lC*[ (9.8642T T 2 dT p p 0 20,3j2.Â».Â«068 ) _( 14 , 2 ,253 Â°;6A9^41 ) o+(;18067 2K9 2 2. T 2 T ^4^ ) p 2 ( 2 . 0 6 5 ! 5 ) ^ ] } ( !Â£. x ) pZO ~ Â“r Z O''*' (58)
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55 Internal energy U is given by dU = TdS Pdv (59) Isothermal integration and substitution of Equations (51) and (58) into Equation (59) yield the internal energy change . U(p,T)U (p T) P n dP . . dp ITT o T dT P P 0 dT C^{i(10.191 + l^Zi2 )ln ^_ (15>432 + 6 JL 02^ )( ~_~ ) + (5< 483 + 2 9 Â° 433 )(p 2 ^)(1.06142 + Â£^^)(p 3 ^)]^[ao.l91 + l^Zll) (15.432+11^)5 +(10 .966 + l^Â£lÂ«2 ) p 2 ( 3. 1 B425 + il2Â£85 )5 3 ] T 0 T 0 T 0 (Â— 1) + Tf (9.8642J^jililjiiliU^g_)(i / 2?2 ? 15 Â’ 432 p T T 2 dT T 2 3 l^)(l^) + (,. 18067 .10^_2 1 90Â«2 )( ^ ) , (2 _ Q6515 _ T dT T T 2 dT *+* q. 3.18425 1 . 0 0 4 2 5 v, , dp 0 ) j ( 1_1 } j T T^ dT p p Q (510) The enthalpy H is found from the definition H = U + PV (511)
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56 Substitution of Equations (51) and (510) into Equation (511) gives the enthalpy, H = U 0 (p 0 ,T) + p o ( fi> + T frr 2 ) P P, dP ( ' 3 T' RT[(l_Â°)_r(ii_) P P PÂ„ dp (_Â£) + C*RTi(in.l91^2lLiZlinnP_u.n/i ?^_ 30 864 9.0441 dT (PP o )(9.1807l 6 449 l :8 Â° 863 )(g 2 p 2 ) + (2.60515T T 0 4.24567 1 . 67375 v ,~3 ~3 tv ' y > (P^Pn) + C (9.8642+illJll)(i4. 23 25+^lHiiI)p 0 y 2 M o 3.0147 (9.1807 + ilÂ£Â£^Z)p2_( 2 .60515 + ll^25)p^](^l) + T[(9.8642T T p 10.191 1.53Â»Â« )( ^ ) . (14j2325 _i L 432_3 .Q1 Â»7 )( < ) + dT dT (9. 180670, 966 ?0433 ) Q6515 3,18423 100425 T dT dp . . (_Â£) ](!!_)} dT p H (512) In the above equations, the property value at the state p,T can be obtained completely in terms of the property at the state P 0 ,T and, where necessary, derivatives such as dp (j j ) of that state.
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57 Pure Liquid Compression Data Bank Critical compilations of the experimental data of compressed liquids are scarce. A recent survey of PVT properties of liquids, as referenced in Chemical Abstracts before the end of 1983, was conducted by Tekac et al.^ and published in the Journal of Fluid Phase Equilibria in 1985. This survey may serve as a source of references to some original papers on PVT properties of compressed liquids. Our present pure liquid compression data bank contains 133 organic and inorganic liquids (including liquefied gases and metallic liquids), 13 molten salts, 17 polymer melts, and 7 deuteriated liquids. They are listed separately in the first column from Table 51 to Table 54. The substance order is based on the Chemical Abstracts system. In the case where the substance has been studied by several authors or by the same author with several investigations, the reference numbers are ordered chronologically. Only those data sets with the pressure ranges greater than 300 bar and temperatures more than two isotherms have been selected. There are 234 total data sets. Several misprints have been found in the printed values of the experimental data in the original articles. They have been corrected in the present data bank. Experimental determination of the volumetric properties of compressed liquids may be divided into two methods: direct measurements and indirect measurements. The major sources of the experimental data in the present
PAGE 71
58 data bank come from the direct measurements of volume change or volume itself as a function of pressure at constant temperature such as from piezometric^Â® or pycnometric methods. ^ 9 Isochoric data 9 ^ are generally not included in the data bank because they are not available in isothermal form. Only a small portion comes from the indirect methods where liquid density can be related to the other physical properties such as ultrasonic velocity, dielectric constant, refractive index, and heat capacity, etc. In the case of the ultrasonic velocity measurement, i Ta p T = F7 + (513) where u is the velocity of sound. Isothermal integration from the reference pressure P Q to the final pressure P yields the density expression.^ P = P 0 + T u dP + T P 2 P dP (514) o r o In the case of the dielectric constant measurements, the theoretical expression of the ClausiusMossotti Equation 29 can be written as n 1 e1 p rn in (515)
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59 where CM is the ClausiusMossotti function 7 ^ 74 and e is the dielectric constant. Similarly, for infractive index measurements the LorentzLorenz Equation^ can be written as 1 n 1 p = TT TT (516) n +2 where LL is the LorentzLorenz analogue 74 of the ClausiusMossotti function CM which replaces e in Equation (515) 2 with n , the square of the refractive index. The direct measurements tend to be more accurate than the indirect measurements. The accuracy of the experimental data varies from 0.01% to 1% in density. Generally, a precision of 0.1% in density corresponds to 1% in pressure or 110% in isothermal compressibility. Results and Discussions The pressure Equation (51) can be used as an equation of state for pure liquids. If the characteristic parameters are known for a specific substance, then the change in pressure due to an isothermal change in density can be calculated. On the other hand, Equation (51) can also be used as a correlation scheme to fit the liquid compression data. Tables 51 to 54 list the characteristic parameters for all of the substances in the present data bank. The results are in excellent agreement with the experimental data for all of the substances over all data ranges. In
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Characteristic Parameters and Comparisons with Liquid Compression Data 60 c0 DC VO rA VO px LA rx px px 03 x ON px NO CD M VO CO 00 C Â• *H O O Z Q. co aON o 3 rH E .c CD O Â•H *H CD O Os FA rH NO <3>Â— i 04 rA <3" o rH NO SO VO rH a A CD <3 o o Q u CD C_> 0 u u o C o c o JD C4 co CJ 0 TD Â•H X o c o o o JO Cl 0 CJ VO CX
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Table 51 (Continued) 61 40 oc rA co CO LA CO St CO OS CO co oo so oo 03 03 O Â•H C C c C 0 CD CD O il JC JC JJ >s JJ JJ 0 JC 03 03 G JJ 2 s: =t Ld 7727 43 160250 51300
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62 <0 0 ce fi CO X 0) Q_ CT c CO cr <3s 0 0 D 1 Â— 1 rH fH 0 CO X u 0 4J 0 U 0 0 O C 0 0 0 r Â”t C <Â— I >N O >, Q. 0 X O 0 4J 0 O 0 o. Â«a: s: Ld
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63 a NO NO On iÂ— I On O IA a LA On O ON ON IA Â• CN CO la a no PA ao la i" PA NO ON A> i~4 CJ X a o L O 3 ' Â— I LU CD a. CD C CO 4> 3 3 a c CO Li 3 CO co CN 3Â• LA PA CD 4> CO u ID o a >, c Â•H AJ PA CO ID "D JC CD a iÂ— i CO Li u 3 ca i c PA CN a i Â— NO CN A' A" On NO CD PA Â• VO _c o P 4.1 ID u 0 a) C ~a c a o >N co 0 <Â— 1 41 C Li ;a CD CD 3 rH C Y u 44> P C CD 0 P s: 1Â— U VO > P o Q O CO A CN A VO ON VO VO Ov A ON A Â•Â— i fH A A r^* H CN i i CN 1 rH Â»Â— 1 CN Â• 1 a Â• a Â»Â— i Â• C3 a co CD CD A CN VO VO A A I A 1 A A Â• CO CD i O I IA ON <3* <3" On CN CN CN CN CO CN CN O Â«Â— i a a A CO NO <3Â‘ ON A o o VO CN NO A A A LA CO Â• Â• Â• Â• o a C3 r Â— 1 O VO NO  Â— 1 PA CO ON vO NO CN NO Â• Â• Â• Â• C3 NO A r^ CN ON CO LA PA A A NO CN CN o A CN 1 Â— 1 <3> A rH CN NO CN Â• Â• Â• Â• CD CO CN A ON ON ON ON N a 4J 3 C co 3 .a p OD o 0 1 CO Â•H C >> Q
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Table 51 (Continued) 64 (p O A SO A CO ON o rH O CN A <} 03 ON A a O o O Â•H Â«H CO i Â— 1 rH rH QC Â»Â— i iH rH rH Â«H rH * Â— ! rH Â•H P O o O O rH SO VO CO C3 a CD CD a o O a VO rH Â•H CN ON o J3 a A A a a Os A A a NO A *H CD N .C L> 0 E 0 Pi L> 0 0 c 0 >N C CD Pi 4> 0 N 0 03 c 0 c 0 C 0 c 0 c CD 03 03 .a 0 N 0 P 03 C c 0 o N C N 0 c c CD co C p C 0 C C 03 03 P p co o 0 CD 0 Â•H a. P C c p D _Q O C3 *D o c 03 03 c Â• Â— 1 O P O Â•H iÂ—i 03 a. CL 0 (p Â£ o P Pi CJ Q_ o O CL p o rÂ— 1 P >N >N 1 03 CD 1 0 p JD Â•H CL CJ H z tÂ— 1 c CL CQ CJ Z A <1* CO A A CO On On A A CD o A ON ON A co A NO rH CO A NO A A NO A A CN rH a
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65 Â• a LA NO NO a <3 CD A' ON On CD A Â«H CM ao CD C'N 4 LA rA A rH <3 AH UN H UN <3 rH c C C 1 Â— i Â•H 0 0 Â•H o CD Q N rH rH JD 1 D C Â•H a 4> CN cn 0 C Jk 0 CD < CJ s: CN
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No. of Data Range Substance C* V*,cc/mol T*,Â°k AAE , % Points T,Â°k P^bar Ref 66 <3 A <3 ON vo O <3 O LA vO ON ON AON 04 CM VO o CM CM CM rÂ—\ CO 00 rH o o CM CD rH r Â— I I Â— 1 Â»H rH iÂ“H rÂ— I rH 1 Â— 1 Â»H rH VO NO CO rÂ— I CD a NO VO PN CO O iH iH CO i Â— 1 H NO rH VO <3 a rH rH i Â— I CM O NO NO o NO rA CD O ON VO a r*N A < Â— 1 O a ON ON CO ON 1 UN LA rH A UN l I UN CM CM iH <H UN ' Â— ( 1 1 1 1 i NO 1 1 1 1 1 1 1 vO 1 Â— 1 O CO i"H iH NO vO <H Â»H rH co 00 i Â— 1 CO Â• CO ON Â• Â• CO ON ON On A Â• Â• A A Â• Â• Â• . rH CO iH a o a CO A A A A CO A A A 00 A A A CO A ON A A A I** <3 Ar^CM ON rH A A ON A <3 i A 1 A A A 1 A i 4J CD ID CO C CD C C 4> ID CO o c a. iH 4) 1 Â— 1 ai <h >. CL a CL >4 a. (D CD ID >, IH X o C iH X a >4 4> u ID O o >, JZ ID Cl DD CD rH JZ 4> 51 O iH Â•iH Cl 41 Ld 1 CD O C Dx (D 1 r*N HH 1Â— < CJ z. r
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67 Â• 0LA CO On o On CD ON NO A A ON H CM A A 05 CN CN CM A a UN o rH CN CN CN A A CN A 00 rH H rH *H H rH H H H H H H H H VO H a *H CN ON CD * rON ON On CN rA CN A CN CN CN CN A CN A CN A A CN 0CO O 4> c <3CN C5 A A ON NO ON co CO CO NO rÂ• *H A CN <3H H A A CN CN A <3o o 2 CL A CN <3NO CN On UN CN ON ON H On N G CO Â•n a. 3 X CN X O X CN CO 1 1 1 0 Â•N c CM O c X CN
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Octamethylcyclo68 03 0 D C C a u I LA 0 _Q 0 0 0 cn u 0 JO Â» 0 Q_ cn c 0 OC 0 0 3C Q o Ura O P C Â• *H O O 2 0_ rv Ld < < X a O O * c_> I CO CSI M3 M3 rH I rA K\ I IA rA CO CM CO la K\ 00 Â• O LA LA ON CO d A as a A ON Â•H <3CN a <3* o A *Â— i *H f ~~ i Â»H o ra o Â«H CN o r^ o o VO a a r Â— 1 a o On CN CO i Â»H lA i M3 Â»H O 1 a 1 o 1 CO 1 pH ON o rA A A A A A CM ON CN CN t r\ rMl 1 A A A A 1 rA I A 1 A i A l 00 1 CO a a a o ON ON rA A A A CN CN pH 00 O <3ON A CN 'd' A CN H CO a CM M3 r Â— 1 CO Â»H A A rA LA A ON 00 CO M3 Â»H rA CO M3 C3 Â•H o CD a o Â»H On CN ON o CO On CO LA o M3 NO CN <3* M3 CN CO LA M3 CO CO CN <3CO rA CO On M3 A LA LA LA lA H CM NO NO A iH Â»H ON ON A NO Â» Â— 1 N n 4> 0 0 c Â•H 0 Q C 1 a rA 2 I rA c pH Cl CO ri 0 0 0 a 1 Â— t O c C c CD 0 0 c 0 u 0 O c c 0 0 0 0 1 0 0 0 0 0 CO 0 a 0 03 c 03 c 0 0 03 C s O CO a a O ZD 0 a 1 1 0 1 u 1 P c c HH C u c
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69 Â• Lpa o VO UN On ON LA ON LA UN NO 0 PA PA MO O CO PA CM M0 NO CM cr rÂ—i rH Â»H rH rH rH i Â— 1 o o CO CO rÂ— I *H VO A CM LA o o a NO VO VO c o a ON o ON CM <* f'N LA 00 LA On NO CD rÂ—i CD CO ON CO CO LA CO O CO PA CO CM * NO N a CJ 0 1 1 03 C rH >N C 0 rH rH C 0 i G C_) CO c 0 >s 03 >, CO C rH X 1 MI ot c 03 Q. 03 C c CJ CL >> a rH 4> a 0 CJ O C CO 03 03 03 >N c 03 1 0 0 a c f4 CO a Â•C C TO X 03 C X Â»H rH TO 0 CO CL 4> 03 CL co CD o C CO o X >, 0 T3 4> Â•H CL T3 0 MI P4 MO CL x: MI X X 4 J 0 CO Q 0 Â•H CD 4J 4> 4> 1 4) 4> 0 0 c X O 1 JC P4 1 03 03 03 rH 03 0 MI MI 0 0 3
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70 4pa a rA a pa X CN <Â— i * rA PA IA IA PA O (A O CO * a CN CN CJ Â• rÂ— I Â• 1 IA Â• a CN co CN Â»H ao rH VO VO LA VO VO O PA PA PA IA IA IA Â»H rH rH Â•H rH Â»H a O PA a o a o O Â—1 o o o LA CO CO CO CN PA LA VO VO r* CN a rÂ» LA LA no PA NO rH IA o O rH NO IA Â• Â• Â• Â• Â• Â• H PA On PA rH LA Os rH LA VO Â«H NO Â»H a NO rH CN co Â• Â• Â• Â• Â• O LA D\ PA VO Â• CN a co CN PA CO *H >Â— 1 1 rH *H Â•H ON I CD a c CO >s x 3 X CO X 3 ud Q i LA Â•v LA 0 c co c o c CD c co CJ 0 "O co X Q. 0 X I c 03 I A16 0 1= Â»N I 0 NO LA >N 0 rH X LA Â•n 0 U  > U c 0 c 0 4> 0 a CD c Â•H 0 0 0 C 0 >N c T3 Q. X X 0 u 0 Â•H CL CD a X X Q 0 nz 0 0 a 1 X i T) o Â»H c i Â•Â» c rH A0 C co co 0 CJ Â•H UJ 1 c Bixylyltoluene 144.750 296.618 612.170 0.1874 24 298353 0.981961
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Bicyclohexyl71 o 03 3 C 43 C o C_) A 03 Q (0 0 0) QC (4 CO O Â•s (U Q. Ol c co q: co co X Q o <4CO O 43 c Â• *H O O 2 0UJ < < x * Ov Ov Ov o O o 1Â—1 iÂ— 1 i Â— i I Â— 1 <Â— 1 iÂ— i VO VO VO ON Ov OV Â»Â— 1 1 IÂ—l 1 *Â— 1 co 1 CO CO Ov Ov Ov . . Â• o a A A A A LA A A 1 A i rA CO i CO 1 CO ON OV ON CN CM CN A A A A A A > Â— i rÂ—\ rH CO A A *H ON A NO CO 1 A i On 1 rH i Â»Â— 1 1 Â•H a O a Â• Â• Â• rH rH rH CO co CO a o o i N C i 03 J C 43 iÂ—i <Â— 1 41 03 X 1Â—1 >, 03 E 1 xi C i Â— 1 iÂ—l CM 4> CD 1 JO 43 >s >s \w03 C 03 o 43 03 1 X X A 1 CO C 03 E Q 03 03 1 iÂ—l a 03 1 > Â— 1 iÂ—l l C C iÂ—l X 03 CD CO CO o o U X c TD iÂ— i Â•H C C Â•H 1 Â— 1 IÂ—l CD c 03 C O CD 03 03 CQ O a C 03 C a> 4> 1 C C 1 X >s 4> C Q. XZ Q. a. u G CD a. Â•V VS i CN CN iÂ—i 03 >v 1 C C 03 iÂ—l CO p C >N o 0 co co 03 rH u 43 TD 03 C CO c "D CL 43 0 co CO c C 4> 2 03 Cl CL 1 Q. 1 03 a CN C i V/ A ON 7Dicyclopentyl4 (3cyclopentylpropyl) heptane 200.999 403.698 660.418 1.1609 119 311408 1.0110337 133
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Table 51 (Continued) 72 a_ a Â•H o *H rH ct Â• a Â• a a c Â•h Â• Â»H Â• Â• CO Â»H rH rH az co P ao 00 CO CO CO CO a a a a a Q X N rH CL Â•Â— i 1 Q. X O  rA o C P CM C Â— ' p p CL s Â— ' p <3* CL 0 rÂ—l 1 rA 03 1 rH 13 1Â— 1 0 >N 0 0 1 1 Â»H X c c P C p Â»H >Â—1 X p 1 CO X 0 c 0 a. >> X 03 4J c 13 1 Â— 1 C3 0 a 0 o 0 13 X X c c 13 c >* 0 C 0 CL 0 JC a > o o 0 C_) 0 CJ 0 CD 0 _Q X >. 13 X u TO 1 CD. 1 SZ 1 sz 1 T3 33 o o Â£Z a a CM fA C U1 1 1 1 w r 1 Â»H Â»H rH ON On ON 228.384 444.701 599.505 1.2509 146 311408 1.0110337 135
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Table 51 (Continued) 73 Â• (*VO ON On o NO VO vO CD 0 pa O a PA r*N PA PA PA 0= rÂ— 1 rÂ— i rH rH P rH rH rH o rÂ— i rH a O a a a o NO VO a a o o a u OÂ’ ON ON a o N CO 0 c JO X p G C u p rH co P G o G 0 CD c 1 >Â» G >> 0 G G o c JZ TD 0 0 o 3 rH CD u C T3 C G CO Q. CO C C a 0 a G 0 O co co p Â•H P 0 0 0 Â•H O c O 0 a G u CO Q P r Â— 1 rH Â•H o 0 CD 1 o 1 O 41 Q 1 ID 0 0 U 1 *U .c c a c G G 3 rÂ— 1 P ZD ZD hrH 1 1 P CO Â•V 0" c r 1 Â•s P f
PAGE 87
74 It'D CC u CO .Q Â•> ID CL cn c CO OC CO D CO X Q o O O Z CL Â•a ID 3 c c o CJ I L/V Q CD a? LU < DC * CJ I D O c co 4> CO JQ 3 CO VO LA VO O CD CD O rH I O CO LA PA I co ov CS Â— 1 CO PA 'H 'H a Al CN 1 1 r Â— 1 i i 1 1 LA l LA PA 1 a rÂ—  1 AO Ad CO CN i Â— I la Â— I CO d CD PA O CO o Ov CN Ov PA PA CD CO CN CO A' CN PA CO i Â— t LA PA PA CN AVO CO d CO o CN a ^ Ov CN CN Â® LA d Ov Ov Ov f" n 0 u 5 : DX c D CJl o (H c D CJl X CD D a X o Lt 3 C c a. o H C 3 D CO X
PAGE 88
Characteristic Parameters and Comparisons with Molten Salt Compression Data 75 c*_ CD QC Li CO JD Â»> CL 0 CT> c 0 oc CD CO Q a cco O 4J c Â• *H o o 2 a. CM CM CM <3Â“ rA lA I CM PA PA rA O M3 On LA CTv O O O Â• Â• Â• *H CM Â•H M3 CO PA IA CM rÂ— ( Â• Â• NO CO Â• CM a VO LA CM VO C D Ll D 0 E 0 CD A O Â•H H 3 ^1 JJ CO iÂ— ( C0 4i Â•H 4> CO CO JD CO Â•H a Â•H JD CD U CO c Â•H C D 4> 4> JD CD O o 3 a_ o_ or CM rH rA A H LA A Â• Â• o O CD D 0 i Â•H 1 i u i Â— i C o rH 1 4> >v o (4 >. Â•H Cl JZ o CL C O Q. 3 0 O Li CO rH 4) Li U CL o C0 CL 0 CO JD 0 U co > Li CL u o (a H L> 4> JD L> Â•H CD 0 (0 CO 1Â— 4> 1Â— a H Ov H vo LA CD fA Â• Â• H a Â«H rH LA a H ON Â• Â• A A vO lA ON LA vo CM CM CO CD Â• Â• rA VO <3* CM rA LA rA A LA rH CO Â• Â• LA LA <3* VO LA rH I I E O i E o D Li iÂ— i D Li Â•H O 0 Â•H O ID C D 4) L> C D JJ O iÂ—i 0 D o ii co CO d~ Ph Q 1= JJ 0) CD CD J> 1Â— JJ
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Tetrapentyl76 <40 ce u CO J3 I 0 oi c 0 oc lA *H rH Â»H 0 Â• Â• Â• Â• Q 3rA rA rA os ro a LA 1 1 1 i Â•s CO LA LA LA 1Â— ao Â«H Â«H rH Â• Â• Â• Â• CM ArA O ON O so OÂ’ rA TO c so 04 Os 0 Â• *H LA <3LA rA 3 O O C 2 Q_ *H JJ C a CM o S? LA CO vt so C_) rv M3 % 3 u rH 3 'H o 0 >Â•H o 0 H o 0 >s Â•H Q_ 0 C 3 P X c D 4J Q. C D j> JC C O jj o rÂ— 1 0 0 o ri 0 0 o rÂ—l 0 Jl o U 0 E (4tl C E Uu C E <4CJ 0 E CL fi E 0 o 0 E 03 o 0 E 0 o 0 E 0 o 0 u JO u 0 fi Q (H 0 H JO (J 0 U JO 4J JJ U 4> 4J J> 4> 0 0 0 0 0 0 0 J> rÂ— u 1Â— P 1Â— 4>
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Table 52 (Continued) 77 it. Â£D 0 CT c 0 a: co 4J CO Q a 4o o CL a? r< Ld < < a * CJ I CD O a i A rA rA LA rA On rA vo CM > 3 Â• H Â•Â— H 3 rH a CL Â•H >N 0 X *H X 0 c o C JZ P 4> C P P ra (4 o p CO 3 o D CO 4> CL E (D P JO E JD p 0 0 E CO o 0 E CO o JO 14 0 u Q Li 0 P Q S 41 P 4> P CD 0 0 0 0 1Â— P 1Â— 4>
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Characteristic Parameters and Comparisons with Polymeric Liquid Compression Data 78 Â• (*LA SO SO NO ACD Â• NO Â• Â• Â• Â• Â• 1 O Â• NO N ' Â— 1 P iH rH iH D Â•H 1 Â— 1 CD CO 0 rÂ— 1 0 Â•P >N Â•H A >N X a J >N O tH c u Â•H X CO X C H CD X C 4J 0 c Â•H CO P c P TD p P 1 Â— C P p CO CD rH Â•H P c CD CD ID CD 0 0 Q JO CD HJ P > O (D >N Q >, X P >N X cn >> CO JO v/ iÂ— 1 O rÂ— i i Â— l a rÂ— i 0 iH Â•H iH CD p >s C O X O c O 0 O X 0 O D Â— 1 0 IH o 3 Q_ CD a_ co Q_ C Q_ CD 3 Oto O o O Â•H U Â•H Â•H CL Q_ _l X CO X
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Table 53 (Continued) 79 tiOv CTv CO LA VO ON Ov ll) a rH rH rH o rH rH rH rH rH CTv CN CN LA CN Ov Ov t" rrÂ— 1 CO NO ON LA rA rH CN * ON On VO o NO CN CN NJ CJ Â• Â• Â• Â• Â• Â• Â• Â• Â• i CN NO CO CN ON CN rA CO LA fÂ—i ON CN rA CN CO rH r*N f'N VO CN rA CN CN CN i CN iH CO 0 1 0 r. O CO i P rH P p Â»H CO rÂ— 1 a >s 0 >N ZN < Â— i 0 0 Â•H r Â— 1 Â•H i Â— i i Â— i 0 iÂ— i rH SZ rH 0 >. rH E ID o O o c >Â» >N p  C p >N CJ S s: 0. 0 xo p 0 0 0 3 P C 0 C 0 (0 1 Â•H p o E c P JO a P C CO C u c p N 1 0 p 0 P o CD o 0) vs CD E JZ p P c sz o P CD o >s a >s rÂ—j P ' Â— Â•P N * c 0 p Â— >> J3 H CO Â•H CO 1 O >, 0 >% 0 >S 0 >N P 3 Â— 1 CO rÂ— 1 ao CO S2 iH E rH CL rH 1Â— 1 E iH 0 to H Â•iH Â•H O O o o to to o 0a. 0Q_ Q_ Poly ( cyclohexyl methacrylate) 131.047 1.06464 1173.20 0.8374 89 395.85472.05 12000 147
PAGE 93
Characteristic Parameters and Comparisons with Deuteriated Liquid Compression Data 80 pa o IA rH CM PA < 3 " CO CO 4 Â— 00 LA ON LA LA LA Â»H Â•Â— H rH LA CD rH rH rÂ— I rH r Â— 1 rH rH Â•Â— 1 cr a LA a a O a CM CD u CD a a o CD a PA a O O CO LA ON o CD LA CD CD LA LA CD Q < 3 * VO CM VO CM ON rH *H rCD X (A PA PA PA < 3 * PA PA PA PA O o  i i i i i 1 i i i hpa PA PA IA PA PA CD JD 3 CO o I s a fi i o <Â— i c*_ O o c U CD O C iH 4 > C CD u s: cA PA TD a i CD C O I CD C ID N C CD JD O U O 3 VO TJ I CD C CD N C CD co CM a i CD c CD 00 x a CD i C ID o c iÂ— t 3 O i Â— I >s O CD IÂ— CD o <
PAGE 94
81 fact, they all appear to be within experimental error. This proves the generality and flexibility of the present correlation. The characteristic parameters for each substance evaluated from the liquid compression data with those from the liquid DCFI data are very close to each other except for the data of Grindley.^" 5 This discrepancy is probably due to the experimental uncertainty of liquid DCFI at low pressure where Grindley's mathematical treatment of the compression data is questionable. The general similarity of the characteristic parameters for each substance indicates the thermodynamic consistency of the present approach. Both the compressibility Equation (41) and the pressure Equation (51) with the same parameters reproduce the crossover behavior of the volumetric properties of compressed liquids discussed in Chapter 3. The characteristic parameters are optimized only in the tabulated data ranges. Thus computations performed outside the data ranges for some substances may be less accurate. This is especially true for those highly polar and associating species where the characteristic parameters are evaluated from a very narrow data ranges or from questionable data sets such as the case of Schornack and Q C Eckert. J The characteristic parameters are evaluated from two isotherms with a temperature span of 20Â°K and the experimental accuracy in density of 0 . 3 ?o only. Though the fitted results are within the experimental errors (less than 3?o in pressure), the characteristic parameters are
PAGE 95
82 probably not as reliable as those from the more accurate experimental results. Even so, extrapolations from the present correlation should be more reliable than from other correlations. This is particularly important in physical properties correlation since experimental data covering the whole state conditions are not always available. There are a number of substances where several sets of characteristic parameters are tabulated together because several sets of measurements exist. For a substance with similar parameter sets, in most cases, any one set can be used as the characteristic parameter set. This is the case for most of the normal liquids where the characteristic parameters are found over a wide range of state conditions. For substances with different parameter sets, the recommended values are characterized with an asterisk. Variations of the characteristic parameters are due mainly to the precision of the experimental results and the insensitivity of the DCFI and pressure changes to parameter variations. It may be that some sets are not equivalent because of limited ranges of measurement as for some polar and associating substances. The recommended characteristic parameters are based on the data range, experimental precision, and accuracy of the measurements. There is a preference in this work for the data of some authors. Generally, they are liquefied noble gases and others from Streett and his coworkers,^Â’^ liquefied natural and petroleum gases from the National
PAGE 96
83 Bureau of Standards (NBS), 60 63 hydrocarbons from Dymond et a 1. , 1 ^ Â’ 1 3 1 Â’ 13 3 long chain hydrocarbons from Doolittle, 138 coal liquids (PSU compounds) from Cutler, et and Lewitz et al., 13 ^ benzene and its derivatives from Gibson and Loe f f ler , 47 Â’ ^ cyclic compounds from Kuss and Taslimi, polar liquids from Kumagai and his coworkers, 78 Â’ 84 Â’ 138 alcoholic liquids from Makita and his coworkers, 91 Â’ 97 liquid acids from Karpela, 95 molten salts from Barton and Speedy, 144 polymer melts from Simha and his coworkers, 147 Â’ 149 and deuteriated liquids from Jonas and his co w orkers 83 Â’ 93 Â’ 1981 94 Group Contribution Method In addition to the molecular corresponding states correlations of the above type, group contribution methods are very useful for physical property estimation. The concept assumes that each molecule is made up of different types of atomic or structural groups and that the molecular properties can be evaluated from contributions of these atomic or structural groups. Unlike the corresponding states correlation, the fundamental assumption is that various groups in a molecule contribute a definite value to the total properties independent of the presence of the other groups on the molecule. 199 There are two general approaches in group contribution method. The first one may be called the propertyadditive approach. It assumes that the groups actually
PAGE 97
84 posses thermodynamic properties or make directly additive contributions to the molecular properties. 15 ^ The molecular properties then can be calculated from the sum of these group properties. M. = I v. M 1 ia a a (517) where v^ a is the stoichiometric coefficient, represents the number of groups or type a in molecule i. This approach has been applied to the liquid molar volume at the normal boiling point 15 ^ and to the liquid heat capacity, as well as to the solution of group methods for activity coefficient. 159 Â’ 160 This group contribution method has also been widely used for ideal gas properties such as heat capacity, entropy, and enthalpy of formations 155 etc., where intraor intermolecular forces play no role. For the DCFI, Equation (517) would yield C . l Z v. C ia a a (518) The other general group contribution method may be called the parameteradditive approach. It assumes that the molecular property can be written as a function of the state variables, x_, and a set of parameters, 0. M. = f(x;0) (519)
PAGE 98
85 where 0 . 1 v ia a 0 a (520) This approach has been applied to the estimation of the critical properties by Lydersen . Evaluation of the molecular parameters from Tables 51 to 54 shows that these parameters appear to be group additive, according to Equation (520). Thus, the DCFI can be written as C i = C*f(pV*,T*/T) (521) and V* l z a v . \l* la a (522) C* i 2 a v. C* ia a (523) Table 55 lists the group parameters that can be added for each group to estimate the crossover parameters for some of the substances. By adopting T*=0.96T C , comparisons of the group contribution correlation with liquid nalkane compression data are listed in Table 56. The results are satisfactory.
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86 Table 55 Group Contributions to Characteristic Parameters C* and V* Group (Attachment) C*(+0.5) V*, cm 3 /mol( + 0.05) CH^ (Linear paraffin) 12.6 26.83 CH^ (nonlinear paraffin) 9.7 26.02 CH^(linear paraffin) 4.3 18.35 Â— C H 2 Â— (nonlinear paraffin) 7.9 16.54 CH^(ring) 5.8 18.39 CH< (nonring) 2.3 12.20 CH< (aromatic ring) 6.8 14.77 >C< 3.4 9.95 >Si< (silicon) 11.9 17.72 >Sn< (tin) 16.4 28.20 F (on nonring) 6.7 12.40 Cl (on nonring) 12.5 21.29 Br (on nonring) 9.1 28.23 C00H (on nonring) 21.2 28.64 COO(ester on nonring) 25.0 21.10
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Group Contribution Correlation and Comparisons with nAlkane Liquid Compression Data 87 0 ) oc * o vo VO * CVJ VO CVJ LA CVJ VO v O VO CVJ o CVJ LA CN 00 CVJ 0V CVJ * O P'S Os O CD LA LA CVJ ON CVJ PA as VO rÂ—\ ao VO A lA co a* a* a* CO VO rH OS CN Os a* OS a LA VO Os OS rH iA VO CN a vO A CO VO CN UJ PA rH CN Os A Os Os CN co CN o VO CN rA CD < iH CO aA LA a* rÂ— j OS 1Â— 1 CN C O' Â• *H vfi O O 2 0 _ VO LA CVJ PA pa vO CN CN iÂ— H O CO rH r Â— 1 i Â— 1 CD a a CO rH ' Â— i 0Â• Â• Â• Â• a ov O SO Os CD a a LA Â• Â• Â• Â• Â• rH a rH o iH CTl c CO CSC rv la i'ov Â• Â— t i <1 X a 4> m Q. o 0) CD a JD O CD X X CD 41 (h 1 1 1 i Ld Q_ c c c c
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88 u03 oc CT\ CM * a A A a * A O no A A CM A NO * fH O A fA A no A fA ON CM lA M3 * MD O CM IA fA LA fA * O A o? Ld < < NO fA CM lA fA On fA A CD A LA CD CO rA o a C3 M3 CO LA C3 LA fA rH MD aH co CO CN CD A rA o CN lA LA 'H LA LA CN rH fA CD CN A iÂ—i NO CN A rA CD iÂ—i rA A lA A LA CO CO A CM o O M3 CN CN 'Â“H A A NO rA rÂ—i CN NO rÂ—  TD 03 D C Â•H C o CJ MD I LA 03 Q CD LCD O P C CD Â• H CM O O 2 Q_ MON CM A I CO A CM CO A A 00 On CM A CM A A CD A A A A I A a A co A A I co A CM A i* A i CD A CM A CM A I A a A CD A A I CO A CM CD CD a i A A A A O A CD A A I CO A CM A r* A i co A CM A r" A A CM A A A A I A A A CD O A I A A A A r~ A I A A 03 a c CD 43 CD JD 3 CD 03 C CD C 0 2 1 C 03 C 03 CD 03 03 c O c c CO 03 03 CD CD o a C a C3 03 co CO 03 03 a u u "D T3 Â•H 4> 03 c a t4 03 a i CD 1 Cj 1 h1Â— 1 c 1 c 1 c C 1 C 03 03 C 03 c 03 CD C CD C CJ co CD CD 03 03 a 03 U c TJ 03 TD 03 CD CO TD CO Â•o CO 43 CO 4 > CO O c X a. P G 03 03 03 a Â•H Q. 1 2 1 2 o Ld C c 1 C c C Group parameters are estimated from these data sets
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CHAPTER 6 APPLICATION OF THE MODEL TO LIQUID MIXTURES Introduction Mixtures of compressed liquids are the most prevalent form of material found in high pressure chemical processes such as hydrogenation, polymerization, and some other energyrelated processes as well as geological systems, including oil and steam reservoirs, for example, the syntheses of ammonia and methanol, petrochemical or coal hydrogenation, the fabrication of polyethylene, etc. Despite their technological importance, the state of knowledge about their compositional and density dependence of the thermodynamic properties of compressed liquid mixtures is still not known in enough detail to be applicable to process calculations. Though there exists a rigorous formula for the reduced bulk modulus of a mixture in terms of the quadratic mole fraction average of the binary DCFI, the present model of Equation (41) does not provide a means for obtaining the unlike pair DCFI. An expedient alternative is to use a onefluid concept^^^ to obtain the mixture characteristic parameters as a function of composition and the pure component characteristic parameters. The pressure equation with mixing rules for the 89
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90 parameters has been applied to all of the compression data in the present compressed liquid mixture data bank. The results with and without a binary parameter are tabulated. For mixtures with negligible excess volume, the simple (mole fraction average) mixing rules work as well for mixtures as does the fitting of the pure components. ^^ 2 This is consistent with the findings of the parameteradditive group contribution described in the last chapter. Derived Thermodynamic Properties Thermodynamic properties of liquid mixtures are completely defined by the binary DCFI. However, statistical mechanics does not provide a means to evaluate these properties except for rigid body systems. 27 The experimental behavior of the like and unlike DCFI can be determined through the simultaneous knowledge of the volumetric properties and activity coefficient. From Campanella, 163 2 v^ dlny^ 1 ' C 11 = vh t RT + x 2 dx 2 V 2 dln ^i 1 " C 12 = V^rT + x id^T dln Y 2 vx jRT + x 2 dxj T,P T,P T,P ( 6 1 ) ( 62a ) ( 62b )
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91 dl n Y 2 (63) VHyRT + X 1 d x 2 T , P where the volume variable of the DCFI is here obtained from the pressure variable. These formal relationships may cause some difficulties in the evaluation of the binary DCFI due to the lack of available good quality experimental information. The alternative is to use the Van der Waals onefluid concept in the corresponding states approach. The mixture is considered to be a hypothetical pure liquid and the reduced DCFI for the mixture is given as ( 64a ) where m ( 64b ) m ( 64c ) T* m ( 64d ) In the case of the binary parameter k . = 0 , the volume vJ parameter reduces to
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92 V* = 2 x.Vf (64e) i From Equation (64), the bulk modulus of liquid mixtures can be written as H Tm = PRTdCjp) (65) and the pressure expression for the compressed liquid mixtures is given as PP irr = K n Â’ p o ) (6 6) where the function g is the integral from p Q to p of the function f in Equation (64a). The pressure dependence of the excess volume at constant temperature and composition is given as 3V E 1 v x i IP" p 2(lC m )RT l p 2(lC.)RT (67) where the subscript i denotes a pure component property. This function can reproduce the variations shown in Figure 38. The partial molar volume can be calculated as
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93 CÂ™ ) vi " K JW~ J l,P t U. 7ii 7ZVK HV~ T ,_N y/w Tm ( MÂ“ ) T,V,N jVi (68) where is the number of moles of component i. For the binary liquid mixtures, the DCFI can be related to the Helmholtz free energy. c 1 pV 3 2 A '11 Â“ x, TFT 3N : t,v,n 2 (69) r 1 pV 3 2 A C 22 = TFT^I T,V,N 1 ( 6 10 ) '12 p \l 3 A Â’RT 3N 1 3N 2 T,V X 1 p V 3 2 A X 2 RT 3N 2 1 V 3P t,v,n 2 x, RT 3 N t,v,n 2 (6lla ) X _l_ pV 3jVA X 1 ^ 3N? t,v,n 1 *1 1 V 3P x7 FT TFT T,V,N 1 (6llb)
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94 Details of the properties from the present correlation are listed in Appendix B. Liquid Mixture Compression Data Bank Experimental information on the volumetric properties of compressed liquid mixtures is important in the fundamental understanding of the statistical theory, structure effects, and intermolecular interactions of solution at high pressurers.^^ However, there are considerably fewer liquid mixture compression data than there are for pure components. The present data bank contains 34 binary systems. They are arranged in the order of increasing complexity of the intermolecular interactions and listed in the first column of Table 61. These include simple liquid mixtures from Streett and his coworkers,^Â’^Â’^Â’Â®Â®Â’^ nalkane mixtures from Dymond et ^ ^ ^ mixtures of various Cg compounds from Matsuo I I Q and Van Hook, mixtures of benzene and its derivatives from Takagi and his coworker, 115 Â’ 116 mixtures of alcohol and hydrocarbons from Ozawa et al., 50 and acqueous solutions of alcohols from Makita and his coworkers.^Â’^ The experimental accuracy of the mixture data is high except for those systems where the volumetric data are derived from ultrasonic velocity measurements.^^ Results and Discussions The extension of the pressure equation to mixtures by using the mixing rules in Equation (64) has been applied to
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Binary Parameters and Comparisons with Liquid Mixtures Compression Data 95 x 0 u NO ON LA ON CO r^oo oo NO CM PA PA PA PA PA LA LA NO NO O 0 0 0 CD G o c s: 0 0 0 0 O TO a c + c co 0 0 o 0 X X u c 4> ID CD X 0 0 o o CL C u X X X a 4J >N CO Â•H 0 1 1 i Q. X X s: c c c >> LjC H o (4 CD c + + + + E + 5: o CD E c 0 0 0 4J + CD + 0 c c c CD c C Ol 0 0 0 >s c CO c O o X o 4> CO o X o X Ph 0 0 a cn HJ Ol H h> X a CD u G co Â•H 1 1 1 c s; < CD 2C c c C On A A NO CO CO ACO
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Data Range No. of Eq(64e) Eq(64b) S V stem T,Â°K P > bar Points A AE , So AAE,?o k..xlOO ref. 96 CO CD PA Â«H M3 M3 LA LA LA V0 CD % CL 0 0 C Â•H X X Â• 4 Â— > a 0 0 0 i X X Â•H
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Table 61 (Continued) 97 Â• rÂ—H CM r" lA ON ON ON O < 13 i Â— 1 p O O o M3 C3 O VO lA LCN (30 rH VO O 00 X rÂ—  p" ON r~ _ 3 Â• Â• Â• Â• Â•H CM P" X 1 JD as rA CM rA CM i LA ON o CM M3 Ld rA 00 ON rA < o LA r^* rA o< Â• Â• Â• Â• Ld dCM co tT< Â• Â• Â• Â• Ld ALA NO ON LCO P C 00 O M3 M3 Â• Â’H csi r 1 O o O O rÂ— 1 CM CM Z Q. vO O M3 CD P LA 00 CM CO o > 0) + SZ P P + d> 03 CO 03 P >s O Â»Â— I CO to c O i Â— 1 3 CO c X C CO C + 4J C p 0 p 03 s: Ld s:
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98 all of the compressed liquid mixtures in the present data bank. Table 61 summarizes the results with and without a binary parameter kÂ—. The predicted results from the simple mixing rules are listed in the fifth column of the table. The results are quite good. There is a definite relationship of the prediction and the accuracy of correlating the pure component compressions. The average absolute percent deviation (AAE?o) increases with increasing complexity of the binary system. For simple liquid mixtures, the agreement between the experiment and calculated pressures is as good as the individual pure component. The ArgonMethane system is the only exception. The relatively large deviation is probably due to experimental uncertainty which is not cited in the original papers. ^ There is a striking feature in the pressure dependence of excess volume in these binary systems. The excess volume is negative at low pressures and rises rapidly with increasing pressure. At pressures above a certain value, there is very little variation of the excess volume on temperature or pressure. The present correlation does reproduce the behavior described above though the decreasing value of SV^/aP  J,x can be faster or slower than the data show. For nalkane mixtures, the deviations tend to increase with increasing difference in the chain length. Use of the volume fraction instead of mole fraction average in the temperature parameter improves the predicted results for all
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99 of nalkane mixtures tested. This is associated with the Van der Waals forces between the segments instead of the molecules. 16 The reduced temperature of a binary mixture is given as T = Z cp.T. . r i 1 1 where ( 6 12 ) ^ = Â— 2 " x . V* l l Z x.V* i = l 1 1 (613) Prom Equations (612) and (613), the characteristic temperature of the binary mixture then can be written as . . T I T 2 (x l' , i +x 2' , ! ) m Â“ x V*T*+x V * T* 12 12 12 (614) The Principle of Congruence^^ works out very well only for this type of mixture. For mixtures of Cg compounds, the present correlation can indicate the presence of order arising from the nature of the packing volume at compressed states. Physically, order corresponds to a cohesion which decreases the volume Â• b # I f n of a liquid. Evaluation of the deviation pattern
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100 reveals the fact that the calculated pressure is systematically lower than the experimental pressure up to highest pressures measured. This is because the experimental volume is consistently lower than the calculated volume at a given pressure, temperature, and composition since the simple mixing rules do not include the order that appears. However, adjustment of mixture volume by taking into account a binary parameter in the quadratic mole fraction average of the volume parameter improves the overall deviations. The results are as good as the pure component correlation . For nonpolar mixtures of globular species, the predicted results are comparable to those of the pure component and are independent of the size difference of the molecules. This is the case of carbon tetrachlorideoctamethylcyclotetrasiloxane 81 in which the latter has a characteristic volume more than three times of the former. For mixtures of benzene and its derivatives, the deviations increase with increasing polarity of the substituents. The large deviation is due probably to the experimental imprecision, which is associated with an anomaly observed in the ultrasonic velocity measurements. For alkanealkanol mixtures, the predicted results are acceptable if the nonideality and experimental inaccuracies are allowed for. The experimental excess volume varies with composition in a complicated manner and nonideality is significant only at pressures below 1 kbar.
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101 The deviation is proportional to the absolute value of the ambient excess volume at the given composition. For mixtures of anilinenitrobenzene, the pure T* and C* values are very similar to each other; the only difference is in V*. Though the predicted results have only about 1 % error, it is not as good as the accuracy of the experiental data. Use of one binary parameter in the composition dependence of volume parameter does improve the result to as good as the pure component. For aqueous systems, the state dependence of the volumetric properties is even more complicated though predictions within 10% error are always obtained. The results with a fitted binary parameter in the composition dependence of the volume parameter are listed in the sixth column of Table 61. For most of the systems, the correlated results are much improved in comparison with the predicted results and one binary parameter is enough. For those systems with complicated state dependence, more than one binary parameter would be required for agreement within experimental uncertainty. Though use of three binary parameters in the composition dependence of the three characteristic parameters would completely recover the same accuracy as the pure component correlation for each composition, predictive capability is missing completely.
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CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS This work has focused on the development of a general and accurate liquid phase equation of state which is based on an unique approach of the statistical mechanical direct correlation function integrals. Two treatments have been adopted to model the liquid DCFI: the corresponding states principle and the group contribution method. The major engineering contribution of the present work is the construction of a threeparameter corresponding states correlation which is based on the detailed analysis of the experimental DCFI in the compressed liquid phase. The molecular corresponding states correlation either in compressibility form (Chapter 4) or in pressure form (Chapter 5) has been shown to be the most accurate in representing the volumetric properties of compressed liquids and liquid mixtures . In the absence of a rigorous sitesite interaction model for the direct correlation function, this work developed an empirical parameteradditive group contribution corresponding states correlation arising from the appearance of the groupadditive crossover parameters. The group contribution corresponding states correlation 102
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103 has been shown to be successful in the prediction of the volumetric properties of compressed liquid nalkanes in which the molecular characteristic temperature can be estimated. However, application to the other substances is still hampered by a lack of knowledge about the characteristic temperature. In practical use, the alternative is to make one compression measurement to determine the characteristic temperature while characteristic volume and DCFI can be estimated from the group parameters listed in Table 53. An important byproduct in the process of the group contribution formulation is the establishment of an extensive data bank for the volumetric properties of compressed liquids and liquid mixtures. The experimental PVT data, either from direct or indirect measurements, have been carefully evaluated and converted to common units, barcm^/molÂ°k. The most important scientific contribution of this work is the finding of the crossover behavior of the compressed liquid isotherms. For every liquid, there exists a unique density where the reduced bulk modulus is independent of temperature. Thus the compressional properties of a real liquid do not behave like those of a rigid body fluid. Furthermore, the assumption of linear density behavior of dense liquid 169 Â’ 170 is not fully correct. In fact, while there exists a transition region of density in which the volumetric behavior of compressed
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104 liquid may be linear in density, below or above this region the corrections for the nonlinear density behavior have to be introduced. Thermodynamic properties of major interest have been derived from the present correlation. For pure liquids, the compressional thermodynamic properties can be predicted from the data along the orthobaric curve. For liquid mixtures, the excess thermodynamic properties can be calculated along with specific mixing rules for the known pure component properties. The correlation method also provides a way to estimate the binary DCFI which can be related to the thermodynamic derivative properties in the fluctuation solution theory. The above conclusions indicate our present concept of compressed liquid properties. There are some suggestions for future work. 1. To rework the form of the polynomial to have all isotherms pass through the crossover point. Currently, extrapolation of the present correlation to the temperature below the freezing point of methane does vary away from this point. The revised formula will be given as m n =2 2 a ii T J_1 (pl) 1 i=l j=l J C*C TTZ* ( 7 1 )
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105 This ensures that all isotherms will pass through the crossover point. Application to low temperatures (TC0.5) would be more reliable with the modified correlation . 2. To investigate thoroughly the pressure dependence of excess volume for compressed liquid mixtures. An empirical correction which could be used to estimate the excess volume of compressed liquid mixtures from that of atmospheric measurement would improve the accuracy of the present correlation, especially for those mixtures with complicated variations of the excess volume. 3. To apply the present correlation to the calculation of partial molar volume and chemical potential and compare with experimental data for compressed liquid mixtures. This would be a strong test of the model's reliability . 4. To optimize the present group parameters and determine the molecular characteristic temperature for the rest of substances. While the relative insensitivity of dense liquid results to the characteristic temperature means the present group contribution corresponding states method may find useful application in predictive work, refinement of the group method would ensure its value.
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APPENDIX A PERCUSYEVICK HARD SPHERE DIRECT CORRELATION FUNCTION INTEGRAL FROM VERLETWE IS ALGORITHM o Verlet and Weis z have presented a simple algorithm to calculate the effective hard sphere diameter d(p,T) from WeeksChandlerAndersen (WCA) perturbation theory. Define a temperature dependent hard sphere diameter as d R (T) 00 [1 u 0 (x) ITT ] dx ( A Â— 1 ) where x=r/a, r is the separation distance, a is the collision diameter, and u Q (x) is the WCA reference potential. Verlet and Weis show that this can be simulated by the empirical expression. A 0.3837+1.0680 R O\'4293+0 Â— ( A Â— 2 ) where 0 = e/kT 106
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107 The density dependence of the WCA values of hard sphere diameter to first order is given by d r d R ( 1 + Y 6 ) ( A Â— 3 ) where 6 = 71 'O'. 21+404.60 (A4) Y = 14. 25n +1.362n 2 0.8751n 3 [to [to [to ( A5 ) to = n I6 r ( A Â— 6 ) it ,3 0 = gPd ( A Â— 7 ) where n is the packing fraction. A few iterations are sufficient to find d. Then the PercusYe vick hard sphere DCFI can be calculated as .pyc _ n ( 2+n ) (4n ) ' hs (77? ( A Â— 8 ) where the superscript PYC is the PercusYevick compressibility equation and the subscript hs is hard sphere.
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APPENDIX B THERMODYNAMIC PROPERTIES OF COMPRESSED LIQUID MIXTURES FROM DCFI MODEL From Equation (64), the DCFI model can be written as C = ~ ~ ~3 a Q + a^p + a 2 P + a^p ( B Â— 1 ) where the reduced density coefficient a^ are a o = 9.8642 10 . 191x 1.5356 t 2 II f Â— 1 CO 28.465 + 30.864X + 6.0294t 2 a 2 = 27.542 32.898X 8.7130x 2 a 3 = 8.2606 + 12.737X + 4.0170x 3 and The pressure equation is given as 108
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109 p = p o p*RT{(pp 0 )C*[ a Q (pP 0 )+ji(p 2 5^) ( B2 ) The change of Hemholtz free energy is given as AA o TT p^irr i p ) C*[a Q lnÂ— P P 0 P (p5 a ) ^Cp 2 t 2 0 )^CpÂ’el)l ( B Â— 3 ) For binary mixtures, the simple mixing rules are given as r = X 1 V I + X 2 V 2 (B4) c* = X 1 C I * X 2 C 2 ( B Â— 5 ) T* = X 1 T 1 * x 2 T 2 ( B6 ) Thermodynamic properties of mixtures then can be obtained through the following relationships.
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110 where 3 2 A 3VÂ‘ T,N 3 P "3V T,N Nh. pRT [lC*(a 0 +a 1 p+a 2 p 2 +a 3 p 3 ) ] ( B Â— 7 ) 3 A / 3 A t,v,n 2 T,N 3 P TFT pv T,V,N 2 = "N>T " RT = y1 [l(Ca o + C*a^)2(C*a o P + C*a 1 P 1 + C*aÂ’p) (1+ r 2 ^^ C i a 2 P 2+2C * a 2 PPl +C * a 2p 2) (1+ ^ + 7 ) P p p i(C*a 3 p , + 3C*a ;5 p 2 p 1+ C* a ^p 3 )(l < _Â£ + ^ + _4) 1(1Â—) 5 P 2 P 3 o K o K o 2 ~3 P P P ( B Â— 8 ) a* = 9.8642 10.191 t 1 3.0712tt 1 a [ = 28.463 + 30.864t 1 + 12.0588tt ;l a 2 = 27.542 32.898x 1 17.426 tt 1 a 3 = 8.2606 + 12.737x^ + 8.0340x1^
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Ill and "! v~ T ! From Equations C B Â— 5 ) and (B6), the partial molar volume be calculated as v ( 3V ) 1 " ^'3TTJ ; T,P,n 2 ( 9P ) = TsrpÂ— v y\r ; T,N PtlÂ”C*(3Qf32P + 32P *+ 3 P ) 1 [l(C*a 0+ C*aÂ»)1 2 (C*a 1 p+C*a 1 p 1 +C*a Â’ p ) (1 )P 1 3 ( r* ^ 1Â”2 p ~ 2 aÂ„p 2 +2C*a 2 pp 1+ C*a^p 2 ) (l + _Â£ + _^)_ i(Cja 3 p 3 + 3C*a 3 p 2 p 1 + C*a'p 3 )(l + ^ + ^4j)] (B9) can
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112 M 1 1 3 A FT " RT 3N. t,v,n 2 = [l(CJa Q + C*a^)]lNÂ£_ [ 1( C*a Q + C*a Â’ ) P n i(Cj ai P o H.C*a 1 pÂ’ + c .a i P o ) J (C I a 2 Po +2C * a 2 5 o 5 o +C * a 2 5 o ) ?( c I*3Po + Â«*a 3 pJp> + C*a'pÂ’)]a!2 ) 2(CJa 1 p+C*a 1 p , +C^a ; [p)(l^Â£)i(C*a 2 p 2 + 2C*a 2 pp 1 + C^aÂ’p 2 )(l^)P j^(C*a 1 p' 5 + 3C*a ;J p 2 p 1 +C*a^p 3 )(ly) (B10) where
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113 3 2 A 3N? 3^ T,V,N, T, V,N. = 5v{ (2C ! a i +C * a S ) ln^[(2CI a ; + C*aJ ) + ^(2C*aiP i ; + 2C*aiPo +2C i a i 5 i +C * a lP a )+ ;C4C I a 2 Po5^2C*a^2 + 2 c *a 2 P i ; 2 + 4C*a^ o p; + C. a2 p 2 ) + 2 (6C*a 3 p 2 p' + 2Ci [ a ^^6C. a 3 P o p; 2 + 6C ! a jPoPi +C * a 3Po >:l(1 Â‘^ ) 2 (2C I a l5l +2C ! a iP +c * a lP>o N 1 (l~)g( 4c i a 2 PPi + 2c f a 2P 2 + 2C * a 2Pl + 4C * a 2^^1 + C * a 2^ Z) P 2 ^"^7^ ~Y2( 6 C*a^p 2 p^ + 2C*ajP' 5 + 6C*a jPp 2 + 6C*a^p 2 p 1 + C*a^ P 3 (1Tj) P ( B Â— 1 1 ) where
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114 2 a Â” = 9.8642 10.191 t 1 3.0712t 1 a = 28.465 + 30.864x 1 + 12.0588 x 2 a 2 = 27.542 32.898t 1 17.426T2 a'j = 8.2606 + 12.737x2 + 8.0340x2 Comparison of Equations (B7), (B8), and (Bll) with the following equations, 9P ~aâ€¢ T,N ^^^.(lx^C 2x x C x^C ) 1 1 11 1 2 L 12 x 2 L 22 ( B Â— 1 2 ) a / a a T T,V,N ) 2 T,N t,v,n 2 R T \r (1 " x i c n _x 2 c i2 ) ( 0 13 )
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115 T , V , N 2 t,v,n 2 RT/1 pV [ T. C ll> ( B14 ) The binary DCFI then can be calculated.
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Appendix C Computer Programs
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o o VOLUMETRIC PROPERTIES OF ARGON USING THE STROBRIGE EQUATION REF: TWU ET. AL. FLUID PHASE EQUILIBRIA, 4( 1980) IMPLICIT REAL*8(AH, 0Z) DIMENSION A ( 1 6 ) DATA A/1. 31024, 3. 80636, 2. 37238, 0. 798872, 0. 198761, 1 1. 47014, 0. 786367, 2. 19465, 5. 75429, 6. 78220, 9. 94904, 1 15. 6162, 86. 6430, 18. 5270, 9. 04755, 8. 68282/ DO 2 1=1, 8 READ (5, *> T PM=0. 6890606D0+2078. 76667D0*( (T/83. 80D0>**1. 59817868D0 11. ODO) TR=T / 1 50. 86D0 TS=1. 2593D0*TR WR ITE ( 6, 101 ) T, PM 101 FORMAT (2X, 2F13. 4/ ) TI1. ODO/TS FI =A ( 1 ) +TI* ( A< 2 ) +TI* ( A ( 3 ) +TI* ( A ( 4 ) +A ( 5 ) *TI*TI ) ) ) F2=A(6)+A(7)*TI F3=A ( 3 ) F4=TI*TI*TI*(A<9)+THMA< 10 ) +A < 1 1 > *TI > ) F5=TI*TI*TI* +A < 14 ) *TI ) ) F6=A( 15)*TI DO 1 N=l, 40 READ (5,*) V RR = 1. ODO/ ( 0. 01341D0*V) RS=0. 3189D0*RR R2=RS*RS R3=RS*R2 R4=RS*R3 R5=RS*R4 Dl=A( 16)*R2 E1=DEXP ( D1 ) F= 1 . ODO+F 1 *RS+F2*R2+F3*R3+F4*R2*E1 +F5*R4*E1 +F6*R 5 DF=Fl+2. 0D0*F2*RS+3. 0D0*F3*R2+F4*E1*<2. 0D0*RS2. ODO 1*A< 16)*R3)+F5*E1*<4. 0D0*R32. ODO*A( 16>*R5)+5. 0D0*F6*R4 OMC=RS*DF+F P=F*RR*13. 41D0*0. 0831434D0*T IF( P. GT. PM) GO TO 1 C=1 . ODOOMC WR ITF ( 6, 5) V, P, C 5 FORMAT (3F 13. 4) 1 CONTINUE 2 CONTINUE STOP END 117
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O Ct o 118 MOLLERUP'3 PROGRAM TO GENERATE THE DCFI DATA OF METHANE PROGRAM METHERM PRINTOUT ISOTHERMS. IMPLICIT REALMS (AH, OZ) COMMON/ONE/AG, AL, BB, BE, DE, GK, TTRP, PTRP, DTRP, TCRT, PCPT, DCRT COMMON/THRE/DPDT . D2PDT2, DLPDLR, DTSDR, DTHDR, DPSIDR, XB1, XB2, XD1, XD2 COMMON/SI X/GP i G, GY, GX, GXX 9 FORMAT ( IX, 3F13. 4) CALL METHAN DO 130 1=1,3 READ (5,*) TT PM=0. 117435675+1909. 40*( (TT/90. 68>**1. 851. 0) IF( TTTCRT) 103,103,104 103 DG=DENGAS ( TT > DL=DENLIQ(TT> 104 DO 120 N=l, 34 READC5, *) W DN=1000. ODO/W IF(TTTCRT) 114,115,115 114 IF(DN. GT. DG. AND. DN. LT. DL> GO TO 120 115 PP = DPDRF (TT, DN) IF(PP.GT. PM) GO TO 130 WRITE (6, 9) W, PP,GY 120 CONTINUE 130 CONTINUE STOP END SUBROUTINE METHAN C NOTE THAT PRESSURES ARE IN BARS, 1 ATM = 1.01325 BAR. C ONE BARLITER = 100 JOULES. IMPLICIT REAL*8 (AH, OZ) COMMON Al, A2, A3, A4, Bl, B2, B3, B4, Cl, C2, C3, C4, C5, C6, Dl, D2, D3, D4, D5 COMMON/ONE/AG, AL, BB, BE, DE, GK, TTRP, PTRP, DTRP, TCRT, PCRT, DCRT COMMON/THRE/DPDT, D2PDT2, DLPDLR, DTSDR, DTHDR, DPSIDR, XB1, XB2, XD1, XD2 COMMON/FIVE/ P, T, DEN, DPDD, E, H, S, CV, CP, W, WK COMMON/EIGHT / DDLDT C CONSTANTS FROM MARSHGAS 9/3/71. WRITE<6, 99) 99 FORMAT (31H Â•Â»** CH4 *** FEBRUARY 22 1983 ) TTRP=90. 68 PTRP=0. 117435675 DTRP=2S. 1472 C PCRT = 45. 956467 3AR TCRT=1.90. 53 PCRT =PSATF ( TCRT ) DCRT=10. 15 AG=. 5 AL=. 5 BB=0. 8 BE=4. 0 DE=1 . 2 GK = 0. 0831434 CJ = 100 WK = 101325/16. 043 A 1 =4. 15452847 A2=4. 77390186 A3=3. 51999044 A4=4. 25737412 Sl=l. 74744656 B2=2. 57679569
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119 33=0. 4653S980 01=5. 10335931 C2=12. 42022748 C3=6. 36124273 Dl = l. 38383919 D2=3. 54713477 D3=l. 37408815 RETURN END FUNCTION PSATF < T ) C METHANE VAPOR PRESSURE VIA PRYDZ/GOODWIN DATA, NOV. , 1970. C NOTE, PRESSURE IN BARS, 1.01325 BAR/ATM. IMPLICIT REAL*8 (AH, 0Z) COMMON/THRE/DPDT, D2PDT2, DLPDLR, DTSDR , DTHDR , DPSIDR, XB1, XB2, XD1, XD2 DATA PTRP / 0.117435675/, TTRP/90. 68/, TCRT/190. 53/, E/1. 5/ DATA A/4.7732553/, B/ 1 . 7665879/, C/0. 5702812/, D/1. 3311373/ 1 XK=1TTRP/TCRT X=( 1TTRP/T)/XK Q=lX IF (Q ) 2,3,4 2 PSATF = 0 DPDT = 0 RETURN 3 W=0 W1=0 GO TO 5 4 W = G**E W1 = E*W/Q 5 DXDT=TTRP/XK/T**2 Z = X*W Z1 = W X*W1 6 FZ = A*X + B*X**2 + C*X**3 + D*Z 7 FI = A + 2*B*X + 3*C*X**2 + D*Z1 8 PSATF= PTRP*DEXP ( FZ ) DPDT=PSATF*F1*DXDT RETURN END FUNCTION DENLIG(T) C METHANE SATD. LIQUID DENSITIES VIA GOODWIN. C THIS FUNCTION REFITTED VIA , DENSATLQ, 4/29/71. IMPLICIT REAL*8 (AH, 0Z) COMMON/EIGHT / DDLDT DATA TCRT/190. 53/,DCRT/10. 15/ DATA A/0.539403/, B/l. 896635/, C/0. 018387/, E/0.88/ 1 X = T/TCRT Z = 1 X IF ( Z. LT. .01) GO TO 7 2 F = B*X**2/Z XP = DEXP(F) 3 DFDX = Â— E* ( 2 X/Z)*X/Z G = Z**0. 36 4 G = A*Z + B*G + C*XP DENLIQ = DCRT* ( 1+G ) 5 G1 = C*XP*DFDX A 0. 36*B*G/Z 6 DDLDT = DCRT *G1 /TCRT RETURN 7 IF(Z. LT. 1. D09) Z = l.D09 G=Z*Â». 36 G=A*Z + B*Q DENLIQ = DCRT* ( 1 . +G )
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n Â° o n o o o 120 Gl = A . 36*B*Q/Z DDLDT = DCRT*G1/TCRT RETURN END FUNCTION TSATF ( DEN ) TSAT(DEN) FORMULA. METHANE, CONSTRAINED TO TRIPLE POINT, IPTS63. YIELDS ALSO THE FIRST DERIVATIVE RSP. TO RHO>DEN/DTRP. EQN. , (TCRT/T1 >/AZ = F ( R ) = U(S)*<1 + A1*LQG(R) + (R1)*W(R>), WHERE, U(S) = ( (Sl )/(STl ) >**(8/3), AZ = < TCRT/TTRP1 > , AND W(R) = A2 + A3*R + . . . + A8*R6. IMPLICIT REALMS (AH, OZ) COMMON/ONE/AG, AL, BB, BE, DE, GK, TTRP, PTRP, DTRP, TCRT, PCRT, DCRT COMMON/THRE/DPDT , D2PDT2, DLPDLR, DTSDR, DTHDR, DPSIDR, XB1, XB2, XD1, XD2 DIMENSION A ( 8 ) DATA A / 0.8142449, 2.4794529, 6.7598384, 41.05600105, 1 133.4449230, 223.4955973, 181.8267154, 58.8359684/ 1 AZ = TCRT/TTRP 1 DSDR = DTRP/DCRT SK = DSDR 1 2 R=DÂ£N/DTRP S=DEN/DCRT IF ( (SD/SK. LT. 0. ) Q=DABS( < Sl > /SIX ) **. 333333 IF( (Sl )/SK. GE. 0. ) Q=( (SD/SK)**. 333333 IF ( Q ) 3,9,3 3 U = G**8 U1 = 8*DSDR*Q**5/SK/3 4 V = R 1 W=0 W1=0 DO 6 K=2, 8 5 W = W A ( K ) *R** ( KÂ— 2 ) Ml Â» W1 + ( K2 ) *A ( K ) *R** ( KÂ— 3 ) 6 CONTINUE G=DLOG(R> F = U*(l + A ( 1 ) *G + V*W) 7 FI = U1 + A ( 1 )* ( U/R + U1*G ) + U*V*W1 + U*W + U1*V*W 8 TSATF=TCRT/( 1+AZ*F> DTSDR=( AZ/TCRT ) *F 1 *TSATF**2 RETURN 9 TSATF = TCRT DTSDR = 0 RETURN END FUNCTION THETAF (DEN) IF S 1, U = AG* ( SÂ— 1 ) **3. IF S A 1, U = AL* ( Sl ) **3. YIELDS ALSO THE FIRST DERIVATIVE RSP. TO RHODEN /DTRP. IMPLICIT REAL*8 (AH, OZ) COMMON/ONE/AG. AL, BB, BE, DE, GK, TTRP, PTRP, DTRP, TCRT, PCRT, DCRT COMMON/THRE/DPDT, D2PDT2, DLPDLR, DTSDR, DTHDR, DPSIDR, XB1, XB2, XD1, XD2 1 S=DEN/DCRT DSDR = DTRP/DCRT IF(S) 9,9,2 2 Q=S1 Q2=Q**2 Q3=Q**3 I F ( G ) 3,8,4 3 U = AG*Q3 U1 = 3*AG*Q2*DSDR GO TO 5 4 U = AL*Q3
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O U o o 121 U1 = 3*AL*G2*DSDR 5 IF (U. LT. 50. ) U=50. XP = DEXP < U ) TS = TSATF ( DEN ) THETAF = TS*XP 7 DTHDR = TS*U1*XP + DTSDR*XP RETURN 8 THETAF = TCRT DTHDR = 0 RETURN 9 THETAF = 0 DTHDR= 1 . E 05 RETURN END FUNCTION PHIF(T) XB = PHI = X*< lEXP(U) >, U = BB + BE/X, YIELDS ALSO DPHI/DR, DPHI/DX, AND D2PHI/DX2. IMPLICIT REAL*8 (AH, 0Z) COMMON/ONE/AG, AL, BB. BE, DE, GK, TTKP, PTRP, DTRP, TCRT, PCRT, DCRT COMMON/THRE/DPDT, D2PDT2, DLPDLR, DTSDR, DTHDR, DPSIDR, XB1, XB2, XD1, XD2 1 X=T /TCRT U=BB+BE/X Ul=~Bt/X**2 U2=2*U1/X 2 XP=DEXP(U) Z=1XP Z1=U1*XP Z2(U2U1**2)*XP 3 PHIF X*Z XB 1 = X*Z1 > Z XB2 = X*Z2 + 2*Z1 9 RETURN END FUNCTION PSIF (T, DEN) XD = PSI = <1W*L0G< l+l/W) >/X. W = (TTH)/(DE*TCRT>. YIELDS ALSO XD1 > DPSI/DX, XD2 > D2PSI/DX2, AND DPSI/DR IMPLICIT REAL* 8 (AH, 0Z) COMMON/ONE/AG, AL, BB, BE, DE, GK, TTRP, PTRP, DTRP, TCRT, PCRT, DCRT COMMON/THRE/DPDT, D2PDT2, DLPDLR, DTSDR, DTHDR, DPSIDR, XB1, XB2, XD1, XD2 1 TH = THETAF (DEN) W = ( TTH ) /DE/TCRT IF < W ) 2,2,3 2 PSIF=1 XD1 ~Q XD2=0 DPS IDR=0 RETURN 3 X=T /TCRT DWDR= DTHDR /DE/TCRT DWDX=1/DE 4 U=l/X DUDX=U/X D2UDX2=2*DUDX/X 5 G=DLOG< 1. ODO+1. ODO/W) WS = 1+W V =Â• 1W*G 6 DVDW 1/WSG D2VDW2 = 1/W/WS**2 PSIF = U*V 7 DPSIDR = U*DVDW*DWDR
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n o o 122 XD1 = U*DVDW*DWDX + V*DUDX 8 XD2 = U*D2VDW2*DWDX**2 + 2*DUDX*DVDW*DWDX + V*D2UDX2 9 RETURN END FUNCTION DPDRF(T, DEN) NOTE, DPDRF = PRESSURE, BAR. REDUCED DERIVATIVE, DLPDLR > R* / ( TCRTTTRP > WÂ— CICRT/T1 >/(TCRT/TTRPl ) 3 F = A(1)*W + A<2)*U**0. 36
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123 DO 4 K=3, 5 4 F = F + A ( K > *UÂ»* < K2 ) 9 DENGAS = DCRT*DEXP
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o o 124 NBS PROGRAM TO GENERATE THE THERMODYNAMIC SURFACE FOR WATER MAIN PROGRAM BLOCK DATA IMPLICIT REALMS (AH, OZ) REAL P, Q COMMON/ACONST/WM, GASCON, TZ, AA, ZB, DZB, YB, UREF, SREF COMMON/NCONST/G ( 40 ) , 1 1 ( 40 ) , UJ ( 40 ) , NC C0MMGN/ELLC0N/G1, G2, GF, Bl, B2, BIT, B2T , B1TT, B2TT COMMON/BCONST /P ( 10 ) , Q< 10) COMMON/ADDCON/ ATZ ( 4 ) , ADZ ( 4 ) , AAT ( 4 ) , AAD ( 4 ) DATA ATZ/2*64. Dl, 641. 6D0, 27. Dl/, ADZ/3*. 319D0, 1. 55DO/, AAT/2*2. D4 $, 4. D4, 25. DO/, AAD/34. DO, 4. Dl, 3. Dl, 1. 05D3/ DATA WM/ 18. 0152D0/, GASCON/. 461522D0/, TZ/647. 073D0/, AA/1. DO/, NC/36/ DATA UREF, SREF/432S. 455039D0, 7. 6180802D0/ DATA Gl, G2, GF/11. DO, 44. 333333333333D0, 3. 5D0/ DATA P/. 7478629, . 3540782, 2*0. , . 007159876, 0. , . 003528426, 3*0. / DATA G/l. 1278334, 0. , . 5944001, 5. 010996, 0. , . 63684256, 4*0. / DATA G/. 53062968 529023D3, . 22744901424408D4, . 78779333020687D3 . 69030527374994D2, . 17863832875422D5, . 39514731563338D5 *, . 33803884280753D5, . 13855050202703D5, . 25637436613260D6 *, . 4321257598141 5D6, . 34183016969660D6, . 122231 5641 7448D6 $, . 1 1 777433655832D7 , . 217348101 10373D7, . 10829952168620D7 25441 998064049D6, . 31377774947767D7, . 5291 1910757704D7 *, . 1 3302577 177S77D7, . 25109914369001D6, . 465618261 1 5608D7 *, . 72752773275387D7 , . 417742461 48294D6, . 14016358244614D7 *, 3 1 55523 1 392 1 27D7 , . 47929666384584D7, . 40912664781209D6 %, . 1 3626369388386D7 , . 69625220862664D6, . 10834900096447D7 $, . 2272282740 1688D6, . 33365486000660D6, . 68833257944332D4 $, . 21757245522644D5, . 26627944829770D4, . 7073041 8082074D5 *, . 225D0, 1. 68D0. . 055D0, 93. ODO/ DATA 11/4*0, 4*1, 4*2, 4*3, 4*4, 4*5, 4*6, 4*8, 2*2, 0, 4, 3*2, 4/ DATA JU/2, 3, 5, 7, 2, 3, 5, 7, 2, 3, 5, 7, 2, 3, 5, 7, 2, 3, 5, 7, 2, 3, 5, 7, 2, 3, 5, 7 2, 3, 5, 7, l,3*4, 0, 2, 0, 0/ END SUBROUTINE BB (T) IMPLICIT REAL*8 (AH, OZ) REAL P, Q C0MM0N/ELLC0N/G1 , G2, GF, 31, B2, BIT, B2T, B ITT , B2TT COMMON/ACONST/WM, GASCON, TZ, AA, Z, DZ, Y, UREF, SREF COMMON/BCCNST /P ( 10 ) , Q(10> DIMENSION V(10) V( 1 )=1. DO 2 1=2, 10 2 V ( I > =V < 11 )*TZ/T B 1 =P < 1 ) +P < 2 > *DLOG ( 1 . / V ( 2 ) ) B2=G( 1 ) B1T=P(2)*V(2)/TZ B2T=0. B1TT=0. B2T1 =0. DO 4 1=3, 10 B1=B1+P< I )*V< 11 ) B2=B2+Q( I )*V( 11 ) S 1T=B IT< 12 >*P ( I ) *V ( I Â— 1 )/T B2T=B2T< I2)*Q< I )*V( 11 >/T B 1 TT=B 1 TT *P ( I ) * ( I 2 ) **2*V ( I 1 ) /T/T 4 B2TT=B2TT+Q( I ) * ( I 2 ) **2*V ( I 1 ) /T/T B 1 TT =R 1 TTB 1 T/T B2TT=B2TTB2T/T RETURN
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125 END SUBROUTINE BASE(D,T) IMPLICIT REAL*8 (AH, OZ ) COMMQN/ELLCON/G1 , G2, GF, Bl, B2, BIT, B2T , B ITT, B2TT COMMQN/BASEF/AB, GB, SB, UB, HE, CVB, DPDTB COMMQN/ACONST/WM, GASCON, TZ, A, Z, DZ, Y, UREF, SREF Y=. 25*B1*D X=1 . Y ZO=(l. +G1*Y+G2*Y*Y> /X**3 Z = Z0>4. *Y*(B2/B1GF> DZ0=(Gl+2. *G2*Y) /X**3+3. *( 1. +G1*Y+G2*Y*Y > /X**4 DZ=DZ0+4. * < B2/B 1 GF ) AB=DLQG < X ) ( G21 . >/X+28. 16666667D0/ X/X+4. *Y* ( B2/B 1GF ) *+15. 166666667D0+DL0G ( D*T*4. 55483D0) GB=AB *Z BB2TT=T*T*B2TT UB=T*B IT* ( Z1 . D*B2 ) /B 1 D*T*B2T HB=Z+UB CVB=2. *UB+ ( ZOi . )*( (T*B1T/B1 )**2T*T*BlTT/Bl ) SD* < BB2TTGF*B 1 TT*T*T ) ( T*B 1 T/B 1 > **2*Y*DZ0 DPDTB=Z/T+D*(DZ*BlT/4. +B2TB2/S1*B1T) SB=UB AB RETURN END SUBROUTINE GG(T, D> IMPLICIT REAL*8 (AH, OZ ) COMMQN/RESF/AR, GR, SR, UR, HR, CVR, DPDTR COMMON/QQQQ/Q, G5 DIMENSION QR(ll), QT(IO), GZR(9), QZT(9> EQUIVALENCE ( QR (3) , QZR ( 1 ) ) , ( GT ( 2) , QZT ( 1 ) > COMMON/NCONST/G ( 40 ) , 1 1 ( 40 ) , UU ( 40 ) , N COMMON/ACONST/WM, GASCON, TZ, AA, Z, DZ, Y, UREF, SREF COMMON/ADDCON/ATZ ( 4 ) , ADZ ( 4 ) , AAT ( 4 ) , AAD ( 4 ) RT=GASCON*T GR< 1 >=0. Q5=0. G=0. DO AR=0. DO DADT=0. CVR=0. DPDTR=0. E=DEXP(AA*D) Q10=D*D*E Q20=l. DOE GR ( 2 ) =Q10 V=TZ/T GT( 1 >=T/TZ DO 4 1=2, 10 GR ( I + 1 ) =QR ( I ) *Q20 4 QT( I )=QT( 11 >*V DO 10 1 = 1, N K.= I I ( I ) + l L=JJ( I ) ZZ=K GP=G( I )*AA*QZR(KÂ— 1 >*QZT(L> Q=Q+QP Q5=G5+AA*(2. /DAA*( 1. E* ( Kl ) /Q20 ) > *GP AR=AR+G( I >*QZR(K)*QZT(L> /Q10/ZZ/RT DFDT=G20**K* ( 1L ) *QZT ( L+l ) /TZ/K. D2F=L*DFDT
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126 DPT=DFDT*Q10*AA*K/Q20 DADT=DADT+G( 1 ) *DFDT DPDTR=DPDTR+G( I )*DPT 10 CVR=CVR+G( I )*D2F/GASC0N QP=0. Q2A=0. DO 20 J=37Â» 40 IF(G(J). EQ. 0. DO) GO TO 20 K=I I ( J) KM= J J ( J ) DDZ=ADZ ( J36) DEL=D/DDZ1. IF (DABS (DEL) . LT. 1. D10) DEL=1 . D10 DD=DEL*DEL EX 1 = AAD ( J36 ) *DEL**K DEX=DE XP ( EX 1 > *DEL**KM ATT =AAT ( J36 > TX=ATZ ( JÂ— 36) TAU=T/TXÂ— 1. EX2=ATT*TAU*TAU TEX=DE XP ( EX2 ) Q10=DEX*TEX GM=KM/DELK*AAD ( J36 ) *DEL** ( Kl ) FC T=GM*D**2*Q 1 0 /DD Z Q5T=FCT* ( 2. /D+GM/DDZ ) ( D/DDZ ) **2*Q1 0* ( KM/DEL/DEL+ ( K1 > *AAD ( J36 ) *DEL** ( K2 ) > Q5=G5+G5T*G( J) GP=QP iG ( J ) Â»FCT DADT=DADT2. *G( J) *ATT*TAU*Q 1 0/TX DPDTR=DPDTR2. *G ( J > *ATT*TAU*FCT/TX G2A=Q2A+T*G( J)*(4. *ATT*EX2+2. *ATT)*Q10/TX/TX AR=AR >Q10*G( J)/RT 20 CONTINUE SR=DADT /GASCON UR=AR Â»SR CVR=CVR+G2A/GASC0N G=Q+GP RETURN END SUBROUTINE DFIND ( DOUT, P , D, T, DPD ) IMPLICIT REAL*8 (AH, 0Z) COMMON/QQGQ/QO, Q5 COMMON/ACONST/WM, GASCON, TZ, AA, Z, DZ, Y, UREF, SREF DD=D RT=GASCON*T IF ( DD. LE. 0. ) DD=1. D8 IF ( DD. GT. 1. 9) DD=1 . 9 L=0 9 L=L+1 11 IF(DD. LE. 0) DD=1. D8 IF ( DD. GT. 1. 9) DD=1. 9 CALL BASE(DD, T> CALL QQ(T, DD) PP=RT*DD*Z+QO DPD=RT* ( Z+Y*DZ ) +Q5 IF(DPD. GT. 0. DO) GO TO 13 IF(D. GE. . 2967D0) DD=DD*1.02D0 IF(D. LT. . 2967D0) DD=DD*. 98D0 IF ( L. LE. 10) GO TO 9 13 DPDX=DPD*1. 1
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127 IF ( DPDX. LT. . 1 > DPDX. = . 1 DP=DABS( 1. PP/P ) IF (DP. LT. 1. D8) GO TO 20 IF(D. GT. .3 . AND. DP.LT. l.D7) GO TO 20 IF ( D. GT. . 7 . AND. DP. LT. 1. D6) GO TO 20 X= ( PPP ) /DPDX IF ( DABS ( X ) . GT. . 1 ) X=X*. 1 /DABS ( X > DD=DD !X IF( DD. LE. 0. > DD=1 . D8 19 IF( L. LE. 30) GO TO 9 20 CONTINUE DOUT=DD RETURN END SUBROUTINE THERM ( D, T ) IMPLICIT REAL*8 (AH. 0Z ) COMMON/ACONST/WM, GASCON, TZ, AA, ZB, DZB, Y, UREF, SREF COMMON/QGQQ/QP, QDP COMMON/BASEF/AB, GB, SB, UB, HB, CVB, DPDTB COMMON/RESF/AR, GR, SR, UR, HR, CVR, DPDTR COMMON/ IDF/A I, GI, SI, UI, HI, CVI, CPI COMMON/FCTS/AD, GD, SD, UD, HD, CVD, CPD, DPDT , DVDT, CJTT, CJTH CALL IDEAL (T) RT=GASCON*T Z=ZB+GP/RT/D DPDD=RT* ( ZB+Y*DZB > +QDP AD=AB +AR+A I UREF /T +SREF GD=AD+Z UD=UB+UR+UIUREF/T DPDT=RT*D*DPDTB+DPDTR CVDCVB+CVR+CVI CPD=CVD+T*DPDT**2/ ( D*D*DPDD*GASCON ) HD=UD Z SD=SB >SR+SISREF DVDT=DPDT /DPDD/D/D CJTT1. /DT*DVDT C JTH=C JTT /CPD/GASCON RETURN END FUNCTION PS ( T ) IMPLICIT REAL*8 (AH, OZ) DIMENSION A ( 8 ) DATA A/7. 8889166D0, 2. 5514255D0, 6. 716169D0 $, 33. 239495D0, 105. 38479D0, 174. 35319D0, 148. 39348D0 S, 48. 631602D0/ IF(T. GT. 314. DO) GO TO 2 PL=6. 35731 18D08858. 843D0/T+607. 56335D0*T** ( 6) PS=. 1*DEXP(PL) RETURN 2 V=*T/647. 25D0 W=DABS( 1. DOV) B=0. DO DO 4 1=1, 3 Z = I 4 S=B+A( I )*W**( (Z+l. )/2. ) Q=B/V PS=22. 093D0*DEXP(Q) RETURN END FUNCTION TSAT(P)
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128 REAL itS P, PS, TO, TSAT, TDPSDT TSAT =0. IF(P. GT. 22. 05) RETURN ft=0 PL=2. 302585*DL0G(P> T <3=372. 83+PL* ( 27. 7589+PL*(2. 3819+PL*(. 24834+PL*. 0193855) ) ) 1 IF(T<3. LT. 273. 15) TG=273. 15 IF ( TG. GT. 647. ) TG=647. IF ( ft. LT. 8) GO TO 2 WRITE(6, 3) ft, P, PP, TG 3 FORMAT ( ) GO TO 8 2 ft=ft+l PP=PS ( TG ) DP=TDPSDT (TG*> IF(DABS( 1. PP/P). LT. . 00001 ) GO TO 8 TG=TG* ( 1 . + ( PPP > /DP ) GO TO 1 3 TSAT=1 G RETURN END FUNCTION TDPSDT (T) IMPLICIT REAL*8 (AH, 0Z) DIMENSION A ( 8 ) DATA A/7. 8S89166D0, 2. 5514255D0, 6. 716169D0 33. 239495D0, 105. 38479D0, 174. 35319D0, 148. 39348D0 *, 48. 631602D0/ V=T/647. 25 W=l. V B=0. C=0. DO 4 1 = 1, 8 Z=I Y=A(I)*W**< (Z+l. )/2. ) C=C+Y/W*<. 5. 5*ZÂ— 1. /V) 4 B=BfY Q=B/V TDPSDT=22. 093*DEXP(Q)*C RETURN END SUBROUTINE IDEAL(T) IMPLICIT REAL*8 (AH, 0Z) COMMON/ 1 DF/AI, GI, SI, UI, HI, CVI, CPI DIMENSION C ( 18 ) DATA C/. 1 9730271 018D2, . 20966268 1977D2, . 483429455355D0 i, . 605743189245D1, 22. 56023885D0, 9. 87532442D0, . 43135538513D1 *, . 4581 55781D0, . 47754901883D1 , . 41238460633D2, 27929052852D3 $, . 14481695261D4, . 56473658748D6, . 16200446D7, . 3303822796D9 *, . 451916067368D1 1 , . 370734122708D13, . 13754606823SD1 5/ TT=T/1. D2 TL=DLOG ( TT ) GI=(C(I)/TT+C(2) ) *TL HI = (C(2)+C( 1 )*( 1. DOTD/TT) CP I =C ( 2 ) Â— C ( 1 ) /TT DO 8 1=3, 18 GI=GI C< I )*TT**( 16) HI=HI+C( I )Â•Â»( I6)*TT**( 16) 8 CPI=CPI+C ( I >*< I6)*( I5)*TT**( 16) AI=GI 1 U I =H I 1
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129 CVI=CPI1 SI=UI AI RETURN END SUBROUTINE SECVIR ( T, VIR ) IMPLICIT REALMS (AH, OZ) COMMON/NCONST/G ( 40 ) , 1 1 ( 40 ) , J J ( 40 > , NC C0MMQN/ELLCQN/G1 , G2, GF, BB1, BB2, BIT, B2T, B ITT, B2TT CQMMON/QQGG/QO, Q5 COMMON/ACONST/WM, GASCON, TC, AA, Z, DZ, Y, UREF, SREF CALL BB(T> V=TC/T VIRBB2 DO 3 J=l, NC IF ( 1 1 ( J) . NE. 0 ) GO TO 3 L=UU(U) VIR=VIR+G( J)*V**(L1 ) /T/GASCON 3 CONTINUE RETURN END SUBROUTINE CORR ( T, P, DL, DV, DELG > IMPLICIT REAL*8 (AH, OZ) COMMON/QQQG/QOO, Q1 1 COMMON/ACONST/WM, GASCON, TZ, AA, ZB, DZB, YB, UREF, SREF COMMON/FCTS/AD, GD, SD, UD, HD, CVD, CPD, DPDT, DVDT, CJTT, C J DLIG=DL IF( DL. LE. 0. ) DLIQ=1. 11. 0004*T CALL 3B(T> RT=GASCON*T CALL DFIND(DL, P, DLIQ, T, DQ) CALL THERM (DL, T) GL=GD DVAP=DV IF(DV. LE 0. ) DVAP=P/GASCON/T CALL DFIND(DV, P, DVAP, T, DQ> IF ( DV. LT. 5. D7) DV=5. D7 CALL THERM (DV, T) GV=GD DELG=GLGV RETURN END SUBROUTINE PCORR ( T, P , DL, DV ) IMPLICIT REAL*8 (AH, OZ) COMMON/ACONST/WM, GASCON, TZ, AA, ZB, DZB, YB, UREF, SREF P=PS ( T > 2 CALL CORR ( T, P, DL, DV, DELG) DP=DELG*GASCON*T / ( 1 . /DV1. /DL) P=P+DP IF( DABS (DELG ) . LT. 1. D4) RETURN GO TO 2 END SUBROUTINE TCORR ( T, P , DL, DV ) IMPLICIT REAL*8 (AH, OZ) COMMON/ACONST/WM, GASCON, TZ, AA, ZB, DZB, YB, UREF, SREF T=TSAT ( P ) 2 CALL CORR (T, P, DL, DV, DELG) DP=DELG*GASCON*T/( 1. /DV1. /DL) T=T*(1. DP/TDPSDT (T) ) IF (DABS (DELG) . LT. 1. D4) RETURN GO TO 2
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130 END SUBROUTINE UNIT IMPLICIT REAL*8 (AH, OZ) DOUBLE PRECISION NT, ND, NP, NH, NNT, NND, NNP, NNH COMMON/UNITS/ IT, ID, IP, IH, NT, ND, NP, NH, FT, FD, FP, FH DIMENSION FFD ( 4 ) , FFP(5>, FFH ( 6 ) , NNT ( 4 ) , NND ( 4 ) , NNP(5), NNH ( 6 ) DATA FFD/ 1 . D3, 1. DO, . 0180152D0, . G16018D0/ , DATA FFP/1. DO, 10. DO, 9. 869232667D0, 145. 038D0, 10. 1971D0/ DATA FFH/2*1. DO, 18. 0152D0, . 23884590D0, 4. 30285666D0, . 42D0/ DATA NNT/1HK, 1HC, 1HR, 1HF/ DATA NND/6HKG/M3 , 6HG/CM3 , 6HMQL/L , 6HLB/FT3/ DATA NNP/6H MPA , 6H BAR , 6H ATM , 6H PSI , SHKG/CM2/ DATA NNH/ 6HKJ/KG , 6H J/G , 6HJ/M0L , 6HCAL/G , 7HCAL/M0L, oHBTU/LB/ DATA Al, A2, A3, A4/8HTEMPERAT , 7HDENSITY $, 8HPRESSURE, 8HENERGY / WRITE<6, 11 > Al 30 WR ITE ( 6, 12) READ (5, *, END=99) IT IF ( IT. EQ. 0) STOP IF ( IT. GT. 4) GO TO 30 NT=NNT ( IT) WRITE (6, 11 ) A2 31 WRITE<6, 13) READ (5, *, END=99) ID IF ( ID. GT. 4 . OR. ID. LT. 1 ) GO TO 31 ND=NND( ID) FD=FFD< ID) WRITE (6, 11) A3 32 WRITE (6, 14) READ <5, *, END=99> IP IF< IP. GT. 5 . OR. IP. LT. 1 ) GO TO 32 NP=NNP( IP) FP=FFP( IP) WRITE (6, 11 ) A4 33 WRITEC6, 15) READ <5, *, END=99> IH IF C IH. GT. 6 . OR. IH. LT. 1 ) NH=NNM( IH) FH=FFH( IH) RETURN 99 STOP 11 FORMAT ( 12 FORMAT ( 13 FORMAT ( 14 FORMAT ( 15 FORMAT ( *ES/MOL, END FUNCTION ENTER UNITS CHOOSE FROM CHOOSE FROM CHOOSE FROM CHOOSE FROM 6=3TU/LB ' ) TTT(T) GO TO 33 CHOSEN FOR ' , A8 ) 1=DEG K, 2=DEG C, 3=DEG R, 4=DEG F') 1=KG/M3, 2=G/CM3, 3=M0L/L, 4=LB/FT3') 1=MPA, 2=BAR, 3=ATM, 4=PSIA, 5=KG/CM2 Â‘ 1=KU/KG, 2=U/G, 3=J/M0L, 4=CAL0RI DOUBLE PRECISION T, TTT, FT, FD, FP, FH, NT, ND, NP, NH COMMQN/UNITS/ IT , ID, IP, IH, NT, ND, NP, NH, FT, FD, FP, FH GO TO <1, 2, 3, 4), IT 1 TTTÂ— T FT=1. RETURN 2 TTT =T +273. 15D0 FT=1. RETURN 3 TTT =T/ 1 . 8D0 FT=. 55555555 5 5556D0
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131 RETURN 4 TTT=(T+459. 67D0>/1. 8D0 FT=. 5555555555556D0 RETURN END FUNCTION TTI ( T > DOUBLE PRECISION T, TTI, FT, FD, FP, FH, NT, ND, NP, NH COMMON/UNITS/IT, ID, IP, IH, NT, ND, NP, NH, FT, FD, FP, FH GO TO (5, 6, 7, 8), IT 5 TTI~T RETURN 6 TTI =T273. 1 5DO RETURN 7 TTI=T*1. SDO RETURN 8 TTI=T*1. 8D0459. 67D0 RETURN END C MAIN IMPLICIT REAL*8 (AH, OZ) DOUBLE PRECISION NT, ND, NP, NH COMMON/UNITS/IT, ID, IP, IH, NT, ND, NP, NH, FT, FD, FP, FH CQMMGN/QGQQ/QO, Q5 COMMON/FCTS/AD, GD, SD, UD, HD, CVD, CPD, DPDT COMMON/ACONST/WM, GASCON, TC, AA, Z, DZ, Y, UREF, SREF COMMON/NCONST/G ( 40 ) , 1 1 ( 40 ) , JU ( 40 ) , NC DATA NS1, NS2/2H M, 2HFT / CALL UNIT NS=NS1 IF ( ID. EQ. 4) NS=NS2 15 READ (5, *, END=9) IOPT, XISO IF ( IOPT. EG. 0) GO TO 9 GO TO ( 101, 201, 301 ), IOPT 101 READ (5, *, END=9) JOPT, Yl, Y2, YI IF(JOPTl) 15,101,103 103 TT=X ISO T=TTT(TT) IF( JOPT. EQ. 2) DGSS=Y1 /FP/T/. 4 IZ=0 CALL BB ( T > PSS= 20000. DW=0. IF(T. IT. TC> CALL PCORR ( T, PSS, DLL, DW ) DGSS=DW IF ( DGSS. EQ. 0. > DGSS=1. 11. 0004*T PSAT=PSS*FP IF(JOPT. EG. 2 .AND. Yl.GT. PSAT) IZ=3 IF(Y1. GT. PSAT) DGSS=DLL IF( JOPT. GE. 3) IZ=3 PIN=Y1YI DIN=P IN PINC=YI/FP DINC=YI*FD 22 IF< JOPT. EQ. 2) PIN=PIN+YI IF( JOPT. EG. 3) DIN=DINtYI IF ( JOPT. EQ. 2 .AND. PIN. GT. Y2) GO TO 101 IF(JOPT. EQ. 3 .AND. DIN. GT. Y2) GO TO 101 IF( JOPT. EQ. 2) PRES=P IN/FP IF( JOPT. EQ. 3) D=DIN/FD 24 CONTINUE
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132 26 27 IF (JOPT. EQ. 3 . OR. (JOPT IF (JOPT. EQ. 1 TSAVE= TTY I . AND. T. LT IF (JOPT. EQ. 1 PSAVE=P INYI . AND. IOPT IF( JOPT. EQ. 2 . AND. PRES IF( JOPT. EQ. 1 . AND. T. GT 2 . AND. PIN. GO TO 26 LT. PSAT) > GO TO 26 23 EQ. 2 .AND. IZ.LE. 2) TT=TT I ( TS ) GT. PSAT/FP .AND. IZ.GE. 2) GO TO 26 TS .AND. IZ.GE. 2) GO TO 26 PRES=PSAT/FP T=TS JOPT. EQ. 1 ) JOPT. EQ. 2) JOPT. EQ. 2) JOPT. EQ. 1 ) IZ=IZ>1 IF (JOPT. EQ. 2) IF( JOPT. EQ. 1 > IF ( I Z. EQ. 1 . AND. IF( IZ. EQ. 1 . AND. IFC IZ. EQ. 2 . AND. IF ( IZ. EQ. 2 . AND. CALL BB(T> I F ( IQPT. NE. 3 CALL GQ ( T, D ) CALL BASE(D, T> RT=GASCON*T PDUM=D*RT*Z+QO IF (JOPT. EQ. 3 .OR. JOPT. EQ. 3) IF(ICPT. EQ. 3 .OR. JOPT. EQ. 3) DGSS=D+PINC/DQ CALL THERM (D, T) U=UD*T*GASCON*FH C=DSQRT ( DABS ( CPD*DQ*1 . D3/CVD ) ) IF ( ID. EQ. 4 > C=C*3. 280833 H=HD*T*GASCON*FH DGSS=DLL DGSS=DW DGSS=DLL DGSS=DW AND. JOPT. NE. 3) CALL DFIND(D, PRES, DGSS, T, DQ) PRES=PDUM DQ=RT*(Z+Y*DZ)+Q5 S=SD*GASCON*FH*FT DPDTX=DPDT*FP*FT DPDD=DQ*FP*FD C0MP=1. D3/D/DQ/FP DDDTL1 . D3*DPDT/D/DQ CP=CPD*GASCON*FH*FT CV=CVD*GASCQN*FH*FT VL=FD/D DQUT=1. 0D03*VL POUT =PRES*FP ROUT=DOUT / 1 7. 857142D0 X0UT=R0UT1. DO GP=DPDD/0. 0831434D0/TT GP 1 =1 . DOGP WRITE(6, 21) TT, POUT, DOUT, GP 1 21 F0RMAT(F9. 3, 3F12. 5) IF(IZ. EQ. 1) WRITE(6Â» 12) 12 FORMAT ( ' * ) IF( IZ. EQ. 1 ) GO TO 23 IF(IZ. EQ. 2 .AND. JOPT. EQ. 2) PIN=PSAVE IFdZ.EQ. 2 .AND. JOPT. EQ. 1 ) TT=TSAVE IFdZ.EQ. 2) I Z=3 GO TO (22, 204, 304), IOPT 201 J0P1=1 PRES= X ISO/FP 202 READ (5, *, END=9) T1,T2, YI IF(T1. EQ. 0. ) GO TO 15 TT=T1 YI T=TTT ( TT ) CALL TCORR(TS, PRES, DLL, DW) D=DLL
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133 IF< . LT. TS ) IZ=0 204 TT=TT+YI T =TTT ( TT ) IF ( TT. GT. T2 > GO TO 202 CALL BB < T > DGSSD GO TO 24 301 J0PT=1 D=X ISO 302 READ <5< *, END=9 > T1,T2Â»YI IF(T1. LE. 0. ) GO TO 15 TT=T1 YI I Z=3 T=TTT GO TO 302 CALL RB(T> GO TO 27 9 STOP END
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o o o 134 CORRESPONDING STATES CORRELATION OF THE DIRECT CORRELATION FUNCTION INTEGRALS OF PURE LIQUIDS MAIN PROGRAM IMPLICIT REAL*8(AH, OZ) DIMENSION P ( 4 > , STEP ( 4 ) , AB<4>, BB<4). NAME (20) DIMENSION TEMP ( 22 ) , TR(22), DR(8>, RRHO(S), DEXP(8), 1DCAL ( S ) Â» DEV (8). ERR (8) DIMENSION VI (8, 22), Cl (8, 22), PI (8, 22) , PI I (8), 1PRES ( 8 ) , V<8),VII<8),CII(8> COMMON/DATA/TEMP, VI, Cl, DEXP, DCAL, VC COMMON/COND/NP, NU, NN, TERM COMMON/PR INT/DDEX (8, 22), DDCA(8, 22) COMMON/PARAM/RHOS, DCFS, TS READ (5,1) NAME 1 FORMAT (20A4) READ (5, 3) NJ 3 FORMAT (12) WRITE (6,4) NU 4 FORMAT (IX, 'NO. OF ISOTHERMS= ',12,//) READ(5, 20) NP, 10 20 FORMAT (212) READ( 5, 7) NPASS, (P(I), 1=1, NP ) , (STEP( I), 1=1, NP ) READ (5, 8) (AB( I >, 1 = 1, NP > , (BB( I ), 1 = 1, NP) 7 FORMAT( 12, 6F10. 0) 8 FORMAT (3X, 6F10. 0) WRITE (6, 101 ) 101 FORMAT (IX, 'PATTERN SEARCH ESTIMATION OF PARAMETERS FROM 1DCFI DATA '/IX, 'CONTROL PARAMETERS WERE SET AS FOLLOWS'/) WR ITE (6, 102) NP, 10 102 FORMAT< IX, 5H NP=, I2/1X, 5H I0=, 12) WRITE (6, 106) NPASS 106 FORMAT ( IX, 15HNUMBER OF PASS=, 12) WRITE (6, 107) 107 FORMAT( IX, 18HINITIAL PARAMETERS) WRITE(6, 108) (I, P( I), 1 = 1, NP) 108 FORMAT( IX, 12, F10. 3) WR ITE( 6, 109) 109 FORMAT( IX, 12HSIZE OF STEP) WRITE(6, 110) (I, STEP(I), 1=1, NP) 110 FORMAT( IX, 12, F10. 3) WRITE ( 6, 111 ) 111 FORMAT (IX, 30HL0WER LIMITS OF THE PARAMETERS) WR ITE (6, 112) ( I, AB < I ) , 1=1, NP ) 112 FORMAT( IX, 12, F10. 3) WRITE (6, 113) 113 FORMAT ( IX, 30HUPPER LIMITS OF THE PARAMETERS) WRITE(6, 1 14) (I, BB(I>, 1=1, NP ) 114 FORMAT( IX, 12, F10. 3) READ (5, 120) TC, VC 120 F0RMAT(2F10. 0) DO 200 11=1, NU READ (5, 300) TEMP ( I I ) READ( 5, 300) VII,CII,PII 300 FORMAT (8F 10. 0) DO 400 KK=1, 8 VI(KK, I I ) =VI I ( KK ) C I ( KK, 1 1 ) =C 1 1 ( KK ) PI(KK, 1 1 ) =P 1 1 ( KK ) 400 CONTINUE 200 CONTINUE
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135 CALL PATERNCNP, P, STEP. NPASS, 2, AE, BB ) CALL DCF I (P. SUM) WRITE (6, 202) NAME 202 FORMAT ( /// /20H THE SUBSTANCE IS : 20A4//) WR ITE < 6, 130 > TC, VC, TS, RHOS, DCFS 130 FORMAT ( // ' TC=',F8.3, ' K'/' VC=', 1F8. 3, ' CC/GMOL'/' T*=', 1F8. 3, ' 'A' / ' D*=',F8. 4, ' GMOL/L ' / ' C*=',F9. 4//) T0T=0. ODO DO 116 J=l, NJ TR(U)=TEMP(J)/TS WR I TE < 6, 1 1 7 ) TEMP < J ) , TR < J ) 117 FORMAT (//IX, 'FOR THE ISOTHERM AT', F7. 2, ' DEG. K, ', 1 'THE REDUCED TEMPERATURE IS',F7. 4//) WRITE<6, 115) 115 FORMAT (2X, 3HN0. , 3X, 5HP, BAR, 3X, 8HV, CC/MOL, 5X, 4HD/D*, 4X, 15HC, EXP, 5X, 5HC, CAL, 7X, 4HDEV. , 5X, 5H7. ERR/) DO 113 1 = 1,8 PRES ( I ) =P I ( I , J) V(I)=VI(I, J) DEXP ( I ) =DDEX ( I , J) DCAL( I )=DDCA< I, J) IF ( DABS ( DEXP < I ) > . LT. 1. OD5 ) GO TO 118 DEV ( I ) =DEXP ( I ) DCAL ( I ) ERR ( I ) =DEV ( I ) /DEXP < I ) *100. DO TOT =TOT +DABS ( ERR ( I ) ) DR ( I >=VC/V( I > RRHO ( I ) =1 . 0D03/VI ( I, J)/RHOS WRITE (6, 119) I, PRES ( I ) , V( I ) , RRHO ( I ) , DEXP ( I ) , 1DCALCI), DEV ( I ) , ERR ( I ) 119 FORMAT (IX, 13, F10. 2, 7F10. 4 ) 118 CONTINUE 116 CONTINUE AVG=TOT /TERM WRITE<6, 201) AVG 201 FORMAT ( / /30H AVERAGE PERCENTAGE ERROR IS =, F10. 4//) STOP END SUBROUTINE PATERN ( NP, P, STEP, NPASS, 10, AB, BB) IMPLICIT REAL*8 C AÂ— H, QZ ) , INTEGER(IN) DIMFNSION P ( NP ) , STEP ( NP ) DIMENSION Bl(4>, B2(4), T(4>, S ( 4 ) , AB ( 4 ) , 33(4) I OF =6 NRD=NPASS L=1 ICK=2 ITTER=0 DO 5 1 = 1, NP B 1 ( I ) =P ( I ) B2( I )=P( I ) T ( I > =P < I ) 5 S( I >=STEP( I )*10. CALL BOUNDSCP, IOUT, AB, BB, NP) IF ( IOUT. LE. 0) GO TO 10 IF( 10. LE. 0) GO TO 6 WR ITE( I OF, 1005) WRITEdOF, 1000) (J, P(U>, U=l, NP ) 6 GO TO 999 10 CALL DCFI (P, Cl ) IF( 10. LE. 1 ) GO TO 11
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136 WRITE! IOF, 1001 > ITTER,C1 WRITE (I OF i 1000) !J, P(J), J=l, NP) 11 DO 9? INRD=1 , NRD DO 12 1=1, NP 12 S ! I ) =S ! I ) / 10. IF( 10. LE. 1 ) GO TO 20 WR ITE( I OF, 1003) WRITE! IOF, 1000) ( J, S ( J ) , U=1 , NP ) 20 IFAIL=0 DO 30 1=1, NP IC=0 21 P! I )=T! I )+S! I > IC=IC+1 CALL BOUNDS ( P, IOUT, AB, BB, NP) IF ( IOUT. GT. 0 ) GO TO 23 CALL DCFI (P, C2 ) L=L+1 IF ( 10. LT. 3) GO TO 22 WRITE (I OF, 1002) L, C2 WRITE! I OF, 1000) ( J, P ( J) , J=l, NP ) 22 IF ! C 1 C2 ) 23,23,25 23 IF( IC. GE. 2) GO TO 24 S(I)=S(I) GO TO 21 24 IFAIL= IFAIL+1 P! I >=T ! I ) GO TO 30 25 T ( I ) =P ( I ) C 1=C2 30 CONTINUE IF( IFAIL. LT. NP) GO TO 35 IF ( ICK. EQ. 2) GO TO 90 IF ( ICR. EQ. 1) GO TO 35 CALL DCFI (T, C2) L=L+ 1 IF ( 10. LT. 3) GO TO 31 WRITE! IOF, 1002) L, C2 WRITE! IOF, 1000) ! J, T! J) , J=l, NP > 31 IF ! C 1C2 ) 32,34,34 32 ICK=1 DO 33 1=1, NP B 1 ! I >=B2( I ) P ! I )=E2( I ) 33 T( I )=B2! I ) GO TO 20 34 C1=C2 35 IB 1=0 DO 39 1=1, NP B2( I )=T! I ) IF! DABS! B1 ! I >B2! I ) ) . LT. < DABS ! S ! I ) )*. 01 ) > IB1 = IB1 + 1 39 CONTINUE IF! IB1. EQ. NP) GO TO 90 ICK=0 ITTER=ITTER+1 IF! 10. LT. 2) GO TO 40 WRITE! IOF, 1001 ) ITTER,C1 WRITE! IOF, 1000) !U, T! J) , J=l, NP > 40 SJ=1. DO 45 11=1, 11 DO 42 1=1, NP
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137 T ( I > =B2 ( I > +SJ* < B2 ( I ) B 1 ( I ) ) 42 P ( I ) =T( I > SJ=SJ. 1 CALL BOUNDS ( T , I OUT , AB , 33 , NP ) IF ( I OUT. LT. 1 > GO TO 46 IF < II. EQ. 11 ) ICK=1 45 CONTINUE 46 DO 47 1=1, NP 47 BKI >=B2( I > GO TO 20 90 DO 91 1=1, NP 91 T ( I ) =B2 ( I ) 99 CONTINUE DO 100 1=1, NP 100 P< I >=7 < I ) COST=C 1 IF ( 10. LE. 0) GO TO 999 WRITE( IOF, 1004) L, Cl WRITE( IOF, 1000) (J, P< J), J=l, NP) 999 CONTINUE RETURN 1000 FORMAT( IX, 5( 17, E13. 6)/> 1001 F0RMAT(//1X, 13HITERATI0N NO. , 1 5/5X, 5HC0ST=, El 5. 6, 20X, 1 10HPARAMETERS ) 1002 FORMAT ( 10X, 3HN0. , 14, 8X, 5HC0ST=, El 5. 6) 1003 FORMAT (/IX, 28HSTEP SIZE FOR EACH PARAMETER) 1004 FORMAT (IX, 13HANSWERS AFTER, 13, 2X, 22HFUNCTI0NAL EVALUATIONS 1//5X, 5HC0ST=, El 5. 6, 20X, 18H0PTIMAL PARAMETERS) 1005 FORMAT (IX, 35HINITIAL PARAMETERS OUT OF BOUNDS ) END SUBROUTINE BOUNDS(P, IOUT, AB, BB, NP ) REAL*8 AB ( 4 ) , BB ( 4 ) , P ( 4 ) I0UT=0 DO 1 1=1, NP 1 IF(P( I). LE. AB( I ) .OR. P( I). GT. BB(I> ) I0UT=1 RETURN END SUBROUTINE DCFI ( P, SUM ) IMPLICIT REAL*8(AH, 0Z) DIMENSION VI (8, 22) , Cl (8, 22) , RH0(8) DIMENSION P ( 4 ) , TEMP(22), DEXP(8), DCAL(8) COMMON/DATA/TEMP, VI, Cl, DEXP, DC AL, VC COMMON/COND/NP, NJ, NN, TERM COMMON/PR INT/DDEX( 8, 22), DDCA(8, 22) COMMON/PARAM/RHOS, DCFS, TS RHOS=P ( 1 ) DCFS=P ( 2 ) TS=P ( 3 ) SUM=0. ODO TERM=0. ODO DO 6 J=l, NJ TT=TS/TEMP ( J) NN=0 DO 10 KÂ»l, 8 IF(VI (K, J). LT. 1. ODO) GO TO 10 NN=NN+1 RH0(NN) = 1. 0D03/VI ( K, J) DEXP ( NN > =C I ( K, J) 10 CONTINUE DO 13 1=1, NN
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138 DN=RHQ ( I ) /RHQS TERMTERM+1 A0=0. 98642D010. 10191D02*TT0. 1 5356D0 1 *TT*TT Al=0. 28465D02*0. 30864D02*TT+0. 60294D01*TT*TT A2=0. 27542D020. 32898D02*TT0. 87130D01Â«TT*TT A3=0. 82606D01+0. 12737D02*TT+0. 40170D01*TT*TT DCAL( I ) = ( AO+DN* < A1 +DN* ( A2+DN*A3 ) ) >*DCFS SUM=SUM+ ( DCAL ( I ) DEXP ( I ) > **2 DDEX ( I / J ) =DEXP ( I ) DDCA< I, J ) =DCAL ( I ) 13 CONTINUE 6 CONTINUE SUM=DSQRT< SUM/ < TERM1. ODO) ) RETURN END
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139 C CORRESPONDING STATES CORRELATION OF THE VOLUMETRIC C PROPERTIES OF COMPRESSED PURE LIQUIDS. C MAIN PROGRAM IMPLICIT REAL*8 ( AHÂ» QZ ) DIMENSION P ( 4 ) ( STEP ( 4 ) , AB < 4 > , BB<4), NAME (20) DIMENSION TEMP ( 34 ) , TR(34>, DR<8), RRH0(8), PEXP(8), 1PCAL ( 8 ) , DEV ( 8 ) , ERR (8) DIMENSION VI <8, 34), Cl (8, 34) , PI (8, 34) , PI I (8) , 1PRES ( 8 ) Â» V(8) CI(KK, I I ) =C I I ( KK ) PI(KK, II)=PII(KK) 400 CONTINUE 200 CONTINUE
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140 CALL PATERN!NP, P, STEP, NPASS, 2, AB, BB) CALL DCF I (P, SUM) WRITE (6, 202) NAME 202 FORMAT !////2lH THE SUBSTANCE IS : 20A4//) WRITE (6, 130) TC, VC, TS, RHCS, DCFS 130 FORMAT ( // ' TC=',F8.3, ' K'/' VC=', 1F8. 3, ' CC/GMOL'/' T*=', 1F3. 3, K ' / ' D*= ' , F8. 4, ' GMOL/L ' / ' C*=',F9.4//) T0T=0. ODO DO 116 J= 1 , N J TR ( J ) =TEMP ( J ) /TS WRITE(6, 117) TEMP ( J ) , TR ( J ) 117 FORMAT (//IX, 'FOR THE ISOTHERM AT', F7. 2, ' DEG. K, 1 'THE REDUCED TEMPERATURE IS',F7. 4) WR ITE ( 6, 134 > PI!1,U), VI(1,U) 134 FORMAT (//IX, 'THE REF. PRESSURE IS',F9. 4, ' 1 BAR,',' THE REF. MOLAR VOLUME IS',F9. 4, ' CC/GMOL'//) WRITE (6, 115) 115 FORMAT ( 2X, 3HN0. , 3X, 5HP, BAR, 3X, 8HV, CC/MOL, 5X, 4HD/D*, 4X, 15HP, EXP, 5X, 5HP, CAL, 7X, 4HDEV. , 5X, 5H7. ERR/) DO 118 1=2,8 PRES ! I > =P I ( I , U> V< I >=VI ( I, J) PEXP ( I ) =PPEX ( I, U) PCAL(I)=PPCA!I, J) IF ( DABS ( PEXP ( I ) ) . LT. 1. OD5 ) GO TO 118 DEV ! I ) =PEXP ! I ) PCAL ( I ) ERR ( I ) =DEV ( I > /PEXP ( I ) *100. DO TOT =TQT +DABS ( ERR ( I ) ) DR ( I ) =VC/V ( I ) RRHO! I ) = 1. 0D03/VI ( I , J)/RHOS WRITE! 6, 119) I, PRES! I ) , V! I ), RRHO ! I ) , PEXP ( I ) , lPCAL(I), DEV ! I ) , ERR! I ) 119 FORMAT! IX, 13, F10. 2,' 7F10. 4) US CONTINUE 116 CONTINUE AVG=TOT /TERM WRITE16, 201) AVG 201 FORMAT ! / /30H AVERAGE PERCENTAGE ERROR IS =, F10. 4//) STOP END SUBROUTINE PATERN ! NP , P, STEP, NPASS, 10, AB, BB) IMPLICIT REALMS ( AH, OZ), INTEGER! IN) DIMENSION P ! NP ) , STEP ! NP ) DIMENSION B1 (4), B2(4>, T(4), S!4), A3 ! 4 ) , BB ! 4 ) I OF =6 NRD=NPASS L=1 ICK=2 ITTER0 DO 5 1 = 1, NP B 1 < I ) =P ! I > B2! I )=P! I ) T ( I ) =P < I ) 5 S ! I ) =STEP ( I ) * 1 0. CALL BOUNDS!P, IOUT, AB, BB, NP ) IF! IOUT. LE. 0) GO TO 10 IF! 10. LE. 0) GO TO 6 WRITE! IOF, 1005) WRITE! IOF, 1000) !J, P! J), J=l, NP )
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141 6 GO TO 999 10 CALL DCF I (PÂ» Cl > I F ( ID. LE. 1 > GO TO 11 WRITEdOF, 1001 > ITTER; Cl WRITE( IOF, 1000) , J=1,NP> 11 DO 79 INRD=1 , NRD DO 12 1=1, NP 12 S< I )=S( D/10. IF( 10. LE. 1 ) GO TO 20 WR ITE( IOF, 1003) WRITEdOF, 1000) (J, S( J), J=l, NP ) 20 IFA IL*0 DO 30 1=1, NP IC=0 21 P( I >=T( I )+S( I ) IC=IC+1 CALL BOUNDS(P, IOUT, AB, BB, NP) IF( IOUT. GT. 0) GO TO 23 CALL DCF I (P, C2 ) L=L+1 IF ( 10. LT. 3) Ga TO 22 WRITEdOF, 1002) L, C2 WRITEdOF, 1000) GO TO 35 CALL DCFICT, C2) L=L+1 IF( ID. LT. 3) GO TO 31 WRITE( IOF, 1002) L, C2 WRITEdOF, 1000) ( J, T( J), U=l, NP ) 31 IF ( C 1 C2 ) 32,34,34 32 ICK=1 DO 33 1=1, NP B 1 ( I ) =B2 ( I ) P( I )=B2( I ) 33 T ( I ) =B2 ( I ) GO TO 20 34 C 1=C2 35 IB 1=0 DO 39 1=1, NP B2 ( I ) =T ( I ) IF ( DABS
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142 40 SJ=t. DO 45 11=1, 11 DO 42 1=1, NP T < I ) =82 ( I > +SJ* ( B2 ( I ) B 1 < I ) ) 42 P ( I ) =T ( I > SJ=SJ. 1 CALL HOUNDS CT, IOUT, AB, BB, NP ) IF ( IOUT. LT. 1 ) GO TO 46 IF( II. EQ. 11 ) ICK=1 45 CONTINUE 46 DO 47 1=1, NP 47 Sl( I > =B2 ( I ) GO TO 20 90 DO 91 1 = 1, NP 91 T ( I ) =B2 ( I ) 99 CONTINUE DO 100 1 = 1, NP 100 P ( I ) =T ( I ) COST=C 1 IFdO. LE. 0) GO TO 999 WRITEdOF, 1004) L, Cl WRITE< I OF, 1000) < J, P< J) , J=l, NP ) 999 CONTINUE RETURN 1000 FORMATC IX, 5( 17, E13. 6) /) 1001 F0RMAT(//1X, 13HITERATI0N NO. , I 5/5X, 5HC0ST=, El 5. 6, 20X, 1 1 OHPAR AMETERS ) 1002 FORMAT ( 10X, 3HN0. , 14, SX, 5HC0ST=, E15 6) 1003 FORMAT (/IX, 2SHSTEP SIZE FOR EACH PARAMETER) 1004 FORMATdX, 13HANSWERS AFTER, 13, 2X, 22HFUNCTI0NAL EVALUATIONS 1//5X, 5HC0ST=, El 5. 6, 20X, 1SH0PTIMAL PARAMETERS) 1005 FORMATdX, 35HINITIAL PARAMETERS OUT OF BOUNDS ) END SUBROUTINE BOUNDS(P, IOUT, AB, BB, NP ) REAL*8 AB ( 4 ) , BB ( 4 ) , P ( 4 ) IOUT =0 DO 1 1=1, NP 1 IF(P(I). LE. ABU) .OR. P ( I ) . GT. BB ( I ) ) I0UT=1 RETURN END SUBROUTINE DCFI(P,SUM> IMPLICIT REAL*8Â• 1
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143 RH0CNN>=1. 0D03/VI (K, J) DR=RHO< 1 ) /RHOS PEXP ( NN ) =P I (K, J) PR=PEXP ( 1 > 10 CONTINUE DO 13 1=2, NN DN=RHO ( I ) /RHOS TERM=TERM+1 A0=0. 98642D01Â— 0. 10191D02*TT0. 15356D01*TT*TT Al=0. 28465D02+0. 30864D02*TT+0. 60294D0 1 *TT*TT A2=0. 27542D020. 32898D02*TT0. 37130D01*TT*TT A3=0. 82606DG1+0. 12737D02*TT+0. 4G170D01*TT*TT BO=RHOS* ( DNDR ) *RK*TS/TT B 1=DN >DR B2=DN*DN+DN*DR+DR*DR B3=DN*DN*DN+DN*DN*DR+DN*DR*DR+DR*DR*DR C0=A0>Al*Bl/2. 0D0+A2*B2/3. 0D0+A3*B3/4. ODO PCAL( I )=PR+BO*< 1. ODODCFS*CO> SUM=SUM+(PCAL< I )PEXP ( I ) ) **2 PPEX(I,J)=PEXP(I) PPCA( I, J)=PCAL( I ) 13 CONTINUE 6 CONTINUE SUM=DSQRT < SUM/ < TERM1. ODO) ) RETURN END
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o o o 144 CORRESPONDING STATES CORRELATION OF THE VOLUMETRIC PROPERTIES OF COMPRESSED LIQUID MIXTURES. MAIN PROGRAM IMPLICIT REALMS ( AH, OZ ) DIMENSION P ( 4 ) , STEP ( 4 ) , AB < 4 ) , BB ( 4 ) , NU ( 8 ) , NAME (20) DIMENSION TC(3>, VC (3), TS(3), VS (3), CS(3> DIMENSION TCM ( 8 ) Â» VCM(8>, TSM(8) , RH0S(8), DCFS (8) DIMENSION XI(8, 3), TEMP(24, 8), TR(24, 8), RRH0(8) 1 , PEXP ( 8 ) , PCAL ( 8 ) , DEV ( 8 > , ERR ( 8 ) DIMENSION VI (8, 24, 8), Cl (8, 24, 8), PI (8, 24, 8) 1, VII('3), CII(S), P 1 1 ( 8 ) , V(8>, PRES ( 8 ) CQMMON/DATA/TEMP, VI, PI, PEXP, PCAL COMMON/COND/NP, NC, NI, NJ, NN, TERM COMMON/PR INT/PPEX( 8, 24, 8), PPCA<8, 24, 3) COMMON /PAR AM /TCM, VCM, TSM, RHOS, DCFS COMMON/CRIT/XI, TC, VC, TS, VS, CS READ (5, 13) NAME 13 FORMAT ( 20A4 ) READ ( 5, 101 ) NP, 10 101 FORMAT (212) READ (5, 102) NPASS, (P ( I ) , 1 = 1, NP >, ( STEP ( I ) , 1 = 1, NP) READ (5, 103) ( AB ( I ) , 1=1, NP), (BB(I), 1=1, NP) 102 FORMAT (12, 6F10. 0) 103 FORMAT ( 3X, 6F 10. 0) WRITE(6, 104) 104 FORMAT (IX, 'PATTERN SEARCH ESTIMATION OF PARAMETERS FROM 1DCFI DATA '/IX, 'CONTROL PARAMETERS WERE SET AS FOLLOWS'/) WRITE(6, 105) NP, 10 105 FORMAT( IX, 5H NP=, I2/1X, 5H I0=, 12) WRITE (6, 106) NPASS 106 FORMAT ( IX, 15HNUMBER OF PASS=, 12) WRITE (6, 107) 107 FORMAT( IX, 18HINITIAL PARAMETERS) WRITE(6, 108) ( I, P( I ), 1 = 1, NP ) 108 FORMAT( IX, 12, F10. 3) WRITE (6, 109) 109 FORMAT( IX, 12HSIZE OF STEP) WRITE(6, 110) < I , STEP ( I > , I = 1 , NP > 110 FORMAT( IX, 12, F10. 3) WR ITE ( 6, 111 ) 111 FORMAT (IX, 30HL0WER LIMITS OF THE PARAMETERS) WR ITE( 6, 1 12 ) (I, AB(I), 1 = 1, NP > 112 FORMAT( IX, 12, F10. 3) WRITE(6, 113) 113 FORMAT (IX, 30HUPPER LIMITS OF THE PARAMETERS) WRITE(6, 114) (I, BB(I), 1 = 1, NP ) 114 FORMAT( IX, 12, F10. 3) READ (5, 1 ) NC 1 FORMATX 12 ) WRITE (6, 2) NC 2 FORMAT (/IX, 'NO. OF COMPONENTS= ', I2/>READ (5,3) (TC(I), VC < I ) , 1 = 1, NC ) READ (5, 3)
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145 READ (5, 3) VII,CII,PII DO 300 KK=1 , 8 VI (KK, JJ, II)=VII(KK) Cl (KK, JJ, II )=CII (KK> PI (KK, JJ, II >=PII (KK) 300 CONTINUE 200 CONTINUE 100 CONTINUE CALL PATERN ( NP , P, STEP, NPASS, 2, A3, BB> CALL DC FI (P, SUM) WR IT E< 6,. 301 > NAME 301 FORMAT (///17H THE SYSTEM IS : 20A4///) TOT=G. ODO DO 115 11=1, NI WRITE (6, 8) CXICII, I), 1 = 1, NC1) 8 FORMAT ( ///IX, ' X1=',2F9. 5) WR I TE < 6, 9 ) TCM (II), VCM (II), TSM (II), RHOS (II), DCFS (II) 9 FORMAT ( ' TCM=',F8. 3, ' K ' / ' VCM=', 1FS. 3, ' CC/GMOL'/' T*=',F9.3, ' K' 1/' D*=',F8. 4, ' GMOL/L '/ ' C*=',F9. 4//) DO 116 JJ=1, NJ( II ) TR ( JJ, 1 1 ) =TEMP ( J J, 1 1 ) /TSM (II) WRITE (6, 10) TEMP ( JJ, II), TR(JJ, II) 10 FORMAT (//IX, 'FOR THE ISOTHERM AT', F7. 2, ' DEG. K, ', 1 'THE REDUCED TEMPERATURE IS',F7. 4) WRITt(6, 14) PI(1, JJ, II), VI <1, JJ, II) 14 FORMAT < // 1 X, 'THE REFERENCE PRESSURE IS', F8. 4, ' 1 BAR, ', 'THE REFERENCE MOLAR VOLUME IS',F9. 4//) WRI7E(6, 11) 11 FORMAT ( 2X, 3HN0. , 3X, 5HP, BAR, 3X, 8HV, CC/MOL, 5X, 4HD/D*, 4X, 15HP, EXP, 5X, 5HP, CAL,7X, 4HDEV. , 5X, 5H7. ERR/) DO 117 KK=2, 8 PRES ( KK ) =P I ( KK, JJ, II) V ( KK ) Â—VI ( KK, JJ, II) PEXP(KK)=PPEX(KK, J J, 1 1 ) PCAL < KK ) =PPCA ( KK, JJ, 1 1 ) IF(DABS(PEXP (KK) ). LT. 1. OD5) GO TO 117 DEV ( KK ) =PEXP ( KK ) PC AL ( KK ) ERR (KK ) =DEV ( KK ) /PEXP ( KK ) *100. DO TOT=TOT+DABS ( ERR ( KK ) ) RRHO ( KK ) = 1 . 0D03/VI (KK, JJ, II )/RHOS( II ) WRITE (6, 12) KK, PRES(KK) , V(KK> , RRHO(KK) , PEXP (KK) , IPCAL(KK) , DEV(KK) , ERR(KK) 12 FORMAT( IX, 13, F10. 2, 7F10. 4) 117 CONTINUE 116 CONTINUE 115 CONTINUE AVG=TQT/TERM WRITE( 6, 201 > AVG 201 FORMAT (//30H AVERAGE PERCENTAGE ERROR IS =, F10. 4//) STOP END SUBROUTINE PATERN ( NP , P, STEP, NPASS, 10, AB, BB ) IMPLICIT REAL*8(AH, 0Z), INTEGER ( IN ) DIMENSION P(NP), STEP(NP) DIMENSION Bl(4), B2 ( 4 ) , T(4), S(4), AB ( 4 ) , BB ( 4 ) I OF =6 NRD=NPASS L=1 ICK=2
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146 ITTER=0 DO 5 1 = 1, NP S 1 ( I ) =P ! I ) E2! I )=P( I) T ( I ) =P ( I > 5 S< I >=STEP< I>*10. CALL BOUNDS!P, IOUT, AB, BB, NP ) I F ( I OUT . LE. O) GO TO 10 IF ( 10. LE. 0) GO TO 6 WRITE! IOF, 1005) WRITE! I OF, 1000) ! J, P ! U > , J=1 , NP ) 6 GO TO 999 10 CALL DCFI (P, Cl ) IF! 10. LE. 1 ) GO TO 11 WRITE! IOF, 1001 ) ITTER,C1 WRITE! IOF, 1000) ( J, P ! J ) , J=1 , NP ) 11 DO 99 INRD=1, NRD DO 12 1=1, NP 12 S(I)=3(I)/10. IF! 10. LE. 1 ) GO TO 20 WRITE! IOF, 1003) WRITE! IOF, 1000) ( J, S ! J ) , J=1 , NP ) 20 IFAIL=0 DO 30 1=1, NP IC=0 21 P ( I ) =T ! I ) +S ! I ) IC=IC+1 CALL BOUNDS!P, IOUT, AB, 3B, NP ) IF! IOUT. GT. 0) GO TO 23 CALL DCFI ! P, C2 ) L=L + 1 IF! 10. LT. 3) GO TO 22 WRITE! IOF, 1002) L, C2 WRITE! IOF, 1000) ( J, P ! J ) , J=1 , NP > 22 IF ! C 1C2 ) 23,23,25 23 IF! IC. GE. 2) GO TO 24 S ( I ) = S ( I ) GO TO 21 24 IFAIl =IFAIL+1 P ( I ) =T ( I ) GO TO 30 25 T ( I ) =P ( I ) C 1=C2 30 CONTINUE IF! IFAIL. LT. NP ) GO TO 35 IF! ICK. EQ. 2) GO TO 90 IF! ICK. EQ. 1 ) GO TO 35 CALL DCFI !T, C2> L=L+1 IF! 10. LT. 3) GO TO 31 WRITE! IOF, 1002) L, C2 WRITE! IOF, 1000) ( J, T( J ) , J=1 , NP ) 31 IF ! C 1C2 ) 32,34,34 32 I CK=1 DO 33 1=1, NP B 1 ! I )=B2! I ) P ( I )=B2< I > 33 T ! I > =B2 ! I ) GO TO 20 34 C 1=C2
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147 35 IB 1=0 DO 3? 1 = 1, NP B2( I )=T( I > IF ( DABS ( B 1 ( I >B2< I ) ) . LT. ( DABS ( S ( I ) >*. 01 ) ) IB1 = IE1 + 1 39 CONTINUE IF( IB1. EQ. NP > GO TO 90 ICK=0 ITTER=ITTER+1 IF( 10. LT. 2) GO TO 40 WRITE! I OF, 1001 > ITTER,C1 WRITfc ! IOF, 1000) ! J, T! J), J=l, NP > 40 SJ=1. DO 45 11=1, 11 DO 42 1=1, NP T( I )=B2( I >+SU*!B2( I >Bl ! I ) ) 42 P ( I ) =T ( I ) SJ=SJ. 1 CALL BOUNDS ( T, IOUT, AB, BB, NP) IF! IOUT. LT. 1 ) GO TO 46 IF! II. EQ. 11 ) ICK=1 45 CONTINUE 46 DO 47 1=1, NP 47 B 1 < I ) =B2 ! I ) GO TO 20 90 DO 91 1 = 1, NP 91 T( I >=B2< I ) 99 CONTINUE DO 100 1=1, NP 100 P < I ) =T ! I > COST=C 1 IF! 10. LE. 0) GO TO 999 WRITE! IOF, 1004) L, Cl WRITE! IOF, 1000) ! J, P ! U ) , J=1 , NP ) 999 CONTINUE RETURN 1000 FORMAT! IX, 5! 17, E13. 6)/) 1001 F0RMAT!//1X, 13HITERATI0N NO. , I5/5X, 5HC0ST=, E15. 6, 20X, 1 10HPARAMETERS) 1002 FORMAT! 10X, 3HN0. , 14, SX, 5HC0ST=, E15. 6) 1003 FORMAT! /IX, 28HSTEP SIZE FOR EACH PARAMETER) 1004 FORMAT (IX, 13HANSWERS AFTER, 13, 2X, 22HFUNCTI0NAL EVALUATIONS 1//5X, 5HC0ST=, El 5. 6, 20X, 18H0PTIMAL PARAMETERS) 1005 FORMAT ( IX, 35HINITIAL PARAMETERS OUT OF BOUNDS ) END SUBROUTINE BOUNDS!P, IOUT, AB, BB, NP ) REAL*8 AB<4), SB (4), P(4> I0UT=0 DO 1 1 = 1, NP 1 IF!P! I ). LE. AB! I > .OR. P ! I ) . GT. BB ( I ) ) I0UT=1 RETURN END SUBROUTINE DCFI(P.SUM) IMPLICIT REAL*8 ! AH, 0Z ) DIMENSION TC ( 3) , VC (3), TS!3>, VS!3>, CS(3), NU ( 8 ) DIMENSION TCM ! 8 ) , VCM!8) , TSM!8) , VSM(8) , RH0S!8) , DCFS(8) DIMENSION VI! 8, 24, 8) , PI ( 8, 24, 8 > , DI J<3, 3 ) , RHO!S) DIMENSION P!4), XI (8, 3), TEMP! 24, 8), PEXP!8), PCAL!3> COMMON/DATA/TEMP, VI, PI, PEXP, PCAL COMMON/COND/NP, NC, NI, NU, NN, TERM COMMON /PR I NT /PPEX ! 8, 24, 8), PPCA!8, 24, 8)
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148 COMMGN/PARAM/TCM, VCM, TSM, RHOS, DCFS COMMON /CR IT/X 1/ TC, VC, TS, VS, CS DIJ(1, 2 ) =P ( 1 ) DI J(2, 1 )=DIJ( 1, 2) DIJ( 1, 1 )=0. DO DIJ<2, 2 ) =0. DO SUM=0. DO TERM=0. DO DO 1 11=1, NI TCM (II) =0. DO VCM (II) =0. DO TSM (II) =0. DO VSM (II) =0. DO DCFS (II) =0. DO DO 2 1 = 1, NC TCM ( 1 1 ) =TCM< 1 1 ) +X I ( 1 1 , I)*TC(I> TSM (II) =TSM ( 1 1 ) +X I ( 1 1 , I)*TS(I) 2 DCFS (II) =DCFS (II) +X 1(11/ I)*CS(I) DO 3 1 = 1, NC DO 3 J=l, NC VCM ( 1 1) =VCM ( 1 1 ) +X I < 1 1 , I ) *X I ( 1 1 , V ) * ( ( VC ( I) +VC < J ) ) /2. DO ) 3 VSM (II) =VSM (II) +X I ( 1 1 , I)*XI (II, J)*( ( (VS(I)+VS(J) )/2. DO) 1*( 1. DODI J( I, J) ) ) RK=0. 0831434D0 DO 6 JJ=1, NJ( II > TT=TSM (II) /TEMP ( JJ, II) NN=Q DO 10 KK=1 , S IF(VI(KK, JJ, II). LT. 1. ODO) GO TO 10 NN=NN+ 1 RHO ( NN > = 1 . 0D03/VI (KK, JJ, II ) RHOS ( 1 1 ) = 1 . 0D03/VSM (II) DR=RHO( 1 )/RHOS( II ) PEXP(NN)=PI(KK, JJ, II) PR=PEXP( 1 ) 10 CONTINUE DO 13 KK=2, NN DN=RHQ ( KK ) /RHOS (II) TERM=TERM+1 A0=0. 98642D010. 10191D02*TT0. 1 5356D01*TT*TT Al=0. 28465D02+0. 3C864D02*TT+0. 60294D01*TT*TT A2=0. 27542D020. 32898D02*TT0. 87130D01*TT*TT A3=Â— 0. 82606D01+0. 12737D02*TT+0. 40170D01*TT*TT BO=RHOS ( 1 1 ) * ( DNDR ) *RK*TSM < 1 1 ) /TT B1=DN>DR 32=DN*DN+DN*DR+DR*DR B3=DN*DN*DN+DN*DN*DR+DN*DR*DR+DR*DR*DR C0=A0iAl*Bl/2. GD0+A2*B2/3. 0D0+A3*B3/4. ODO PC AL ( KK ) =PR+BO* ( 1 . ODODCFS ( 1 1 )*C0) SUM=SUM+ ( PCAL ( KK ) PEXP ( KK ) ) **2 PPEX ( KK, JJ, II) =PEXP ( KK ) PPC A ( KK, JJ, II) =PCAL ( KK ) 13 CONTINUE 6 CONTINUE 1 CONTINUE SUM=DSQRT( SUM/ ( TERM1. ODO) ) RETURN END
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BIOGRAPHIC SKETCH YungHui Huang was born on July 20, 1953, in Taipei, Taiwan, Republic of China. He was educated in the public school system in Taipei and graduated from National Central University in June 1976 with a Bachelor of Science degree in chemical engineering. After serving two years for his country as a technical lieutenant in the Chinese army and working one year as a fulltime teaching assistant in the National Central University, he began graduate studies at the University of South Carolina where he received a Master of Science degree in chemical engineering in May 1982. He joined the Department of Chemical Engineering, University of Florida, in August 1982, where he has been a research assistant, working towards his Ph.D. degree. In 1985, he spent three months in the Instituttet for Kemiteknik, Danmarks Tekniske H^jskole, Lyngby, Denmark. 159
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Professor of Chemical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Associate Professor of Chemical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Gerry K. Lyberatos Assistant Professor of Chemical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Samuel CL Colgate Associate Professor of Chemistry
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. William Weltner, Jr, Professor of Chemistry This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 1986 /Xxy a. & 1^6 Dean, College of Engineering Dean, Graduate School

