DIFFUSION AND MASS TRANSFER
IN
SUPERCRITICAL FLUIDS
by
PABLO G. DEBENEDETTI
Ingeniero Quimico
Universidad de Buenos Aires
(1978)
S.M.
Massachusetts Institute of Technology
(1981)
SUBMITTED TO THE DEPARTMENT OF
CHEMICAL ENGINEERING
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
December 1984
O
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
The author hereby grants to M.I.T. permission to reproduce and to
distribute copies of this thesis document in whole or in part..
Signature of author ...................
i.....
......
En i eering
Department of Chemic
eeebr 1984
NJ .............
Professor Robert C. Reid
Thesis Supervisor
Certified by ..................................---
Professor Ulrich W. Suter
Thesis Supervisor
_. -·
%.- .- ...............
Accepted
by
...............................
1.ASSAiHUSETcThLN-i?
Chairman, Departmental Committee
OFTECHNOLOGYraduate
Studies
on Graduate Studies
FEB 1 3 1985
LIBRARIES'
ARCHIVES
DIFFUSION AND MASS TRANSFER
IN
SUPERCRITICAL FLUIDS
by
Pablo Gaston Debenedetti
Submitted to the Department of Chemical Engineering
on December 18 th, 1984 in partial fulfillment of the
requirements for the degree of Doctor of Philosophy
ABSTRACT
The different pressure dependence of fluid density and viscosity in going
from the dilute gas to the dense fluid state gives rise to kinematic
viscosities which, in the near supercritical region (1<Pr<4 ; 1<TrY1.1)
are exceptionally low.
The density gradients which exist in any mass
transfer situation will, in the presence of a gravitational field, cause
buoyancy-driven currents which, for a given geometry and Reynolds number,
are two orders of magnitude higher than in ordinary liquids (the comparison
being based upon the ratio of characteristic buoyant and inertial forces).
The diffusion coefficients of benzoic acid and naphthalene in supercritical
SF6 , and of benzoic acid and 2-naphthol in supercritical CO2 were measured
with a hydrodynamic technique. Analysis of the data suggests that hydrodynamic behaviour at the molecular level is approached as a high viscosity
limit. When experiments were conducted in the presence of bouyant currents,
significant mass transfer enhancements were observed.
The solution to
the problem of diffusion in a finite rectangular duct is presented in
analytical and graphical form.
An analysis of diffusion in the light of irreversible thermodynamics
leads to the new concept of infinite dilution fugacity coefficient, and
to an accurate and simple expression for the composition dependence of
the fugacity coefficient. The form and asymptotic behaviour
of this
expression have interesting thermodynamic implications.
The behaviour of the system CO,-benzoic acid at solute infinite dilution
was studied by molecular dynamics simulation of the motion of 107 CO,2
and 1 benzene molecules, respectively, modelled as rigid polyatomics.
The different symmetry of CO2 and benzene required the implementation
of two different algorithms within the same computer program. The binary
diffusion
coefficient
and its temperature and density dependence, as
well as solvent velocity and C-C radial distribution functions were calculated. Numerical problems were encountered when electrostatic forces were
superimposed upon the site-site Lennard-Jones potential due to the orientation-sensitivity of the coulombic interactions.
Thesis Supervisors : Dr. Robert C. Reid; Dr. Ulrich W. Suter
Title : Professor of Chemical Engineering; Professor of Chemical Engineering
1
To Silvia
To my Father and Brother
To the memory of my Mother
To my family in Rome
To Professor Reid
2
ACKNOWLEDGEMENTS
It
and
to express my sincere
is a pleasure
who
organizations
I am indebted
helped me
thanks to the many people
throughout this work.
Specifically,
to:
Professor Robert Reid, who taught me the rigorous beauty of Thermodynamics, and without whose help and encouragement I would not be venturing
into the academic world.
can be summed
My respect for him as a human being and scientist
up in one word:
Master,
is what he has been to me
which
Professor Ulrich Suter, whose enthusiasm and insight were a constant
source of inspiration
Professor Howard Brenner, for many a delightful discussion on matters
technical and not so technical, and for his help and advice concerning
my future career
Professors
Brady, Robert
John
Brown and William Deen, members
of
my Thesis Committee, for their valuable help throughout this work
Professor
Jack
Longwell, who suggested
the experimental technique
in the course of a Friday afternoon seminar
Mario Da Silva and Dick Jankins of the Chemical Engineering Machine
Shop, for their patience and skill in transforming clumsy drawings into
working equipment
William Schmitt, for his inexhaustible good will and spirit of collaboration
Sanat Kumar and Thanassis Panagiotopoulos, for their uncommon insight
and willingness to share thoughts, ideas and speculations
Glorianne Collver-Jacobson, for her help in typing part of the thesis
The National
Science Foundation and
the Nestle
Products Technical
Assistance Co. for their financial support
Howard Bernstein, for his friendship, which I truly value
Bach and Mozart, for their musical company
My family, for their constant love, support and understanding
My
Father, who
taught me
the meaning and importance of goals and
guidelines in life
Silvia, who gave so much and asked for so little
years.
during three long
Her love was my constant source of strength and joy.
2.a
TABLE OF CONTENTS
Page
ABSTRACT
1
ACKNOWLEDGEMENTS
2
LIST OF FIGURES
6
LIST OF TABLES
13
1.
15
SUMMARY
1.1
1.2
1.3
1.4
1.5
Physical Properties and Mass Transfer Mechanism
Hydrodynamic Experiments
Experimental Results and Data Analysis
Diffusion and Irreversible Termodymamics
Molecular Dynamics Simulations
15
27
33
45
51
73
2.INTRODUCTION
3.
2.1
Perspective
73
2.2
2.3
2.4
2.5
2.6
Diffusion in Dense Fluids: Theoretical Difficulties
Theoretical Approaches
Diffusion in Dense Fluids: Computer Simulations
Diffusion Coefficients: Experimental Approaches
Mass Transfer
76
76
86
89
92
2.7
Objectives
92
PHYSICAL PROPERTIES AND NATURAL CONVECTION
3.1
3.2
3.3
4.
5.
Mass Transfer and Natural Convection; Hydrodynamics
Physical Properties in the Supercritical Region
Flow Regimes
APPARATUS AND EXPERIMENTAL PROCEDURE
94
94
96
103
108
4.1
4.2
Introduction
Apparatus
108
108
4.3
Experimental Procedure
112
DIFFUSION IN RECTANGULAR DUCTS
115
5.1
5.2
5.3
5.4
Velocity Profile; Fully Developed Flow
Diffusion; Conservation Equation
Solution
Cross-Section Averages
3
115
118
120
130
TABLE OF CONTENTS (continued)
Page
6.
HYDRODYNAMIC EXPERIMENTS; RESULTS AND DISCUSSION
6.1
6.2
6.3
6.4
7.
DIFFUSION AND IRREVERSIBLE THERMODYNAMICS
7.1
Forces
7.2
Relationship Between Thermodynamic
and Phenomenological Approaches
The Kinetic Conversion Factor
7.3
8.
and Fluxes
DYNAMICS OF INTERACTING BODIES:
8.1
8.2
8.3
8.4
8.5
8.6
9.
Diffusion Coefficients; Results
Discussion
Effect of Natural Convection
Sensitivity Analysis
THEORETICAL FUNDAMENTALS
Kinematics of the Rigid Body
Dynamics of the Rigid Body
Integration of the Equations of Motion
Statistical Treatment of Diffusion
Velocity Distributions
Equipartition and Virial Theorem
Sample Size; Boundary Conditions; Cutoff Radius
Time Considerations
Start-up; Rescaling; Relaxation Runs
Interatomic Potentials
10. MOLECULAR DYNAMICS SIMULATIONS: RESULTS AND DISCUSSION
10.1
10.2
10.3
10.4
10.5
Velocity Distributions
Radial Distribution Functions
Diffusion Coefficients
Simulations with Coulombic Interactions
Compressibility Factors
10.6 Summary
11. CONCLUSION
140
156
186
189
198
198
COMPUTER SIMULATIONS; TECHNICAL ASPECTS
9.1
9.2
9.3
9.4
140
200
204
212
212
216
217
221
226
228
234
234
240
244
246
276
276
284
291
303
309
311
313
TABLE OF CONTENTS (continued)
Page
APPENDIX
1
CUBIC EQUATIONS OF STATE
315
APPENDIX
2
EQUIPMENT DESIGN: EXPERIMENT DESIGN AND
CALCULATIONS
320
.Flat Plate Design
Sample Equilibrium and Diffusion Calculations
Experimental Errors
Density Profiles
ChemicaJs Used in the Experiments
A2.1
A2.2
A2.3
A2.4
A2.5
320
322
324
326
328
APPENDIX
3
C ij
APPENDIX
4
COMPUTER PROGRAMS
349
Computer: Technical Details
Computer Program EQUIL
349
350
370
372
375
378
380
382
403
A4.1
A4.2
A4.3
A4.4
A4.5
Subroutine
Q
Subroutine PUT
Subroutine START
Subroutine TURN
Subroutine BOLTZ
Computer Program LINALB
Computer Program NEUTRAL-2
A4.6
A4.7
A4.8
A4.9
APPENDIX
COEFFICIENTS FOR EQUATION (5.27)
5
A5.1
A5.2
A5.3
A5. 4
THE INTEGRAL ENTROPY BALANCE IN IRREVERSIBLE
THERMODYNAMICS
Mass Balance: Conservation Equations
Thermodynamic Definitions
Energy and Entropy Relationships
Integral Entropy Balance
330
406
406
407
407
409
NOTATION
411
REFERENCES
420
5
LIST OF FIGURES
Page
1.1
Schematic pressure-density diagram for a pure substance showing
the region of interest in supercritical extraction
17
1.2
Dimensionless isothermal compressibility of CO,
2 , as modelled by
the Peng-Robinson equation of state
18
1.3
Dimensionless
by
19
1.4
Dimensionless isothermal compressibility of SF6 , as modelled by
the Peng-Robinson equation of state
20
1.5
Dimensionless isothermal compressibility of SF6 , as modelled by
the van der Waals equation of state
21
1.6
Density,
isothermal
compressibility
of CO 2, as modelled
the van der Waals equation of state
viscosity
and kinematic
viscosity
of CO 2 at 310 K, as
1.7
22
,
a function of pressure
Comparison of physical properties of air, water, mercury (at
298 K and 1 bar), and CO 2 (at 150 bar and 310 K).
25
Relative
importance of natural convection of constant Reynolds numbers
1.8
Cylindrical duct geometries for which forced vertical laminar
flow is impossible under supercritical conditions due to natural convection
26
1.9
Schematic diagram of flat plate for hydrodynamic experiments
28
1.10
Effect of natural convection on the apparent diffusion coefficient and mass transfer rate; CO2-benzoic acid; 318 K,
30
160 bar
1.11
Geometry for the rectangular duct problem
32
1.12
Relative saturation versus inverse Graetz number; solution
valid for a
1/3 and high axial Peclet numbers
34
1.13
z-averaged local Sherwood number versus inverse Graetz number;
solution valid for a < 1/3 and high axial Peclet numbers
35
1.14
Experimental binary diffusion coefficients as a function of
solvent molar density
36
1.15
Diffusion coefficients for te
1.16
Diffusion coefficients for the naphthalene-SF6 system
39
1.17
Diffusion coefficients for the 2-naphthol-CO2 system
40
6
benzoic acid-SF6 system
38
LIST OF FIGURES (continued)
Page
1.18
Diffusion coefficients for the benzoic acid-CO2 system
41
1.19
Diffusion coefficients of benzoic acid versus SF6 fluidity
43
1.20
Hydrodynamic and power law isothermal behaviour of the
diffusion coefficient as a function of fluidity
44
1.21
Universal part of the composition dependence of the thermodynamic transport coefficient () that will give rise to a
48
composition-independent
D
2
1.22
Linearity of the kinetic conversion factor versus solute
mole fraction relationship. Benzoic acid-CO2 , 280 bar,
308 K. Peng-Robinson equation of state
50
1.23
Pressure and temperature dependence of the exponential correction factor for the fugacity coefficient of a solute in
a supercritical fluid (benzoic acid-CO2 )
52
1.24
Geometry of the benzene and CO2 molecules used in molecular
dynamics simulations.
53
1.25
Euler angles
55
1.26
Two-dimensional periodic boundary conditions
57
1.27
Kinematic description of two linear molecules
58
1.28
van der Waals, coulombic and total intermolecular interaction
energy for two CO2 molecules resulting from the sum of nine
elementary atomic interactions
59
1.29
Maxwell-Boltzmann and computed velocity distribution; # of
60
samplings
= 64; <T> = 310.3 K
1.30
Maxwell-Boltzmann and computed velocity distribution; # of
samplings = 67; <T> = 318.6 K
61
1.31
Maxwell-Boltzmann and computed velocity distribution; # of
62
samplings
1.32
<T> = 337.4
K
Maxwell-Boltzmann and computed velocity distribution; # of
samplings
1.33
= 67;
= 64; <T> = 342.3
63
K
Effect of temperature upon the carbon-carbon radial distribution function; p = 13.87 mol/lt
7
65
LIST OF FIGURES (continued)
Page
1.34
Effect of density upon the carbon-carbon radial distribution
function;
66
<T> = 315 K
1.35
Ensemble generating technique
67
1.36
Effect of temperature upon the mean squared displacement
versus time relationship
68
1.37
Arrhenius plot for the computed diffusion coefficients of
69
benzene
in CO2
1.38
Effect of density upon relaxation (short time) and diffusive
long time) behaviour; benzene-CO2; <T> = 310K
70
2.1
The Slattery-Bird correlation: a two-parameter corresponding
states approach
80
3.1
Dimensionless isothermal compressibility of C02 , as modelled
by the Peng-Robinson equations of state
98
3.2
Dimensionless isothermal compressibility of C02, as modelled
by the van der Waals equation of state
99
3.3
Dimensionless isothermal compressibility of SF6 , as modelled
100
by the Peng-Robinson
3.4
3.5
equation
of state
Dimensionless isothermal compressibility of SF6 , as modelled
by the van der Waals equation of state
101
Density, viscosity and kinematic viscosity of CO2 as a function 102
of pressure,
at 310 K
3.6
Comparison of physical properties of air, water and mercury at
ambient conditions with those of CO2 at 150 bar and 310 K.
Relative importance of natural convection at constant Reynolds
numbers
104
3.7
Hydrodynamic regimes for vertical cylindrical duct flow
105
3.8
Cylindrical duct geometries for which vertical forced laminar
flow is impossible under supercritical conditions
107
4.1
Equipment flow sheet
109
4.2
Flat plate assembly for hydrodynamic experiments
111
8
LIST OF FIGURES (continued)
Page
5.1
Rectangular duct geometry
5.2
Velocity
5.3
Relative saturation as a function of modified inverse Graetz
number, for various aspect ratios
138
5.4
Local Sherwood number (z-averaged) as a function of the modified inverse Graetz number, for various aspect ratios
139
6.1
Diffusion coefficients of benzoic acid in SF6
149
6.2
Diffusion coefficients of naphthalene in SF6
150
6.3
Diffusion coefficients of benzoic acid in CO2
151
6.4
Diffusion coefficients of 2-naphthol in CO2
152
6.5
Experimental diffusion coefficients as a function of solvent
density
153
6.6
Experimental diffusion coefficients as a function of solvent
reduced pressure
154
6.7
Experimental Enskog-Thorne factors
165
6.8
Experimental, theoretical and regressed Enskog-Thorne factor
for benzoic acid in SF 6
166
6.9
Experimental, theoretical and regressed Enskog-Thorne factor
167
profile
for naphthalene
6.10
as per Equations
(5.1) and (5.2); a = 1/3
benzoic acid
168
in CO 2
Experimental, theoretical and regressed Enskog-Thorne factor
for 2-naphthol
117
in SF 6
Experimental, theoretical and regressed Enskog-Thorne factor
for
6.11
116
169
in CO 2
6.12
Experimental diffusion coefficients as a function of reciprocal 178
viscosity at constant temperature (hydrodynamic test)
6.13
Linear (hydrodynamic), power law and transition regimes for the 181
dependence of the diffusion coefficient upon fluid viscosity at
constant temperature
6.14
Schematic density profiles for a dilute binary mixture in which 190
solute dissolves into the solvent from a plane (=O) of constant
(equilibrium) composition
9
LIST OF FIGURES (continued)
Page
6.15
Effect of natural convection on the apparent diffusion coef318 K and
ficient and mass transfer rate: benzoic acid-CO2
191
160 bar
7.1
Universal part of the composition dependence of *pfor various
weight ratios
203
7.2
Composition dependence of the kinetic conversion factor from
infinite dilution to saturation, as modelled by the Peng-
207
Robinson
equation
for C0 2 -benzoic
of state,
acid,
at 308 K
and 280 bar
7.3
Fugacity
coefficient
of benzoic
in C02 , at 380 K and 280 bar,
210
as a function of solute mole fraction, calculated with the
Peng-Robinson equation of state (EOS), and with the exponential
decay expression (K)
7.4
K values
for the CO2
benzoic
acid system,
as a function
of
211
temperature and pressure
8.1
Euler angles
213
8.2
Kinematic and dynamic characteristics of a linear rigid body
220
8.3
Maxwell-Boltzmann
9.1
Two-dimensional periodic boundary conditions
237
9.2
Nearest image location in a three dimensional cubic cell
239
9.3
Elementary generator of face centered cubic lattice
241
9.4
Cube-sphere site-site distance test
243
9.5
The shifted force potential
251
9.6
True and shifted Lennard-Jones potentials; rc/o = 2.4
253
9.7
Normalized error for the shifted force potential
254
9.8
Normalized force error; shifted force potential
255
9.9
Kinematic description of two linear molecules
263
9.10
van der Waals, coulombic and total energy for a binary CO,
2
interaction
264
velocity
distribution;
10
N = 107
229
LIST OF FIGURES (continued)
Page
9.11
van der Waals, coulombic and total energy for a binary CO2
265
9.12
van der-Waals, coulombic and total energy for a binary CO,
2
266
interaction
interaction
9.13
Absolute value of van der Waals, coulombic and total force
a binary
9.14
for
267
for
268
for
269
CO2 interaction
Absolute value of van der Waals, coulombic and total force
a binary CO2 interaction
9.15
Absolute value of van der Waals, coulombic and total force
a binary CO2 interaction
9.16
Coulombic force components for a binary CO2 interaction
270
9.17
Coulombic force components for a binary CO2 interaction
271
9.18
Coulombic force components for a binary CO2 interaction
272
9.19
Lennard-Jones force components for a binary CO
2 interaction 273
9.20
Lennard-Jones force components for a binary CO
2 interaction
2714
9.21
Lennard-Jones force components for a binary CO2 interaction
2'15
10.1
Geometry of benzene and C02 used in the simulations
277
10.2
Theoretical
sample
10.3
10.6
velocity distributions; <T(tr)>=318.6K;
280
size = 67
Theoretical and cdmputed velocity distributions;
sample size = 70
<T(tr)=321.4K;
281
Theoretical and computed velocity distributions; <T(tr)>=337.4K;
sample
10.7
279
size = 64
Theoretical and cmputed
sample
10.5
278
i
Theoretical and computed velocity distributions; <T(tr)>=310.3K;
sample
10.4
and cmputed velocity distributions;<T(tr)>=309.3K;
size = 64
282
size = 67
Theoretical and computed velocity distributions; <T(tr)>=342.3K;
sample
size = 64
283
11
LIST OF FIGURES (continued)
Page
10.8
Qualitative features of the radial distribution function
286
10.9
Effect of density upon g(r).
Density = 10.53 mol/lt,
<T> = 3 1 4.9K, sample size = 58; Density = 7.42 mol/lt,
<T> = 316.5K, sample size = 53
289
10.10
Effect of temperature upon g(r); density = 13.87 mol/lt.
290
<T> = 304.2K,
sample
size = 53;
<T> = 329.8,
sample
size = 56
10.11
Ensemble generation for test-particle "experiments"
292
10.12
Temperature dependence of the mean squared displacement versus
time relationship. Density = 10.53 mol/lt
295
10.13
Arrhenius
296
plot for four different
simulations
at a density
of
10.53mol/lt
10.14
Density dependence of the mean squared displacement versus time 299
relationship. <T(tr)>=309.3K, p=10.53 mol/lt; <T(tr)>=310.3K,
p=7.42 mol/lt
12
LIST OF TABLES
Page
2.1
Comparison of physical properties of air, water and super
critical CO2
75
2.2
References and experimental conditions for Figure 2.1
79
2.3
Experimental studies of diffusion of organics in supercritical fluids
90
5.1
Coefficients of the eigenvalue equation
124
5.2
Eigenvalues of equation (5.26)
125
5.3
Expansion coefficients for cross-section averages
137
6.1
Diffusion coefficients of benzoic acid in supercritical
sulfur hexafluoride
141
6.2
Diffusion coefficients of naphthalene in supercritical sulfur
hexafluoride
142
6.3
Diffusion coefficients of benzoic acid in supercritical carbon
dioxide
143
6.4
Diffusion coefficients of 2-naphthol in supercritical carbon
dioxide
144
6.5
Equilibrium solubility of benzoic acid in SF6
145
6.6
Equilibrium solubility of naphthalene
1 46
6.7
Equilibrium solubility of benzoic
6.8
Equilibrium solubility of 2-naphthol
6.9
Lennard-Jones and critical parameters for solutes and solvents
1 60
6.10
Enskog-Thorne analysis for the SF6 -benzoic acid system
1.61
6.11
Enskog-Thorne analysis for the SF 6 -naphthalene system
162
6.12
Enskog-Thorne analyses for the C0 2 -benzoic acid system
163
6.13
Enskog-Thorne analysis for the C0 2 -2 naphthol system
1 64
6.14
Stokes-Einstein analysis for the SF 6 -benzoic acid system
174
6.15
Stokes-Einstein analysis for the SF6 -naphthalene system
175
13
in SF 6
acid
in CO2
in CO 2
147
1148
LIST OF TABLES (continued)
Page
6.16
Stokes-Einstein analysis for the C02 -2 naphthol system
176
6.17
Stokes-Einstein analysis for the CO 2 -benzoic acid system
177
6.18
Proportionality constants for infinite dilution diffusion
coefficient correlations
183
6.19
Average values of the ratios between predicted and experimental diffusion coefficients for different hydrodynamic
correlations
184
6.20
192
Effect of natural convection upon mass transfer rate and
apparent diffusion coefficient; C0 2 -benzoic acid; 160 bar, 318 K
9.1
Site-site potentials in molecular dynamics; previous work
248
9.2
Site parameters for Slater-Kirkwood equation
258
9.3
Calculated site Lennard-Jones parameters; Slater-Kirkwood
equation
259
10.1
Diffusion coefficients corresponding to Figure 10.13
298
10.2
Diffusion coefficients corresponding to Figure 10.14
298
10.3
van der Waals and coulombic interactions for Figure 10.15
306
10.4
Empirical site-site parameters for CO2 simulation
308
10.5
P-V-T performance of Slater-Kirkwood parameters
310
Al-1
Parameters of cubic equations of state
319
A2-2
Buoyant stability calculations
329
14
SUMMARY
1:
The
two
technical
types
of
feasibility
constraints:
of a chemical process is determined by
thermodynamic
(equilibrium) limitations and
kinetic (rate) limitations.
In supercritical fluid extraction, the thermodynamic constraint defines
the maximum
achievable solute concentration in the supercritical fluid,
for any given temperature, pressure, and condensed phase composition.
From a
kinetic point of view, on the other hand, the objective is
to understand and, eventually, predict, the rate at which mass is transferred
from the condensed to the supercritical phase.
The
present work addresses
several problems related to kinetic (or
rate) aspects of supercritical fluid extraction, specifically:
the
·
influence of physical properties and their different behaviour
at slightly supercritical conditions in determining the rate and mechanism
of mass transfer in supercritical fluids
the development of a hydrodynamic technique to measure true and apparent
binary
diffusion coefficients in supercritical fluids, where "apparent"
denotes diffusion coefficients measured in the presence of buoyant forces
the interpretation of experimental data, and the use of hydrodynamic
*
theory to analyze diffusion in supercritical fluids
an analysis of diffusion in the light of non-equilibrium thermodynamics
·
the
use
of
molecular
dynamics to study the equilibrium structure
and binary diffusion in mixtures under supercritical conditions and infinite
dilution (i.e., conditions whereby solute-solute interactions are negligible).
1.1:
PHYSICAL PROPERTIES AND MASS TRANSFER MECHANISM
A supercritical fluid is rigorously defined as one whose temperature
and pressure are both above their critical values.
though,
we
restrict our attention
In the present context,
to the region bounded by 1<Tr<1.1,
and 1<Pr<4 (these are, of course, approximate numbers), where
the rate
of change of fluid properties such as the density, specific heat at constant
15
pressure,
viscosity,
etc. with respect
to temperature and pressure is
high and gives rise to behaviour which is unique to this relgion.
The
Figure
above
1.1,
defined
supercritical
region
is shown shematically
in reduced-density reduced pressure coordinates.
in
Some of
its peculiarities and their relevance to mass transfer wi 1 be discussed
below.
The dimensionless isothermal compressibility,
KI
T
is
defined
change,
at
=
aln P)
temperature.
and is shown
pressure,
Robinson
(1.1)
T
as the relative density change per unit relative pressure
constant
of state,
and
=ia~ne)
for
(1976)
CO2
It
in Figures
and
equations
SF6 ,
of
can
be calculated
f om an
1.2 to 1.5 as a functio~
for
state
the
van
(see
der
Appendix
Waals
of temperature
(1873)
1).
equation
and
iK'T tends
Pengto
1
at low pressure (ideal gas region), 0 at high pressure (den e fluid region),
and diverges at the critical point, where matter is infinitly
compressible.
The fact that K'T is finite at low pressure and infiniteIat the critical
point gives rise to states of matter whose density is co nparable to that
of an ordinary liquid, and which, simultaneously, are m re compressible
than a dilute gas.
region.
CO2
As
an
Such properties are characteristic of the supercritical
example,
at
318
K
and
100
bar
(Tr =
1.05;
Pr
=
1.36),
is - 280 times denser but almost three times more compressible than
at 318K and 1 bar.
From the point of view of mass transfer, it is the different rates
of change of density and viscosity in going from the
deal gas to the
dense fluid region that give rise to unique behaviour un er supercritical
conditions.
of
C02 (T
Figure 1.6 shows the density, viscosity and ki ematic viscosity
= 304.2
of pressure.
its
K; Pc = 73.8
bar)
at 310
K (Tr = 1. 2) as a function
Although the density of supercritical CO, is liquid-like,
viscosity,
being virtually pressure
independent a
low pressures,
is less than an order of magnitude higher than the cor esponding dilute
gas value.
by
roughly
This is due to the fact that, although both qulntities increase
an
order of magnitude near Pc, the
finitf compressibility
of the dilute gas gives rise to an increase of approxirlately two orders
16
2.0
u,
0c
I
U
V
1.
O.
0
1.0
0.1
10.0
Reduced Pressure
FIGURE 1.1: Schematic pressure-density diagram for a pure substance showing
the region of interest in supercritical extraction.
17
m
-;
4
0.
0
LU
I.
W
( -
2
c
0
we
0
100
200
PRESSURE ( Bar)
300
FIGURE 1.2: Dimensionless isothermal compressibility of C02 , as modelled
by the Peng-Robinson equation of state.
18
I-
-
J
4
130
m
3
C.:
I-
2
L
_J
I
0
Cr
mo
Cow
0
100
200
300
PRESSURE (Bar)
FIGURE 1.3: Dimensionless isothermal compressibility of C0 2 , as modelled
by the van der Waals equation of state.
19
_
_I
-i
l
I
I
I
Tr -1.02
C)
4
w
1.04
0O0
2
1.08
SF6 (P-R)
1.10
Y)
0
w
L)
0
wI
nv
!
0
_
I
I
I
_
·
50
100
--
150
PRESSURE ( Bar)
FIGURE 1.4: Dimensionless isothermal compressibility of SF6 , as modelled
by the Peng-Robinson equation of state.
20
-Iye
I
%.
I
I
I
CD
C-
n
3
2
.L-
0
iI
I
D
W
Cr
.
-L
0
50
I
-1
100
150
PRESSURE ( Boar)
FIGURE 1.5: Dimensionless isothermal compressibility of SF6 , as modelled
by the van der Waals equation of state.
21
r t ulrl
FIGURE.6.i.:Densait, viscosity
and kinematic viscosity
a function of
pressuref
o
CO
C2
22
at
3,
of magnitude in density when the pressure is raised up to values slightly
No such behaviour is displayed by the viscosity.
lower than Pc.
Consequently, the kinematic viscosity under supercritical conditions
This is clearly shown in Figure 1.6.
is very low.
Liquid metals, which
combine moderate viscosities with high densities, are normally associated
with
kinematic viscosities.
low
extremely
This property, as will
be
shown below, attains even lower values in the case of supercritical fluids.
The relevance of
these facts to mass transfer
is best illustrated
in the case of duct flow under the combined influence of a pressure gradient
We consider the situation whereby an incompressible, Newtonian
and gravity.
fluid,
developed laminar flow conditions, flows inside a
fully
under
duct whose walls are coated with a solute that dissolves into the fluid,
under the action of a concentration gradient (thermodynamic phase equilibrium
at
exists
the
aforementioned concentration gradient
The
interface).
will give rise to a density gradient which will, in turn, alter the velocity
profile.
When
concentration
the
(and density)
changes are small,
it
is an admissible approximation to expand the density about the pure fluid
value in terms of solute concentration, consider linear terms exclusively,
and neglect the composition dependence of other properties (i.e., viscosity).
This
approximation) gives rise
(Boussinesq's
simplification
to
the following dimensionless momentum balance, (see Section 3.1).
2
(V+) v
Re
where Gr
and Re
If
V+n
( = [(2R)3g
( = 2R <v>/v),
we
now
r
_ g, Gr
Ap/p]/v2)
is the Grashof number for mass transfer,
the Reynolds
introduce
(1.2)
O
number.
the natural scales for viscous,
inertial and
buoyant forces,
Viscous forces - n <v>/2R
Buoyant
forces
(1.3)
(1.4)
- 2R g Ap
2
Inertial forces - <v>
po
(1.5)
the physical significance of the parameter GrRe
Gr
Re2
(2R g Ap) (<v> 2 po)
(n<v>/2R)2
(n<v>/2R) 2
2
(<v> 2 po)
23
-
follows immediately,
Buoyant forces
Inertial forces
(1.6)
Consequently, if different fluids flow inside identical ducts under
mass
diffusive
the
conditions at any
convection
kinematic
given Reynolds number, and
(Ap/p), the relative
comparable density changes
assuming
natural
transfer
importance of
(buoyant forces) scales inversely as the square of
Thus, fluids with
of the fluid in question.
viscosity
low kinematic viscosities (i.e., supercritical fluids) can develop appreciable
buoyancy driven flows even with small density gradients.
If
we
now
compare
(Figure 1.7) the properties of air, water, and
(298K, 1 bar) with those of supercritical
mercury at ambient conditions
CO 2 at 310K and 150 bar (Tr = 1.02; Pr = 2.03)
in the light of the previous
We note the fact that,
some important consequences arise.
discussion,
for v, CO2 has the lowest value.
The fourth column is the ratio of buoyant
to inertial forces at constant Reynolds number and duct geometry, scaled
with the corresponding value for water.
convection,
therefore, is more
The relative importance of natural
than two orders of magnitude higher in
a supercritical fluid than in ordinary liquids.
When mass transfer in the fluid phase is the controlling step, therefore,
the usual design relationships' for mass transfer in packed beds
(Gupta
and Thodos, 1962; Wilson and Geankoplis, 1966; Williamson et al., 1963)
are
either
unsuitable
on account of the absence of a
Grashof number,
or do not cover (Karabelas et- al., 1971) the low Schmidt numbers (- 10)
which characterize
diffusion of light organic
solutes in supercritical
fluids.
Figure
1.8 shows the
importance of natural
convection
laminar duct flow under supercritical conditions.
itself
is not
limited to supercritical conditions,
in vertical
Although the figure
the values of
the
physical properties used to construct the actual curves are typical of
supercritical
fluids
(molecular weight -
(v) and of diffusion of
small organic molecules
100) in supercritical fluids (Sc = v/ ).
In Figure
1.8, D is the duct diameter, and L, the coated length along which diffusion
takes place.
For
any
value
of
p/p
(interface-bulk density difference divided
by mean density), the region lying above and to the right of the given
curve represents tube geometries for which vertical forced laminar flow
is impossible due to the presence of natural convection.
24
For example,
lo0-
4
-
Pr
tLI
4
Hg
I
,H2 0
Hg
10-4
104
I 1 Hg
IO3
^
0oz I
H2 0
1(-34
I
10
Air
IH2 0
SCF
I
10
10-6t iH2 0
104
I0
I
I SCF
10
I
PAir
Hg
0-5
?
i7
ISCF
i
-2
IAir
I
IAir
p(Kg/m 3 )
10 - 3
o-0.
· r(Nsm
-2
)
V(m2 /s)
(YH2 0/z)2
of physical properties of air, water, mercury
FIGURE 1.7: Comparison
(at 298K and 1 bar), and C0 2 (at 150 bar and 310 K). Relative
importance of natural convection at constant Reynolds numbers.
25
-I
10
(D )
-2
I0
-3
10
2
4
6
8
10
12
14
D (mm)
FIGURE 1.8: Cylindrical duct geometries for which forced vertical laminar
flow is impossible
under supercritical conditions due to
natural convection.
26
given a relative density change as small as 10 3, any aspect ratio greater
than 10- 3 makes
it impossible to attain forced laminar flow in an 8mm
vertical duct.
Figure
1.8
is
valid
for
If the parameter Gr.Sc(2R/L)
10 - 2
<
Sc
(2R/L)
< 1
and
vertical
flow.
(a scaled mass transfer Rayleigh number)
exceeds 104, forced laminar flow is impossible (Metais and Eckert, 1964);
this criterion has been
Gr- Sc- 2R/L <
104
used to obtain
Figure 1.8.
The
condition
is a necessary but not sufficient criterion for forced
laminar flow, so that the region lying below and to
the left of each
curve does not, by itself, guarantee laminar flow.
1.2:
HYDRODYNAMIC EXPERIMENTS
The
experimental
technique
involved
fully developed
laminar flow
of a supercritical fluid in a horizontal rectangular duct (Figure 1.9),
the bottom surface of which was coated with the solute of interest.
The amount of solute that, at steady state, precipitates, upon decompression, from a measured amount of solvent during the course of a carefully
timed experiment is, for a given temperature, pressure (and hence equilibrium
solubility of the solute in the supercritical solvent), flow rate, and
duct geometry, a function of the binary diffusion coefficient.
The determination of a diffusion coefficient thus involves (at least)
one
equilibrium
experiment,
where
the
solubility
of
the
solute
in
the
supercritical fluid is measured, and a diffusion experiment.
In
a diffusion
experiment
(Figure
1.9)
a brass
plate
(4)
is tightly
fitted into an enclosure made up of two aluminium hemi-cylinders
(1,2);
fluid flows inside the resulting channel (3).
The test section (6) is made by casting molten solute and carefully
machining and polishing after solidification.
a
section
section
The plate also
contains
(5) where laminar flow is allowed to develop, and an outlet
(7).
Fluid
by-pass of the test section is prevented by a Viton gasket
(8) which forces the plate against the upper surface (9'), and by
the
labyrinth seal (9) which results when the hemi-cylinders are forced together
(10,11) and Teflon
tape
is placed between
27
the upper and lower mating
-
i
5
,
I
·
6
. .
.
!f//J
/
.
.
..
7
.. _
7
//////JJ,
P-j-
I
8
8
i
I
-
I
_
II
_
I
I0
10
-
--
_
J
0if
_J--
r
_
I
.
2
1i
e.
1
0i
1I
_
r--
m
,
I
I
J-L..
12
FIGURE 1.9: Schemactic diagram of flat plate for hydrodynamic experiments.
28
surfaces
(9).
The whole
assembly
inches) stainless steel pipe.
12 is
1 and
notched:
the pressure
is tightly fitted inside a 5 cm
(2
0-ring 13 provides sealing, while 0-ring
in 3 is thus
equal
to the pressure
outside
2.
For dilute solutions, the density decreases monotonically away from
the surface
if
M,/M 2 > V/V
and
viceversa.
molecular
In
(1.7)
2
Equation
weights, V
(1.7),
M,
and M 2 are
is the solute
the solute
and solvent
partial molar volume, and V2
the
solvent molar volume.
The
above inequality (see Section 6.3 for derivation) was satisfied
under all of the experimental conditions
addition,
of
the
tested
(see Appendix
2).
In
the equilibrium solubility increased with temperature for all
systems investigated at every
experiments
were conducted.
This
value of the pressure for which
implies that the solubilization was
endothermic under all experimental conditions.
Consequently, when channel 3 was horizontal and constituted the duct's
bottom surface, the flow was unaffected by gravity, and true binary diffusion
coefficients were measured, as explained above, by weighing the amount
of solute that precipitates, upon decompression, from a measured quantity
of solvent, at steady state.
1 and 2
Buoyant effects were introduced by rotating
(Figure 1.9) inside the steel pipe.
gave rise to different results, which
The same experiment then
provided qualitative information
on the importance of natural convection in mass transfer with supercritical
fluids.
The
CO2
at
results
of such an experiment,
160 bar and 318K,
are shown
for benzoic acid diffusing
in Figure
The dotted line (mass transfer rate)
1.10
(see also Table
in
6.20).
in Figure 1.10 does not extend to
00 since the diffusion (00) experiment was done at a different flow rate,
and mass transfer rates are a function of fluid velocity.
of natural convection, as well as
when
using
The importance
the potential for experimental error
hydrodynamic techniques, can be seen from the fact that a
650% increase in the apparent diffusion
rotation of the flat plate.
29
coefficient results from a 900
40
10
9
8
30
7
E
0
E
E
20
5
N
Cx
10
3
:)
!
n
"0
10 20
30 40
50
60
70 80 .90
a (degrees)
FIGURE1.10: Effect
of natural
convection on the
coefficient and mass transfer rate;
318 K, 160 bar.
30
II
lapparent
diffusion
C0 2 -benzoic acid;
The actual calculation of a diffusion coefficient from experimental
measurements implies knowledge of the solution to the problem of diffusion
from a constant composition source plane into a rectangular duct where
axial fully developed laminar flow of an incompressible fluid takes place.
For the geometry shown in Figure 1.11, the problem to be solved can
be written in dimensionless form,
+
+
v
,
2c
= (-2) Pe
+
rC 2C)
X +2
+
+
x
ax
+
2 +
ay
a2c+
]
(1.8)
az
where
a = b/a
B = L/2a
x+ = x/b
y+ = y/b
z+
(1.9)
= z/b
c + = 1 - c/ci
v + = v/<v>
Pe
= <v> L/
with ci,
<v>,
the interface solute concentration, L, the coated length, and
the cross section average velocity.
Equation (1.8) can be simplified
by performing an order of magnitude analysis, defining
- b - y
and
using
(1.10)
the following empirical
expression for the velocity profile
(Shah and London, 1978)
v
v
=[1- ]
max
(l [
b
[
I
a
(1.11)
m = 1.7 + 0.5 a 14
n = 2
a < 1/3
n = 2 + 0.3 (a -
where
1/3) a
(1.12)
1/3
Equations (1.12) were obtained by fitting Equation
finite difference solution of the momentum balance equation.
31
(1.11) to the
.,
i
Y~
2b
[
K
_mZ
_
FIGURE 1.11:
--
Geometry for the rectangular duct problem.
32
From order of magnitude considerations, diffusion in the axial (x)
and
transverse (z) directions can be considered negligible with respect
to axial convection and diffusion away
from the source plane (i.e., in
the y direction), once the actual values of the shape factors and physical
properties are taking into account (a,
, Pe; see Chapter 5).
The problem then becomes, for a < 1/3,
+
+
-
(25
Equation
and
+
+2
_c
m
+2ac
)
ax+
m + 1
-
4B
1
3a Pe
[1 - (al+l 1)m]
2c
(1-13
(1.13)
+
(1.13) can be solved by variable separation, series expansion,
integration
of
the resulting two dimensional solution across
+
transverse (z ) direction.
1.12 and 1.13.
the
The solutions are shown graphically in Figures
The full expressions are given in Chapter 5.
The quantities
plotted in these figures are defined as
<r> = <c>/ci
Sh-
(1.14)
4b
(1 + a)
1
k d(z/a)
(1.15)
<v> b 2
Xo = x9/
(1.16)
i.e., the relative saturation, the z-averaged local Sherwood number based
on the hydraulic diameter and a modified inverse Graetz number.
The important point is that <r>, the experimentally measured quantity,
is a function of a( = b/a) and X.
Thus, for a given aspect ratio, flow
rate, coated length and channel height, <r> is a function of the diffusion
coefficient.
In
an
experiment,
then,
the
ratio
<r>
is
determined
by
measuring c and ci in a diffusion and an equilibrium experiment, respectively,
and
calculating
1.12, to obtain
1 .3:
X
from
finally,@,
the mathematical expression plotted
from X,
in Figure
x (= L), <v> and b 2 .
EXPERIMENTAL RESULTS AND DATA ANALYSIS
The experimentally measured diffusion coefficients are shown in Figure
1.14
as a function of solvent molar density; the same data are plotted
33
z
0
0.4
I--
ASPECT RATIO
cf)
.I
0.2
.125
.2
I
w
.25
cr
.33
I
I
0.0
0.5
_
_
II
1.0
XD
<v>b 2
FIGURE 1.12:
Relative saturation versus inverse Graetz number; solution
valid for a
1/3 and high axial Peclet numbers.
34
Sh
xD
<VPb2
FIGURE 1.13:
inverse Graetz
axial Peclet
high
and
c 1/3
z-averaged local Sherwood number versus
number; solution valid for a
numbers.
35
I
I
I
I
I
I
C
I
I
I
I
I
I
I
i
.
I
.
Temperature (K) -
·
Solvent
SF6
Solute
Benzoic Acid
o
0
SF6
SF6
Benzoic Acid
Naphthalene
a
SF6
Naphtholene
328.2
338.2
318.2
328.2
+
CO2
Benzoic Acid
318.2
v
CO2
Benzoic Acid
o
CO2
2-Nophthol
K
CO2
2- Naphthol
328.2
308.2
318.2
I
I
1
..- 1"
, " -..
0
hbhb
m
B
<i
io-5
I
-
5I
I
I
I
I
I
.......
I
I
I
10
I
15
,
I
I
20
I
I
25
p (mol/4t)
FIGURE 1.14:
binary diffusion coefficients as a function
Experimental
of solvent molar density.
36
separately for each system in Figures 1.15 - 1.18.
The use of density
instead of pressure as an independent variable is a consequence of the
molecular approach to diffusion, whereby this phenomenon is seen as the
macroscopic
consequence
of collisions
(interactions), whose
frequency
(for a given binary system) is a function of the molecules' average velocity
(temperature)
and number
density
(i.e., number of molecules per
unit
volume).
The
density
range
over which
experiments were made for any given
system was too small to allow generalizations as to the observed linearity
of the data when plotted
in log D vs. p fashion.
The measured diffusion coefficients of benzoic acid were always lower,
at any given density, than the corresponding values for naphthalene
SF6 )
and
benzoic
2-naphthol (in C02), both of which are larger molecules
acid.
This
(in
than
suggests a possible dimerization of benzoic acid
in the fluid phase, a behaviour experimentally observed in CC1, and CC1 3H
(I'Haya and Shibuya, 1965), and in its vapour, C 6 H2,,,CCl
et al., 1966).
and C 6H 6 (Allen
This hypothesis is also consistent with the exceptionally
high temperature dependence of the diffusion coefficient of benzoic acid
at constant density (see Figure 1.18), which shows an activation energy
(6.9 Kcal/mol) in good agreement with the experimentally measured values
for the dimerization of benzoic acid in cyclohexane (6.4 Kcal/mol), CCl4
(5.5
Kcal'/mol),
1966).
and
its
own
vapour
phase
(8.1
Kcal/mol)
(Allen
et
al.,
These considerations can only be taken qualitatively, since the
overall diffusion coefficient is related in a non-linear way to the dimerization constant.
Although
benzoic acid
association in the fluid phase has
not been
measured in either SF6 or C02, the published equilibrium constants (Allen
et
al.) give rise to high associated
fractions
(69% at 303K
in CC1,
for example; see Chapter 6 for detailed calculations), making the dimerization
hypothesis at least plausible.
At
any
given
density,
the temperature dependence of the
diffusion coefficients is higher than the T
1/2
observed
hard sphere prediction.
However, for three of the four systems investigated, the quantity rnDT 1
was
found
suggesting
acid-CO2
to be fairly
a
constant over
hydrodynamic
the range
of conditions tested,
(Stokes-Einstein) description.
data, on the other hand,
37
The benzoic
exhibit peculiarities which
seem to
m
*.
.
.
.
I
.
I
.
-
*
I
:
I
CM
I
I
0
I
I')
to
0
/
u
.
0
a)
4i
iZ
I
co
4l
0o
LU
m
n)
N
N
0
a)
C
4-4
U,
I
-
ED
Lid
O)
4J
.-
In
0o
eo
U,
4H
I
f,
-A
f-I
;L
I
(D
i
I
-
CM
I
I
I
IO
m
(
LO)
W3)
I
S01
N1
38
[
[[[
-~~~~[
[
I
.
w
I
I
^
.
.
I
I
I
.
.
cN
0
O
a0
c)
CI
4.J
Nb
-
I
O
E
o Q
0co
4I
0
a)
--o
-r-I
U%
,-I
-
(0
,-4
l
I
*I
I
nl[
I
I.
--
-
I
I--
I
tI
.0-1
( S/ZW)
0
39
L)
_~~~..
iiii
_
II
_
I
I
I
C%
NM
-
00 0o
cO
o
0
N
0
e
.0I
-C
aJ
-I
)O
Z
IC
z0
a)
I
-
0E
A
40
r.
C:
(1
4r
cx
.]
4-
I
m
0
c-
u
(S
I
I
I·
I·
I
I'
U)
W3) (rG01l
40
LO
Ij
11
N
0CJ
4
an
4.J
ci
0)
0o
I
N
04-
a)
O
t-i
0O
E
a,
o
4.4
co
.rq
(o
a,
0:
Ln
I-
0
(s/Z==)
Lo O
cO
I
41
arise from a highly temperature-sensitive
cannot, therefore, be expected to show
effective molecular size, and
constant nDT l values, as will
be explained below.
From Tables 6.1l4,6.15 and 6.16, it can be seen that nDT
' is constant
to
within
standard
deviations
(expressed as percent of
the mean), of
4.2% (benzoic acid - SF6 ), 9.2% (naphthalene - SF6 ), and 6.5% (2-NaphtholC02 ), respectively.
The Stokes-Einstein equation (Einstein, 1905)
D
kT
kT
=
relates
(1.17)
the diffusion coefficient
of a
spherical Brownian
particle of
radius a to the temperature and viscosity of the surrounding fluid, when
no slip exists at the interface.
= f
nDT
Equation (1.17) implies the basic concept
(1.18)
[size]
-1
or, in other words, for hydrodynamic behaviour, a plot of D vs. n
should
yield a straight line through the origin with a slope proportional to T.
Figure 1.19 shows the benzoic acid - SF 6 results plotted in this manner.
As can be seen, the lines have
deviation
from
strictly
small but finite intercepts, indicating
hydrodynamic
behaviour, a
fact already noted
by Feist and Schneider (1982) in connection with their studies of diffusion
in supercritical CO,.
The general picture that emerges is sketched in Figure 1.20. Hydrodynamic
behaviour is approached at high viscosities; deviations from this limiting
behaviour can be correlated (but not understood) by means of empirical
power law relationships of the type D a n
1971).
(a < 1) (Hayduck and Cheng,
Supercritical viscosities fall roughly in the range 0.04.n50.1 cp
for 1.1Pr4
and 1Tr-1.06,
which corresponds to
110
-3
n- 1 -2.5
in the
units of Figure 1.20 (a typical liquid viscosity is also shown for comparison).
The
exact point at which hydrodynamic behaviour breaks down (point
c) cannot, at present, be predicted from first principles for any given
system.
However, from Figure 1.20 it can be concluded that predictive
correlations
based
on
the Stokes-Einstein equation
(Wilke and
Chang,
1955; Scheibel, 1954; Reddy-Doraiswamy, 1967; Lusis-Ratcliff, 1968) will
42
_
_ _
I
I
I
15
E
U 10
60)
K
In,
5
0
I
__
_
I
O
2
I
103 X ( p-I)
FIGURE 1.19:
Diffusion coefficients of benzoic acid versus SF6 fluidity.
43
/
/%
D
ICp
io3 ?- (/ P 1)
FIGURE 1.20:
and power law isothermal behaviour of the
Hydrodynamic
diffusion coefficient as a function of fluidity.
44
overestimate
diffusion coefficients in supercritical fluids.
This was
indeed found to be the case (see Chapter 6) when each of the above correlations was
applied to diffusion of naphthalene in supercritical CO2
and
ethylene (Iomtev and Tsekhanskaya,
1964), benzene
in supercritical
CO2
(Swaid and Schneider, 1979), as well as 2-naphthol in supercritical
CO2 , and naphthalene and benzoic acid in supercritical SF6
(this work),
with the single exception of the Wilke-Chang expression for the naphthaleneethylene system.
In addition, at high enough viscosities
(or, equivalently, at high
enough pressures for any given temperature), the quantity nDT-
1
approaches
a constant value; geometrically, this is equivalent to saying that, at
small
-1
n
values, the curve
Ocb is well approximated
by the line Ob.
As an example, the measured diffusion coefficients of benzene in supercritical
CO 2 (Swaid and Schneider,
1979) give rise to an nDT-
1
value that is constant
to within a 4.6% standard deviation (expressed in percentage of the mean)
when n
0.04 cp, irrespective of the temperature and pressure.
Thus, at high enough viscosities, hydrodynamic behaviour is approached,
and
this fact can be used to extrapolate experimental data by assuming
constancy of
1.4:
DT1
.
DIFFUSION AND IRREVERSIBLE THERMODYNAMICS
Kinetic
approaches
view
gradients in concentration.
diffusion as
a phenomenon resulting
from
From the point of view of irreversible thermo-
dynamics, on the other hand, chemical potential gradients, and not concentration gradients, constitute the appropriate driving force for diffusion.
The
rationale behind this approach
is that the uniformity of chemical
potential (for any given species) throughout a system in which no impermeable
boundaries exist is a condition for thermodynamic equilibrium.
if
the deviations from this
Consequently,
condition are small enough to justify the
assumption of local equilibrium, it is plausible to assume a restoring
(driving) force proportional to chemical potential gradients.
This relation-
ship between fluxes and thermodynamic driving forces can indeed be proved
(see Appendix 5) without any additional assumption other than the local
validity of thermodynamics within an overall non-equilibrium situation.
the
From
standpoint
irreversible thermodynamics, then, we
of
can
write,
for isothermal binary diffusion with no viscous dissipation (see
Chapter
7),
11 =
where j,
a V ,
-
(1.19)
a mixture chemical potential per unit
is a solute mass flux,
mass,
MlM1
M1
a,
and
the
(1.20)
M2.
transport coefficient.
corresponding
be rewritten using the definition of
, the
Equation
(1.19) can
Gibbs-Dubem equation,
and
the definition of 01, the fugacity coefficient,
1
=
0
= xl
Vl1 + (1 -
f
= xl
P
obtain
to
1
V
Vp1 -
M
(1.21)
VU2
xl) V
(1.22)
2
(1.23)
1
solute flux and solute
relationship between
a mathematical
mole fraction,
[1 +
-a
Al
MIn
this
Equation
of
x
T,P
into Equation
[ 1 +
(aln 0l/
][1x +
1
]. 1
M 2 (1
-
X
(1.24)
approach, however, the transformation of
thermodynamic
(1.19)
a term,
n
1+(aln
(1.24) has resulted in the introduction
aln xl)T
], which
phenomenological descriptions of diffusion.
is absent
in
the usual
Of the many equivalent phenom-
enological expressions, we choose (Bird et al., 1960)
C2 M
M
P
12
y
where c
is the total molar
and$
therefore, reads,
t
1 2,
a RT
1 - x
Ic
x1
x
(1.25)
The relationship between a
concentration.
112/
M+
M2
1 46
12
2
X,
(1.26)
It
(D1 2)
is customary
to define
a "thermodynamic" diffusion coefficient
with the same units ofC@1 2,
and such that it will equal9
1 2
in
ideal mixtures, in which case
an x+l )T,P
[1
1
(1.27)
or, in other words, when ~ is composition-independent.
=DD12
2
[1
Consequently,
+ (lnalnx
x 1iT,
)TP ]
(1.28)
or
2
D12
(
=
D1 2 is normally
than
2.
xl
MM
assumed
2
(1.29)
(Reid et al., 1977)
Thus, at any
independent D
1
to be less composition-dependent
given temperature and pressure, a
composition-
imposes upon a the requirement that it depend on x
as
the dimensionless function
(x
=c()
1+ c(o)
(
V ) dx
1/2
(130)
x1
1/2
M2
.30)
1 -x1
where V is a molar volume, and Vi, a partial molar volume.
Whereas the numerator is very specifically dependent upon the binary
system under consideration,
solute
mole
fraction
and
the denominator is a universal function of
solute/solvent weight ratio.
The numerator
being a finite number. we notice that 4p(xl) (and hence a) vanishes at
x
- 0 and x
*+ 1, in agreement with the fact that p
an undefined quantity at x
+ 0 and x
(and hence V)
is
+ 1.
Figure 1.21 is a plot of the reciprocal of the denominator of Equation
(1.30)
(i.e.,
that D
2
the universal
part
of the
is composition-independent).
i exhibits a maximum value M2 /4 M.
47
composition-dependence
of
At a mole fraction of M/(M,
a such
+ M2 ),
2.5
A
U
I
I
I
I_
Im
2.0
M
1.5
M
1.0
0.5
0.0
!
i
,I
-1
0.0
0.5
-_I
1.0
Xl
FIGURE 1.21:
Universal part of composition dependence of the thermodynamic
transport coefficient (a) that will give rise to a compositionindependent
D
2.
The left hand side of Equation (1.27) represents a conversion factor
between "kinetic" and "thermodynamic" diffusion coefficients, whose theoretical implications are discussed in Chapter 7.
esting
consequences
of
Here, some of the inter-
the calculated composition dependence of this
function will be outlined.
The explicit form of [1 + (aln cl/
any given equation of state.
xl)Tp]
xln
can be obtained from
For two-parameter cubic equations of state
with composition-independent combining rules (see Chapter 7 and Appendix
1), a highly non-linear and complicated expression results (see Equation
(7.27)).
However, when plotted as a function x
(from infinite dilution
to saturation) Equation (7.27) is a straight line of negative slope and
unit y - intercept,
1
+
an
X TP
=
1 - K x
(1.31)
or, in other words,
,(x=,T,P) =
(O,T,P) exp [-K(T,P)x,]
(1.32)
where
~l
4,(x)
= lim
X1
(1.33)
+ O
This functionality is shown graphically in Figure 1.22 for the benzoic
acid-CO 2 system at 280 bar and 308 K between infinite dilution and saturation,
and
is representative of what appears to be a general feature of the
thermodynamic
behaviour of binary systems consisting of a non-volatile
solute and a supercritical fluid (see Chapter 7).
Quite apart from the simplicity of Equation (1.31) relative to Equation
(7.27), the parameter K has a physical meaning and an apparent asymptotic
behaviour
that
the linearity
could have
interesting thermodynamic implications.
implied by equation
where the factor [ 1 + (ln
,/Dln
(1.31) is extrapolated
xl)Tp]
K = 1/ x(l.s.)
0
where x
=
jlexp[
If
to the point
equals zero, we obtain
(1.34)
- x,/ x(l.s.)]
(1.35)
(l.s.) is the composition (mole fraction) of the mixture when
49
.I
1.00
I
I
I
I I·I
I
UI
U
I
I
U
I
X
xIL1-:
C
0.95
m
0.90
m
<-EeC
+
Ira-
I
0.000
I
·
0.001
-
0.002
I
I
0.003
- II
0.004
x
FIGURE 1.22:
Linearity
of the kinetic conversion factor versus solute
mole fraction
relationship.
Benzoic acid-C0 2, 280 bar,
308 K. Peng-Robinson equation of state.
50
it reaches its limit of stability at the given T and P (Modell and Reid,
1983).
Equation (1.35), although obtained
from a linear extrapolation
that may not be valid up to the limit of stability, suggests a "natural"
scaling for the concentration, in analogy with the idea of corresponding
states.
Furthermore,
when
the K values for a given system are plotted as
a function of temperature and pressure
K approaches a high
(Figure 1.23), it appears that
pressure limit which is independent of temperature
(at 280 bar, the K-values
in Figure
1.23 are within
6.5%
of the mean).
We may summarize by saying that, according to Equation (1.32), the
fugacity
(¢1,
coefficient
the
is the product of a
composition-independent term
infinite dilution fugacity coefficient), and an exponential
and explicit composition correction which is not only small, due to the
small values of x
, but appears to approach a high pressure limit which
is independent of temperature.
1.5:
MOLECULAR DYNAMICS SIMULATIONS
Molecular dynamics is the numerical solution of the many body problem
and
the use
of
statistical mechanics to
interpret the results
(i.e.,
evolution in time of the velocities and coordinates of the bodies (molecules)
whose motion is being studied).
In this work, 107 CO2 and 1 benzene molecules were considered (Figure
1.24) and treated as rigid polyatomics (i.e. point centers of force with
no internal degrees of freedom), with pairwise additive atom-atom interactions:
U1
where
=
iI jJ
i and j denote
U..
sites
(1.36)
(atoms) belonging to bodies
(molecules) I
and J, respectively.
The dynamic simulation of this system of interacting molecules required
the
solution of equations for the translational and rotational degrees
of freedom.
For the former, Newton's
51
second law of motion was written
600
400
K
200
0
0
50
100.
150
200
250
P (Bar)
FIGURE 1.23:
dependence of the exponential
Pressure and temperature
correction factor for the fugacity coefficient of a solute
in a supercritical fluid (benzoic acid-C02 ).
52
0
0
C
0
H
H
H
12(
H
H
H
FIGURE1.24:
Geometry of the benzene and CO2 molecules used in molecular
dynamics simulations.
53
as two coupled first order equations,
= m-
1
(1.37)
f ({x}, {e})
(1.38)
v
-
where m, x and v denote the molecule's mass, center of mass coordinates
and
velocity,
respectively,
the
molecules'
and
sites,
and
is
the
function
a
f,
force,
of
is
the
sum
forces
translational
the
e} instantaneous configuration of the system.
rotational
of
x}
on
and
The rotational
equations for the solute, a non-linear molecule, are
1
e =
where
I-1
1
K
1
({x},
e})
(i = 1,2,3)
(1.39)
ew
(1.40)
2 =
i denotes one of the body's principal directions, and Ii and Ki
are therefore the it h principal moments of inertia and the torque component
along
a
principal direction
the i
of
function
the
(which, as was
the case with
system's instantaneous configuration).
f,
is
In Equation
(1.40), e and e denote, respectively a vector and a matrix whose elements
are the time derivative and the instantaneous value of the four CayleyKlein
parameters (Goldstein, 1981) (with appropriate signs and ordering
in the latter case), and w contains the principal angular velocity components.
The Cayley-Klein parameters represent a non-singular kinematic
description of the rotational degrees of freedom of the rigid body, and
replace the Euler angles (Figure 1.25), which give rise to singular equations
(Murad and Gubbins, 1978), unsuitable for numerical applications.
For the linear solvent, on the other hand,
where
w
I-1 K (x},
i
w x 1
(1.41)
e})
(1.42)
I is the line's moment of inertia, K is the torque, and 1 is a
unit vector parallel to the line.
Equations
(1.37),
whereas Equations
(1.38),
(1.41) and
(1.42) are frame-invariant,
(1.39) and (1.40) refer to the body's principal axes
of inertia, and imply a linear coordinate transformation at each integration
54
z
y
FIGURE 1.25:
Euler angles.
55
step.
to
The resulting system of 1297 of
the configuration dependence of
quations is highly coupled due
forces and torques, and was solved
numerically via an Euler predictor, trapezoid corrector algorithm.
The 108 molecules were placed in a unit cell of space-filling geometry
and
cubic shape, and periodic boundary conditions were used throughout
(Figure 1.26).
Although C02 has an appreciable quadrupole moment, the introduction
of electrostatic
forces into the model was abandoned after simulations
with van der Waals forces and point monopoles exhibited large temperature
and pressure fluctuations.
This behaviour was ascribed to the stiffness
caused by the highly orientation-dependent effective electrostatic interactions, and is illustrated in Figures 1.27 and 1.28 where the relative
orientation of two CO2 molecules is defined and the effective dimensionless
(i.e., U/kT, T = 300 K) intermolecular Lennard-Jones, electrostatic and
total energies (which result from the corresponding interatomic pairwise
additive
energies), are plotted against carbon-carbon separation.
The
salient features of Figure 1.28 are the orientation-sensitivity of the
electrostatic
potential,
and
the effective short range
electrostatic
interaction arising from elementary long-ranged Coulombic interactions.
Velocity
distributions
for the
solvent molecules corresponding
to
four different run average translational temperatures,
<T(tr)>
=
2<KE(tr)>
3Nk
(1.43)
with tr denoting translation, KE, kinetic energy, and N = 107, are shown
in
Figures
1.29-1.32.
Maxwell-Boltzmann)
where
The continuous
line is the
theoretical (i.e.,
prediction, and the points correspond to <AN(v)>/Av,
<AN> is the average number of molecules having velocities within
+ Av/2 of v, with Av equal to 5% of the total velocity range considered,
namely,
0 < v* < 3, with
v
v
v
(1.44)
1/2
Vrm
vmss
(3k<T>)
m
i.e., three times the root mean
square velocity.
The number of times
the ensemble's velocities were analyzed to arrive at <AN> is indicated
56
bI
b
|1
I
I
I
I
ia02
&Of3
b
I
I
I
FIGURE 1.26:
,5
b4
b
|b6o
buI
bb
I
I
Two-dimensional periodic boundary conditions.
57
l
0
z
0
J
C
0
FIGURE 1.27:
Kinematic description of two linear molecules.
58
a I-
I
I
I
4
2
0
I
II
i
I
[
5
5
15
10
15
10
C-C DISTANCE(Angstroms)
C-C DISTANCE (Angstroms)
·
·
-
0
10 F
a- 900
a - 450
-2
5
' 450
*y450
6 450
- 450
0
-4
I
I
6
I
I
I
10
15
5
C-C DISTANCE(Angstroms)
FIGURE 1.28:
6 450
I
10
15
C-C DISTANCE (Angstroms)
van der Waals, coulombic and total intermolecular interaction
energy for two CO2 molecules resulting from the sum of
nine elementary atomic interactions.
59
0.3
0.
I
-I
I
I
o0
a
310.3K
0. a
CE
Z0>
'h '
0.1
O
0.0
I
I
I
0
500see
1000
v (m/s)
FIGURE 1.29:
Maxwell-Boitzmann and computed velocity distribution; #
of samplings = 64; <T> - 310.3 K.
60
I
0.as
. 20
0. 15
-
-I
I
I
II
I
I
-
<T > 318.6K
-
-
0 10
Z
"I'
e .0s
0.00
I
e
-
--
I
see
I
L
1
leee
v (m/s)
FIGURE1.30:
Maxwell-Boltzmann
of samplings
and computed velocity
= 67; <T> - 318.6 K.
61
distribution;
#
U
I
I~
I
I
as15
<T > - 337.4K
0.15
10
O. 10
-
0.05
-
0.00
-
I
__·,
-
0
I
__
6 (
I
ie00
v (rams)
FIGURE 1.31:
Maxwell-Boltzmann and computed velocity distribution;
of samplings
=
67;
<T> -
62
3 37.4 K.
#
.e S
I
1
I
I
I
I
I
0. 2e
E
' 342.3K
.15
%-f
0.10
0.05
0.00
I
_I
0
,,
I
see
-
-n a'
le00
la
v (m/s)
FIGURE 1.32: Maxwell-Boltzmann and computed velocity distribution;
of samplings
64; <T> - 342.3 K.
63
in each case.
Simulations were either started from a unimodal velocity
distribution (i.e., all molecules assigned their equipartition velocities
with random orientations), or from the end of a previous simulation.
The radial distribution function for the central carbon atom of the
CO2 molecule is shown in Figures 1.33 and 1.34 as a function of density
At the moderate densities considered, fluid structure
and temperature.
is almost exclusively limited to a nearest neighbour shell whose radius
The mild secondary peak disappears at higher temper-
is approximately 4A.
atures and lower densities.
Diffusion
coefficients were calculated
by creating an
ensemble of
solute "experiments" shifted in time and computing squared displacements
at corresponding instants with respect to the respective origins (Figure
1.35).
This
test-particle
approach
(Alder et. al., 1974) allows the
computation of ensemble averages from the simulation of the motion of
a single solute molecule.
displacement
The long time behaviour of the mean squared
versus time relationship yields the diffusion coefficient
which is simply 1/6 of the slope of the resulting straight line (Einstein,
1905) (or, more precisely, the slope is 2dD, where d is the dimensionality
of the displacements).
The
temperature
dependence of
the squared displacement history
is
shown in Figure 1.36, and the results of four different simulations are
plotted
in Figure
1.37 in Arrhenius fashion.
energy should be compared to the 10.9 KJ mole
Schneider's
at
two
data
The regressed activation
-1
obtained from Swaid and
(1979) for diffusion of benzene
different temperatures.
in supercritical CO2
There being only one solute molecule,
the temperature dependence of properties can, at best, yield semiquantitative
numbers since the very concept of temperature implies a statistical distribution of velocities.
The isothermal density dependence of the squared displacement vs. time
relationship,
shown in Figure
1.38, displays interesting
trends.
The
zero-displacement limit of the linear relationship defines a relaxation
time, which, for a Brownian sphere, is given by (Chandrasekhar, 1943)
T
m
(1.45)
where n and a are the viscosity of the medium and the radius of the sphere,
64
1.5
1.0
g(r)
0.5
0.0
0.
5.
10.
C - C Distance ( Angstrom )
FIGURE
1.33:
Effect of temperature upon the carbon-carbon radial distribution
function; p
13.87 mol/lt.
65
I N
I.D
1.0
g(r)
0.5
0.0
0.
10.
5.
C-C Distance( Angstrom)
FIGURE 1.34:
Effect of density upon the carbon-carbon radial distribution
function;
<T>
315 K.
66
i
Time
n
"4
L'In
I.-
OD
m
0
I
si)
r :
M-
In
x\
r
II
LS., --
i:
i,jt
(rjjj -)
t
FIGURE 1.35:
Ensemble generating technique.
67
)
10.0
o<
A
N
5.0
V
0.0
0.0
0.5
1.0
1.5
Time (I 0 -1 2 Sec)
FIGURE 1.36:
Effect of temperature upon
versus time relationship.
68
the mean
squared displacement
__
__
__
I
I
I
4
E- 14.8 KJ/mol
Cn
p -IO0.53mol/It
3
E
0
1E
2
-
I
-
I
I
I
2.5
3
3.5
( K-I)
IO3 T-
FIGURE 1.37:
Arrhenius plot for
the
of benzene in CO2
69
computed
diffusion
coefficients
10.0
C<Z
N
A
5.0
V
0.0
0.0
0.5
1.0
1.5
Time ( 10- 12 Sec)
FIGURE
1.38:
Effect of density upon relaxation (short time) and diffusive
(long time) behaviour; benzene-C02 ; <T> - 310 K.
70
respectively.
Furthermore,
the short-time
)
(i.e., t <<
behaviour of
<r2> can be obtained from a truncated time expansion which, when squared
and ensemble-averaged, yields
<r2> = (3kT) t 2
(1.46)
At a given temperature, (i.e. same initial slope), then, decreasing
the density (i.e., the viscosity) causes an increase in
, which dominates
the short-time behaviour, only to give rise to a higher diffusion coefficient,
(the computed diffusion coefficients are 1.396x10l4
as expected, for t >
cm 2 /s at 10.53 mol/lt
4
and 1.649x10
cm2/s at 7.42 mol/lt).
Relaxation
times calculated from Equation (1.45) with experimental C02 viscosities
o
(.05cp
to
at
H
310 K,
atom
90 bar,
distance)
12.2
for a,
mol/lt)
yield
and 2.49
values
(5.5
A
x
(i.e.,
center
10 13sec)
of mass
in excellent
qualitative agreement with T values obtained from the simulations.
A very interesting theoretical question is raised by the fact that,
although
linear
only
in
the
behaviour
an
simulations
the mean squared displacement exhibits a
at long times, the relationship
asymptotic
law approached
DT
1
= f
at high viscosities.
[size] is
This
apparent
paradox can be explained by noting that the long time relationship between
<r2 > and time can be derived without postulating any explicit form for
the
hydrodynamic
drag.
Alternatively
(Chandrasekhar, 1943),
2
from the Langevin equation, the limits <r >
(t
-
2
t
(t -
starting
0) and <r2> - t
) can again be obtained without postulating any form for the drag
coefficient,
, although the drag term itself is, in this approach, pro-
portional to the particle's velocity
tionality constant,
We conclude,
(with an as yet undefined propor-
).
therefore,
that,
if
is non-linear
in n (i.e.,
B -
n 6,
for example), the Stokes-Einstein equation (or, more precisely, its form,
i.e., nDT
= f
[size]) would not describe physical reality;
in spite
2
of this, though, the short and long time limits of <r > would, of course,
still be parabolic and linear, respectively, and the fundamental relationship
between <r2 > and D at long times would still be valid.
71
The
breakdown
of
hydrodynamic
behaviour in supercritical fluids,
then, is associated with a "hydrodynamic" drag that can best be explained
in
terms of a power
law relationship between the drag coefficient and
viscosity.
Using the activation energy calculated from the log D vs. T 1 plot
(Figure 1.37), we
10.53
mol/lt from
can estimate a diffusion coefficient at 313.2 K and
the value obtained at 309.3K and the same density.
This number (1.5 x 10- 4 cm2/s) is to be compared with the value obtained
by
graphical
interpolation of Swaid and Schneider's data at
the same
temperature (2.05 x 10 4cm2/s).
The molecular dynamics prediction is 36.7% lower than the experimental
value.
This is an encouraging result, given the facts that no adjustable
parameters were used in this work, and that the ensemble-generating technique
involved just one solute particle.
72
2:
INTRODUCTION
2.1:
PERSPECTIVE
Research in the general area of supercritical fluids has grown considerably over the past few years
(Ely and Baker, 1983).
of this technology have, by
aspects
far, received
The thermodynamic
much more attention
than the transport aspects.
From the point of view of transport, the goal is to understand and,
eventually, predict, the rates at which mass and/or heat are transferred
This requires knowledge of the transport
within the dense fluid phase.
properties of
the mixtures
involved, as well as of the particular flow
situation being considered.
The present work addresses several aspects of mass transfer in supercritical
fluids.
The first question that must be answered, therefore,
is the relevance of the problem itself, i.e., why (if at all) is mass
transfer in a supercritical fluid any different from, say, liquid phase
mass transfer?
Part of this answer is, of course, obvious:
the physical properties
of supercritical mixtures are different from liquid properties (see below).
by
This,
itself, would justify
interest in the problem, at least from
the point of view of property measurement, estimation and correlation.
In
region
addition,
give
rise
the unique properties of
in the supercritical
fluids
to mass transfer mechanisms which are
different from the corresponding liquid phase case.
qualitatively
Thus, what is required
in order to understand and predict mass transfer rates in supercritical
fluids is a study of the physical properties and of the peculiar convective
mechanisms
Such
an
that
arise
as a
analysis has been
consequence of those physical
done for heat
transfer
properties.
(Nishikawa and
Ito,
1969; Harrison and Watson, 1976; Hauptman and Malhotra, 1980; Shitsman,
1974; Nishikawa, et al., 1973; Kakarala and Thomas, 1980; Hall, 1975),
but not for mass transfer.
In this work, physical property measurement and computer simulation
have been done for binary diffusion of aromatic compounds in supercritical
fluids.
In addition, the importance of natural convection in mass transfer
73
with supercritical fluids has been analyzed in the light of fluid properties,
and observed experimentally.
In Table 2.1, the physical properties of a supercritical fluid (CO2
at 150 bar and 310 K) are compared with those of air and water at ambient
conditions.
We notice, in the first place, the very low kinematic viscosity
As outlined
of the supercritical fluid.
in Section
1.1 and explained
in Chapter 3, this is the reason why, at any given Reynolds number, natural
convection
plays a much more
important part
within a supercritical fluid
mechanism
in the overall transport
than it does
in the case of a
liquid or a gas.
In the second place, the Schmidt numbers corresponding to diffusion
of typical organic solutes (molecular weight - 102) are roughly two orders
of magnitude
lower in a
supercritical fluid than
in a typical liquid,
whereas Prandtl numbers are comparable.
We can therefore complete the answer
to the question posed at the
beginning of this section by noting that a supercritical fluid is simultaneously
as dense as a liquid, more
compressible than a dilute gas,
possesses a kinematic viscosity that can be lower than liquid metal kinematic
viscosities,
Prandtl numbers similar to those of liquids,
and Schmidt
numbers two orders of magnitude lower than the corresponding liquid values.
This
is
certainly more
than enough to justify the study of transport
in supercritical fluids.
Although
of
the
the
existence
present work,
properties
of supercritical fluids are a consequence
of the critical
point, the analysis,
is entirely classical.
throughout the
Critical phenomena (Stanley, 1971)
such as the divergence of the specific heat at constant pressure or the
isothermal
compressibility of a pure substance at
its critical point,
or the vanishing of diffusive fluxes at mixture critical points (Tsekhanskaya,
1968) require entirely different theoretical approaches (Pfeuty and Toulouse,
1978; Ma, 1976).
Non-classical behaviour, however, whereby fluctuations
of some characteristic quantity (the order parameter) grow without limit,
is restricted
to very narrow
regions
were not explored in the present work.
74
close to criticality, and these
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2.2:
DIFFUSION IN DENSE FLUIDS:
THEORETICAL DIFFICULTIES
Mutual and self diffusion coefficients for low pressure gaseous systems
can be estimated to within - 7.5% accuracy (Reid et al., 1977) from purely
The starting point in this case is the Boltzmann
theoretical expressions.
transport equation (Hirschfelder et al., 1964; Chapman and Cowling, 1970;
Huang, 1963; Pauli, 1981).
equation
derived
for
the rate
by Boltzmann
The latter is a non-linear integro-differential
of change of the distribution
in its original
function and was
form by considering only binary
interactions of point particles and introducing the molecular chaos hypothesis, whereby the position and velocity of a particle are uncorrelated.
The Boltzmann
transport equation admits a
stationary solution, the
Maxwell-Boltzmann distribution (see Chapter 8).
The Chapman-Enskog method
is a successive approximation approach to the solution of the Boltzmann
transport equation, and yields expressions for the transport coefficients
after considerable effort.
As soon as fluid density becomes such that
the mean free path is comparable to molecular dimensions, higher order
interactions and the finite molecular size must be taken into account.
In addition, collisional transfer (i.e., the energy and momentum transfer
that
a
takes
place
progressively
instantaneously
in a hard
sphere collision) becomes
important transport mechanism which
is not considered
in the Boltzmann equation.
As a result, there exists no accurate kinetic theory of dense fluids
that will allow, as with the dilute case, an accurate prediction of the
transport coefficients.
Some of the theoretical and experimental approaches
to the problem will be briefly reviewed.
2.3:
THEORETICAL APPROACHES
Corresponding states arguments have been applied to the correlation
and
extrapolation
of
self-diffusion data.
dimensional analysis arguments
In this
case, one can use
(Hirschfelder et al., 1964) to show that
the reduced self-diffusion coefficient of a spherical non-polar molecule,
76
D+ = D
m 1
(//2
£)
-1
(2.1)
must be a function of the reduced temperature and volume,
D+ = D+ (v+,T+ )
(2.2)
where
v+
- 3
(2.3)
T + = kT/e
(2.4)
v
and
In
Equations
to
(2.1)
a
(2.4),
e are
and
a
characteristic
length
and
a characteristic energy, m is the molecular mass, and v, the molecular
volume (i.e., total volume divided by number of molecules).
though
Even
not
may
the
detailed
functionality
known, this approach
be
provides,
implied by Equation
(2.2)
in principle, an extremely
powerful technique for data correlation and extrapolation.
Equation (2.1) can be transformed, again through dimensional arguments,
by writing
a -
(kTc/Pc)1 /3
(2.5)
- k Tc
(2.6)
m = ML
k = RL
(2.7)
1
(2.8)
where L is Avogadro's
number,
D M1 / 2 Pc1 /3 (RTc) 5
D+
which
to arrive
= D
at D +
(Tr, Pr)
is the usual starting point
(2.9)
for corresponding states approaches
to diffusion.
The
extension
of these
ideas to binary diffusion
outset, a degree of arbitrariness
parameters.
implies, at the
in the definition of mixture critical
Thus, Slattery and Bird (1958), define geometric mean critical
temperatures and pressures,
(2.10)
?C12- (PC
- 1 PC2 ) 1 / 2
TC1 2 (Tc1 TC2 ) 1 / 2
(2.11)
77
whereas a more common approach is to define
TC12
3 +
PC12= [1/2 (TC/P) 1
TC12
with
as per Equation
1/2 (T 2 /P
(2.11).
2)
1 3 ]3
(2.12)
The denominator of
Equation (2.12)
is equivalent to a Lorentz-Berthelot mixing rule for a,
1
-+-
2 (al
=
012
(2.13)
+ 02)
The mixture molecular weight is usually defined as a modified harmonic
mean,
+1(2.14)
Ml2
From
Ml
M2
this
short
it should be apparent
discussion,
ideas can be extremely useful
corresponding-states
that, although
in correlating self-
diffusion data, the extension to binary diffusion can, at best, be considered
as a convenient empiricism, since the definitions of mixture parameters
are somewhat arbitrary.
a
of
two-parameter
approach
Furthermore, for highly non-spherical molecules,
(a,
, or
Tc,
the system's behaviour.
describing
is not,
Pc)
As an example, we
(1958) correlation, obtained
Slattery-Bird
in general,
by statistical
capable
consider the
analysis of
experimental data in the light of a corresponding states approach,
(2.15)
= 3.882 x 10 4 T 1.823
PD
r
with
1/2
5/6
T
5/6
c
(2.16)
P 2/3
c
and mixture parameters as per Equations (2.10), (2.11), (2.14).
(2.15)
and
(2.16)
2
are dimensional, with
the pressures
Equations
in atmospheres
Equation (2.15) was originally derived for self-diffusion
and D in cm /sec.
in dilute systems, and extended to dilute binary systems by the authors.
When several dense fluid data (Table 2.2) are plotted according to
Equations
(2.15),
it can be seen
(2.16),
(2.10),
(2.11)
and
(2.14)
(see
Figure
2.1)
that deviations are not associated with high pressures
78
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A
A
A
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--
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--- ID
A
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A
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a,
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79
co
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C.
c
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0
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a,
a
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Cu
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a
a
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>s
s
C
c
E
N
0
X:
,_I
I
IU-l
I
10-2
10-3
10o-4
x
O
H2 H2
CH 4 - N
w * -'4
-' '"b
4
N2
-
C2H 6
0
C0 2 - C 6 H6
o
C0 2 - n-propylbenzene
'
C02- 1.2,3trimethylbenzene
10-5
v
C2 H 4 - nophthalene
v
C02 - nophthalene
*
N2
- naphthalene
I
I
10
I
10- 6
_
10
1'
T/Tc
FIGURE 2.1: The Slattery-Bird correlation: a two-parameter corresponding
states approach.
80
In particular, diffusion
but with the nature of the solutes and solvents.
of
aromatics is poorly represented
N2 -naphthalene
H2
-
system.
690
CH
bar),
- C 2H 6
(69
to 690
bar)
and
CH4
- N2
(7 to 172 bar) are well correlated in terms of this two parameter
the
to
(1960)
for
of
(69
hand, Berry and Koeller's
the other
data
equation.
N2
On
correlation, except for the
by the
The C02 -naphthalene data include Tsekhanskaya's (1971) studies
at mixture
vanishing of diffusion coefficients
critical points
which are, a priori, beyond the scope of Equation (2.15) (see Chapter 7).
corresponding states arguments, therefore,
Two-parameter
should be
used with extreme care in dense binary systems, and must be considered,
Mixture composition is not
at best, as helpful correlating guidelines.
taken
such as Equations
into account by relationships
(2.10), (2.11),
(2.12) or (2.14), which can only be applied at infinite dilution.
The extension of corresponding states ideas to substances that depart
from
strictly
conformal
behaviour has received
considerable attention
recently.
In one approach, a reference substance of well-known properties
is
and
used,
to the reference
conformality with respect
substance is
forced through the introduction of state-dependent shape factors (Murad
and
Gubbins, 1977), which are either calculated iteratively or obtained
from empirical fits. For mixtures, pseudo-critical properties are introduced;
this leads to the appearance of cross interaction parameters, which must
be regressed from experimental transport data (Murad and Gubbins, 1981).
Although interesting in principle, this approach is far from standardized
and
requires
extensive iterative calculations or
empirical values for
the shape factors and interaction coefficients.
Alternatively,
(2.9)
are
expanded
the
reduced
coefficients
transport
in Taylor series about
a known
using a third parameter as an expansion variable.
(i.e., Equation
(reference) value,
This idea is derived
from
Pitzer's (1955) original work, so that the acentric factor is the
most
commonly used third parameter.
a modified version of this approach
Teja
(1982) has recently proposed
for binary diffusion in liquids by
Two diffusion coefficients are needed
considering two reference fluids.
(of different solutes in a given solvent), and the resulting expression
still contains an adjustable parameter.
81
The
in
problem of calculating density-dependent diffusion
dense
fluids
is
still
unsolved.
The
Enskog
(1921)
coefficients
theory
is
still
widely used in spite of the fact that, in most cases, it does not predict
the observed trends.
smooth
spheres,
In this approach, molecules are considered as hard
whereupon
the problem again becomes tractable,
since
The main result
only binary collisions can exist in a hard sphere fluid.
of this theory, in its unmodified form, and for self-diffusion, can be
written as
(PD)
X_1~~~
~(2.17)
(P~~~D)°~~~
- X
indicates the dilute limit, and X is the factor by
where superscript
which collision frequency in a hard
responding number
sphere fluid differs from the cor-
in a fluid composed of point particles and is given,
for a hard sphere fluid, by
X = 1 + 0.625
bo-V
1
+ 0.115 (boV-1 )3
+ 0.2869 (boV-1 )
+
.
(2.18)
where bo is the molar second virial coefficient,
=
bo
( 3
)L
(2.19)
and V, the molar volume.
in Chapman and Cowling
A derivation of Equation (2.18) can be found
(1970) and follows
from geometric arguments by
taking into account the increase in collision frequency due to the finite
molecular
size as well as the decrease
in collision
frequency due to
the "shielding" effect that close-packed molecules exert on each other
by blocking incoming molecules.
Enskog's
ideas were extended to binary
(1-2) diffusion
by Thorne,
to arrive at
pD12
1
(2.20)
with
X12
1 +
where n
2 no3
(8 - 3a/o12)
1n2
+
a3
(8-3 a2 /o1 2 ) + ...
(2.21)
and n2 are number densities, and the hard sphere diameters are
combined according to Equation (2.13).
82
As can be seen from Equation (2.18) or its binary equivalent (2.21),
Enskog's theory predicts diffusivities which decrease with density more
rapidly
than
-1
p
.
In addition, since X and X12 are both greater than
unity, we must have, according to this theory,
(pD) < (pD) °
(2.22)
for diffusion of
aromatics
(Swaid and Schneider, 1979), as well as
in supercritical CO,
2
aliphatic
always.
halogenated
and
experimental measurements
However,
in He
aliphatic hydrocarbons
(Balenovic et
al., 1970)
are not only consistently above the Enskog prediction, i.e.,
pD
(pD)O
but,
>
exp
pD
>
(2.23)
(pD)O
Enskog
in addition, result
in ratios greater than unity for several data
points, a fact that cannot be explained by the Enskog theory.
The modified Enskog theory is an attempt to preserve the simplicity
of the basic relationship (i.e., Equation
the idealizations implicit in Equation
(2.17)), while correcting for
(2.18).
This is done, following
Enskog, by postulating that, in the hard sphere equation,
-1
- 1
bo V
where
(2.24)
z is the compressibility factor, the pressure be replaced by the
"thermal pressure", to arrive at
X bo V -1
V
P
RT
-
T(Tf)
(2.25)
In this approach, bo is now obtained from
T
()
RT
(1
+ X bo
v-1)
(2.26)
V
and the limit of X bo as V tends to infinity.
When P-V-T data, are expressed
as
P = VRT [1
+ pB(T) + p2 C(T) +
83
...
]
(2.27)
we obtain,
M (aBT)
bo
(2.28)
V
The
modified Enskog theory (Hanley et al., 1972; Hanley and Cohen,
1975) has been used successfully for simple pure dense fluids.
In essence,
it is an ad-hoc correction of the original theory that takes into account
the actual properties of a fluid by replacing the hard sphere expression
for X (Equation (2.18)) by Equation (2.25), where X is calculated
P-V-T data.
is an
from
Tne extension of the modified Enskog theory to binary diffusion
interesting possibility, although, from the previous discussion,
it should be evident that such an approach would constitute a semiempirical
modification of an existing theory, rather than a predictive method based
on first principles.
The
so
called van der Waals picture of transport
(Dymond and Alder, 1966), whereby
via
a
a
spherically
in dense
fluids
molecules are considered to interact
symmetric potential consisting of a hard core plus
long-range attractive tail, has served as a basis for a significant
amount of recent work.
empirical
correction factors have
sphericity.
"composed
In order to preserve this intuitive general picture,
been introduced to account for non-
Thus, the idea of a rough hard sphere fluid (Chandler, 1975)
of
spherical
instantaneous
particles which collide
collisions are
capable of
impulsively, and
these
changing the angular momentum
of a particle as well as the linear momentum" leads to a correction factor
A, (the "roughness factor"), such that
DRHS
A DSHS
(A < 1)
(2.29)
where RHS and SHS denote, respectively, rough and soft hard spheres (i.e.,
spheres with and without rotational-translational coupling, respectively).
The roughness factor is used as an adjustable parameter.
arguments
it follows that A should be
a given system.
From equipartition
independent of
temperature for
In spite of the obvious empiricism of Equation (2.29),
the rough hard sphere approach has received considerable attention (Bertucci
and Flygare,
1975; De Zwaan and Jonas, 1975; Fury et al., 1979; Chen,
1983) and has been widely used for correlating purposes.
Hydrodynamic
of a Brownian
theory
sphere
(Einstein,
of radius
1905) predicts,
for the diffusion
a in a fluid of viscosity
n, at a temperature
T, a diffusion coefficient given by
D
where
kT
6Tran
(2.30)
the coefficient 6
to a no-slip boundary condition,
in the opposite case whereby the sliding friction coefficient
and becomes 4
vanishes.
corresponds
It has long been known that an equation of the form of (2.30),
ice.,
nDT
can
be
fluids.
used
= f [size]
(2.31)
to correlate and understand
molecular diffusion in dense
This implies the very remarkable concept of hydrodynamic behaviour
at the molecular level, and is the basis of numerous empirical correlations
which
have been used with varied success in the study of diffusion in
liquids
(Wilke and Chang, 1955; Scheibel, 1954; Reddy and Doraiswamy,
1967; Lusis and Ratcliff, 1968).
For non-spherical molecules of realistic
shape, hydrodynamic theory cannot, in general, provide a predictive equation
due to the difficulty in evaluating the drag.
However, Equation (2.31)
is extremely useful for correlating purposes.
The hydrodynamic limit can also be analyzed in the context of hard
sphere theory (Dymond, 1974), according to which
D - T1/2 (V - 1.384 V)
1/2
n
T2
(V - 1.384 V)
a
-1
(2.32)
(2.33)
a
a
(2.33)
where V
is the hard sphere close-packed molar volume, and V is the molar
volume.
It follows that
-1
Dn T
in
-
(2.34)
The hard sphere fluid, then,
with
hydrodynamic
theory.
hydrodynamic
behaviour at
the molecular
agreement
shows
-11
dependence
level.
of fluid viscosity predicted by Equation
The temperature
(2.33), though, is
obviously inconsistent with experimental evidence, whereby liquid viscosities
85
are
To
strongly
remedy
regressed
(i.e.,
this,
from
activated)
decreasing
functions of temperature.
temperature-dependent hard sphere diameters have
experimental
data
(Fury
et
al.,
1979;
Chen,
1983),
been
but
this is obviously in contradiction with the concept of a hard sphere.
From this brief survey, it can be concluded that the prediction of
transport coefficients in dense fluids is, at present, an unsolved theoretical
problem.
Existing
theories can, at best,
serve as useful guidelines
for data analysis and correlation.
2.4:
DIFFUSION IN DENSE FLUIDS: COMPUTER SIMULATIONS
The
calculation
of transport coefficients via computer simulations
can be done using molecular dynamics, a technique first introduced almost
thirty years ago (Alder and Wainwright, 1959; Wainwright and Alder, 1958).
In this approach, the dynamics of a finite number of molecules is simulated
by integrating the classical equations of motion, and the resulting evolution
in time of the positions (and orientations for non-spherical molecules)
and velocities (including angular velocities in the case of non-spherical
molecules) of the molecules is interpreted statistically.
The method, as described above, is deterministic.
for the calculation of
is necessary
properties can,
in addition to
The dynamic aspect
transport properties; equilibrium
the dynamic approach,
be obtained
from
statistically generated configurations from which averages can be calculated.
The efficient generation of configurations with their appropriate weighing
factors is accomplished by using the so-called Monte Carlo method, first
introduced over thirty years ago (Metropolis et al., 1953).
This represents
the other important technique in the area of computer simulation of fluids,
and will not
be considered here since,
by
its very nature,
it cannot
be used to calculate transport properties.
Two theoretically equivalent methods can be used to compute diffusion
coefficients.
d
dt
relates
The Einstein expression
<[r(t)
the
- r(O)]2>
(2.35)
= 2 d D
diffusion coefficient of an
ensemble of molecules
to the
slope of the ensemble-averaged, squared displacement versus time relationship
86
long
at
discussion),
be
can
In Equation
times.
(2.35)
corresponding
d is the dimensionality
along
calculated
a
within
line,
for derivation
(see Chapter 8
a
plane,
the latter
to r, i.e.,
or,
as
in
and
the
present
work, in three dimensional space.
The calculation of a diffusion coefficient, then, involves following
an ensemble of molecules during a time long enough
the motion of
for
the ensemble-averaged squared displacement to grow linearly with time.
can calculate diffusion coefficients using the
one
Alternatively,
time-correlation formalism (McQuarrie, 1976), whereby
D
=
-
<v(t)
v(O)>
(2.36)
dt
0
In Equation
(2.36), v
is, again, a d-dimensional vector.
Equation
(2.36) can be derived from Equation (2.35) (McQuarrie, 1976); both methods
are therefore theoretically (though not computationally) equivalent.
In this approach, then, one computes the time integral of the ensembleaveraged
velocity
starting from a given
autocorrelation,
(arbitrary)
initial instant, and continuing until the autocorrelation decays to zero,
whereupon the integral is invariant.
The method therefore falls into the category of computer "experiments"
(Gubbins
et
al.,
1983).
This apparently contradictory classification
follows from the fact that, given some initial conditions, the temporal
evolution of the system is determined but cannot be known a-priori; the
computer then performs the simulation (the "experiment"), whose results
are finally analyzed and interpreted.
The
main
problems
associated
with the molecular
fall into two very different categories.
additive
potentials,
the duration
dynamics approach
In the first place, for pairwise
of a simulation
(given an event of
fixed duration to be studied) is a quadratic function of the sample size.
With
current
computers,
representing - 10
-
10
3
problems are
limited to simulations
seconds of real time, and sample sizes smaller than
molecules (or, more generally, 103sites
fluids).
The
tractable
in the case of molecular
This first type of limitation is technical rather than fundamental.
relative
performance
of several computers
applications is discussed by Ceperley (1981).
87
for molecular dynamics
A much more fundamental limitation stems from the lack of basic knowledge
in
the
field
of
intermolecular (or interatomic) potentials.
et al. (1981) summarize
functions
which
considered
He-He and Ar-Kr are
and
our present state of knowledge about potential
by considering four categories.
are
Maitland
quantitatively
Class
I includes "functions
accurate".
Ar-Ar,
Kr-Kr, Ne-Ne,
the only members of this group, with He-Ar, He-Kr
He-Xe described as
"probably of this quality".
Class II includes
potential functions obtained by means of reliable methods; in this category,
however, the potentials "have not been extensively tested on a wide range
of
properties".
Members of this group
include Hg-alkali metal,
inert
gas-alkali metal and some inert gas-inert gas (He-Ar, He-Ne, for example)
potentials.
Class III includes potentials ..." which result from serious
attempts to describe the interactions of non-spherical polyatomic molecules".
The
authors
recommend
procedures
to tackle the problem
(inclusion of
point monopoles, for example) but conclude that "... the most convenient
representation of this anisotropy has not yet been established".
in Class
IV, the
authors
Finally,
include "... the determinations not of full
potential energy functions but merely the parameters that enable a model
potential
function
to best
fit selected
data", and conclude that, at
best "... this procedure offers a way of smoothing or of modestly extrapolating the data at hand. To invest such potentials with more value than
this can lead to confusion".
The fact that the technique (i.e., molecular dynamics) is well developed,
whereas
the
fundamental
input to the simulation
(i.e., the potential
functions) cannot, at present, be determined with anything even approaching
the same degree of confidence, makes
limited
at
best.
the predictive use of the method
In the present context, the word
predictive should
be considered incompatible with the very concept of an adjustable parameter.
It
is not surprising,
contribution of molecular
therefore, that by far the most significant
dynamics to date has been the study of model
fluids rather than the predictive calculation of properties for specific
substances.
Thus,
important phenomena such as the
long-time tails in
the velocity autocorrelation function (Alder and Wainwright, 1967; Alder
and Wainwright, 1970) the existence of a phase transition in a hard sphere
system (Alder and Wainwright, 1962), or the equation of state for a hard
88
sphere or hard
disk solid
(Alder et al., 1968) have been successfully
studied with this technique.
In fact, in some cases (long-time tails),
the simulated behaviour displayed hitherto
unknown features which were
thus first "observed" in a computer simulation and only later predicted
as a general feature of dense fluid behaviour.
The
generality
and usefulness of such model
fluid simulations are
in sharp contrast with the necessarily empirical and restricted information
gathered from specific fluid simulations with ad-hoc potential parameters
(Stillinger
of
and
interatomic
Rahman, 1974).
and
Progress
intermolecular
in the fundamental knowledge
forces should lead, eventually, to
a completely predictive approach.
2.5:
DIFFUSION COEFFICIENTS; EXPERIMENTAL APPROACHES
In the light of the previous discussion, it follows that experiments
must necessarily
dense fluids.
play a fundamental role
In the particular case of supercritical fluids, the critical
pressures of
the solvents of
C 2 H,; 37.5 bar for SF6,
which
in the study of diffusion in
interest
(73.8 bar for C02 ; 50.2 bar for
for example) result in high pressure operation,
makes the study of diffusion under supercritical conditions more
difficult, experimentally, than the study of diffusion in liquids.
Previous studies of diffusion in supercritical fluids are summarized
in Table 2.3.
The weight loss method is a static technique whereby the
instantaneous mass of a suspended pellet of diffusing solute is related
to the cell geometry, equilibrium solubility of the solute in the solvent,
diffusion
time and binary
diffusion coefficient
through an analytical
expression resulting from the solution of the appropriate diffusion problem.
In view of the importance of natural convection in supercritical fluids
(see Chapter 3), the assumption of a stagnant solvent may lead to errors.
The use of supercritical chromatography to study diffusion coefficients
represents
an
Taylor, 1954).
application
of
Taylor dispersion
theory
(Taylor, 1953;
The diffusion coefficient of a solute injected as a pulse
in trace amounts
into a capillary where a solvent circulates in laminar
flow can be calculated from
capillary's exit.
the solute's concentration profile at the
The diffusion coefficient is then a function of fluid
89
=r
%.O
_
kD
/_
-
co
o
Y
:Y
a
I.,
)
a)
L.
'4
0·I
(1)
E
E
'0
C
'0
C
0
0
a
a)
C-
E
C
C
C)
U
U
C)
C
C
C
I
(4
CO
_s
0
co
O
0
0
9,
0
0
0
'0
la
a
a)
c
_
0
.1-4
.- 4
·(·(
aC
Co
U
9-4
(U
r-4
.- 4
s
4O
a
3
3
C,) c
CO
>
>
>
Ca
Ca
0
o
0
0
L
.C
0
L.
.5
0
S.c
CO
,
C
(U
O3
0co
_
4)
0
0
0
0
C
(U
(U
cm
CO
C
C/
Co
bo
C
ca
*1
U
_r
_
C
4O
EE
O
a"
1C..
co
ON
L
o
v
a)
E
O
co
t
OI
L.
a)
-
E-
C)
0
co
cd
c
0)
aCl)
a
s:
0
.
_
co
M
1-4
C.
c)
0
rT4
C.
0.
0)
0
0
C)
co
.
0
C
0co
0
o
0
O o
4)
0-C4
oa
:
'4
0o
S
'4
'
.,-
.4
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a
a
a
C)
C
a
C)
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4
4)
4
.C
CZ
ba
3
S
o~
S
0
(
0
cvI
C:
(o
cVI
C:
(U
vI
.:
(o
00
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4
U
C)
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C
.
CL
CL.
CL.
0 0
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C)
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C'
O
O
O
0
o
O
(D
cl c.
Yo
9-4
ac40
co
cx
o
co
0
w
co
09
'.
F
0
N
0M
VI
E
VI
co
VI
EI
VI
Ln
o
0
o
O0
)O
0
C
ao
Co
bo
t0o
(U
C)
C)
4
--
.,4
c
(U
0
a,
a,
co a
a
.
o
O
t0
a
VI
VI
VI
VI
0
VI
t-
co
a,
(n 5N)
r
v.94
.0
C
(1)
I_
l-,
E- X4
E
0r9
N
N
VI
EVI
o
0O
0
M,
VI
EVI
co0
0 0
CY )
oN
VI
E
VI
co
a
VI
-
VI
VI
VI
VI
VI
VI
F
E-
Cv
E-
CM
0_
W
C
N
0
C
4
0
0
0N
co
O
0m
(o
*-4
4)
4)
c0
4
4)
o
0
4
CO
,Y
O
(U)
CL
C
SC
0
0(U
.- 4
cr
4)
13
0
a
C
C
a) .0
N
0I
0
4)
a
F-I
D
>9
u)
f.
0
C
0
C
C
I
C
O
C)
ci
C)
0
0
a4
t0
(UL0
C3
co
&
C
4
f-
.C
c)
0.
I
N
C
(0
0
s
a
O
CC
CQ
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I
e
Ci
C)
-
Wr
0
L)
a0
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D
oCII
I
C)
I
J.)
(L)
O
a
M
C)
C
0
0
C)
C8
90
I
oC
a.
B
ou
velocity, duct radius, concentration profile peak width and solute retention
time.
This technique can be classified as hydrodynamic, since a well characterized flow situation must exist for the theory to be applicable.
effect
of density non-uniformities
across the duct's
The
cross section can
always be minimized by modifying the duct's aspect ratio (i.e., increasing
its length to diameter ratio), making this technique preferable to diffusion
cell approaches.
The use of chromatographic techniques to study binary diffusion has
been applied to gaseous (Balenovic et al., 1970) as well as liquid systems
(Ouano,
1972).
A comprehensive review can be found
in an
article by
Maynard and Grushka (1975).
Hydrodynamic techniques can be broadly classified as Fickian or phenomenological.
This
means
that diffusion coefficients are obtained from
the solution of a differential equation relating diffusive and convective
transport in a well-characterized flow situation.
Not only is concentration
considered as the driving force for diffusion (see Chapter 7), but the
resulting transport coefficient is an average value, resulting from the
assumption of composition-independence.
always be used at infinite dilution.
Thus, hydrodynamic methods should
This condition can always be approached
with chromatographic techniques (detector sensitivity being the limiting
factor)
but
needs
to be
carefully checked when
diffusion occurs from
a source of given composition.
In
light
scattering
techniques,
(Burstyn and Sengers,
1982; Saad
and Gulari, 1984), on the other hand, the decay rate of the autocorrelation
of
scattered
light
intensity is measured at various
by
means of a suitable signal detection scheme.
scattering angles
The scattering angle
is related to the wave number of concentration fluctuations through Bragg's
equation, whereupon the diffusion coefficient is obtained from the relationship.
r
where
D
2
2q
(2.37)
r is the autocorrelation decay rate and q, the wave number.
technique
has
been used to study binary diffusion
91
The
in liquid mixtures
both
away
from
(Czworniak et
al., 1975) and in
(Burstyn and Sengers,
1982) the critical regime.
The technique is accurate and does not give rise to the experimental
problems
associated
with hydrodynamic methods.
In
particular, natural
convection phenomena are entirely absent due to the time scales involved.
In addition, the concentration dependence of D,2 can be measured unambiguously
by performing experiments at various different concentrations.
Finally,
the technique has important theoretical implications, especially in the
critical region, which make its use mandatory in the study of non-classical
dynamic critical phenomena (Enz, 1979).
2.6:
MASS TRANSFER
Little systematic work (Brunner, 1984) has been done on mass transfer
into a supercritical fluid in practical situations (packed beds, liquid
columns,
stirred tanks, etc.).
As explained in Chapter 3, significant
enhancements in mass transfer rates are to be expected whenever transport
in the supercritical phase is rate-limiting.
This
is an important and
interesting problem, and should receive increasing attention in the future.
2.7:
OBJECTIVES
In the previous sections, a brief review was presented on the current
status of theoretical and experimental approaches to the study of transport
phenomena in dense fluids.
Within this general picture, the main objectives
of this work can be summarized as follows:
* to understand the role of physical properties in determining both
the rate and mechanism of mass transfer in supercritical fluids
* to measure binary diffusion coefficients of different organic solutes
on supercritical fluids and interpret the results
* to study binary diffusion using molecular dynamics computer simulations
The experimental technique
work
(also
is
hydrodynamic.
(see Chapter 4) selected in the present
Although
a hydrodynamic method) and
both cromatographic
peak
broadening
light scattering yield more accurate
92
results, the flat plate method, by allowing for the introduction of buoyant
effects, made it possible to verify experimentally some of the qualitative
predictions made in Chapter 3 in connection with the analysis of physical
properties and natural convection.
important insights were gained in this way, future studies
Although
separate the mass transfer and property measurement aspects of
should
the problem with carefully designed hydrodynamic techniques, and, ideally,
light scattering used to study the former and latter problems, respectively.
more
As
data
In
hydrodynamic
inevitably, benefit
fluids will,
supercritical
approach.
available, the understanding of diffusion
become
in
from a more fundamental
the interesting theoretical implications of
particular,
ideas in the supercritical regime
should prove to be more
fruitful than the current emphasis on rough hard sphere theory.
The
prediction of
simulations
transport properties through the use cf computer
is presently limited by
the lack of
fundamental knowledge
in the area of interatomic and intermolecular potentials.
In the present
work, the approach chosen was somewhat different from the usual procedure:
the simulations referred to specific molecules, yet no adjustable parameters
were used.
The results are encouraging, though accurate prediction (with
no adjustable parameters)
is still more
case of molecular fluids.
a goal than a reality
in the
In addition, the study of infinite dilution
binary interactions poses severe problems related to computer speed and
memory (see Chapters 9 and 1O); the results obtained with a test-particle
approach (Alder et al., 1974), again, are encouraging, but they should
be considered only as a step in the right direction, which enables the
study of a difficult problem without the need for a supercomputer.
is not t
This
say, however, that the statistical significance of the answers
would not improve from either a large solute ensemble or longer simulations
with completely independent "experiments" (see Chapter 10).
lations,
however,
require,
far more
were available for this work
93
Such calcu-
powerful computer resources than
3.
PHYSICAL PROPERTIES AND NATURAL CONVECTION
3.1 MASS TRANSFER AND NATURAL CONVECTION;HYDRODYNAMICS
Mass
ties.
transfer in a fluid is inseparable from
Under the influence of gravity
,
density non-uniformi-
density gradients give rise to
natural convection currents , the relative importance of which is determined by fluid properties.
For
steady laminar flow of an incompressible Newtonian fluid under
the influence of gravity and an imposed pressure gradient,
n V2 v + p g - VP = 0
(3.1)
A second component (solute) will diffuse into the fluid if, for example, the latter is in contact with a surface from which the solute dissolves.
The
resulting concentration gradient will
give
rise
gradients which will, in turn, alter the velocity profile.
to
density
This coupling
between mass and momentum transfer can be analyzed in those cases where
concentration (and density) gradients are small.
about the pure fluid value
Expanding the density
in terms of solute concentration, and trun-
cating after the linear term,
p = p
)
+
(C -
c)
P
-
m(
-
co)]
(3.2)
c T,P
Substituting into Equation (3.1),
n V2 v + g po [1 - Bm c] - VP = 0
where co=O has been used.
the
density
and
Equation (3.3)
neglecting
changes
(3.3)
has been obtained by linearizing
in viscosity.
This decoupling..of
the mass and momentum balance equations is known as Boussinesq's approxi94
mation and is obviously restricted to small density gradients.
(3.3), let us apply it to flow in a duct of
To illustrate Equation
radius R and non-dimensionalize it by defining
1 = P + Po g h
(3.4)
2
n+ = fl/p
o <v>
(3.5)
v+
r
<v>
where
is
the
(3.6)
-
v/<v>
(3.7)
= c/c i
average
velocity
fluid
in
the
duct,
ci
is the
solute
concentration at the duct boundary, where phase equilibrium is assumed,
and h is the height of a plane of constant hydrostatic pressure measured
along the direction of gravity (g' is a unit vector collinear with the
direction of gravity)
(3.8).
Vh = -g
into Equation
Substituting
as length
(3.3) and non-dimensionalizing, with R
scale,
2
(V+)2 v+ - V+n+
g'(Gr )
Re
Re
(3.9)
2
We now define Ap as the difference in fluid density at the interface
and in the bulk
(pure solvent), and
introduce the natural scales
for
buoyant, viscous and inertial forces,
(3.10)
Bouyant forces - 2R g Ap
n <v>/2R
(3.11)
Inertial forces - <v>2 Po
(3.12)
Viscous forces
The physical significance of the parameter Gr Re-2
tely,
95
follows immedia-
Gr
(2R g Ap)(<v>2 p)
22
(n <v>/2R)2
2
Re
2
(n <v>/2R)
(<v>
pO)
2
Buoyant forces
(3
Inertial forces
13)
This ratio, then, can be used to investigate the scaling behaviour
of
bouyant
relative
forces.
In
the present context, this means
comparing the
importance of natural convection in different fluids.
It
is
obvious that such comparisons should be made at equal Reynolds numbers.
If different fluids flow inside identical ducts under diffusive mass
transfer conditions at any given Reynolds number, and assuming comparable
density changes
(Ap/p), the relative importance of natural
convection
scales inversely as the square of the kinematic viscosity of the fluid
in question.
Thus, fluids with low kinematic viscosities
can develop
appreciable buoyancy-driven flows even with small density gradients.
3.2 PHYSICAL PROPERTIES IN THE SUPERCRITICAL REGION
The role of the kinematic viscosity in determing the relative importance of natural convection has already been shown.
Supercritical fluids
have exceptionally small kinematic viscosities as a consequence of the
very different behaviour of density and viscosity in going from the dilute
gas to the dense fluid region.
In what follows, attention will be focused on the region 1 < Tr <1.1
1 < Pr < 4 , where most of the changes associated with the passage from
the dilute gas to the dense state occur.
The dimensionless isothermal compressibility is the relative change
in density per unit relative change in pressure, at constant temperature,
K'
T
gln p
i
=
al
gin
V ) = p KT
-Tdln PT
(3.14)
For a fluid whose volumetric properties can be adequately described
by
a cubic
equation of state, for which
is (Schmidt and Wenzel,1980)
96
the most
general formulation
RT
RT
V-b
K
a
2
(3.15)
+ uVb + wb
V
can be written as
T
K'
T
z(-K)
(2 + u)[1
1
1 -
(3.16)
- z(1 - K)]
1 + UK + WK
K
where u,w are numerical constants which depend on the particular equation
of state being used (Appendix 1) and K=bV- 1.
compressibility
is
unity
for
an
ideal
The dimensionless isothermal
gas,
zero
for
an
incompressible
fluid, and infinite at the critical point.
Figures
SF6 ,
to
in the region
equations
is
at
3.1
3.4
show
the
behaviour
1.01 < Tr < 1.1,
of state.
At
of this
quantity
for
CO0
2
.1 < Pr < 4, for two different
318 K and 100 bar
(Tr =
1.05
and
cubic
; Pr = 1.36 ), C02
280 times as dense but almost three times more compressible
318
K and
1 bar.
compressibility
This
unique
combination
of
high
density
is one of the distinguishing features
than
and
high
of supercritical
extraction (Paulaitis et al.,1983).
Of
limit
more immediate
concern here is the fact
of the dimensionless isothermal
that the low pressure
compressibility
gives
rise
to
roughly two of the three orders of magnitude by which the density changes
in going from
atmospheric to
supercritical conditions.
This is
shown
in Figure 3.5, where the density, viscosity and kinematic viscosity of
CO 2
at
310K
(Tr
= 1.02)
are
plotted
as
a function
of
pressure
from
1
to 200 bars.
The viscosity, on the other hand, shows a very mild pressure dependence at low
and increases
density
by
(ideal gas viscosities are
a factor
of roughly
supercritical viscosities are less
pressure-independent)
6 in the range
.8 < Pr < 1.6.
than an order of magnitude
Thus,
higher
than ideal gas viscosities.
The
leads
combined
effect of liquid-like
to an exceptionally
density and moderate
low kinematic viscosity.
The
viscosity
supercritical
region, then, can be viewed as an interesting transitional domain, where
some fluid properties attain unique values, not necessarily intermediate
97
-''urn
_
I
I
__
I
I
I
ITr. 1.02
u)
4
1.04
C.
0J
C02 (P-R)
1.06
Cr
w
2
1.08
m
0I-
1.10
Q0
w
U)
MI
0
I
_J
II
I
,,·
0
100
200
I
300
PRESSURE ( Bar)
FIGURE
3.1
Dimensionless isothermal compressibility of CO2 , as modelled
by the Peng-Robinson equation of state.
98
- I-
IC
-i
4
LU
3
0
2
UL
3:
0
0
1'
LU
C,
0
LU
0:
0
0
200
100
300
/
PRESSURE (Bar)
II/
FIGURE 3.2 : Dimensionless isothermal compressibility of C0 2, as modelled
by the van der Waals equation of state.
99
/
/
/"
/'I /
I/
/
FIGURE 3.3 : Dimensionless isothermal compressibility of SF., as modelled
by the Peng-Robinson equation of state.
100
ot
-I
'
Il
'I
I
Y
ye
4
-1
3
~J
-
0
0
IM
a-
2
'-j
0
w
0
C)
Cr)
0
w
or
I
%v
. I __
0
I
L
50
_
I
100
_
I150
PRESSURE ( Bar)
FIGURE 3.4 : Dimensionless isothermal compressibility of SF., as modelled
by the van der Waals equation of state.
101
I0
uro
Cn
-E
'I
x
,I'.
3
on .
ua
C)
,
0xo
0-
.-
0.
P (bor)
FIGURE 3.5 : Density, viscosity and kinematic viscosity of CO2 as a function of pressure, at 310 K.
10n
between the ideal gas and the liquid extremes.
If we now compare (Figure 3.6) the properties of air, water and mercury at
298 K and
1 bar with
those
of supercritical
C0 2 at 310 K and 150
bar in the light of the previous discussion, some interesting consequences
arise.
We
note the fact that, for v, CO0
2 has the lowest value
than for mercury, a liquid metal).
The
(lower
fourth column is the ratio of
buoyant to inertial forces at constant Reynolds number and duct geometry,
scaled with the corresponding value for water.
The relative importance
of natural convection, therefore, is more than two orders of magnitude
higher in a supercritical fluid than in ordinary liquids.
It
is
important
to note
that
this
phenomenon
is
independent
of
the
free convective currents that originate very close to the critical point
as a consequence of the diverging fluid compressibility.
3.3 FLOW REGIMES
The extent to which natural convection controls the overall transport
mechanism in a supercritical fluid is well
illustrated in the case of
vertical flow inside ducts.
Figure 3.7, adapted for mass transfer from heat transfer theory (Metais and Eckert,1964) shows the possible regimes that can exist for vertical flow inside a cylinder under the combined influence of buoyant forces
and pressure gradients.
This
figure summarizes available experimental
and theoretical knowledge, covers the cases of forced and free convection
both aiding and opposing each other, and is valid for 10-2 < Sc D/L <1.
As can be seen from Figure 3.7, the hydrodynamic regime can be characterized with two parameters: the Reynolds number and the product of the
aspect ratio times the Rayleigh number
(which, for mass
transfer,
is
equivalent to the product of the Schmidt and Grashof numbers).
The lami-
nar-turbulent
shown
a dashed
transition in the forced and mixed
area
of finite thickness.In
regimes
is
as
the free regime, the transition
occurs at an abscissa value of - 109.
For Ra(2R/L) < 109, then, increasing the Reynolds number (at constant
abscissa value),
leads to a laminar-turbulent
transition.
Increasing
the abscissa at constant Reynolds number, on the other hand, give3 rise
103
P,%-l
lU'
I..
^. -I_
-
_
10 Z
-P,
LILt
Hg
PHg
4
10-4
lo
IHg
103,
H20
lr3
SCF
102 .
H2 0
10
Air
10-5i
O-4
I
H2 0
-o6
SCF
I
H2 0
4
Air
I0-I
Air
Hg
10 -
10-5
io
4
I0I
I
Air
p(Kg/m 3 )
FIGURE
3.6
Io-6
0-2
SCF
10-8.
(Nsm2)
v(m 2/s)
(vH20/V)?
: Comparison of physical properties of air, water and mercury
at ambient conditions with those of C02 at 150 bar and 310K.
Relative importance of natural convection at constant
Reynolds numbers.
104
6
10
10'
4
3
10
2
10
100
Ra (-R)
L
FIGURE3.7 : Hydrodynamic regimes for vertical
105
cylindrical
duct flow.
to
a forced-free
transition.
It is very
important to
forced laminar region is bounded on all sides.
things,
in
This means, among other
to attain forced laminar flow beyond an
that it is impossible
abscissa value of
notice that the
104, no matter how low the Reynolds number is, or,
other words, that Ra(2R/L) < 104 is a necessary but not sufficient
condition for forced laminar flow.
This
above and to
of each curve represents
the right
the area lying
in Figure 3.8, where
criterion has been used
geometries for which
laminar flow is impossible in supercritical operation (notice the v and
Sc values).
The parameter in Figure 3.8 is the relative density change.
Thus, even with negligibly small density changes (10-3) and aspect ratios
(10-3), forced laminar
flow cannot be attained for D
are shown dotted for D/L
> 10-1
> 8 mm.
Curves
since, for Sc = 10, this is the upper
limit for the validity of Figure 3.7.
For packed bed flow, the large contribution of buoyant forces suggests
that the usual mass transfer correlations are unsuitable for design purposes when supercritical fluids are involved if the controlling resistance
lies
in the supercritical
phase. Correlations which take
into
account
buoyant forces have been published (Karabelas et al., 1971), but do not
cover the low Schmidt number range characteristic of diffusion of small
(MW - 100) organic solutes in supercritical fluids.
106
-I
10
(%)
-2
I0
-3
10
2
4
6
8
10
12
14
D (mm)
FIGURE 3.8 : Cylindrical duct geometries for which vertical forced laminar
flow is impossible under supercritical conditions.
107
4.
APPARATUS AND EXPERIMENTAL PROCEDURE
4.1 INTRODUCTION
The
experimental aspects of the hydrodynamic technique used in the
present work will be discussed in this Chapter.
The theoretical basis
will be presented in Chapter 5.
In essence, the technique
involves laminar flow of a supercritical
fluid within a duct of well-characterized geometry.
A solid solute dis-
solves into the fluid from a surface at the duct boundary.
Diffusion
coefficients and mass transfer rates can then be calculated from a knowledge of the flow'rate,
duct geometry and equilibrium solubility
of the
solute in the supercritical fluid at the same temperature and pressure,
and from the measurement of the amount of solute that, at steady state,
precipitates upon decompression from a known amount of fluid.
4.2 APPARATUS
A schematic flow sheet of the experimental appartus is shown in Figure
4.1.
The solvent gas
is compressed
by diaphragm compressor K
J46-13411) and pumped from a gas cylinder
(Aminco
(TK2) to a 2-liter autoclave
(TK1) whose pressure is maintained by an on-off controller (indicator-controller
PIC)
acting
on
pressure regulator, V1
the
(Matheson Model
Matheson Model 3064 High
sations.
electric
4 High
Pressure Regulator)
drive
Pressure
(M).
A
manual
Regulator
or
eliminates downstream pul-
This is essential for hydrodynamic experiments.
The pressurized solvent
coil CL,
compressor's
which
is
immersed
is preheated to the desired temperature in
in a 24
in.x
18
in.x
18
in. water
bath,
B1,
whose temperature is maintained by heater-circulators A1 and A2 (Thermomix
1460, accurate to within 0.010 C).
The pressure is displayed on a panel-
mounted guage, PI (Heise guage, 0 - 400 bar, accurate to 0.5 bar).
the
diffusion
tube,
is
a
12
1/2
in.
long
2in.
Sch
160 316
C2,
stainless
steel pipe with threaded ends, inside of which is located a flat plate
where the diffusional process occurs.
108
All tubing up to C2's inlet connec-
A
41
)
U,
*I
3
r-4
as
H0
4-4
4.J
V'
0)
rz
-H
'-4
iL
u
109
tion is 1/8 in. 316 stainless steel.
The outlet section is 1/4 in. 316
stainless steel.
A hydraulic jack, J (American Scissor Lift), raises B1 so that, during
an experiment, C2 is immersed in water; B1 is then lowered to allow easy
to C2,
access
be opened between experiments to replace the
which must
flat plate.
The partially
saturated fluid emerges from C2
to valve V2.
(JL) up
jacketed line
and flows
through a
Bath water is circulated
through
the jacket by pump P to maintain the outlet line isothermal and prevent
solute precipitation. V2 (30VM 4882, GA, Autoclave Engineers) is a 1/4in.
manual regulating valve which controls the flow and reduces the pressure
down to atmospheric.
In operation, it is maintained at least 200 C above
the solute's melting
oint in order to avoid clogging due to solid accumu-
lation.
The precipitaed solute is collected in two glass wool packed U-tubes
immersed in an ice bath (B2, a 1500ml beaker); the solvent flows through
rotameters R (Matheson R 7640, Series 603 and 604), and the total amount
in a dry test meter (DTM, Singer 802).
is integrated
The temperature
at C2's inlet/outlet, and at the DTM's outlet can be read in a panel-mounted digital indicator (TI).
A U-tube manometer (U) provides an accurate
reading of the solvent's pressure at DTM outlet conditions.
A
secondary
allows separate
(purge) line is also shown in Figure
4.1; its design
purging of the upstream, downstream and inlet sections
of the equipment.
Also
shown in Figure
4.1 is C1, a lin.OD x
316 stainless steel nipple which increases the holding cpacity
by TK1.
The
17in.1
provided
C1 and TK1 are both protected by separate rupture discs (X1,X2).
actual
supercritical
experiment
fluid inside
involves fully developed laminar
a horizontal rectangular duct
flow of a
(Figure 4.2),
the bottom surface of which is coated with the solute of interest.
The
brass plate (4) is tightly fitted into an enclosure made up of two aluminum hemi-cylinders (1,2); flow occurs inside the resulting 1in. x 1/8in.
rectangular channel (3).
(5) where laminar flow
section
The plate contains three sections:
is allowed to develop, a
(6), and an outlet section
(7).
a section
1in. w x 3in. 1 test
The test section
is made
by
casting the molten solute and carefully machining the surface after sol110
In . .I
6
5
8
FIGURE 4.2 : Flat plate assembly
7
8
for
111
hydrodynamic experiments
idification.
Fluid by-pass of the test section is prevented by a Viton
by
(8), which forces the plate against the upper surface, and
gasket
the labyrinth seal (9) which results when the hemi-cylinders are tightened
(10,11) and Teflon tape
surfaces
is placed
between the upper and lower mating
(9).
The whole assembly is tightly fitted inside C2.
13,
by O-ring
while
O-ring
12
is
notched:
the
Sealing is provided
pressure
in
3
is
thus
equal to the pressure outside 1 and 2.
When channel 3 is horizontal, there are no buoyant effects and true
binary diffusion coefficients can then be determined (this is not true
in general; for the
this work,
systems and experimental conditions considered
however, this statement is valid in all cases; see Chapter
6 and Appendix 2 for theoretical development and
variable
buoyant forces can
The same experiment
C2.
in
proof).
be introduced by rotating
Arbritrarily
1 and 2
should then give rise to different
and information can be gathered on the relative
inside
results,
importance of natural
convection in supercritical fluids.
As already mentioned above, the calculation of binary diffusion coefficients requires knowledge of the eqvuilibrium solubility of the solute
in the supercritical fluid under the same conditions of temperature and
pressure.
Equilbrium
solubilities
are measured
in separate experiments.
The
configuration of equipment is the same as that shown in Figure 4.1, except
for the fact taht C2
steel vertical
is replaced by a
column packed with
in. OD x 12in. 1 316 stainless
the solute under
is also replaced by a smaller, cylindrical water bath.
investigation.
B1
The supercritical
fluid thus flows upward through the bed and, at low enough soovent flow
rates
The
(see below), emerges from the column saturated with
experimental procedures
for equilibrium and diffusion
the solute.
experiments
will now be discussed.
4.3 EXPERIMENTAL PROCEDURE
C2
is partially pressurized and B1 raised
to its working position.
Partial pressurization is attained by opening V1
112
(with V2 closed) and
closing the on-off valve immediately following V1 once the desired pressure has been attained.
period
of 2-3 hours, during which
tained.
rium
The experiment is only started after a minimum
thermal uniformity inside C2 is at-
It is important to select the pressure for this thermal equilib-
period in such
a way that only a negligible amount of solute is
dissolved, otherwise,
the test section's geometry will
be altered even
before the experiment begins.
TK1 is always kept at pressure at least 20 bars higher than the desired value, since V1 can only control if there is a pressure drop across
its body.
C2
pressurized, and the heated regulating valve V2
is then
opened and manually operated to attain a constant flow rate.
The precipi-
tating solute is collected in a pair of U-tubes, the contents of which
are unimportant, since at least 15 minutes
to attain
are allowed
a steady state while fluid is flowing at
temperature and pressure.
continuously
throughout
for the system
the desired rate,
The recirculating pump is started and operates
the duration of both this start-up stage and
the actual experiment.
The' back-up U-tubes are removed, and the DTM initial reading is then
recorded.
The actual U-tubes are the connected and an
accurate timing
of the experiment is started simultaneously.
It is imperative that, throughout the experiment, the flow rate be
kept as constant as possible.
In general, flow characteristics are good
when the equilibrium solubility of the system under study is less than
-10 - 4
mole
fraction.
Above this value, the flow is more uniform the
lower the melting point of the solid.
Temperature
and pressure readings are taken every 5 minutes.
The
duration of the run is determined by the need to collect at least 10mg
of solute (see Appendix 2 and Chapter 6 for a more detailed discussion).
Weighing
is done on a Mettler H51-AR
balance, accurate
to
100
g.
Detailed calculations on the flat plat design, and on the duration, accuracy and limitations of the experiment are given in Appendix 2.
Equilibrium
experiments are
conducted in
an entirely similar way,
but neither accurate timing nor constancy of flow rate is important in
this case.
The thermal equilibrium stage is also much shorter on account
of the abscence, in this case, of stagnant layers of fluid
113
inside the
extraction column.
In equilibrium experiments, on the other hand, solute
precipitation inside the outlet lines is of more concern than in diffusion
experiments, since the fluid is now saturated with the solute (as opposed
to - 15-20% saturated in a typical diffusion experiment).
Flow rates for equilibrium experiments must be low to guarantee solvent
saturation at
the column's outlet.
DTM conditions (Kurnik, 1981).
114
Typical values are 0.8
lt/min
at
5.
DIFFUSION IN RECTANGULAR DUCTS
5.1 VELOCITY PROFILE; FULLY DEVELOPED FLOW
The basic geometry of the problem is shown in Figure 5.1.
We consider
fully developed laminar flow in the axial (x) direction inside a rectangular duct of height 2b (-b
y
b) and width 2a (-a
z
a).
The velocity profile can be expressed empirically as (Shah and London,
1978):
yl
yjn
V
_
=
II m
)
I [1 ]
[C1-
v
b
I
(5.1)
a
max
-1.4
m = 1.7 + 0.5a
n = 2
ca < 1/3
n = 2 + .3(a - 1/3)
a
>
(5.2)
1/3
where
a = b/a
Figure 5.2
(5.3)
is a plot of Equation (5.1).
The expressions (5.2) were
obtained by matching the finite difference solution of the momentum balance equation to the empirical form (5.1) .
-=
-.
From Equation (5.1) we obtain, upon integration over the duct cross
section,
v
m+1
C
<v>
n+1
)
m
( -
n
l
) [1 -
Y
(
b
115
Izl
n
) ] [
-
(-)
m
(5.4)
a
2b
2a
FIGURE5.1 : Rectangular
duct geometry
116
0.
Y
Q
-0.
-2.0
-3.0
0.0
2.0
3.0
-3.0
-2.0
0.0
2.0
3.0
FIGURE 5.2 : Velocity profile as per Equations (5.1) and (5.2); a - 1/3
117
where
b a
ab<v> =
[1 .
I
] I1
(
-
(
) ] dy dz
(5.5)
a
b
00
5.2 DIFFUSION; CONSERVATION EQUATION
The steady state conservation equation for a diffusing solute
2
ac
vax
2
2
ac
ac
= D ( -+
2
ac
+ -
-
ax
ay
2
az
2
)
where c is the solute molar concentration and D, the binary
coefficient.
is
(5.6)
diffusion
If we now non-dimensionalize by defining,
B = L/2a
= L/2b
+
x = x/b
y = y/b
(5.7)
z = z/b
c = (c - ci)/(co
- ci)
v = v/<V>
Equation (5.6) becomes
Pe
(
a
+ ac
+
), v ax
a
2+
=
+
2+
ac
2+
ac
(5.8)
ax
+
2
ay+
118
2
az
+
2
Pe = <v>L / D
(5.9)
In the above expressions, L is the relevant axial dimension, which
will later be identified with the coated (or heated, in the equivalent
heat
transfer
problem)
length; c i is the solute
interface, where
(still undefined) solvent-solute
assumed, and c
is the solute concentration at x
concentration at
the
phase equilibrium
is
0- (which will even-
tually be equated to zero).
Dividing through by
( aPe/2B
), and taking into account that, with
the definitions (5.7),
Y ~
0(1)
y+
O-1
z
x
(a
-
(5.10)
)
0(25/a)
we have the following scales,
-2
x - convection
(a/26)
~ (2a/aPe) - 5 x 10
y - diffusion
x - diffusion
2 x 10
-
~ (a/2aPe)
-
-6
2 x 10
-
8
(5.11)
-5
z - diffusion
where
presently
= 1/8,
(2ca/Pe)
= 3, Pe - 104 have been used, corresponding to the case
considered
Appendix 2 ).
x 10
( D
-
10- 8 m 2 /s;
<v>
10-3
m/s;
L ~ 10-1 m;
see
So, from order of magnitude considerations, we can neglect
axial and transverse diffusion.
This conclusion is by no means general.
The problem now becomes, using Equation (5.4),
119
2 ac2
+2) ac+
(2E+ -
ax+
(
.
)
-
aE+2
m-1
[1 - (calz+)
m
4a
A(z+ ) =
(5.12)
= A(z+ )
m+1
3aPe
(5.13)
i+ - 1 - y+
with boundary conditions
c+(x+,2)
= 0
ac+
(x+,o)
c+(O,E
corresponding
to a plane
(5.14)
= 0
--v
a
+ )
= 1
= 2b; y = -b) from
(
which
the solute
dissolves
into a Newtonian fluid moving inside the duct under steady laminar conditions.
(
The fluid contains no solute at the entrance, and the upper plane
= 0; y = b) is impermeable
to mass
transfer.
Thermodynamic
equilibrium
is assumed at the source plane.
5.3 SOLUTION
Equation (5.12) can be viewed as a two- dimensional (x+ ,
with a coefficient, A(z+), that depends on a third dimension.
+) problem
Solutions
to the two- dimensional problem must therefore be integrated across the
third dimension's domain (-1 < az+ < 1).
We now separate the two- dimen-
sional problem into an axial and a "radial" part,
c+ = H(W+ ) . X(x+ )
120
(5.15)
and rewrite Equation (5.12)
2
1
1
dX
d H
1
(5.16)
+
A(z )
X
dx
+
2
-
H( 2+
dE
+
+
2
The full "radial" problem then becomes
2
dH
__- + y2 H ( 2
+
-
;+
2
) =
(5.17)
2
dE+
H(2)
= 0
(5.18)
dH
(-
)
+
= O
o
o6
i.e., we have a homogeneous problem in
+.
We postulate for H a series
expansion,
H =
a
+
(5.19)
i=O i
whose coefficients a i will obey some recurrence relationship to be found
from Equation (5.17), while the first boundary condition will give rise
to
an eigenvalue
problem.
Also, from the
second boundary condition,
we immediately obtain, upon differentiating Equation (5.19),
a1
= 0
Equation (5.17), in terms of the postulated expansion, becomes
121
(5.20)
2 )E + [12a4
+ (2al- a)Y2] +2 +
2a2 +(6a3 + 2aoY
+ [20a5 + (2a2 -al)Y2 ]E+ 3 + ...
(5.21)
n-2
+ [n(n-1)an + (2an-3 - an-4)y2]+
from which we obtain
a 2 /a
= 0
a3/ao
a4/a O
y 2 /3
=
= y 2 /12
a5 /a o =
(5.22)
a6/ao = y4/45
a7 /a o = -
Y4/84
a8/a o = Y4/672
6
ag/a o = -
/1620
As can be seen from the generic term in Equation (5.21), an/ao becomes
a polynomial in even powers of Y for n
a
>
12. Thus, we can write
I
i
--ai
(5.23)
ao
ai
=
C
2j
Y
(5.24)
j=l 1ij
where, for example, C 6 ,2 = 1/45; C6,j = 0 (j
2 ).
The first boundary
condition in Equation (5.18) gives rise to the eigenvalue equation which,
122
in the light of Equation (5.24), becomes
1 +
co
i
i=1
or, in a more convenient
CO
2
2j
C
j=1
Y
(5.25)
=0
ilj
form,
2j
1 +
i
Y
2 C
j=1
=
0
(5.26)
i ,
i=1
Eigenvalues up to 10 in magnitude require an equation of degree 40
or higher in Y.
cient
is a'80
The last a'i containing a non- vanishing Ci,20 coeffiterms in Y4 0 through y 5 0 ).
(which contains
for i up to 80 and j up to 20 are shown in Appendix 3.
been
said,
(5.24),
we
conclude
for eigenvalues
that
Equation
10, is
=
i
coefficients
From what has
the computational equivalent of
a
The
Ci,j values
of
2j
26
C
Y
j=1
Equation
(5.27)
i,j
(5.26) are listed
the first five eigenvalues, in Table 5.2.
in Table 5.1, and
The axial problem has the
formal solution,
X = (const.)
The
two-
dimensional
exp [-A(z + ) y2 x + ]
(5.28)
solution, therefore, can be written,
in its
most general form, as
+
C (x+,
+ )
W
=
C exp[- A(z + ) Y
n
2
2
x+][1
n
+ a
5+ + a
1,n
n=1
123
i+
2,n
+ ...]
(5.29)
Table 5.1 :
Coefficients of the Eigenvalue Equation (*)
n
Tn
1
-1 .333333333333
2
.2793650793651
3
-. 2319063652397
4
.1027637853035
5
-.2828429817471
7
-.7204830258760
8
.7420607052348
9
-.5992478391541
10
.3895904572009
x 10
- 2
x 10 -
4
x 10-8
x 10-10
x 10-12
x 10-
14
-.2082920733589 x 10-1 6
11
12
.9319185747388
13
-.3540455208018
x 10 - 19
x 10 -21
.1156354935000 x 10- 23
14
15
-.3281648836384
x 10 -26
16
.8167183808737
x 10 -29
17
-.1797000439336
18
.3520715535377
19
-.6181214828232
=
x 10-31
x 10-34
x 10-37
.9781035465679 x 10- 40
20
2n
Y2 4n
-1
.5302430465802 x 10-6
6
(*) 1 +
x 10
0
n=1
124
Table 5.2 : Eigenvalues of EqLLation(5.26)
.9546676
2.9743079
4.9810344
6.9845839
8.9876252
125
where n is an eigenvalue index, and a
indicates the result of Equation
i,n
(5.24), with Y -
n (i.e., for the nth eigenvalue).
From the x+ - related
boundary condition, we obtain
1 -I
[ 1 +
C
a' a'
n-1 n
a'
1,n
+ 2+
...
2,n
]
C H'
(5.30)
n-1 n n
Since the form of Equation (5.17) implies that the eigenfunctions,
H' ,are orthogonal with respect to
the weighting
function
(2&
-
&+Z),
n
we can write (Arpaci,1966)
2
(2+-
2 ) H' d
+
n
0
C
-
where
(5.31)
Hn is
I
H
simply Equation (5.19) divided by a,
I
1
- 1 + a
n
i
1,n
2
a
+ a.
(5.32)
2,n
The numerator and denominator of Equation (5.31) will now be transformed.
For the numerator, we write
2
2
d Hn
+
2
Yn Hn (2X+ - + )
(5.33)
O
d2 f
and integrate,
126
2
2
+"| Hn
+-5+
) d+
-Yn
+
(
)
d
(25
od
d+
J
-2
2
(5.3 4)
to obtain, finally,
2
-2 dHn
-Yn
=
-
d+
Hn (2&+ _-
+2)
d
(5.35)
+
+2
0
where
dHn
(5.36)
0
dE
+
o
has been used.
For the denominator, we multiply Equation (5.33) by
dHn/dYn,
dHn
.
dYn
d Hn
+
2
n
dHn
Hn
2
(25+ -
)
O
(5.37)
dYn
de
and integrate, to obtain
I
dHn
dHn
dYn
d&+
12
dHn
10
dE+
O
d
d
+
-(
dE+
cdHn
) d +
+
dYn
(5.38)
2
2
Yn
(H)
2
(25+ -
+2 ) d +
dYn
0
127
-
0
--
where use has been made of the following fact
'
2
2 ' dHn
Yn Hn
Yn
-
' 2
d(Hn)
(5.39)
dYn
2
dYn
Multiplying Equation (5.33) by Hn and integrating,
2
dHn I
2
dHn
2
+
dI
2
(
Hn
o
)
de
d+
+
n)2
'
(2&+
(H n )
2
n
2
-
) d+
(5.40)
= O
+
0
0
now differentiate both sides of Equation (5.40) with respect to
We
Yn and divide by 2,
2
1
dHn
dHn
2
1*
Hn
2
dYn
dE+
o
2
*
'
1
d
dH
dYn
de
+
1
o
2
dY
2
dH
d
2
d
n
+
0O
(5.41)
2
2
f'2
2
2
d nfi(Hn)
d (H
dYn
+ Yn i (Hn) (2&+ - + ) d + +n
2
2 (2+
-
+2 ) d
+
O
O
.
Using the two "radial" boundary conditions to eliminate the second
term, plus the fact that
d
t
(
dYn
dS +
1*
2
dHn
)
dHn
.
- 2 d& +
d
-
1.
dHn
dHn
dHn
d
. - 2
. -
dYn
d
+
d
we can rewrite Equation (5.41)
128
+
d
.
+
(5.42)
dY n
2
1
dHn
dH n
2
dY n
d
I
I
dH n
S
+
J d& +
0
d
dHn
-.
-
dS+
dYn
d&+ +
0
(5.43)
2
2
(Hn) ( 2
+ Yn
+
-
+2) d+ +
(Hn) (2+ -
-
) d+
0
dY n
2
O
O
Subtracting Equation (5.43) from Equation (5.38),
I
,
1 dH n dHn
2
2
....
_
+
2 dYn d
n
8
2
(Hn) (2E+ -
+ ) d
(5.44)
+
o
0
Equations (5.35) and (5.44) constitute the desired expressions, which,
when substituted into Equation (5.31), yield
1
dHn
2
d
O
2
Yn
Cn -
t
1
-2
(5.45)
I
dHn | dHn
2Yn dYn
2
d
|
+
dHn (2)
dYn
dn
2
where (dHn/dE+ ) o - O has been used.
Although the
transformations are
non-trivial, Equation (5.45) is much easier to use than Equation (5.31).
Substituting Equation (5.45) into Equation (5.29),
I
c+
(x+.E+,z
+)
-2
H
1
--
n-1
where (dHn/dYn)
x+
exp[ - A(z + ) Y2
~nX]
[¥n (dHn/dYn)
]
(5.46)
2
means that the derivative expression should be calcula2
129
ted formally and evaluated at
+
2.
Equation (5.46), together with the definition of Hn (Equation (5.32))
and the numerical values of the coefficients Ci,j (Equation (5.24), Table
5.1)
to
and
the
of
the eigenvalues, Yn
+
two- dimensional
(x ,
(Table 5.2),
+) problem.
constitute the solution
The siplification whereby,
following order of magnitude arguments, lateral diffusion was neglected,
has resulted in a two-dimensional solution that must be integrated across
z+ , instead of a full three-dimensional problem.
5.4 CROSS-SECTION AVERAGES
We define a cup-average concentration,
-1
C 4ab<v>
<c(x)> -
]
2a
2b
J
J c v dy
-2b
dz
(5.47)
-2a
or, equivalently, using the definition of c+ (Equations (5.7)),
<c+(x+)>
with
S
- [ 4ab<v>
(5.48)
c+ v dS
-]
denoting the duct's cross section.
Using
Equation
(5.4)
plus
the fact that, for a < 1/3, n - 2,
1/a 2
m+l
<c>
..
m
3
1
. 2
.
l - (1- +) 2][l - (alz+l) ]
ab
I
4ab
-1/a
0
d
dz +
~~~~~~~~~~
k3 .49)
which can be rewritten, taking into account the symmetry of the z+ problem, as
130
-1/a
2
3(m+1)
.+ ) 2[
[1 - (1-
a .
<0>
- (z
+ ) M]
d
+
(5.50)
dz +
4m
0
0
Using Equation (5.46), this becomes
<c+(x+)> (5.51)
1/m
2
[1-(1I
-3a(m+)
2m
[
-
(z
+)
+
2 '
) ]Hn
exp[- A(z+) Yn x+] dz+
]
d~+
n-1
Yn (dHn/dYn),2
0
0
The form of the z+ integral can be made more explicit by using Equation (5.13), to obtain
2
2
A(z+) Y
+
x
1
mYnXo
VA~~
~
3(m+1)
~
~
1 -
~
+
(z
-
(5.52)
)
where X, is a modified inverse Graetz number,
x D
Xo
(5.53)
b
2
<v> b
and
z+l
has been replaced by z + since integration is over positive values
of the variable only.
Defining
2
,
XO (m,n)
2 m Yn Xo
-
the z+ integral becomes
131
--3 (m + 1)
(5.54)
1/a
a
M
[ - (az+)
] exp [-A(z+ ) Yn x3+
dz+
=
O
1/a
Xo(m,n)
i
a
-
(z+)
] ex
[-
-X
mn)
] dz+
(5.55)
=
[1 - (az+ ) ]
0
1
Xo(m,n)
m
) exp [ -
(1 -
-
J
m
(1 - n )
+
We now consider the
] dn
n(Xo,m)
integral in Equation (5.51),
2
2
(25+
-
t+
)Hn
dE +
I
(
Yn dHn/dYn
+
-2
E+
(5.56)
Yn dHn/dYnj2
0
0
which, taking into account the definition of Hn, can be expressed, after
integration, as a series
2
-
f(2
1+
rn
2'
1
- + )HndE+ = -
rn
;
4
,
,
2a_2,,-aJ-3,
[ - + X (2/J)(
n)]
3
(5.57)
j =4
0
where a1 ,n
=
has been used, and rn is defined below,
r n ' Yn dHn/dYn2
1 32
(5.58)
Taking into account the expansion, Equation (5.24),
c
-1
j
2
-
aj-2,n)
j-4
and the
-
(5.59)
j=l
c
I
2J (2 a
2j
J Ck,j Yn
2
k-1
i
c
k+l
rn -=
;
-1
XJ
j-14
2k
1 (2 Cj-2,k - Cj-3,k) Yn
2
(5.60)
k=1
+ integral becomes, finally,
82
4/3 +
26
(2&+ - &+2)HI
J
-1
j
2
j-4
X (2 Cj-2,k
-
2k
Cj-3,k)
k-1
dS+ -
2
Gn
80 k1
26
X
2
0
Yn
k-1
2j
J Ck,j Yn
j-1
(5.61)
where
the summations contain
in Section 5.3.
<c
N
I
=
2m
as discussed
Equation (5.51) now reads
-3(m+1)
+ (x + )>
their computational limits,
n(Xo,m) Gn
(5.62)
n-l
where N is the number of eigenvalues used.
For a pure solvent at the
inlet, such as we are presently considering, the cup- average relative
saturation and <c+> are related by
<c+> + r = 1
(5.63)
r = <c>/c i
(5.64)
133
so we can write
3(m+1)
N
I n(Xo,m) Gn
n=1
r - 1 +
2m
This
n-1
is the expression used
cients.
For
is
a
only
(5.65)
to calculate binary diffusion coeffi-
a given aspect ratio
function
of X.
(i.e., m),
Given <v>
the relative
(solvent flow
saturation
rate), x
(coated
length) and b (duct width), then, the measured r is only a function of
D.
Finally, an expression for the local Sherwood number will be derived.
We define a mass transfer coefficient, k,
ac(x,z)
D
=
a
(-
2
- <c(x)>]
k [ci
(5.66)
b)
or, after nondimensionalization and rearrangement,
kb
1
ac+
(5.67)
D
<c+> +a
(
' 2)
The relevant length parameter is 4 times the hydraulic radius
4(Cross Section)
4b
4 (4ab)
I rh
(5.68)
A:
Wetted Perimeter
4
1 + a
2(2a + 2b)
Bc +
1
Sh(x+,z+) - -
4b
k
1 +a
<c+>
(5.69)
+
2
1 + a
D
The z+ - averaged Sherwood number, therefore, is
134
1
aCZ+,X+)
4
Sh(x+) -
d(z/a)
1 + a
<C+>
+
(5.70)
2
0
From Equation (5.46) we obtain
ac
+
m
-
a+
2
2
X (1 /
-2
rn) . exp [-A(z+) Yn x + ] .
c
j-l
n-
j-1
j 2
1
2k
Cj,k Yn
(5.71)
k=1
We now define
1
1
An
-
=
nr
(5.72)
80 k+l
k-l
Bn'
80
k-1 26
I k 2
X Ck,j
k-1
J-i
26
2j
jCk, j Yn
j-1
2j
(5.73)
Yn
where, again, the computational limits have been used in the summations.
The final expression is, therefore,
16m
N
I An
Bn
**n(Xom)
n=l
(5.74)
Sh - N
3(m+1)(1+a)
I Gn · n(Xom)
n-l
with
1 35
2
exp[-A(z+)Yn x+] d(z/a) =
exp [-
X0 (m,n)
] d
-* n(X,m)
(5.75)
(1 - rm)
0
0
The numerical values of An, Bn, Gn can be found in Table 5.3. Figures
5.3 and 5.4 are plots of Equations (5.65) and (5.74), which constitute
the solution to the problem of diffusion in rectangular ducts at high
Peclet numbers and low aspect ratios.
On and
n, though well behaved, are non- analytic and must be evaluated numerically.
136
Tab'e 5.3 :
n
Expansion Coefficients for Cross- Section Averages
1
-.6242144449211
2
.1916121727370
-.1136094212907
3
4
5
B
A
-. 8717135423055
1.
62136522176
-2.606320953153
-5
.4394766591177x10
196572.6180829
o3212078027422xl
0-10 26634903493.79
137
G
-. 5970396020662
-. 4033326872425x1
-1
-. 1209851681885x1 o
.5196364776233x1
.4744797120123x
10
1
-2
-2
z
0 0.4
I-
I(/)
F 0.2
aJ
tr
I
.
I
0.0
0.5
,
I
1.0
XD
cv> b2
FIGURE5.3 :
Relative Saturation as a function of modified inverse Graetz
number, for various aspect ratios.
138
Sh
x D
cV,b2
FIGURE
5.4 :
Local Sherwood number (z- averaged) as a function of the
modified inverse Graetz number, for various aspect ratios.
139
HYDRODYNAMIC EXPERIMENTS; RESULTS AND DISCUSSION
6.
6.1 DIFFUSION COEFFICIENTS; RESULTS
Diffusion
coefficients
for four
were measured
hydrodynamic technique explained in Chapter
using the
coefficients
are shown
of solvent density.
in Figures
were
6.3,
and
6.4
as
a function
equation of
densities were obtained via
4); solvent
on cubic
1976;
(Peng and Robinson,
state for SF 6
1 for a discussion
see Appendix
6.2,
The experimentally controlled variables were tempera-
ture and pressure (see Chapter
the Peng-Robinson
6.1,
The results
4.
The measured diffusion
are summarized in Tables 6.1, 6.2, 6.3 and 6.4.
ties
different systems,
equations
of state);
CO 2
obtained from the International Thermodynamic Tables
densi-
of the
Fluid State (Angus et al., 1976).
The approximation whereby fluid density is calculated without taking
into account solute concentration is only justified for dilute systems.
Equilibrium solubilities for the four systems investigated were measured
with the flow technique described in detail in Chapter 4.
librium solubilities
maximum
at
338
are listed
solute weight fractions
K
and
120
bar for
in Tables
6.5, 6.6, 6.7 and 6.8.
under experimental
benzoic
acid
in
Measured equi-
SF 6 ;
conditions
0.34%
at
The
%
(0.02
328 K and
120
bar for naphthalene in SF6 ; 1.06% at 328 K and 200 bar for benzoic acid
in C02 ; 0.28% at 318 K and 250 bar for 2-naphthol in CO2 ) are extremely
low.
Under these conditions, the infinite dilution assumption introduces
no analytically detectable
error.
Example
calculations of a diffusion
coefficient and an equilibrium solubility can be found in Appendix 2.
Diffusion is the macroscopic manifestation of collisions at the molecular level.
The time dependence of the mean
displacement of a
given
ensemble of particles with respect to some arbitrary initial configuration
is a function of molecular velocity (temperature) and packing (density).
Diffusion
coefficients
(see Chapter 8) can
be obtained from the time
evolution of the mean squared displacement of an ensemble of molecules,
and should therefore be interpreted in terms of the relevant variables,
i.e., temperature and density.
140
I
a
I
f
a,
II
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Lr%
M
I
I
'.o
%O
-
-
W
C
a)
O
C.
C4
C.
H
0
.:
L,
0
L
a
H
-Cl
E-4
o
X-,.C
zH
0
H
03
0)
a
a,
t.
0.
L
C0.
0zc.3
a) g Z
z
H
C)
rC
C
.0
*
**u
_
0
ED
O
Z~
0t~
QOO
:
CD
kCO
az
z
s
-4
)
*
C:
C
o
(s
Ct
A
,
.54
L
40"
c
1o
o
Cu
02
C
4
:
:
E-
ca
141
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11
11
11
0
C:
0
z
E4
,-
::I
00
0 .4sI
-I
0.
-
0
U)
o
a)
Cu
oo
E.
CO
-I
00
0
0
C)
H
-:
C
ct
'-
U)
CL
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o
C
142.
0
C
L-
.
O
L
oa
C
a
_
'a
I
I
I
I
I
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Illl
I
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a E
u O
r
OY
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·
ciM
0
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ax
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ci,
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0
L
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C
CC
a)
.
H
0.
L)
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o
0_ c)
Ez
04 0 =
'0zra
a0
_
D
2'-
4
co
r=
-_-
O
Fe
O
a
a
_
0
-I
0H )
C)
U)
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0
0-
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,-
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a
a
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ci
cci
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am
9-'
D0.
cr,
CZ.
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I
()
·c
3L:
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L-
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Of 4
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C..
C)
bo
9-
--
O0
EL.
b
a)
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0
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H
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=
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0
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a
a)
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lll
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0
0)
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a)
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a)
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cci
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9-
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cn
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to
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CM
m
144
V
('
C.
C-.
O
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Q
0CC
^
.HO
_
-
CO
i
COa 02 NC)0N,
'.0
_- r 0_ ,
U
bO *.CO
0
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U)
s'
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a
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0-)
o
4-)
4.
*
O a:-
C-.
0LA
C
V.
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U)
a)
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-C
I
ct
03
CO
0
U)
X
C
V
E
C)
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C
02
o
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EvX
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N
0
C)
TABLE
T
(K)
6.5 :
EQUILIBRIUM
SOLUBILITY
OF BENZOIC
ACID IN SF6
104x
P
(bar)
(mole fraction)
328.2
65
1.194
328.2
80
1.491
328.2
338.2
338.2
338.2
120
65
80
120
1.825
1.646
2.076
2.803
145
TABLE 6.6
: EQUILIBRIUM
SOLUBILITY
OF NAPHTHALENE
IN SF6
10 3 x
T
P
(K)
(bar)
(mole fraction)
318.2
65
1.978
318.2
80
2.152
318.2
120
2.445
328.2
65
3.184
328.2
80
3.513
328.2
120
3.91 4
146
TABLE 6.7 : EQUILIBRIUM
T
(K)
SOLUBILITY
P
OF BENZOIC ACID IN CO 2
103x
(bar)
(mole fraction)
318.2
160
2.341
318.2
200
3.580
328.2
160
2.495
328.2
200
3.864
147
TABLE 6.8
: EQUILIBRIUM
SOLUBILITY
TABLE6.8 EQUILIBRIUM
SOLUBILITY
OF
NAPHTHOL
OF 22 NAPHTHOL
IN
CO,
IN CO^,
1 4x
T
P
(K)
(bar)
(mole fraction)
308.2
150
4.460
308.2
200
5.408
308.2
318.2
318.2
250
165
250
5.910
5.662
8.655
148
,
N
'o
o
'a
.H
O
N
0
0
Ef
O
~Q.O
(
+1
0
LA
UD-H
0·
( S/W3)
149
sr1
I
0o
CJ
o
0
a
O~4-
a
F4
c
C\j
L
_N
-
(Sz
150
w
)
0
vb
L
_
-
-
0
oC
Q0)
,
co
N
_C.
Q
.7'O
If)
_
W~~..
Q
.,
O
.H
;D
0
( S/
LO
W3) C01
,~ ~~
r
151
I
II I
I
I
I
I
IIII
A .
I
V
I
I -
C
N
0o 0o
O
N
0
r4
rl
Ut
C),
I
0
an _
Sc
a
0ca
4H
r1
z
O
0)
E
%ooQ0
0c
_2ca
I
0)Q
I
I
(S/ZU3)
I
.rl
Co
I
PC4
If
152
0
I
-
I
0
01
Ir
114,
_
I
I
I
I
1
I
I
I
I
I
Solvent
SF6
SF6
SF6
SF6
CO2
CO2
CO2
CO2
I
I
I
Solute
Benzoic Acid
Benzoic Acid
Naphtholene
Naphthalene
Benzoic Acid
Benzoic Acid
2-Naphthol
2-Nophthol
-
|
I
·
Temperature (K) -
328.2
338.2
318.2
328.2
318.2
328.2
308.2
318.2
10-4
U
10
I
I
n
_
m
I
5
I
W
I0
I
n
I
n
B
I
m
I
I
B
15
I
m
I
I
I
B
I I
B
20
I
I
I
I
II
I
I
25
p (mol/4t)
FIGURE 6.5 : Experimental diffusion coefficients as a function of solvent
density
153
__
I
I
Solvent
o
SF6
Solute
Benzoic Acid
SF6
Benzoic Acid
O
SF 6
a
SF6
+
C02
Nophthoa lene
Naphthalene
Benzoic Acid
Benzoic Acid
0
C02
x
C02
*
0)
E
U
O
_h
R1
I
=O
+
+
_
_
2-Naphthol
2- Nophthol
T(K)
328.2
338.2
318.2
328.2
318.2
328.2
308.2
Tr(-)
318.2
1.05
1.03
1.06
1.06
1.05
1.08
1.01
x
I
I
l
2
3
4
FIGURE 6.6 : Experimental diffusion coefficients as a function of solvent
reduced pressure
154
5
Pressure,
on the other hand, is only relevant for any particular
system insofar as density is a function of pressure at any given temperature.
This can be clearly seen from the fact that the measured diffusion
coefficients are only slightly higher than typical binary diffusion coefficients
in
liquids
at comparable temperatures and densities, but at
ambient pressure.
For dilute systems, such as the ones presently considered, corresponding states arguments can be invoked in support of the use of solvent
pressure as an
reduced
The
independent variable
(Paulaitis et al.,
data are therefore summarized both as a function
of fluid
1983).
density
(Figure 6.5) and solvent reduced pressure (Figure 6.6).
At constant temperature, low density diffusion coefficients are inversely proportional to fluid density.
(Chapman and Cowling,
ically
1970).
This result can be derived theoretFurthermore,
since
the
pressure
and density of an ideal gas are directly proportional at constant temperature, a logarithmic plot of diffusion coefficients versus pressure approaches a limiting slope of -1 at low densities (Paulaitis et al., 1983).
No
equivalent simple relationship exists at high pressure, and the use
of a semilog scale in Figure 6.6 is simply a matter of convenience. Similarly, there exists no accurate theory that will predict the isothermal
dependence of
density
below in
be discussed
log
D
vs. p
diffusion coefficients in dense fluids, as will
connection with the Enskog
relationship
suggested
by
Figures
6.1,
theory.
6.2
and
The
6.4
linear
(i.e.,
systems for which more than two points per isotherm at least at one temperature were measured) has been reported by other researchers who studied
diffusion in supercritical fluids
(Swaid and Schneider, 1979; Feist
Schneider,
case as well
studies,
any
1982).
In the present
though, the ratio
given
isotherm was,
of the maximum
at most,
and
as in the above mentioned
to the minimum density
for
three; the smallness of this number
suggests that the observed linearity should be interpreted with caution.
Even though benzoic acid is a smaller molecule than either naphthalene
or 2-naphthol,
the measured
diffusion coefficients of benzoic acid
in
SF6 were smaller than those of naphthalene in SF6 , and the measured diffusion coefficients of benzoic acid in CO, were smaller than those of 2naphthol in C02 , at the same density (and slightly higher temperature).
155
These observations suggest fluid phase association of benzoic acid,
a possibility that will be discussed below in detail.
6.2 DISCUSSION
No rigorous kinetic theory for dense fluids exists that will
an a priori calculation of transport properties.
of Boltzmann's
allow
A perturbation solution
transport equation constitutes the basis of the Chapman
Enskog expressions for the transport coefficients in dilute fluids (Chapman and Cowling, 1970; Hirschfelder et al., 1964).
equation
Boltzmann's transport
(Huang, 1963; Pauli, 1981) contains three fundamental
assump-
tions:
- molecules
are points and hence have only translational degrees
of
freedom
-
position and velocity of
the
a molecule
are uncorrelated
(molecular
chaos assumption)
- only binary collisions are considered
At
high densities, each of
occuring
(i.e., transfer
in addition, collisional transfer
plausible, and,
less
these assumptions becomes progressively
during an encounter) must
be taken into account
(Chapman and
Cowling, 1970).
In spite of the fact that it cannot reproduce important experimental
trends, Enskog's dense gas theory (Enskog, 1921) is widely used for correlating purposes, although never in a predictive way.
fairly
common to report
experimental data in terms of deviations from
(Enskog) behaviour
theoretical
It is, in fact,
(Balenovic et
al., 1970, for
example).
The simplicity and plausibility of Enskog's assumptions and of the predicbehaviour
ted
(i.e., a
density correction factor whose
reciprocal
is
linear in density) are chiefly responsible for this rather unusual situation.
In this approach, a dense atomic fluid composed of smooth hard spheres
is
considered.
since hard
The
assuption of molecular
chaos is maintained,
and,
sphere collisions are instantaneous, only binary collisions
are taken into account.
156
As
generalized
by Thorne to binary diffusion, the main
result of
Enskog's theory for dense fluid diffusion is (Chapman and Cowling, 1970)
o
pD
1
- (pD
) X12
12
12
(6.1)
where superscript o denotes the low density limit, and
3
nloa 1
X12 =
+
n,
n2,
ro2
n2
(
6
with
3
01+4a2
)
01+02
a01 and
401+02
(
.
) + ...
6
2
(6.2)
01+a2
denoting, respectively, solute (1) and solvent
(2) number density and hard sphere diameter.
When
experimental
values of
(6.1) can still be used, with
(pD1 2 )o are not available,
Equation
(pD12 )o calculated from Chapman -
Enskog
dilute gas expressions (see below).
For infinite dilution of n
(solute) in n2 (solvent), Equation (6.2)
can be rewritten as
3
Irno2
X12
=
4s+1
1 +
(
)
6
s+1
(6.3)
s
where
a1/02
0
n is now the solvent number
density.
Equation
(6.3)
provides
a rational basis for correlating purposes, with 02 as an adjustable parameter, and
Lp
n
-
(6.4)
M
with L, Avogadro's
number and M,
solvent molecular weight.
157
Since the
(4s+1)/(s+1) varies only from 1 to 4 when s
factor
(Brownian solute),
solute) to
o2
varies from 0
(point
is a better choice for a regression
In addition, for non-spherical solutes, s retains more phys-
parameter.
ical significance than a 2 .
The low density limit of pD was calculated from the Chapman- Enskog
expression (Bird et al., 1960)
1/2
O
)
1
·22.2646x
+1
M
0
1
2
a1 2
M2
) ]
(6.5)
d
D is in cm2 /s, p in g/cm 3, a in A, and T in K.
where
An average
deviation
of 7.5 % was found when 114 experimental low pressure diffusion coefficients
were
compared
(Reid et al., 1977).
the corresponding Chapman-Enskog
with
The collision integral,
prediction
d, was found from Table
5-2 of Bird et al.(1960), where the 12-6 Lennard-Jones potential function
has been assumed.
The following combining rules were used for the inter-
molecular potential parameters
1/2
E12 = (1
2)
(6.6)
°1 2
The
i
and ai
=
(a1 + 02)/2
values are listed
in Table 6.9 and were
calculated
from the following expressions (Tee et al., 1966)
Tc
1/3
= 2.3551 - 0.087w
a (-)
(6.7)
Pc
=
0.7915 + 0.1693w
(6.8)
kT c
where P
The
is in atmospheres, T
in K, and a in A.
experimental, calculated
and regressed values of X12 are shown
158
of
squared
with
02 as
deviations
between X12
adjustable parameter.
(experimental) and
Equation
(6.3),
In the present context, experimental
(Chapman-Enskog) low
divided by theoretical
observed pD
means
The regression was done by minimizing the sum
6.10 to 6.13.
in Tables
pressure
The s values used in the regression were calculated as the ratio
pD.
of the respective (Tc/Pc)1 /3 quantities.
The experimental X12 values are plotted in Figure 6.7 as a function
of fluid density, and the calculated, regressed and experimental values
for each system are shown in Figures 6.8, 6.9, 6.10 and 6.11.
As mentioned
above, the Enskog-Thorne expression
(Equation
(6.3))
predicts a linear increase of X12 with fluid density, which, physically,
means that, at constant temperature, the diffusion coefficient decreases
with density more rapidly than p-1 .
Deviations from this predicted behaviour have frequently been reported
in the literature
O'Hern
and Martin,
(Swaid and Schneider,
1955, for
1979; Balenovic
et al.,
1970;
example) and interpreted in terms of the
positive correlation of molecular velocities found by Alder and Wainwright
in their molecular dynamics work (Alder and Wainwright, 1967).
it is very qualitative in nature, the currently accepted
Although
interpretation
of Alder and Wainwright's results and of observed experimental behaviour
is that a large solute particle will " gather a 'cloud' of carrier molecules which move with it; since collisions by carrier molecules originate
within this 'cloud', there is reduced net momentum transfer and therefore
a reduced retardation of the particle..." (Balenovic et al., 1970).
This deviation from the Enskog prediction can, in some cases, lead
to X12 values which are less than 1 (Swaid and Schneider, 1979; Balenovic
et al., 1970), a fact which cannot be predicted by Equation(6.3).
The experimental X12 values shown in Figure 6.7 reveal some interesting trends.
In the first
> 1 and increasing with p
the
solvent,
whereas
creasing X vs. p trend.
same
carrier
place, theoretically predicted
behaviour
(X
) is displayed by both systems where SF6 is
the C0 2 -2-naphthol
data do show
a slightly de-
Balenovic et al.(1970) note that, whenever the
gas exhibited both types of
behaviour, X
decreased with
p for large solute/solvent ratios ( He - C3 H8 ; He - C4H1O ) and increased
with p for small solute/solvent size ratios ( He - Ar ), in qualitative
159
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163
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164
*
*
*
_
N
I
- I~~
I
I*
I
I
I
SF6 -
Benzoic Acid
*
I
I
I
~
I
I
~~~~~
I
I
o SF 6 - Naphthalene
2- Naphthol
o C02A C0
3
-
2
Benzoic Acid
IC
/
/
/
X12
2
/
/
a
t
/ /
/
0
I
0
I
_
E5
I
I
I
I0
.
III I_
15
p(mol/. t)
FIGURE 6.7 : Experimental Enskog-Thorne factors
165
I
I
I
__
I
-
I--
20
0,%
I
o
I
_· _
I
Aft
4. L
1
_
__
I
I
experimental
z& calculated
s = 1. 233;
2
=- 4.752
0 regressed
s = 1.248;
2
= 4.20A
SF 6
1.8
_ _
3ic Acid
-
_I
1.7
X
n
1.6
0
0
1.5
0
1.4
0
I
5
FIGURE
6
I
I
7
8
P ( mo4/tt)
LI
9
I0
6.8 : Experimental, theoretical and regressed Enskog-Thorne factor
for benzoic acid in SF6
166
I
_
I
I
I
1.9
I.E
1.7
I-
calculated
A
1.6
s = 1.295;
2
=4.752A
* regressed s = .296; o'2 =3. 15A
1.5
SF6 -
m
Naphthalene
0
CX
X
1.4
=,,m=m
0
1.3
0
1.2
1.1
0
0
I
I
7
8
,,
I
9
p(mot/It)
I
I0
10
II
FIGURE 6.9 : Experimental, theoretical and regressed Enskog-Thorne factor
for naphthalene in SF6
167
_
__
__
·
I
__
I
I
0
2.3
0
2.2
2.1
2.0
1.9
N
I
8
X
1.7
m
0
1.6
o experimental
& calculated s
1.5
1.4
*
0
regressed
s = 1.587; o-2 =3.95
CO 2 -Benzoic
1.3
L
_
15
3.762 A
1.557; 2-
A
Acid
l
I·
I
JI
16
17
18
19
p( mo /Tt)
FIGURE 6.10 : Experimental, theoretical and regressed Enskog-Thorne factor
for benzoic acid in CO 2
168
1.9
1.8
1.7
I
cm
I
0
1.
0
-
-
1.
I.'
~ :50
* regressed
CO2 -2
I
17
17
FIGURE 6.11
0
0 experimental
a calculated
I
I
s -1.648; Cr2 3. 762
s 1.671; 2
2.45 A
I_ _ a..
N apntnol
is
'p
toa
p (mot/at)
Experimental, theoretical
and regressed Enskog-Thorne
2
-naphthol
factor
in
for
CO 2
169
agreement with Figure 6.7 and the interpretation of Alder and Wainwright's
The very high X12 values for benzoic acid
dynamics results.
molecular
in CO2 , on the other hand, can be explained if high pressure associatphase occurs, a possibility discussed
ion of benzoic acid in the fluid
It should also be noted that X12 values for benzoic acid in SF6
below.
are also significantly high, again suggesting association.
Furthermore,
X is an explicit function of density (and not of temperature); the fact
that the intermediate density data for benzoic acid show a marked temperature dependence at constant density (see Figures 6.7 and
corresponding
data
points,
with
p = 17.16 mol/lt
318.2
to
cases)
in both
K,
160
when
bar,
plotted
and
.10;intermediate
328.2
K,
as experimental
200
bar,
X vs. p
again suggests a temperature- dependent dimerization equilibrium of benzoic acid in the fluid phase.
This possibility is further substantiated
below in connection with the temperature dependence of the measured diffusion coefficients.
From
the
it can be concluded that
previous discussion
the Enskog
theory overpredicts the X12 correction factor, a fact already observed
researchers
by other
1979).
(Feist and Schneider, 1982; Swaid and Schneider,
More significantly, though, this theory predicts a linear increase
of X with density, and X values which are always greater than one.
Cor-
rection factors which decrease with density (Figure 6.7; 2-naphthol-CO2
system)
or
are less
the theory.
accounted for by
the
Enskog
approach
(Swaid and Schneider,
than unity
1979) cannot
be
This is an unfortunate situation, since
is the only rigorously derived predictive theory
for transport in dense fluids.
Modified versions of the Enskog theory,
on the other hand, are really ad-hoc modifications of a hard sphere theory, and have no rigorous theoretical basis.
A reliable molecular theory
of transport in dense fluids is lacking, as discussed in connection with
the computer simulations.
From
Figure
6.5 and Tables
6.1, 6.2, 6.3 and
6.4, an
interesting
feature regarding the diffusion coefficients of benzoic acid in SF6 and
CO 2
arises.
The
measured
diffusion coefficients of benzoic acid are
lower, at any given density, than those of naphthalene (in SF 6 ) and 2naphthol (in C02 ), both of which are larger molecules than benzoic acid.
In each case, moreover, the temperature was
170
slightly higher in the benzoic
This suggests association of benzoic acid in the fluid
acid experiments.
phase,
a
behaviour
and Shibuya,
experimentally observed
in CC14
and CHC1 3
1965) and in its own vapour, C 6 H 12 , CC14 and C6 H 6
(I'Haya
(Allen
et al., 1966).
The dimerization of benzoic acid can only be invoked as an explanation
of observed behaviour if it is quntitatively significant.
For the dimeri-
zation equilibrium, we write
2A = A 2
(6.9)
I_
or, in terms of the non-associated fraction (1-x) and the total concentration of benzoic acid, [AC],
(x/2) [A°
(x/2)
K
_
{(l-x) [A]}
2
[A] (1-x)
(6.10)
2
which can be rewritten as a quadratic equation
2
x
1
-
x ( 2 +
) + 1 = 0
(6.11)
2K [A ° ]
Using K = 3660 l/mol (Allen et al., 1966), which corresponds to benzoic
acid
CC1,
of
7x10
tration
in
at
-~
303 K , and considering a total benzoic acid concenmol/l
as representative of
the actual
conditions, we obtain a 31% unassociated fraction
increases
to 61.9% at 333 K
(K = 710).
(i.e.,
experimental
1-x),
which
The dimerization constants of
benzoic acid in carbon tetrachloride (Allen et al., 1966) are intermediate
between the corresponding cyclohexane (5830 1/mol at 308.2 K; 1210 l/mol
at
333.2 K) and benzene (462 1/mol at 303.2
values.
at
333.2 K)
Even for the lowest K (i.e., in benzene, at 333.2 K), association
is still significant (1-x
Under
ficient
K; 150 l/mol
these
0.849).
circumstances, the
very concept of
a diffusion coef-
as a molecular property is questionable, since results
171
simply
refer to a "molecule" that does not exist in reality and, moreover, the
"effective size" of this molecule could be extremely temperature dependent.
The
temperature
dependence of the measured
diffusion coefficients
is interesting, since it provides useful insights into dense fluid behaviour, as well as a further confirmation of fluid phase association of
benzoic acid. Hard sphere theory (Dymond, 1974) or its ad-hoc modification, rough hard sphere theory (Chandler, 1975) predict a T 1/
2
of diffusion coefficients
modified
rough
hard sphere
at constant density,
theory by allowing a mild
the sphere's diameter).
(slightly
dependence
in
temperature dependence of
In the hydrodynamic approach, on the other hand,
the Stokes - Einstein expression is used as a starting point,
kT
D
(6.12)
--
6nan
where D is the diffusion coefficient of a Brownian sphere of radius a
n and temperature T, when no
in a continuum of viscosity
between the particle and the continuum.
slip exists
Since the viscosity of liquids
is a very strongly decreasing function of temperature, often correlated
in the Andrade form (Andrade, 1930),
n - A exp(B/T)
(6.13)
it follows that diffusion coefficients in dense fluids should, according
to this approach, exhibit a stronger tempreature dependence than predicted
by hard sphere theories.
Equation (6.12) can be rewritten as
nD
1
-
kT
(6.14)
-
6rwa
172
The quantities on the left hand side are all experimentally measuraThe right hand side, however, depends on the validity of the no-slip
ble.
and on
assumption
particle
spherical
a, which
is unambiguously defined only for
(6.14) will
Consequently, Equation
(molecule).
be used here to predict diffusion coefficients.
a truly
not
The constancy of nDT- 1
for any given system, however, has important consequences that will be
discussed below.
The quantity nDT-1 is shown in Tables 6.14, 6.15, 6.16 and 6.17 for
of
each
the systems
For SF 6 as a solvent(1.24
investigated.
-1
both systems exhibit fairly constant nDT
dynamic arguments may be relevant.
< Pr < 2.1),
values, suggesting that hydro-
As discussed above, the nDT- 1 value
for benzoic acid is slightly lower than for naphthalene, again suggesting
association.
The CO 2 - 2-naphthol system (Table 6.16) also exhibits a fairly constant nDT- 1 value.
The CO2 - benzoic acid data (Table 6.17) show a pro-
nounced temperature dependence, as can be seen from Figures 6.3 and 6.6.
the postulated fluid
Although
phase dimerization has not
effective activation
in CO 2 , an
energy for
diffusion can
been measured
be obtained
from the two coefficients measured at the same density and two different
This value (6.9 Kcal/mol) is in
(318K, 328K, Pr = 1.61).
temperatures
good qualitative agreement with the experimental values for the dimerization of benzoic acid in cyclohexane (6.4 Kcal/mol), CC
noted,
however,
that the overall
(5.5 Kcal/mol),
(Allen et al., 1966).
and its own vapour phase (8.1 Kcal/mol)
be
4
It must
diffusion coefficient is related
in a non-linear way to the dimerization constant, so the above considerations should only be taken qualitatively.
Figure 6.12 is a plot of the measured diffusion coefficients versus
reciprocal solvent viscosity.
Only those isotherms for which three points
were measured are included in the figure, since two points always define
a line.
A plot such as Figure 6.12 is a stringent test on whether diffu-
sion in any given system can be described by an
nDT- 1
equation of the form
f(size), which implies, for any given system, a zero intercept
and slopes
proportional to T
(for a temperature-independent molecular
size).
For the benzoic acid-SF 6 system (see Table 6.14), the isotherm has
173
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o Nophthalene -SF 6
5
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103 71-1 (
·,
·,
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I
3
14 P-1)
FIGURE 6.12 : Experimental diffusion coefficients as a function of reciprocal viscosity at constant temperature (hydrodynamic
test)
178
a 1.07 intercept and a 5.57 slope
(in the figure's units) when the 65
bar data point (which is two standard deviations away from the mean nDT- 1
value) is not considered; a 2.34 intercept and a 4.46 slope are obtained
from
a least
squares regression including all
338 K, the intercept is .43, and the slope, 6.38.
is
1.03, whereas
at 328 K).
the slope
ratio
is 1.145
three data points.
At
The temperature ratio
(disregarding
the 65 bar point
The higher intercept represents less than 15% of the lowest
diffusion coefficient measured
for this particular system.
For benzoic
acid in SF 6 , then, Figure 6.12 suggests that hydrodynamic behaviour is
a
reasonable,
if
not entirely accurate, description of reality.
The
fact that the slope ratio is higher than the temperature ratio can be
explained by postulating a temperature dependent association of benzoic
The
acid in the fluid phase.
postulated association,
though,
cannot
be high, since the temperature dependence of the measured diffusion coefficients for this particular system is not as pronounced as in the benzoic
acid-CO2 case.
The other systems shown in Figure 6.12 exhibit a behaviour that cannot
be described
mathematically
by a relationship
of the form
nDT - 1 = f(size),
since the intercepts are clearly comparable to the actual measured diffusion coefficients.
Feist and Schneider (1982) analyzed diffusion coefficients of benzene,
phenol, naphthalene and caffeine in supercritical CO 2 at 400 C and, from
a D
vs.
n- 1
plot
concluded that the Stokes-Einstein relation did not
apply, since the intercepts were non-zero.
Feist and Schneider correlated
their data with a power law relationship D - n- a ( a<1 ), as proposed
by Hayduck and Cheng (1971).
However,
when
experimental data for
six
binary systems
consisting
of an aromatic solute and a supercritical fluid were analyzed, the quantity nDT-1 was found to be remarkably constant (these systems will be discussed below; the average nDT- 1 values and the corresponding standard deviations expressed as
percentiles of the mean
Furthermore,
and Schneider's
Feist
are shown
in Table
6.19).
data, as read from their published
plot (no tables are provided in the paper) show nDT- 1 values with maximum
and standard deviations
(in percent of the mean) of 31% and 12.4% for
benzene, and 24.6% and 9.2% for naphthalene.
179
The above discussion and the small observed deviations in nDT- 1 values
suggest a general picture, shown in Figure 6.13. Hydrodynamic
behaviour
is approached at high viscosities; deviations from this limiting behaviour
can
be
correlated
relationships
Supercritical
1
.
1
of
(but not
understood)
the
D
type
n-a
by
(a <
viscosities fall roughly
means
1)
of
empirical
(Hayduck
and
power
Cheng,1971).
in the range 0.045nS0.1
PrS4 and 1TrS1.0 6 , which correponds to 110-3
law
cp
for
n-152.5 in the fluidity
units of Figure 6.13 (a typical liquid viscosity is also shown for comparison).
The
exact point at which hydrodynamic behaviour breaks down
(point
c) cannot, at present, be predicted from first principles for any given
system.
However, from Figure
6.13 it can be concluded that predictive
correlations based on the Stokes-Einstein equation (Wilke and Chang,1955;
Scheibel,1954; Reddy-Doraiswamy,1967; Lusis-Ratcliff,1968) will overestimate diffusion coefficients in supercritical fluids.
In
addition, at high enough viscosities
(or, equivalently, at high
enough pressures for any given temperature), the quantity nDT-1 approaches
a constant value; geometrically, this is equivalent to saying that, at
small
n- 1 values,
the curve Ocb is
well approximated by the line Ob.
As an example, the measured diffusion coefficients of benzene in supercritical CO2
(Swaid and Schneider, 1979) give rise to an
DT- 1 value that
is constant to within a 4.6% standard deviation (expressed in percentage
of
the mean) when nO.04
At
cp, irrespective of the temperature and pressure.
high enough viscosities, hydrodynamic
behaviour
is
approached,
and this fact can be used to extrapolate experimental data by assuming
constancy of nDT
1.
the systems and equations tested (see detailed discus-
sion below).
The possibility of having non-hydrodynamic behaviour and, simultaneously, a linear mean squared displacement versus time relationship is discussed in Chapters 1 and 10.
In the present context, it should be pointed
out that this simply means that the "drag" has a power law
dependence
on the viscosity.
From Figure
6.13 it can
values should increase
tha
curve Ocb
ith
also be seen that the experimental
viscosity at any
lies above the straight lines
180
given temperature,
corresponding to
nDT- 1
since
constant
/
I'
D
103 -7
FIGURE
(F
P')
6.13 : Linear (hydrodynamic), power law and transition regimes
for the dependence of the diffusion coefficient upon fluid
viscosity at constant temperature
181
This behaviour was observed in all of the systems shown in
nrD values.
Table 6.19.
The general picture that emerges from the above discussion will now
be tested by discussing the application of predictive correlations based
Hydrodynamic
on the Stokes-Einstein relation to the supercritical regime.
expressions should, according to Figure
6.13,
overestimate
coefficients by some factor which is not, at present,
diffusion
predictable a-prio-
ri.
ad-hoc
Several
the Stokes-Einstein relationship
of
modifications
have been introduced for engineering use with the purpose of extending
its use to molecules of arbitrary shape, while preserving its basic form
(nDT - 1
f[size]).
=
All of these correlations have the form
=
DI2
where
D1 2
the
is
1 in fluid
(6.15)
dilution diffusion coefficient of molecule
infinite
2 (of viscosity
K (T/n 2)
n2 at a temperature
T).
Values
of K are listed
in Table 6.18 for four different correlations.
The Wilke-Chang, Scheibel, Reddy-Doraiswamy and Lusis-Ratcliff correlations were tested for three of the four systems investigated (CO2 -benzoic acid was not included since this system, as explained above, cannot
be studied quantitatively without knowledge of the dimerization constant),
as well
as for the C0 2 -benzene
(308
and Schneider,
1979); C0 2 -naphthalene
bar),
and Tsekhanskaya,
(Iomtev
5 308 K; 66
P
328 K; 80
T
(308
T
P
160 bar),
328 K; 83
1964) and C2 H4 - naphthalene
P
(Swaid
304
(285 5 T
304 bar), (Iomtev and Tsekhanskaya, 1964) systems.
The mean values of 105 nDT- 1 (cm2UP/sK) and the corresponding standard
deviations
(in percentages of the mean)
are shown in Table 6.19.
for the systems
In the benzene-C02 experiments, chromatographic
peak broadening was used, whereas weight loss
cell of
investigated
within a stagnant diffusion
well characterized geometry was used in the naphthalene studies
of Iomtev and Tsekhanskaya.
te and reliable, but there is
analysis.
The chromatographic technique is more accurano way of introducing this into the present
It must also be pointed out that, although the standard devia182
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tions
(Table 6.19) are quite small, the constancy
improved away from the solvent's critical point
of
DT- 1 invariably
(for example, at 308.2
K, deviations of up to 32% with respect to the overall mean nDT-1
value
occur in the C0 2 -naphthalene experiments; the maximum deviatio at 328.2
K
is
In addition, nDT- 1
19%).
increased with
fluid viscosity in
all
cases. as anticipated above.
The molar
volumes
at the normal
boiling point can be estimated by
other means than through the use of an
equation of state.
If the Le
Bas additive volume method is used (Le Bas, 1915), the resulting K values
(Table 6.18) were, in all cases, within 3% of the K values used here.
The Le Bas method, though, cannot be used for SF6 systems, since it is
inaccurate for the estimation of molar volumes of simple molecules; for
C0 2 ,
the recommended
value of 34 cm 3 /mol was used when Le Bas-based K
values were calculated (Reid et al.,1977).
Several conclusions can be drawn from Table 6.19.
In the first place,
all correlations overpredict observed diffusion coefficients by a considerable
amount, as anticipated in
connection with Figure 6.13, with the
single exception of the Wilke-Chang expression for the ethylene-naphthalene case.
For diffusion in C0 2, the Wilke-Chang expression is consistently
less in error.
The ratio
of observed to predicted diffusion coefficient cannot be
excpected to remain constant over a wide range of conditions (see Figure
6.13 and the corresponding discussion).
with
an association factor
of 0.565
Thus, the Wilke-Chang expression
(obtained by averaging
the
three
C0 2 entries in Table 6.19 under the Wilke-Chang column) gives a diffusion
coefficient which is only 5.9% higher
than the experimental value for
benzene in C02 at 313 K and 80 bar (Swaid and Schneider,1979), but, at
160 bar, it overcompensates and the estimate is 14% lower than the experimental
value.
the Wilke-Chang
reasonable
Once
these limitations
equation
are understood, though, use of
with an association factor
engineering estimates
of 0.565
leads to
of diffusion coefficients of aromatic
hydrocarbons in CO2 .
For SF6 as a solvent, both the Scheibel and Lusis-Ratcliff expressions
give similar predictions, and the correction factors are smaller, under
the
experimental conditions, than
either the Wilke-Chang or the Reddy-
185
correction factors.
Doraiswamy
Overpredictions in the benzoic acid case
are exceptionally high; this is consistent with the previous discussion
and the added effect of the postulated fluid phase association of benzoic
acid: the effective molecular size would then be larger
than indicated
by the value of the size parameter used to calculate the K value in each
case (Table 6.18).
The situation can be summarized by saying that, at present, no reliable predictive method exists for the calculation of diffusion coefficients
If one experimental value of D
in supercritical fluids.
12
(and therefore
- 1
) is available, it can be used to estimate diffusion coefficients
DT
of
in the same system
- 1
value.
nDT
400
P;
This
under different conditions by assuming a
recommended for n <
extrapolation technique is not
its accuracy increases with fluid
constant
viscosity, that is to
say,
the experimental diffusion coefficient should be measured at the highest
possible viscosity, and the constant nDT- 1 assumption should not be used
below the recommended minimum n.
The use of predictive correlations based on the Stokes-Einstein equation
is
The correction factors are
not recommended.
always significant,
and can only be generalized after analyzing large amounts of data.
emerging trends, moreover,
are extremely system specific
The
(at any rate
solvent specific, since all of the solutes considered were aromatic compounds).
When
such data
exist, useful rules
can
be formulated;
they
should never be used over a wide range of conditions, as discussed above.
One
such recommendation
an association factor
in supercritical
C02 .
is the
use of the Wilke-Chang
-= 0.565 for diffusion of
expression with
aromatic hydrocarbons
It is obvious, however, that the way in which
this number was arrived at cannot be called predictive.
6.3 EFFECT OF NATURAL CONVECTION
The importance of natural convection in mass transfer with supercritical fluids has already been covered in Chapter 3.
As discussed in Chapter
4, buoyant effects can be introduced by performing the hydrodynamic experiments with
the two hemicylinders
(Figure 4.2) rotated at an
angle
with respect to the horizontal inside the high pressure steel enclosure.
186
a
Before
analyzing the results
of such experiments, the
criteria for
the development of stable and unstable density profiles will be analyzed.
In the present context, stable signifies that density increases uniformly
in the
direction of gravity, and
vice-versa.
Thus,
a stable
profile
will not, by itself, lead to natural convection.
We
imagine a plane interface where fluid is saturated with solute,
which diffuses into the bulk solvent under the influence of a concentration gradient.
Let us denote the solute mole fraction by x,
and distan-
ces from the interface, measured along a line perpendicular to the interface, by E.
The molar volume, molecular weight
and
density at
= 0
are then
V(O)
M(O)
=
xl(O) V
+ [1-xl(O)] V 2 = V 2 [1+x
1 (o) (VI/V2 -1)]
(6.16)
x(O)
+ [1-xl(O)] M 2
(6.17)
M2
M
M2 [1+x(O)
(M1 /M2 -1)]
1) ]
[1 + x1 (O) (Mi/M2 -
P(O) =
(6.18)
V 2 E 1 + X1 (0) (V 1 /V
Away from
2
the interface
1)
(E
]
-
a),
on the other hand, we have pure
solvent
V(W)
V2
(6.19)
M(X)
M2
(6.20)
()
- M2/V2
(6.21)
For dilute solutions, such as we are presently considering (see Tables
6.5-6.8), we may assume
Va - V 2
(6.22)
f(x)
187
whereupon the density ratio between interface
simply,
and bulk fluid becomes,
1 + x(O)(M 1/M2 - 1)
p(O)/p(-) =
(6.23)
1 +
1
()(V
1 /V 2
-
1)
from which we conclude
p(O) > p(-)
(M 1/M 2 ) > V/V
<=>
Furthermore,
x(O)
(6.24)
2
in Equation (6.23) is just
Equation (6.24) is a general criterion,
i.e.,
a parameter, so that
whenever the solute to
solvent molecular weight ratio is higher than the corresponding partial
molar ratio, the local density is higher than the pure solvent density,
regardless
volume.
of the composition dependence of the solute
This criterion is valid for
partial
molar
dilute solutions, where Equation
(6.22) is an accurate description of reality.
Although Equations
(6.23) and (6.24) are useful,
the analysis
be pursued further to investigate the full density profile.
In
must
general,
we can write
1
dp
dp
.
-=(
p(-)
dE
dx 1
.
dx,
1
) .
(6.25)
p(*)
dE
where (dxl/dE) is a monotonically decreasing function of
If
state.
A
M/M
B
V/V2
-
2
we now define
- 1
(6.26)
- 1
(6.27)
and neglect the composition dependence of B
for
at steady
dilute
solutions
(which is a valid assumption
but, together with Equation
188
(6.22), needs
to be
modified otherwise), we obtain
1
A - B
dp
M'/M"
- V/V2
(6.28)
p(a)
2
dx I
(1+Bxl)
[
1 + (V,/V2 - 1) ]2
Thus, we conclude that, for dilute solutions,
M1 /M2 > V/V
2
> density decreases monotonically away from the interface
M1 /M2 < V/V
2
=> density increases monotonically away from the interface
Schematic profiles are shown in Figure
6.14.
For stable
profiles,
then, the interface should constitute the bottom of the rectangular duct
if the solute to solvent molecular weight ratio exceeds the partial molar
volume
ratio,
and
the
top
in
the
opposite
presently considered, stable profiles were
case.
In
all
of
the
cases
developed with the interface
at the bottom, and, consequently, the source plane constituted the bottom
of the rectangular duct; natural convection was introduced by rotating
the hemicylinders away from this equilibrium configuration (see Appendix
2 for detailed calculations).
The results of such experiments are shown, for benzoic acid diffusing
into CO0
2 at 160 bar and 318 K, in Figure 6.15 and Table 6.20.
The dotted line in Figure 6.15 does not extend to 0 ° since the diffusion experiment was done at a
different flow rate.
The
importance of
natural convection, as well as the potential for experimental error when
using hydrodynamic techniques,
can be seen from the fact
that a
650%
increase in the apparent diffusion coefficient results from a 900 rotation
of the flat plate.
6.4 SENSITIVITY
ANALYSIS
Detailed examples of an equilibrium and a diffusion calculation are
given in Appendix 2.
The accuracy of the measured diffusion coefficients
and the sensitivity to the
various sources of
experimental error will
be discussed in this section.
A diffusion experiment involves the measurement of r, the fractional
saturation at the exit of the test section
(this, in turn, implies the
determination,-in a separate experiment, of the equilibrium solubility
189
p/p(O)
B<O
A>B; B>O
I
I
FIGURE
6.14
: Schematic density profiles for a dilute binary mixture
in which a solute dissolves into the solvent from a plane
(E = O) of constant (equilibrium) composition
190
A
J%
f%
4U
I
30
E
U
(/
0
E
20
cx
0
U
O
10
n
'0
10
20
30
40
50
60
70
80 .90
a (degrees)
FIGURE
6.15
: Effect of natural convection on the apparent diffusion
coefficient and mass transfer rate : benzoic acid-CO2
@ 318 K and 160 bar
191
-
W
EI(I
a
a:
U
<
H
O
0
cro
t=;
Z H
0) 0
Z N
e z
i-
t-
I
zo
e
u
o
C[-
z
[.-,
C'2
O
03
U"N
H
O1 C
HI
:d
§
cr
ul rCt
z
X 0
03
n
CC
0
ZI
0
o
a
0 I.0
Ct
Z
0
o
1-I
o:
z-·
aH
rX
E
(a
*J
cu
D_
D~I
N:
ET
-cc
0
0
1-.
or~~o
a)~~~~~
CM ,MG)
X
..IL
co
E
o0
.
-
~
4
(1)
~~~~~~
CN
a
~~N
0- 0-4
C)
E
192
of
the
solute in the supercritical fluid at
pressure).
For
a given
aspect
ratio,
the same
r is a function
temperature and
of
the parameter
X,, defined in Chapter 5 as a modified inverse Graetz number,
X0 = L D / <v> b2
The relationship between r and X
shown graphically
length, <v>,
in Figure
is given by Equation
In
(4.65), and is
Equation (6.29), L is the coated
the mean fluid velocity, and 2b,
D is calculated from X,
tion
4.3.
(6.29)
the duct height.
Since
the sensitivity of D to errors in the determina-
of r is given by
dD
dD
dXo
dr
dXo
dr
<v>b 2
dXo
(6.30)
L
dr
from which we obtain,
din D
dln X
1
dln r
dln r
dln r
(6.31)
dln X0
The qualitative form of
without recourse
dimensionless
to algebra, once it
length,
so
unity asymptotically as X0
in Figure 5.3.
measured
the dependence of r upon X
that r must
+
.
can be deduced
is realized that X
grow monotonically
is simply
a
and approach
The initial part of this curve is shown
The important conclusion is that the sensitivity of the
coefficients to experimental errors
in the
determination
of
r grows without limits as r approaches unity, or, in other words, experiments should be conducted at the lowest possible value of X
(and hence
at low relative saturations).
From Equation (6.29) it can be seen that X
can be reduced by conduct-
ing the experiments in a short duct at high fluid velocities (the duct
193
height is fixed by practical design considerations and will not be considered a variable).
It will now be shown that there are physical constraints
which dictate the optimum operating conditions.
The optimum fluid flow rate is determined by the ability of the flow
regulating valve to maintain a constant flow under operating conditions,
i.e., under conditions where solute deposits are melted by contact with
the heated valve and pass through a tapered orfice, designed exclusively
In the present case, a maximum flow rate of
for fluid flow.
per
minute
at
ambient
conditions could be attained with
2 liters
satisfactory
controllability.
The coated length, on the other hand, cannot be reduced indefinitely
without altering the physics of the problem, i.e., without making axial
diffusion significant with respect to axial convection and vertical diffusion
from the order of magnitude
(this follows
(Section 5.3)
analysis
where it was shown that axial convection scales as L- , axial diffusion
as L- , and vertical diffusion as L ).
constraint; rather, a
However,
limited
This, in itself, is not a physical
different analytical solution would
be required.
the
possibility of
shortening the coated section is in fact
by a
very different
constraint: the duration of a run
becomes
impractically long at very low values of r, given the limitation on <v>.
As an example, consider an equilibrium solubility of 10- 3 mole fraction.
Then, at 15% saturation, the mole fraction at the test section's
outlet
is
1.5x10
,
which amounts to
for a 2 standard liters
solute molecular weight
1.34x10-5 solute moles
per minute solvent flow rate.
3
of 130, this means 1.74x10-
per minute
For a typical
g/min, or roughly
half an hour to collect 0.05 g of solute, an amount of solute that would
give rise to less than 2% error in weighing, (this follows from the reproducibility of the empty U-tube weighings, which was found to be approximately 0.001g).
Under typical conditions, r is roughly 18%, so the poten-
tial for accuracy
improvement by shortening the
coated
length
cannot
be materialized without on-line analysis (which would eliminate weighing),
since weighing requires runs that soon become impractical due to their
duration, when due account is taken of the absolute necessity of maintaining as constant a flow rate
as possible throughout the run
(it would
take 1.5 hours to collect 0.05 g at 5% saturation, for example).
194
With
the equipment described in Chapter 4, then, the minimum amount of solid
that must be collected for accurate weighing, and the maximum flow rate
that can be maintained with adequate controllability are the two factors
which
determine the accuracy
of the measured
7x10-5
If we now substitute typical values ( D
cm; <v>
0.1
cm/s;
L = 7.62
cm) we
obtain
X
Furthermore, the local slope is given by Ar/AX0
dln D/dln r
(r.AX 0 )/(X O .Ar)
diffusion
cm2 /sec; b = 0.15785
= 0.21,
-
coefficients.
and
r = 0.19247.
0.58, so that, finally,
= 1.56
(6.32)
or, in other words,
AD
Ar
1.56 1--1
--I
i
D
(6.33)
r
In Equation (6.33), Ar/r includes all possible sources of experimental
error
affecting the measured
value of r, in
both the equilibrium
and
the diffusion experiments, such as weighing errors, gas measuring errors,
etc.
The actual
in Appendix 2.
value
of IAr/ri for
a typical experiment is derived
In the present context, the objectives are the derivation
of Equation (6.31) and the calculation of the numerical coefficient in
Equation (6.32).
From Appendix 2, therefore, we obtain 10% as a conservative estimate
for
Ar/r,
which means that errors in the determination of r
uncertainties of
Equation
and X0
lead to
+ 16% in D.
(6.31) states the fact that, since D is obtained from X 0,
is linear in D, the relative errors in D caused by inaccuracies
in the measurement of r are simply the relative errors in X
same cause.
Equation
The final step
(6.29) for D.
due to the
in the determination of D involves solving
The determination of <v> introduces a new source
of error. The fluid velocity is obtained as follows (see Appendix 2 for
an example)
195
Moles of solvent
<v> =
Solvent molar volume
x
Run duration
In
Equation
(6.34)
Duct cross section
(6.34), the inaccuracies in the determination of the
amount of solvent gas have already been included in Ar/r, so we can write
A <v>
AT
<V>
T
T
<v>
where
AV
=+
T
V
(6.35)
V
is the run duration, and V, the solvent molar volume under experi-
mental (i.e., high pressure) conditions.
A typical run lasts
~ 15 seconds,
30 minutes; the timing has an uncertainty of
since (see Chapter
which steady flow is maintained
4) a run is started
after
a period
during
for at least 15 minutes, whereupon the
actual U tubes are connected and tightened, an operation that lasts approximately 15 seconds.
The solvent molar volume was determined from the International Thermodynamic Tables of the Fluid State (Angus et al., 1976) for C02 , and for
the Peng-Robinson equation of state
(Peng and Robinson, 1976) for SF6 .
The error in this case is less than 5% (this figure would be much higher
if runs had been done close to the critical point of the solvent; see
Tables 6.1-6.4 for actual run conditions).
From the previous discussion, therefore,
IA<v>/<v>I - 0.0083 + 0.05 = 0.0583 - 0.06
(6.36)
so that, finally,
AD
-
1.56x0.1 + 0.06 = 0.216 - 0.22
(6.37)
D
There are, in addition to the above, two sources of error that are,
however, very difficult to quantify.
196
In the first place; erroneous num-
bers result if the angle a (Figure 6.15) is not zero.
of the flat plate after a run was
Visual inspection
always conducted, and any diffusion
runs where the solute was etched at any detectable angle were rejected.
Although great care was taken to guarantee a
O0 under operating condi-
tions, experimental errors are inevitable and, though small, are difficult
to quantify.
In the second place, experiments were conducted inside a closed tube,
with no possibility for visual observation.
two
series
rotameters
(see Figure
The flow was monitored in
4.1) at atmospheric pressure, but,
except for the constancy of pressure and the fact that, at steady state,
the flow through the rotameters reflects the molar
there was no way
high pressure flow,
of quantifying the instantaneous constancy
flow.
197
of
fluid
DIFFUSION AND IRREVERSIBLE THERMODYNAMICS
7.
In the previous chapters, diffusion has been treated in a phenomenoThis characterization refers to these approaches
logical, or kinetic, way.
which view diffusion as a phenomenon resulting from gradients in concenFrom the point of view of irreversible thermodynamics, however,
tration.
chemical potential gradients, and not concentration gradients, constitute
the appropriate driving force for diffusion.
This represents an entirely
different interpretation of physical reality; moreover, as will be discussed
below, kinetics and thermodynamics predict opposite behaviour at mixture
critical points.
The
fundamentals and
some of the consequences of the thermodynamic
approach will be discussed in this chapter.
FORCES AND FLUXES
7.1
The integral rate of entropy creation
in a closed, isolated binary
system can be written as
d
dt
P sdV =
J
1
i
r
v
k
dV
(
TdV
k
-jl
·-dV
(7.
This relationship is derived in Appendix 5.
responds to viscous dissipation, and
The first integral cor-
ik is therefore a stress tensor.
The second integral corresponds to internal heat fluxes, with
heat
flux vector,
j,
a
species 1
mass flux, and p
a local
a mixture chemical
potential per unit mixture mass,
--
uM
M,
M2
(7.2)
M2
The third integral corresponds to diffusion.
is
simply
a
statement
of the fact that closed,
Equation (7.1), then,
isolated, macroscopic
systems approach stable equilibrium through irreversible processes involving
viscous dissipation, internal heat fluxes and diffusive currents.
198
Although viscous dissipation can be incorporated into what follows,
restrict our attention to the other two dissipative
we shall henceforth
processes.
dt
psdV
=
as
(7.3)
I [Z Ji Xi]dV
-
each conjugate J-X pair can be seen to represent the product of
where
a
written
(7.1) can be formally
Equation
flux
(J)
(7.1) and
and
a
force
driving
convenient
(X).
Thus,
from
Equations
(7.3), we can write
J
X
VP
+-
Mass flux (jl)
-- ("force")
VT
Energy flux (q In this context,
way)
or
iJl) +-+-
therefore,
diffusion
("force")
is linked
potential gradients, whereas
to chemical
(in an as yet unspecified
in kinetic, mechanistic
phenomenological approaches, this thermodynamic quantity is replaced
by concentration gradients.
We next address the question of the relationship between forces and
fluxes.
If the gradients are small enough, the problem can be treated
in the linear approximation, which, in its most general form, reads
J
i.e., each flux
=
LX
(7.4)
is a linear combination of all forces.
This implies,
in addition to the diagonal contributions, energy fluxes driven by chemical
potential
gradients,
and mass
fluxes driven by temperature gradients.
These phenomena have been experimentally verified (Dufour and Soret effects,
Onsager's
respectively).
reciprocity
theorem
(Onsager, 1931) proved
the symmetry of the matrix L (the "conductance", or transport coefficient
matrix),
aJ.
(i
J.
=
(7.5)
199
Considering
mass
and
energy
VFP
VT
fluxes
in
the
-
VT
light
of
Equation
(7.4),
we write
J1 = - A' T
-
VP
q -
j
6 =
ST
C
=
T
a V
B'
(7.6)
VT
- D'
= -
VP - Y VT
(7.7)
with
where
(7.8)
the last relationship follows from the symmetry of
the transport
coefficients.
The
- (7.8).
one
present analysis can carry us no
further than Equations
(7.6)
As is always the case with arguments based on thermodynamics,
obtains
useful
relations between quantities of interest, but not
actual values for the properties under study.
Before turning our attention
to the behaviour of the actual coefficients (a in particular), the profound
significance of
Equations (7.6) -
(7.8) should be appreciated.
We may
summarize this by stating that, when viewed as a particular case of dissipative processes,
- diffusion appears to be driven by chemical
potential (as opposed to
concentration) gradients.
- diffusion can occur as a consequence of temperature gradients and energy
fluxes as a consequence of chemical potential gradients
- the cross proportionality constants (off-diagonal terms in the conductance
matrix) are equal.
7.2
RELATIONSHIP BETWEEN THERMODYNAMIC AND PHENOMENOLOGICAL APPROACHES
If we consider, for simplicity, isothermal diffusion with no viscous
dissipation, we can write
Jl = -a
(7.9)
V
200
We next use the definition of
, the Gibbs-Duhem equation, and the definition
of fugacity coefficient,
1
VP
1=
0
fl
1
Ml VP2
Vp VP,
= x
V
= x1 P
(7.10)
+ (-x 1 ) V 2
(7.11)
1
(7.12)
to obtain
aRT
I[I+
On
alnx,
the
reported
other
as
gradients.
i
(ln)l
M1
hand,
T,P
[
x'
diffusion
proportionality
]
1
M2
(1-x)
Vx
-
coefficients
constants
(7.13)
are phenomenologically
between flux and concentration
Of the many equivalent expressions, we choose (Bird et al.,
1960)
J
=-
c 2 p~~~~
M1 M 2
VX
(7.14)
Equating the last two expressions, we obtain, after some rearrangement,
2
1/2]
+M
1-x1/2
M2 '1I
RT
[ '(J'
M~C
Sinceg
2
has units [Length2/time]
( ln 1i)
(7.15)
and a, [Mass x Time/Length3 ] it
is customary to define a "thermodynamic" diffusion coefficient that will
equal9)w2 in ideal mixtures, in which case (Modell and Reid, 1983)
A1
aln x1 T,P
= RT
(7.16)
or, in other words, when
is composition-independent.
Consequently,
CD
1 + (aln
am
=
12 an 12
)T,P
2
_ D1
RT
( ln
3=lx )T,P
(7.17)
or
1/2
D1 2 = M
c
I·.
(1-x 12
X,
'
M
M
2
( 1l-x,
x
201
) 1/2
(7.18)
Little
assumed
D
that
1977).
sition
be said
can
Taking
about
is
2
the actual
less
of D
2
composition-dependent
of a, that
or a.
than
It
12
is normally
(Reid
to the limit, we can now explore
this assumption
dependence
values
et
al.
the compo-
to satisfy
is, we force
2
1/212
1/2
aC(X1 )
[ (1-X)
c(x 1 )
+
XI
MS
(
M
)
>
f
(xI)
(7 19)
1-x1
or
-2
a Ix1/2
x)[
[
a = c(x1 )
In
other
I
+
XI
l-xM
M2
words,
1/2
)1 /2
f(T,P,
solute,
a concentration-independent D 1 2
solvent)(720)
(7.20)
imposes upon
the
requirement that if depend in x, as the dimensionless function
1
(x)
+
1
-
(V-
)dx
o
=
(l-xi)
(7.21)
2
1/21
1/2
X
M2
1-XI
where V is now a molar volume.
vanishes at x
Since the numerator is finite, we note that
and x
+ 1,
in agreement
with the fact that
- 0 and x
quantity at x
+ 1.
(and hence V)
-
0
is an undefined
The numerator in Equation (7.21) is very
specifically dependent upon the solute and solvent under consideration.
Moreover, it will yield different numerical values depending on the equation
of
state
used
Since, however,
in its evaluation.
the numerator is a
finite number, we can focus our attention on the denominator, a universal
function
is
shown
of
in
the
Figure
weight
7.1,
ratio and mixture
where
of x for various values of M/M
the
2
.
function
composition.
is
plotted
At a mole fraction x
Its
as
behaviour
a function
= M 2 /(M1
+ M),
W has a maximum value of M2 /4 M.
Thermodynamic
and
phenomenological
approaches to diffusion differ
in one fundamental aspect, which goes well beyond unit conversion.
202
For
=
-
-L
I
I
2.5
I
2.0
M2
1.5
A' 1.0
0.5
0.5
m
0.0
u
-
e--
L
r
i
I
-I
II
--
I
0.5
0.0
,
II
1.0
X1
FIGURE 7.1:
Universal part of the composition dependence of
weight ratios
203
for various
a
mixture,
binary
one of
the many
equivalent ways
stability
in which
criteria can be expressed is (Modell and Reid, 1983)
(N/N)RT
[1
>o
)(
+·
(7.22)
aln xl T,P
N
aN, T,P,N 2
When the term in brackets becomes zero, the mixture reaches its limit
of stability and a new phase is formed.
of stability.
Critical points are stable limits
According to Equation (7.13), then, diffusive fluxes vanish
at mixture critical points even when finite concentration gradients exist.
same
(the
would
be
true
for
limit
any
stability,
of
but
critical
only
points, being stable, make experimental verification simple).
The
vanishing
fluxes at mixture critical
diffusive
of
points has
been experimentally observed, (Tsekhanskaya (1968), Tsekhanskaya (1971)),
and the results have often been reported in the light of Equation (7.14),
that
is,
as vanishing
phenomenological
diffusion coefficients.
From
the previous discussion, it is obvious that phenomenological formulations
of
diffusion
this
coefficients
experimentally
(i.e., Equation
observed behaviour.
(7.14)) cannot account for
At least
in the limit where
vanishing thermodynamic driving forces coexist with finite kinetic driving,
forces, therefore, it appears that diffusion can be explained as an entropy
generating relaxation process rather than as a kinetic phenomenon.
7.3
THE KINETIC CONVERSION FACTOR
The behaviour of the conversion factor [1+(aln;l/alnxl)Tp], henceforth
called kinetic conversion factor, will now be analyzed.
It will be assumed
that the mixture under consideration can be described, in its equilibrium
properties, through a cubic equation of state, for which the most general
formulation is (Schmidt and Wenzel, 1980)(see Appendix 1).
PRT
V-b
a
V 2 + uVb + wb2
(7.23)
204
In Equation
(7.23),
quantity,
V is a molar
u and w are numerical
constants,
and they vary with the particular equation of state being used.
Mixture
parameters a and b are obtained from pure component parameters by means
of
suitable mixing
and combining rules, of which the most popular are
(with i and j denoting
a =
x.x.
1
it
and jth components,
respectively,
a..
1j
b = x.b.i
(7.24)
(a.a.)1 /2
a.
[1-k . (1-6.)]
where
ij
ij
(aia
1
ij
kij, the binary interaction coefficient, is the single adjustable
parameter
once
an equation
of
state
is selected,
and 6ij
is Kronecker's
delta.
The methods
pure component parameters are summarized
for obtaining
in Appendix 1, where tabulated values of v and w can also be found.
From the general thermodynamic relationship (where V is now extensive)
V
RTln
i =
R^ln
f
-[
-P
~ )T,V,N[i]
N~il
VRT] dV -
(7.25)
RTln Z
where N[i] means constancy of all mole numbers except for Ni, we obtain,
with
A = a P/(RT)2;
1
In q i =
(Z-1)
Aik
aik P/(RT) 2 ; Bi = biP/RT,
- In (Z-B) +
A
B/u 2 - 4w
2 k xkAik
A
and B = bP/RT,
B
(7.26)
Z
In
+
B (u-/u2 -4w)
Z+B ( 22
)
which is valid for any cubic equation of state with mixing rules as per
Equation (7.24) (provided both u and w are non-zero) and any composition
and
density-independent
in Equation (7.26)).
combining rule
(note that Aik
is not defined
From this we obtain, after considerable rearrangement
205
L1+(nx, ) Te
Z + B (-/U2-4w)
2
=2 +I-Z)
B4Vu2-B
TP
|nx,
l
[
+ B (u + / u 22- 4w )
2
2 (Bn+-ZJ
+(B,-B2)
-
+
xA
k k
(
j
k
ABI
B
B2
(nB -
Z)
- w) +U4( 2-J4w][Z+U
[Z+B
U-][Z+
2
][Z
2
(7.27)
where
c = (Al-A,2)B3
+ A(B,-B2 )B
{A1x1 (2B,-B2 ) - A 2B(1l-x,) + A 12 [B1 (2-3x,) - B 2 (1-Xl)]}B2
-
(7.28)
B
A+(uZ+2wB)[1-(Z-B)]2 -2[1-(Z-B)]
b-b2
A B
--BB)
(7.29)
A - (2Z + uB)[1-(Z-B)]2
(7.27)-(7.29)
Equations
are
only
applicable
to binary mixtures.
It is hardly worth emphasizing that these expressions are, in principle,
highly nonlinear in composition; in fact, x, appears not only in a nontrivial explicit form, but also, implicitly, through Z, A and B.
However,
for all of the dilute mixtures considered, the calculated kinetic conversion
factor is a linear function of x,
from infinite dilution to saturation,
and for all of the equations of state tested (see below).
Figure 7.2,
for C02 -Benzoic Acid at 308K and 280 bar (Peng-Robinson equation of state,
The significance of
k.. = .0183) is typical.
13follows
at once from integration
follows at once from integration
(alnx,)T,P
l(Xl) =
,
= 1 - K x,
this observed behaviour
(7.30)
(7.31)
exp (-Kx,)
where K is composition-independent, and
1
-
limrn
X1+
(7.32)
l (x)
206
I- I~~
1.00
I
I
U'
'I
I
I
I
0.
x
a
0.95
0-
.
0.90
I
I
0.000
0.001
r-
I
-
---
L
0.002
L
C
~I
LI
0.003
I
I
--·
I
I
0.004
x
FIGURE 7.2: Composition dependence of the kinetic conversion factor from
infinite dilution to saturation, as modelled by the PengRobinson equation of state, for C0 2 -benzoic acid, at 308
K and 280 bar
207
In
Equation
been substituted for the least-squares-
(7.30), 1 has
Also,
regressed y-intercept, which in all cases, was greater than .999.
0 at low values of x has been verified
the composition-independence of
an example, for C02 -Benzoic Acid,
As
in all cases.
-6
is 7.527 x 10
calculated
when x = 10
-10
and
the Peng-Robinson
10
-6
when
x
7.525
x
Since
the left hand
at 280 bar and 308K.
= 10,
an
K has
physical significance.
interesting
side of Equation (7.30) vanishes at limits of stability, we write
-
where x(i.s.)
at
unstable
k x
(s.)
=
(7.33)
0
is the solute mole fraction at which the mixture becomes
the given
T and P.
In writing
Equation
(7.33)
the assumption
of a linear behaviour up to the limit of stability has been made; this
is an obvious
idealization, but a useful one in the present context.
Equation (7.31) now becomes
l (xl,T,P) =
l
(7.34)
(T,P) exp [ - xl/x,(k.s.)]
Within the limits of the idealization implicit in Equation (7.33), therefore,
Equation (7.34) suggests a "natural" scaling of concentration, in analogy
with the idea of corresponding states.
Figure 7.3 is a plot of the fugacity
coefficient of benzoic acid in supercritical C02 (308K; 280 bar; kij =.0183),
infinite dilution
from
to saturation, calculated with Equation
(7.26)
and with the simplified exponential relationship (Equation (7.31)).
This
remarkable agreement was observed
(CO2 -benzoic acid; C0 2 -2-naphthol; SF6
Pr
3.4, 1.01
Tr
-
in all of
the cases tested
napthalene; SF 6 -benzoic
acid;
1.1; Peng-Robinson and Soave-Redlich Kwong equations
of state).
K-values for the C02 -Benzoic Acid system are shown
The regressed
The binary
in Figure 7.4 as a function of temperature and pressure.
used in Figure 7.4 were obtained by minimizing
interaction coefficients
the
sum
of
the
absolute
values of log[x(kij)] -
log[x(experimental)]
were taken from
The experimental solubilities
for each temperature.
Kurnik (1981). The resulting values (used in Figure 7.4) are: 0.013856
(@308.2 K), 0.010308
(@318.2 K), -0.003336
208
(@ 328.2 K), -0.01272
(338.2
K).
Figure 7.4 suggests that the fugacity coefficient becomes composition
independent at low pressure.
This
is in agreement with the concept of
infinite dilution fugacity coefficient, introduced above.
More interestingly,
though, K values approach a high pressure limit (at 280 bar, all K values
are within 6.5% of the mean).
If this behaviour is general, and K values
can be predicted or correlated, this could simplify high pressure phase
equilibrium
calculations, apart from the intrinsic theoretical interest
that such a trend would have.
K was found to be relatively insensitive to k.. in all cases.
For
example, at 308 K and 120 bar, for the benzoic acid-C02 system, K=66.81477
for
k..
=
.013856
and K = 68.13017
(obtained by regressing experimental
solubilities),
for k.. = 00.
1j
The fugacity coefficient, then, is the product of a composition-independent
term
(l,
the
infinite dilution fugacity
coefficient), and an
exponential composition correction which is not only small, due to the
small values of x,
but appears to approach a high pressure limit which
is independent of temperature.
209
I
I
I
~~I
I
I
I
I
I
I
7.4
7.2
<('0
o
7.0
6.8
-
I
0.000
I
I
0.001
0.002
I
0.003
I
0.004
X1
FIGURE 7.3: Fugacity coefficient of benzoic acid in C02 , at 308 K and
280 bar, as a function of solute mole fraction, calculated
with the Peng-Robinson
equation of State (EOS), and with
the exponential decay expression (K).
210
600
_
L
I
L
--1I -
I
I
I
I
I
I
I
I
JI
50
100.
150
200
II
400
K
200
0
I
0
I
IE
I__ - -i
-l
250
P (Bar)
FIGURE 7.4: K values for the CO2 benzoic acid system, as a function temperature and pressure.
211
DIFFUSION AND MASS TRANSFER
IN
SUPERCRITICAL FLUIDS
by
PABLO G. DEBENEDETTI
Ingeniero Quimico
Universidad de Buenos Aires
(1978)
S.M.
Massachusetts Institute of Technology
(1981)
SUBMITTED TO THE DEPARTMENT OF
CHEMICAL ENGINEERING
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
December 1984
Q
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
The author hereby grants to M.I.T. permission to reproduce and to
distribute copies of this thesis document in whole or in part.
Signature of author
....................
...
Department of Chemica lngineering
bember 1984
Certified by...................
··
..........
Professor Robert C. Reid
Thesis Supervisor
/f&4 4. 41
Professor Ulrich W. Suter
Thesis Supervisor
Accepted by .............................
...
.............
..........
*...
_
Chairman, Departmental Committee
NlvaO6FAaUUR I
iNSTITUTE
on Graduate Studies
OFTECHNOLOGY
FEB 1 3 1985
LIBRARIES
ARCHIVES
8
DYNAMICS OF INTERACTING BODIES : THEORETICAL FUNDAMENTALS
The dynamic simulation
knowledge
of
the
laws
of a system of
interacting bodies requires
of classical mechanics for
its implementation,
and of statistical mechanics for its interpretation.
These
will
be
discussed
in this Chapter.
The following sections
constitute a convenient summary, where the theoretical basis of the work
is presented in a
concise and coherent form.
outset, however, that
the discussion, although
It must be said at the
self-contained,
is
by
no means exhaustive, and references are given throughout to texts and
articles dealing with each of the topics in detail.
8.1 KINEMATICS OF THE RIGID BODY
A rigid body is an idealization.
as a set of particles such
is constant.
mics,
It can be defined,when discrete,
that the distance between
any two of
them
In what follows, molecules will be idealized as rigid polyato-
that is, as point centers of force with no
internal degrees of
freedom.
In general, a rigid body has six degrees of freedom, this being the
minimum number of independent quantities that must be specified in order
to define uniquely the position of the system under investigation (Landau
and Lifshitz,1982).
Three degrees of freedom pertain to the body's transla-
tional motion,and can be chosen as the coordinates of its center of mass
as measured from an origin
(x,y,z) fixed
in space
(the laboratory
or
inertial reference frame).
The other three degrees of freedom pertain
to the body's rotational motion, and describe the instantaneous angular
orientation o
a set of axes rigidly fixed to the body (x',y',z') with
respect to the set of axes fixed in space.
A possible choice of angular
coordinates, known as Euler angles(8,p,4), is illustrated in Figure (8.1).
A vector X is transformed from the fixed to the moving frame
(the
latter being rigidly fixed to the body) by the linear relation
A X = X
212
(8.1)
z
y
FIGURE 8.1 :
Euler angles
213
where A is an orthogonal transformation matrix, which , in terms
of Euler angles, becomes
A
=
[ cosncoso-cososinesin'-
cosnsin+icosicosqcose
-sin'csos-cospcossin
-sinsino+coscoscos
cosnsin
-sine cos~
cose
sine sino
sinosinO
(8.2)
The explicit form of A can be easily obtained as the product of the
,~ and A.
three elementary transformations defined by
The angular
velocities can be expressed, in the x ,y ,z system, in
terms of 4,$,O, as follows (Goldstein,1981)
w
+ e cosi
= $ sine sin
x
w
W
=
sine cosp - e sins
=
cose +
(8.3)
z
where,again,Equations
(8.3)
can be
considerations.Solving for 4,,$,
derived from elementary
we obtain
a = w , cos
w t sing
-
x
= w
geometric
y
,- (1/tge)[w , cosi + w
z
y
, sina]
(8.4)
x
4 = (1/sine)[w , cos$ + w , sinO]
y
The expressions for the rate of change of
x
and
diverge for small
0, making Euler angles unsuitable for numerical integration of the equations
of motion.
To overcome this problem,we introduce the so-called quaternions
or Cayley-Klein parameters (Goldstein, 1981;Murad and Gubbins,1978)
21"4
e
=
o
l
-coscos
2
]
2
= sin -cos [
el
(8.5)
e2
sine sin[
e3 = cos
"]
sin [j]
which satisfy
3
2
E ei = 1
(8.6)
i-o
Differentiating Equations (8.5), and after considerable rearrangement,
we obtain (Murad and Gubbins, 1978)
1
e =
(8.7)
ew
where
T
= [eo;e1 ;-e2 ;e 3]
(8.8)
; o]
(8.9)
wT=
w
,;w;w
x
-el
e =
eo
-e3
L-e
2
y
z
-e2
-e3
eo
-e3
e2
el
-eO
el
-e2
e1
eo
e3
1
(8.10)
From the definitions (Equations (8.5)) of the Cayley-Klein parameters,
we obtain
- (el+e-2e -e
2 3 )
A =
-2(eoe 3-e le 2)
2(e1e3+eoe
2)
2(eoe
2(e1 e3 -eoe2)
3 +el e2)
(e2+e2-e
-e
1
3
)
2(eoe1 +e2 e 3)
(e 3 +e-e
-2(eoe1 -e2 e3 )
2
-el2)
2(3-e2))]
215
(8.11)
Equation (8.7) is the quaternion equivalent of the Euler-based Equations
(8.4), but is singularity-free, and thus provides a convenient and consistent
kinematic
of the rigid body, which
description
can be
easily adapted
to numerical integration of the equations of motion.
If a rigid body has a shape that justifies idealizing it as a line,
the Euler angle or quaternion representation
cannot be
has only two rotational degrees of freedom.
line
used,
since
a
The implications of
this fact will be discussed below.
8.2 DYNAMICS OF THE RIGID BODY
The equations that describe the rotational motion of the rigid body
assume a particularly concise form when the set of moving axes (x ,y ,z )
in direction with the body's principal axes of
coincide
'
inertia, which
will now be defined (Landau and Lifshitz, 1982).
Assuming
the rigid
body to
be a discrete assembly
of masses,
the
j,kth component of the inertia tensor can be written as
',2
m[(xi) 6jk - xjxkJ
Ijk =
where m
denotes the generic discrete masses,
is the jth
over all such masses, and x
(8.12)
the summation
(j=1,2,3) coordinate
extending
of the
local
j
mass with respect to a set of axes fixed to the body but otherwise undefined.
The tensor defined by Equation (8.12) is symmetric and can always
be diagonalized.
The particular choice of
(x',y',z') that diagonalizes
the inertia tensor defines the principal directions; the diagonal elements
are the principal moments of inertia.
The kinetic energy then becomes,
KE= (m)v
--
2
2
+ 1
-
2
[ 1(w
216
1)
2
+
'
2 (w 2
2
) +
'
3 (w 3
)
2
]
(813)
(8.13)
where v is the translational velocity of the center of mass,
total
mass
of
the body,and
are
1,2,3
the
prin'cipal directions.
m is the
As
can
be seen from Equation (8.13), the rotational contribution is now formally
similar to the translational term.
The equations of motion can now be written , as follows. For translation,
in the inertial reference frame,
= (m)-lf
(8.14)
and for rotation, in the principal reference frame,
.9
Wi
where r
I
= (Ii ) - 1l Ki
(8.15)
is the position vector of the center of mass with respect to
the inertial reference frame,f is the external force,i denotes the ith
(i=1,2,3) principal direction,and w',K' and Ii are therefore the ith
i
principal angular velocity, torque and moment of inertia, respectively.
Equations (8.14) and (8.15) are general, and are obviously independent
of the kinematic description of the rigid body.
8.3 INTEGRATION OF THE EQUATIONS OF MOTION
The force f in Equation (8.14), and the principal torque
component
K' in Equation (8.15) are functions of the instantaneous position and
i
orientation of all bodies in the system. Integration of these equations,
therefore, requires expressions for the rate of change of the rotational
coordinates, which will necessarily be explicit in the coordinates themselThus, the expressions that will be introduced (the kinematic equations)
ves.
are necessarily less general than the dynamic equations ((8.14) and (8.15)).
For a non-linear body, with quaternion kinematics, we have
217
v = (~m)- l
f({r},{e})
(8.16)
x=v
i
= (Ii)-1 Ki({r},{e})
(i=1,2,3)
(8.17)
e =
1
e w
The translational equations have been written
as two coupled first
order equations, rather than as a single second order expression (Equation
(8.14)).
Also, in Equations (8.16), f({r},{e})
denotes the dependence
on all position vectors and quaternions of the force (the same is true
for the torque in Equations (8.17)).
It is
important to notice
that Equations
(8.16) are calculated
in
the inertial frame, whereas, in Equations (8.17), i denotes a principal
direction.
The torques are
calculated in the inertial frame
converted to principal torques using the quaternion-explicit
A (Equation
and then
form
of
8.11))
AK
K
Linear bodies have two rotational
(8.17) must therefore be modified.
(8.18)
degrees
of
freedom;
Equations
In this.case, the rotational equations
become
w
(I)-1 K
=
(8.19)
i-=w xl
where
I is now the non-vanishing principal moment of inertia and 1
is
a unit vector, co-axial with the linear body, which specifies the instantaneous configuration of the line.
218
Equations
(8.19)
are hence conveniently
do
not require a coordinate transformation, and
integrated
in
the
inertial
frame
throughout.
The structure of the rotational Equations for a linear body follows
from the fact that, for such a geometry,
I1 = 12
(8.20)
I3 = 0
w3 = 0
i.e., two principal moments of inertia are equal and the third one vanishes; also, a line cannot rotate about its axis, and w must always lie
on a plane perpendicular to the line's axis(Figure 8.2).
The actual
integration of Equations (8.16) and (8.17) (or (8.19))
can be implemented in many different ways, according to the particular
numerical
order
the translational equations, a second
algorithm selected.For
chosen, with
predictor-corrector method was
an
Euler
predictor
and explicit trapezoid correctors,
x (n+l) = x(n) +
t v(n)
v(n+1) = v(n) + (m)-1
(8.21)
(6t/2)[f(n+l) + f(n)]
(8.22)
x(n+1) = x(n) + (6t/2)[v(n+l) + v(n)]
where - denotes predicted values, and
translational and rotational
f
(8.23)
is calculated from the predicted
coordinates.
In Equations
(8.21)-(8.23),
the superscripts identify the integration step. For non-linear molecules,an
explicit second order predictor-corrector method was also chosen,
219
S
V1
V
at
,'1CW 2
V
V
V
J
WIV
V,
V,1
)
A %-i}
FIGURE8.2 : Kinematic and dynamic
body
characteristics of a linear rigid
220
e(n+ l )
=
e(n) + (6t/2)
wi
e(n)(w')(n)
(8.2 4 )
1
(8.25)
(n) + (n)
(6t/2)[(Ki)(n+l) + (Ki)(n)]
(i=1,2,3)
e(n+l) = e(n) + (6t/4)[e(n+1)(w')(n + 1) + e(n)(w')( n)]
(8.26)
where torques are first calculated in the inertial frame and then converted
to principal torques using the quaternion-explicit form of the transformation
matrix.
Predicted torques are calculated
from
predicted
coordinates
and then converted from the inertial to the moving frame with the predicted
transformation matrix.
For
linear
molecules,
an
implicit predictor-corrector scheme was
chosen,
l(n+ l ) =
(n) + 6t[w(n)
x
(n) ]
.(8.27)
w(n+l) = w(n) + (I)-1 (6t/2)[K (n+l) + K(n)]
l(n+i)
= l(n)
+ (6t/2)[w
(
n+ l) x
1 (n+
l
) + w(n) x
(8.28)
1 (n)]
(8.29)
where, as in Equation (8.25), predicted torques are calculated from predicted
positions and orientations.
Equation (8.29) is implicit as a consequence
of the vector product in the right hand side; 3 simultaneous equations
in
1 (n+1)
result.
8.4 STATISTICAL TREATMENT OF DIFFUSION
The stochastic approach to diffusion originated with Einstein's work
221
on
the
theory of Brownian motion
(Einstein, 1905).
Many
alternative
derivations of the basic relationships have since appeared, but Einstein's
approach
is still the simplest and most
general.
An illuminating and
comprehensive treatment of the general area of stochastic processes can
be found in Chandrasekhar's review article (Chandrasekhar, 1943).
We
2,
and
focus our attention on a mixture of particles of species
consider
the limiting
1 and
case where the concentration of species
1 is so small that we can neglect 1-1 interactions.
We consider a time
interval, AT, with the following characteristics: AT is small with respect
to observed, macroscopic time,
but large enough so
that the
velocity
of
independent
velocity
at a time
any
particle
at
a time
t is
from
its
t-AT,
<v(t+dA).v(t)> = <v(AT).v(O)>
=
O
(8.30)
If we now identify particles with molecules, and AT with a characteristic
time for a molecular interaction, then, upon observation at time intervals
ALIT,
the system must be described statistically rather than in a deterministic
way.
At
thermal
during an
equilibrium,
the displacement, 6, of type
interval AT, projected upon any
subject to a probability distribution,
arbitrary direction,
x,
is
1, which must satisfy
01( 6 x) = 1(-6x)
_1(6x)
1 molecules
d(6x)
(8.31)
= 1
where symmetry follows from a zero net flux condition, and normalization
implies that molecules must
be found somewhere in space.
The function
01 itself is defined in such a way that the number of molecules of species
1 that,
and
6
during
an
interval AT, experience x-displacements between
x + d(6x), is given by
dN1 (6x)
= N1
222
1( 6 x) d(6x)
(8.32)
6
x
where N1 is the total number of type 1 molecules under consideration.
Let
f l ( x,t) be the
z,y-averaged number of particles
per unit volume at a time t and location x.
of species
1
Then, we write
+00
I
fl(x,t+A-)
fl(X+6xt)
41( 6 x) d(6x)
(8.33)
-0o
Equation (8.33) assumes that the kinematic description of the system
at a time t + AT is independent of the system's history prior to time
t.
This is the single major assumption in Einstein's derivation, and,
in the language of stochastic processes, is equivalent to saying
that
the evolution in time of fl(x,t) is a Markow chain (van Kampen, 1981).
Expanding both sides of Equation (8.33) in Taylor series,
f l (x,t) +
fl AT +
+x
rfl(x,t)
+ fl
6
(6X)2+
+ 1 a
ax
2
...]
1(6x)
d(6x)
(8.34)
x2
and taking into account that
+WD
+O
Jfl(x,t)
1(6x)d(6x) = fl(x,t)
+
)
6
( 6 ) d(6x)
x
((x
14(6x) d(6x)
6
( 6 ) d( 6 x)
=
f l (xt)
(8.35)
0
(8+36)o
x)
we obtain
+X
at1
at
AT (6x)2 1(6x) d6x)
a2
a-x2
L2At
-m
223
(8.37)
Since
1 is a probability distribution, Equation (8.37) can be rewritten
as
a2 fl
<62 >
f
at
2MA
ax
(8.38)
2
Equation (8.38), on the other hand, is a species-1 conservation equation,
so we may at once write,
D12
<62>
x
(8.39)
28AT
or, taking into account that, since x is arbitrary and
2
2
<6x> = <6y >
2
1 symmetric,
<6 >
<6z>
3
(8.40)
=
<62>
D12
Equation
(8.41)
6-T
(8.41) is the fundamental relationship to be used in the
calculation of diffusion coefficients via molecular dynamics.
It is important to realize that at no point in the present derivation
was the assumption of Brownian particles (i.e., particles so large that
species 2 becomes a continuum relative to species 1) introduced.
Although
Einstein himself derived Equation (8.39) while considering Brownian motion,
the major
advantage
of this approach
is precisely the generality and
plausibility of the assumptions on which it rests.
The short-time (t <
with
this treatment.
AT)
An
behaviour of the system cannot be described
analytical equation covering
the
whole
time
range can be obtained for the special case of Brownian particles.
By splitting
the force on
a Brownian particle into a deterministic
frictional force, proportional and opposed
to the
particle's velocity,
and a stochastic component representative of random collisions with the
224
molecules that make up the continuum, (species 2) we can write a stochastic
differential equation (Langevin's equation)
v = -v
(8.42)
+ A(t)
from which we obtain (Chandrasekhar,1943)
<6> a [t - 1 - exp(-at)]
For t
(8.43)
A,
4
<
(8.44)
t
> a
O, expanding the exponential up to second order and simplifying,
and for t
d~>
a t2
(8.45)
It is obvious that, in writing Langevin's equation, we are introducing
restrictions which are not present
important
in the original
Einstein
The long and short-time limits, however, have a fundamental
treatment.
significance
that
is independent of
the assumptions built into their
derivation.
For a generic particle (molecule), we write
a2 6
as
6(t
t
-)
+
(-)
-
+
(8.46)
at2 o 2
at
For t
t2
O , we can drop quadratic terms.
Squaring, ensemble averaging,
and using equipartition,
<62> = 3kT t2
m
(8.47)
where m is the mass of the particles over which the averaging is done.
225
Without making any assumptions on the relative size and mass of the
particles under consideration, we can say that, for times which are short
with. espect to the characteristic inter-particle interaction time, the
dynamics is deterministic, since it can be described by Equation (8.46),
and then <62>at2.
Since
(8.46) is time-reversal-invariant, we
Equation
a reversible or deterministic regime.
can call this
Irreversibilities associated with
molecular motion, on the other hand, are characterized by <62> a t, and
we conclude that the onset of irreversible, stochastic behaviour, requires
amount
a finite
1
time,
characteristic of interparticle interaction
If species 2 can be regarded as a continuum, we can readily identify
times.
B-
of
with
At
(Chandrasekhar, 1943). Each degree of freedom, then, contributes
to the entropy over a time greater than a characteristic relaxation time.
Entropy
is meaningless
for t << A.
In Equation (8.42), and assuming the Brownian particles to be spherical',
then
is given by Stokes' law,
B6nna
m
(8.48)
where n is the viscosity of the medium, a is the radius and m, the mass
of the particles.
8.5 VELOCITY DISTRIBUTIONS
We define a distribution function by saying that the number of molecules
of species
i which,
at
time t, have velocities within
an interval dv
of v and positions within an interval dr of r, is given by
dNi(r,v,t)
fi(r,v,t) dr dv
(8.49)
where Ni is the total number of species i molecules in the system under
investigation.
At
equilibrium,
fi is independent of position and
226
time-invariant, so we write
fi(v)dv
ni
(8.50)
where ni is now referred to unit volume and the integral is three-dimensional
and extends over all possible velocities,i.e.
from -
to +
for each
component.
The particular functional dependence that corresponds to equilibrium
is the Maxwell-Boltzmann distribution,
fi(v)
=
ni [2T
For a fixed volume, V,
ex -
2kT
(8.51)
containing Ni molecules of species i, the
number of molecules having velocities within a range dv of v is given
by
dNi
dv
V fi
4
V v2 fi dv
(8.52)
or
[mi
dNi
dv
4
Ni
F2kT
2
3/2
3
m iv 2
v
(8.53)
exp[2kT-
which is maximized for
1/2
has
aroot
mean square velocty,
(8.54)
The distribution fi has a root mean square velocity,
1/2
Vrms
[
=
227
3i
1/2
1kT
/2 = <v2 > 1/2
(8.55)
a mean velocity,
<v
=
fNi j4fv fi dv =[
(8.56)
m-]
01O
and a standard deviation,
<
2 1/2
>
-1
=
[N
As will
iJ 0
2
4Nrv[v
1/2 =
2
-
[(
)3-8
1/2
f dv ]
<v>
i
(8.57)
,r
mi
be explained later, setting the root mean
square velocity
to unity is a convenint way of defining a simulation time scale (having
previously
defined
a simulation length scale; see Chapter 9).
Then,
Equation (8.53) becomes
dNi
dv
1.5
4(3/2)
2
1/2
Ni v exp(-1.5
2
v )
-
w
(8.58)
2
= 4.146 Ni v
exp(-1.5
2
v )
Equation (8.58) is plotted in Figure 8.3, with Ni
107, which is
the number of solvent molecules used in the simulations.
8.6 EQUIPARTITION AND VIRIAL THEOREM
For a system
of N bodies with
degrees of freedom per body, the
equipartition theorem (Huang, 1963) can be written as
228
100
50
zs
0
O.
1.
2.
V/Vrms
FIGURE 8.3 : Maxwell-Boltzmann velocity distribution; N-107
229
3.
H
Pi
> = kT
api
(8.59)
<q-P
H
> = kT
aqi1
where Pi and qi (i = 1,...,EN) are generalized momenta and coordinates,
and H is the system's Hamiltonian (i.e., its energy expressed as a function
For smooth spherical or point bodies with
of coordinates and momenta).
= 3 and q and p are simply the position
no rotational degrees of freedom,
For rigid bodies,
and linear momentum vectors, respectively.
= 6 and
q is a vector whose six components are the three cartesian coordinates
of the center of mass with respect to an inertial reference frame and
the three components of the rotation vector (Landau and Lifshitz, 1982)
along
the body's principal directions, while
components
is a
the cartesian components
are, respectively,
vector whose six
of
the
body's
linear momentum in the inertial frame, and the principal angular momentum
components.
Conjugate momenta and coordinates satisfy Hamilton's equations,
aH
aqi
i
(
= 1....N)
(8.60)
8p i
api
Restricting
'
qi
qi
(i = 1, ..... EN)
to the translational degrees
our attention
of freedom,
we consider the sum
3N
I
i1
qi Pi = r
(8.61)
3N
3N
+
= I qi Pi
I Pi qi
i=1
Time-averaging,
230
i=1
(8.62)
r dt
<r>lim
T'
= lim
1
~
T
T
T
r(T)-r(o)]
(8.63)
0
but, since r is not unbounded, we must have
<
= O
(8.64)
=>
which simply states the fact that thtetime average of a derivative of
a finite quantity vanishes.
Making into account Equations (8.64), (8.60), and (8.59),
3N
3N
Pi qi
<
i1
>
=
-
<
3N
1
H
>
qip
Pi
i-1
-3NkT
(8.65)
- 2<KE(t)>
where KE(t) is the translational kinetic energy.
The left hand side
of Equation (8.65) is the time average of the sum, over all bodies, of
the scalar product of external force and position.
This can be written as the sum of two terms,
3N
3N
Pi
< I
i > = -P
r.n dF + <
i=1
The first term is the contribution of
bounding
surface,
I
i=1
qi fi >
the collisions
(8.66)
against
the
not the coordinates of a molecule,
with r denoting
but a location on the bounding surface, n
is a unit normal vector, and
F denotes the bounding surface. The second summation, then, is the contribution of intermolecular forces, f.
We can transform the surface integral,
-231
-P
r.n
dF
-P
=
(V.r) dV
I
-3PV
=
(8.67)
and rewrite Equation (8.65),
3N
+ < C qi fi >
2 <KE(t)>
3PV
=
(8.68)
1=1
Dividing by 2<KE(t)>,
3N
<
3PV
qi fi>
1 +
2<KE(t)>
(8.69)
2<KE(t)>
or, equivalently,
3N
<
Z=
N
qifi>
<
>
=+ 1
1 +
2<KE(t)>
_-q
j=1
(8.70)
2<KE(t)>
If the interparticle forces are pairwise additive, the summation
can be expressed in a computationally convenient way,
3N
I qi
N
i
N
f
-
i=1
j=1
N-1
-
f ji
j.
j=1
i*j
N
X
=
i i
ij.(ai
-s
(8.71)
j-i+1
i.e., as the sum of the products of interparticle forces and interparticle
separations.
Equation (8.70) then becomes,
N-1
<
Z
=
N
I
Lj.
(qi
1 +
- j)>
(8.72)
2<KE(t)>
232
where fij is the net force on particle
is a vector directed from j to i.
i exerted
by particle
j, and (qi- j)
The double sum is therefore positive
when interactions are, on the average, repulsive, and negative when they
are attractive.
233
9.
9.1
COMPUTER SIMULATIONS; TECHNICAL ASPECTS
SAMPLE SIZE; BOUNDARY CONDITIONS; CUTOFF RADIUS
The goal of molecular dynamics
of matter.
is the study of the bulk properties
This is done by following the motion of an ensemble of molecules
and interpreting the "results" (i.e., the evolution in time of velocities,
forces, positions, orientations and torques) statistically (see Chapters
8 and 10 for fundamentals and results, respectively).
The first question that must be addressed, therefore, is the number
of molecules
that will
studying bulk matter,
as possible.
be considered in the simulation.
Since we are
it is obvious that this number should be as high
For pairwise additive, continuously differentiable potentials,
the simulation is simply a repetitive procedure whereby, at each step,
every molecule
"scans" all other molecules
in the system, and, due to
pairwise additivity,
F.
-1
where F
f..
=
ji
(9.1)
J
is the net force on the ith molecule, and f..
-1J is the force exerted
by the jth molecule on the ith molecule.
In the presently considered case, however, molecules are really rigidity
constraints, and the forces are
interatomic, so that Equation
(9.1) is
really computed as
Fi = I
j~i
where n and
molecule,
I
nEi
fn E=
now denote atoms.
each
f
integration
(9.2)
jfi -
tcj
Thus, for N molecules with n atoms per
step
involves, in principle, S elementary
evaluations
234
N(N - 1 )n 2
21
-
S
(9.3)
As will be explained below, the density of the simulated fluid has
a
strong
present
influence on the average duration of a
purposes,
or,
at
any
rate,
at
a
given
"scan", but,
density,
the
for the
duration
of
a simulation is a quadratic function of the sample size and of molecular
complexity, as measured by the number of sites per molecule.
The
other
factor
influencing the duration of a simulation
is the
integration step, which is determined by numerical accuracy considerations.
The run duration is only a linear function of the number of integration
steps.
Consequently, it
is the number and complexity of the molecules
that determines the size of the problem (given an event of fixed duration
to be simulated).
Even
to N -
with
0(103),
supercomputers,
with
n between
-11
events that last - 10
currently solvable problems
1 and 5, and simulations
are limited
are used
to study
seconds, at typical liquid densities (Ceperley,
1981).
In the present case, N = 108, with one solute and 107 solvent molecules.
For C02 as a solvent and C6 H 6 as, a solute, S = 54891 elementary evaluations
per integration step, or a total of - 1.7 x 108 evaluations per simulation
(for a 3000
step simulation).
The choice of appropriate boundary conditions is the next important
question that must be addressed.
A possible choice would be to enclose
the N molecules inside a perfectly reflecting rigid surface of arbitrary
shape.
This approach, however,
spherical
the
boundary of radius R and an effective
fraction
of molecules
general,
the
with
boundary
the
suffers from a major
fraction of
surface
interacting with
drawback.
interaction range
the surface
is 3Ar/R.
molecules that, at any given time,
is directly
For a
Ar,
In
interact
proportional to the product of
the effective interaction range and the surface-to-volume ratio, which,
in turn,
a
typical
is inversely proportional to some characteristic length.
molecular
dynamics simulation at
characteristic length can be estimated as
235
liquid-like densities,
For
the
1/3
1000 molecules x 5x10
-5
3
40 A
100
m /mol
6.02xl0 2 3molecules/mol
which means
is -
boundaries
molecules interacting with the rigid
that the fraction of
3x10 6
times
than
higher
in a
1 cm 3 macroscopic
system,
for any given effective interaction range.
This problem can be overcome by using periodic boundary conditions,
which have been used in molecular dynamics since the method was originally
1958; Alder and Wainwright, 1959).
(Wainwright and Alder,
proposed
In
this approach, computations take place in a unit cell which repeats itself
which enters
"image"
it is replaced by an
eaves the cell,
Whenever a molecule
in space.
This is shown
opposite boundary.
through the
in
Figure 9.1 for a two dimensional system with a quadrangular unit cell.
Thus,
periodic
conditions not only
boundary
influence of the boundary surface, but,
infinite,
presently considered case, the unit
is three-dimensional.
conditions,
a
assumption.
cell is a cube,
In
the
since the problem
Because of the periodicity imposed by the boundary
molecule
and
its images are never
moved
in considering the interaction between any
Instead,
the artificial
in effect, treat the system as
an arbitrary periodicity
with
although
eliminate
independently.
two molecules
(see Figure 9.1), only the closest of all possible a-b pairs must
taken into account (a-b4 or b-a5
a-b
be
in Figure 9.1; the choice between these
two is arbitrary and one may conveniently assign a number to each molecule
and, by convention, look at interactions between i and the possible images
of j, including
Periodicity
j itself,
thus
if i < j, and viceversa).
introduces an
important limitation:
because, as
explained above, a molecule and its images are not independent, the range
of the
intermolecular (or, in the present case,
cannot be infinite.
interatomic) potential
This follows from simple geometric reasoning.
particle a (Figure 9.1) and let the cell side have unit size.
Consider
Coordinates
can
be measured from any arbitrary origin: let us place the origin at
the
cell's corner nearest
and
horizontal
to a,
and denote vertical
coordinates by x.
236
coordinates by y,
Then, it is obvious that, whenever
I
I
I
I
I
a
lot
b
b
Ib
&,02
Vfr
V-r-
I
I
10_
4A4
b
b5
Il
lI
I
I
0I
I
---
FIGURE9.1 : Two dimensional
periodic
237
boundary conditions
1Xb -
Xal
> 1/2,
and
only
when
this
condition
is met,
an
image
b i of
b exists such that
IXb -
x a l > IXbi - Xal
(9.4)
Similar considerations apply to the y (and z, in three dimensions)
coordinates,
so we
conclude that, to avoid
double counting between a
molecule and its images, the range of the interparticle potential cannot
exceed half the cell side's length.
In the presently considered case,
a and b would represent molecules, (or, more specifically, their centers
of mass).
Having
interactions
located
the closest pair, and only then, are atomic
into account.
taken
Since, for N =
108, there
are 5778
pairs, and, in three dimensions, 27 possible images per pair, it follows
that an efficient algorithm for locating closest pairs without
calculating all
possible distances
is mandatory.
actually
Such an algorithm is
shown in Figure 9.2, for a cubic cell.
The actual cutoff radius is determined more conservatively than the
above
considerations
would suggest.
When simulating the
dynamics of
rigid polyatomics, the cutoff distance must obey
c
- ( a
(9.5)
b)
where the a's denote the maximum possible distance between a molecule's
center of mass and any one of its sites.
Thus, this conservative criterion
guarantees that the closest possible distance between atoms of molecule
a and atoms of molecule b, when
of
its
images, will
b has been discarded in favor of one
never equal the cutoff radius.
Since
any finite
cutoff distance represents a distortion of physical reality, the extreme
form of Equation
the expense of
(9.5) may have to be relaxed whenever a is large, at
introducing slight inaccuracies in the program.
In the
present work, a cutoff radius of 7.4 Angstrom was used throughout
(see
Figure 10.1 for molecular geometry).
fundamental about
There is, of course, nothing
the unit cell.
the cubic
shape of
It merely provides a simple geometric description, and
238
X1
FIGURE 9.2 : Nearest image location in a three dimensional cubic cell
239
a convenient start-up
are
arranged
in a
lattice "backbone": the molecular centers of mass
face-centered cubic lattice.
This structure can be
generated by spatial repetition of a unit cube containing occupied sites
in
four
different
positions
(Figure 9.3).
in a cube of unit size which contains 13
one depicted
32,
108,
in Figure 9.3.
256,...,
The
simulation takes place
elementary units such as the
Thus, molecular dynamics
413, ... etc. molecules are
common.
simulations with
The details of
the start-up procedure are discussed in Section 9.3.
Periodic and bulk matter are obviously not equivalent concepts, although
this assumption is built into the choice of periodic boundary conditions.
At present, however, there is no better approach to the problem of simulating
bulk matter using a small number of particles.
An interesting question
arises when we consider the possible influence of the unit cell's geometry
upon the results.
Although systematic studies have not been made, this
problem should be of particular concern when molecular dynamics is used
to study fluid-solid phase transitions, for example, where symmetry considerations play an important role.
Thus, one can imagine simulations taking
place in polyhedric space-filling unit cells, or, even more simply, in
deformed
cubic cells
(parallelepiped cells, for example; Theodorou and
Suter,1984).
In principle, one would hope the computed relaxation, transport and
equilibrium properties of fluids to be independent of these considerations.
This is strongly suggested by the good agreement with experiment obtained
so
far
with
molecular
dynamics, unless we assume
that cubic symmetry
is more than an accidental choice, and disordered, isotropic matter can
only
be
described
but fascinating
with
particular choices of symmetries, an unlikely
possibility.
TIME CONSIDERATIONS
9.2
In the area of molecular modelling of matter, computer time considerations
are
of
fundamental
importance.
Without an
efficient code, the
simulations quickly become prohibitively expensive, and, in extreme cases,
too long even in the absence of computer time cost constraints.
The core of a molecular dynamics simulation, as explained in section
240
X1
Xl
X2
X3
1
2
3
4
FIGURE 9.3 : Elementary generator of face centered cubic lattice
241
9.1,
is
the
"scanning"
procedure, which
is repeated N(N-1)n 2 /2 times
(Nn)2 /2 times.
Hence, all efforts to improve
per step, or for large N, -
code efficiency should be focused on this part of the program.
The
of the N(N-1)/2
optimization
nearest image scans was
already
Having located the nearest image,
discussed and is shown in Figure 9.2.
n2 site-site interactions must now be computed.
The first step, therefore,
is to check whether the particular site-site distance considered falls
Important
time savings follow
within
the cutoff radius.
sphere
scan (Figure 9.4) is implemented.
if the cube-
In this approach, the site-
site distance is only calculated if each of the components of the sitesite separation is less than rc
in absolute value.
This is equivalent
to constructing a cube of side 2 rc whose center of symmetry coincides
with one of the sites (one eighth of which is shown in Figure 9.4), such
that all sites falling outside its boundaries are rejected.
This cube,
then, is the geometric equivalent of a necessary but not sufficient condition
for a site-site separation to be
rc.
The sufficient condition, a concentric
sphere of radius rc, is then introduced, but only after having rejected
all sites which do not obey the necessary condition.
In addition, since
a distance is the square root of the sum of three squares, the sufficiency
condition is tested with respect to the square of the cutoff distance,
so that a rejected
site does not cause the unnecessary computation of
a square root.
The last
part of each elementary "scan" involves the actual calculation
of a force and, if needed, an energy.
(see Section 9.4) would make
expressions
the program prohibitively slow
To overcome this problem, the force and potential are
and inefficient.
tabulated
Explicit calculation of the resulting
at the beginning of the program.
There are as many
tables
as there are different types of site pairs (i.e., 0-0, C-C, 0-C for pure
CO02 simulations for
a sufficiently
per table).
fine
example).
The
range 0
< r
rc
is divided into
"grid" (typical simulations employ -
104
elements
The site separation is then assigned an integer value, I,
according to
(9.6)
I = INT (M*R/RC)
242
z
O,0, rc
Y
x
FIGURE 9.4 : Cube-sphere site-site distance test
243
where
M
is
the
number
of
elements
into
divided, 6000 in the present work.
interpolation is not necessary
which
the
range
0
r
rc
is
Since M is such a high number, linear
and the force and potential calculation
are therefore simply table look-ups.
To summarize, the N(N-1)n 2 /2 "scans" per integration step constitute
the rate-limiting step of the simulation, and have been optimized by
* logical nearest image search (i.e., with no algebraic operations)
(Figure
9.2)
* cube-sphere site-site test (Figure 9.4)
· potential and force tabulation
The influence of density upon the duration of a run follows directly
from
the
above considerations, once
cutoff radius,
it is realized
that, for a given
the number of sites within a sphere of radius rc scales
as the fluid density.
The higher the density, then, the higher the propo-
rtion of sites for which computations have to be made, or, in other words,
the lower the proportion of cube-sphere time saving rejections.
9.3
START-UP; RESCALING; RELAXATION RUNS
There are two different start-up modes:
ation.
placed
In
strict start-up, the
in a
velocity
face-centered
components
cubic
are assigned
START, in Appendix 4).
centers of
strict start-up and continumass of
lattice, and random
to them
the molecules are
orientations and
(see subroutines
PUT, TURN,
The magnitude of both angular and linear velocities
are the equipartition values, i.e.,
v
w
start-up
start-up
= (3kT/m) 1 / 2
(9.7)
= (2kT/I)1/2
(9.8)
where Equation (9.8) corresponds to the case of a linear molecule.
For
the general case,
[w
start-up
i
=
kT/I.)
1
1/2
C i = 1,2,3 )
(9.9)
where i denotes one of the principal directions.
is used most of the time
(i.e., for all of the actual simulations and
most of the "relaxation" runs).
and
angular
The continuation mode
Here, the initial configuration (cartesian
coordinates) and velocities
(translational and rotational)
are read from a file, generated at the end of
the previous run, where
this information is stored.
"Relaxation" runs are shorter (typically 500 to 800 steps) than the
actual
to
simulations.
reach
an
During a "relaxation"
equilibrium
state either
run, the system is allowed
from a
highly ordered condition
(strict start-up), or from the final state of a previous run at different
conditions (continuation).
Whereas
the
density
in any given simulation
a length scale (/simulation
is fixed by defining
length units) and the number of molecules
in the cube, the temperature fluctuates, since it is given by the kinetic
energy
<KE(t)> = 3NkT
(9.10)
2
= [2(N-1)
<KE(r)>
(9.11)
2
2 < KE(r) + KE(t) >
<kT>
where
+ 3] kT
-
(9.12)
(5N+1)
(N-1) linear solvent molecules and
1 non-linear
solute molecule
have been assumed, and KE(r) and KE(t) denote, respectively, rotational
and translational kinetic energy.
coincide
with
Since
there is no reason why T will
the desired value, velocities
(both linear and angular)
are repeatedly rescaled during a relaxation run, so as to force the system's
configuration to equilibrate at the desired temperature.
The rescaling frequency is set at the beginning of the run with ten
steps between rescalings a typical value.
as follows:
Rescaling factors are calculated
let T* be the desired temperature; then, the instantaneous
total kinetic energies will, in general, be different from their equipartition
245
values
2(KE(t))
3NkT
=
1
(9.13
=
1
(913)
2(KE(r))
3NkT
+
3
(9.14)
3N
Since kinetic energies are quadratic in velocites, each linear velocity
-1/2
, and each angular velocity component
component is rescaled by a factor a
is rescaled
by a factor
1/2
1/2
vi(n+l)
= vi(n)
(a)
wj(n+l)
= wj(n)
(1)
i =1,
...,
(9.15)
1/2
9.4
j = [1, ...,
2(N-1)
(9.16)
+ 3]
INTERATOMIC POTENTIALS
The
assumed
form of
the interatomic
(or intermolecular) potential
is an input to any molecular dynamics simulation.
Most
point
of
the
centers
symmetric
early
of force
potentials,
work
involved the
idealization of molecules as
interacting via pairwise additive,
such as the square-well
spherically
(Alder and Wainwright,
1959), hard sphere (Wainwright and Alder, 1958), or Lennard-Jones (Rahman,
1964)
models.
Three-body
interactions were
first taken
into account
(Haile, 1978) via a triple dipole Axilrod-Teller potential.
Two different approaches regarding intermolecular potential parameters
were followed in these studies.
In the first approach, no attempt was
made to simulate the behavior of any particular fluid; rather, the abstract
Lennard-Jones
models,
and
or hard
their
sphere fluids, for
properties
example, were
considered as
investigated (Verlet, 1968; Alder and
Wainwright, 1962; Alder et al.,1974).
These early idealizations, however,
were also applied to the study of specific atomic fluids (Rahman, 1964)
due to the inherent plausibility of the spherically symmetric, one-center
246
assumption for this particular case.
The
simulation
of rigid
polyatomic molecules
(i.e., point centers
of force with no internal degrees of freedom), requires the specification
of appropriate interatomic potentials.
of
the
In addition, choosing the shape
polyatomic implies abandoning the abstract
approach whereby
an
idealized fluid (i.e., Lennard-Jones, for example), is simulated in favor
of a more concrete and realistic description of a given particular fluid.
What
is gained in detail and predictive power
is lost in generality.
Although the specificity of the problem imposes severe restrictions upon
the
choice
of
interatomic
is, in this case, even more
parameter estimation.
potential parameters,
limited
than for
fundamental knowledge
intermolecular potential
Table 9.1 lists some of the approaches that have
been used to select appropriate potential parameters.
We may summarize the situation by saying that molecules without significant electrostatic effects are usually modelled as multi-centered polyatomics
with sites interacting via pairwise additive Lennard-Jones-type potentials;
parameter
selection is far from standardized, with
still widely used.
The lack of fundamental significance for the site-
site parameter values
et
fitting techniques
al., 1977) and C-C
is clearly shown from the fact that F-F (Singer,
(Murad and Gubbins, 1978) parameters had
to be
significantly altered (Nose and Klein, 1983) to fit the volumetric properties
of CF,.
Electrostatic effects are even more problematic.
Here, the choice
is between a more fundamental description involving localized point monopoles, or the use of multipole expansions.
The first of these approaches
(Rossky and Karplus, 1979) will be discussed below; apart from stability
considerations, long-range Coulombic forces are not,'inherentlyappropriate
for
use with periodic boundary conditions.
The
use of multipoles, on
the other hand (Murthy, et al., 1981) is a contradiction in terms, since
it assigns certain features to the potential on an a-priori basis, the
simulation of which is the very essence of the molecular dynamics approach.
In this work, site-site Lennard-Jones potentials were used, although
the
introduction of coulombic interactions, as will be discussed below,
247
0.
LA
00
b.-
0o
Co
C4
0
U)
0
o
-
0)
U)
C)
0
a,
C)
a.,U)
.0
b-
a)
bO
r-
(L)
C
bO
U1)
4)
o
0
oO
0
., 0
0
CX
Cr)
i-I
o
H
:3
C-,
0
a)
0:
0ou~
0
C-)
4)
,4
a)
0.
:5
0.i-I
H
03
.-
H
C)
o
C.-
C)
0
a0
0
0 ) >0
0
o
4o
C',
'o
C
0 C)
4)
a
4-4
)l
V
o
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OC
EH
40
C: C=
2 4H U)
.
affi
0
co
E--,
-o
i-
v)
0
0a
U)o
U) 4)
EC/)
0
a
)C
co
Ua)U
o0 - E
c'
c)
C
H
a) -H ) 0
o
O *H L U) U)
4)
)
I H
, .4O
a
I
c: a) U V U)
C'
C
oo
a) 10
L C
4
)U)
)4
, PL
V E L0
.
0
o
.H C
a
4U) U)
C- *H C0
c0 , C,L 0-i
C\M-4
0 H
248
CL.
U)
0
.s
-r0
a )·
4)
.,
O
,
0
(L)
c
a'
0 -HS
CO
I
a)
3
, q- 0
0
3 "
00
U) LO
oH
)L
I
L
-.
0 4) 0
4
U) U) Cl
· C, I
CM CL X
C -C C
4 C0
C',
aCLd'
0
U)
U) :
Ua))CL
4) 4) 4) a
-4
· -· -
o
) .
CM Y)
*·
t C1
OL.
-:.
C', -i
N C\Jm O..
C,
,
LN
C'
<
>
bO
-H
~
) C
was also studied.
patible
Since the use of periodic boundary conditions is incom-
infinite
with
range
potentials,. truncation
is necessary.
In
a simulation where all forces are conservative i.e., where
au
f =
with
(9.17)
U an appropriate potential, energy
f any force and
conserved except for numerical inaccuracies.
is inherently
Therefore, truncation
must always be done with this constraint in mind (unless the cutoff radius
is increased to a point where this effect is negligible, with the consequent
sharp increase in computation time).
The shifted force potential (Street
et al., 1978), shown schematically in Figure 9.5 and used in this work,
is defined as follows
F
m
=_dU
dr
+ dU
dr
r
r
(9.18)
c
F
=0
m
r > rc
r
(9.19)
c
Um - -JFm(r)
dr
(9.20)
where, for the Lennard-Jones case,
= 4[
F
=24
)12
2
6
-
]
(2
(9.21)
(9.22)
and therefore,
249
rc
m
rc
r
[r
r
r
]
o
rc
c
(9.23)
F=m21 a
Fm
2((r)
and Um,
(C)
c
1
r
both through
r
rc
r
3]
the definition
(9.24)
I
(Equations
(9.18) and
(9.20))
and through the explicit expressions (Equations (9.22) and (9.24)) satisfy
the energy conservation condition, i.e. Equation (9.17).
The Lennard-Jones force has a minimum of
1 /6
a
(9.25)
= 1.244455
7
whence
I Fmi n
=
I
2.396429
(9.26)
/a
so that a percent normalized force error can be defined
F1
iF
100
IFinI
1
= 1001.49
((a )
Fmini
-
rc
- 2 (
rc
)13
(9.27)
Similarly, we define a percent normalized potential error,
100
= 400 (a )
c
The
error
ith Equation
_ (a )
+ 6(r _ )
c
c
in the shifted force
(9.18),
)c
)
(9.28)
c
is independent of r,
and is only a function of a/rc.
250
2
in ageement
The error in the
SHIFTED
U
POTENTIAL
POTENTIAL
FORCE
FORCE
SHIFTED
(Streett
1978)
et al.,
al., 1978)
(Streett et
IUm
I
I
r
U
]
.)
I
I
Fm
246
FrLJ
Fm
- Ur + Ur
Fm
O
rar c
0or
r >rc
r > rc
r
Um m -fFm(r)dr
co
FIGURE 9.5 : The shifted force potential
251
(2
)13 (u)7)
potential, on the other hand, depends on r/rc as well as on a/rc.
The
9.6,
a
=
r/a
for
of
function
in Figure
and shifted Lennard-Jones potentials are shown
true
and
2.4,
r/a,
for
Equation
2.3
< r/a
is plotted
(9.28)
< 3.05,
covering
in
the
Figure
9.7
as
range
used
in
Equation (9.27) is shown in Figure 9.8.
the simulations.
As can be seen, very small errors result from using shifted potentials
2.4, and no discontinuities are introduced that would violate
for roia
Truncation does, however, give rise to computed compressi-
Equation (9.17).
bilities
which are
slightly higher
than would result from an
infinite
potential, since a weak attractive background has been artificially removed.
distribution
the radial
If
molecules,
spherical
for
function
taken
can be
as 1 for r > rc, then,
the attractive tail can be easily
introduced
by adding analytical correction to Equation (8.72), namely
long-range
where
n
is
4
compressibility
correction
molecules
or spherical
.
N
r
n (
a)
dr
(9.29)
ar
2(1<KE(t)>)
r
c
number
a
density, and
U is the
intermolecular potential.
An angular-average potential should be used in the above expression for
non-spherical molecules.
- U/kTd
fe
d4
Given the largely empirical way in which interatomic potential parameters
have been selected in the past (see Table 9.1), it was one of the objectives
of the present
parameters
1959;
work to use a
theoretical approach rather
to fluid properties.
Equation
The Slater-Kirkwood
(9.32)) allows the
252
calculation of
than to fit
equation (Pitzer,
site-site parameters
0.20
0.00
.. -0.20
D~~~~~~~~~
1.0
1.5
2.0
2.5
r/o
FIGURE9.6 : True and shifted Lennard-Jones potentials;
253
rc/o
=
2.4
MW
O
0o
erroror poteial
e e
FSUs-,tro
de
er
.Ifted force ot
II
I
I
I
I
I
3.0
3.5
40
00
20
A
v
-.0
1.5
I
,
I
2.5
2.0
.
rc / o
FIGURE 9.8 : Normalized force error; shifted force potential
255
for
the
general
additional
condition (Equation
at a distance
a
ij
r
Lennard-Jones
interaction between sites i and
; an
(9.33')) imposes the minimization of
equal to the sum of the van der Waals
Uij
radii
C
12
r
~~~~U
6
iS
1_3.i~~~~~~~~~~~(9.31)
(a)
as follows,
365
ij
a.
a.
1/2
(9.32)
2 1J (rio
io + rj3,0)
(9.33)
1/2
+ ()
C..
aij
ij
is the polarizability
where a
in ]P, ro
is the van der Waals radius,
in , and N is the outer shell effective number of electrons. From Equations
(9.21) and (9.31) we obtain
a..
1/6
(9.34)
id)(C
C2.
(9.35)
Eit 44ai
ij
a..ij
The Slater-Kirkwood approach has been widely used in the molecular modeling
of polymeric materials (Suter, 1979).
From Equation
(9.31) it is obvious that the method assumes, at the
outset, a form for U...
This represents a problem for molecules where
13electrostatic
effects
are important
since there is no theoretical way
electrostatic effects are important, since there is no theoretical way
256
of
these effects
decoupling
and van der Waals-type
to yield rigorous, a-priori electrostatic
site-site parameters.
For example, knowing the
experimental quadrupole moment of CO 2 does not mean that one can model
this
by superimposing
molecule
to a
the corresponding point monopoles
type potential with Slater-Kirkwood parameters, since the
Lennard-Jones
latter already incorporate electrostatic effects through the polarizability,
albeit in an oversimplified way.
This, of course, means that the explicit
inclusion of electrostatic forces with experimentally measured parameters
cannot, at present, be done without simultaneously fitting
the Lennard-
Jones parameters so that the overall fluid behavior reproduces some arbitrarily
selected property (generally pressure). For C02 -benzene simulations,
the site
in
parameters for use
Table
(9.33) are
listed
calculated Lennard-Jones parameters are listed
the
The length parameters a were reduced by .15
in Table 9.3.
lations.
and
9.2,
in Equations (9.32) and
in the simu-
this change are treated in Chapter 10, where
The effects of
the actual results are presented and discussed.
Apart from its linearity,
The simulation of CO2 is not a simple problem.
which requires
implementation of
the
a different kinematic description
than for the generic rigid polyatomic, this molecule has a
moment
quadrupole
(Murthy et al. 1981).
significant
In the present work,
it was
attempted to take this into account by introducing appropriate point
located
monopoles,
at
the Lennard-Jones sites.
Severe numerical problems
originated as a consequence of this, and they are discussed in Chapter
10. Here, the general aspects of long-range interactions and their molecular
dynamics implementation will be treated, with CO2 as the specific example.
The measured quadrupole movment of CO2 (Buckingham and Disch, 1963)
is
-1.43448
effect
that
corresponding
x 10- 3 9
Cm 2 ; this represents a considerable electrostatic
should be taken
into consideration
value
-4.67
for
a C-O site separation
N2
is
x 10
(as a comparison, the
Cm 2 , or
307%
smaller).
For
= 1.23 A, the quadrupole moment corresponds to
partial charges (in electronic charge units)
Z C = + .5912
(9.36)
Z
= -
.2956
257
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o
co
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4r
m
Ln
t-
Ln
0
Z
E-.
00'
0
C
Ct:
!
t:
fcc
E
_
O
a
-
D.
e
0
.- 4
E
O
O
L)
0
O
O
L
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U
258
- - -
*
CCj
N
0co
a
0r
m
I
Cj
0
0
0C)_
·-
E
0
I-cc
M
E
0 0
L.
z
m
rvr
M
0
C
u
0
a.
E-
InIu
Ln
r.O
El
U)
0
Oql
lid
,.-I
C-,
:
0
-
D
C-#
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m
-L
0
_
0
CC
0
'
W
aI,
E,-
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E
0'
-4
t-
'0
oo
'.0
Ln
0)
N'
Ln
a)
C
O
.-
11
O
_I
0U
co
o1)
r.3
z
O
LA
C
0
CL
ko
0
Y
O
CCU
Y
Uo
O
*
Mf)
*
0' '.o
0
m
0
co
0s
0A
N
'.
L
C\
t'-
a ,-I
.0
L)
N
pc
L^)
_v
*
0
*
l% -
.0*
*
m
ID
:D
-
-
I
a)
L
.
0
o
.- 4
to
E
E
o
0
0
:
o
-
I-
0
1-1
1-I
L)
C)
u)
o
0
L0
I
0
I
W
'
*
259
It is not a-priori necessary to locate these charges exactly at the
However, not doing so requires the
sites.
Lennard-Jones
introduction
"switching function" (Stillinger an Rahman, 1974) that
of an arbitrary
in such a way that opposite charges
modulates the electrostatic forces
do not collapse on each other.
The electrostatic contribution to the intermolecular potential energy
2
such as C0 2 , is given
molecule
for a three-site
3
3
· ,,,1-
Z
by
Z
E'
0
i=1
j=1
(9.37)
ij
where e is the electronic charge, and Co, the permittivity of free space.
e
= 1.603592
C
= 8.854 x 10 -
(9.38)
x 10 19 Coul
12
J-1 Coul 2 m-1
(9.39)
To study the relative importance of Coulombic interactions, we concentrate on two CO2 molecules, whose relative positions are defined in Figure
9.9.
Figures
9.10
to
9.12
show the
total, Coulombic and Lennard-Jones
energy, as a function of the carbon-carbon separation for different relative
orientations.
The energy has been normalized with kT, with T = 300 K.
The absolute value of the total, Coulombic and Lennard-Jones forces are
shown in figures 9.13 to 9.15, as a function of carbon-carbon separation,
Force has been normalized with kT/1,
for the same relative orientations.
with
= 1.23
, and T = 300 K.
The normalized Coulombic and Lennard-
Jones components are shown in Figures 9.16 to 9.21, for the same relative
orientations.
The
force components represent
forces exerted
I, and are referred to the (x,y,z) coordinate set (Figure 9.9).
parameters used correspond to Equation (9.36) and Table 9.3.
260
by J on
Potential
As explained above, van der Waals and electrostatic potentials cannot
be superimposed without altering the parameters; however, the conclusions
that follow from Figures 9.10 to 9.21 will be used in a qualitative sense
and are valid for any realistic set of parameters.
The most
important conclusion to be drawn from Figures 9.10 to 9.21
is
the virtual
a
conservative
vanishing
of
pairwise
figure; as can be
are very weak already at r - 7.5
tion.
).
interactions
seen from
for
r > 9
the Figures,
(this
is
interactions
This fact is independent of orienta-
Effective fields and forces around any given molecule would vanish
even more rapidly in the presence of more than one molecule, but it
s
remarkable that this feature is already clearly displayed for the elementary
binary interaction.
It follows immediately that, because of this cancellation, the effective
electrostatic potential between two multi-site polyatomics is short-ranged
and
can
be
modelled
with periodic
boundary conditions.
This is done
by defining a shifted site-site coulombic potential,
e
U
Z.Z.
1 j(
=
1
1)
r
r
(9.40)
U
m
=0
r > r
c
and the corresponding force,
e2Z.Z.
F
m
=
1
4we0
r
1_
2
c
(9.41)
F -0
m
r> r
c
this potential gives the correct force for r
(9.17).
<
rc, and satisfies Equation
An unshifted Coulombic potential truncated at rc, on the other
hand, gives rise to an infinite (impulsive) force which is totally unphysical
and which, if overlooked, can lead to severe errors in energy conservation.
A
shifted force potential can also be defined; as with van der Waals
261
forces, such a potential would satisfy Equation (9.17) and hence energy
conservation.
used.
(9.40) and
(9.41) were
10.
Another
the
the present work, Equations
The results of the inclusion of electrostatic forces are discussed
in Chapter
is
In
interesting feature that emerges from Figures 9.10 to 9.21
qualitatively
different behavior of electrostatic and van der
Waals interactions with respect to molecular orientation.
value of Y and 6, changes in a and/or
For any given
give rise to important changes
in the electrostatic part of the interaction, whereas the van der Waals
contribution
is much
less sensitive to orientation.
This
can be seen
for the energy (Figures 9.10 to 9.12) and for the cartesian components
of
the
force
(Figures 9.16
of
the
difficulties
to 9.18).
This is a possible explanation
encountered in simulations where
were included.
262
coulombic forces
O
z
0
J
C
0
FIGURE 9.9
Kinematic description of two linear molecules
263
I
I I
I
1.0
10
0.5
5
0.0
w
MJ -0.5
z
0
-1.0
-5
I
i
I
_
5
10
15
5
C-C DISTANCE (Angstroms)
10
15
C-C DISTANCE (Angstroms)
r
I
I I
6
2
4
O
0
0
0u
2
0
-1
I
5
10
C-C DISTANCE (Angstroms)
FIGURE 9.10
:
I
5
15
I
. I
10
15
C-C DISTANCE(Angstroms)
van der Waals, coulombic and total
CO2 interaction
264
energy for a binary
!
5
10
5
15
I
'
I
0
10
-2
5
0
I
-4
i
lI
'I
5
10
5
15
9.11
:
10
15
C-C DISTANCE (Angstroms)
C-C DISTANCE (Angstroms)
FIGURE
15
C-C DISTANCE(Angstroms)
C-C DISTANCE(Angstroms)
I
10
van der Waals, coulombic and total energy for a binary
CO2 interaction
265
Ir
I
I
I
I
10
l
I
Im
10
5
0
U,
0
i
-5
i
*
5
I
10
5
0
-5
II
15
I
I
5
I
10
-
l
15
C-C DISTANCE (Angstroms)
10
15
C,
w
z
iU
0
10
0
10
5
15
C-C DISTANCE (Angstroms)
C-C DISTANCE (Angstroms)
FIGURE 9.12
:
van der Waals,
C02 interaction
coulombic and total
266
energy for a binary
I
4
I
I
a
3
4
2
2
1
0
0
5
10
-!
5
15
10
15
C-C DISTANCE (Agstroms)
C-C DISTANCE(Angstroms)
10
1.5
w
U
5
1.0
0.5
0.0
0
5
10
15
5
C-C DISTANCE (Angstroms)
FIGURE
9.13
:
Absolute
10
15
C-C DISTANCE(Angstroms)
value
of van der
Waals, coulombic and total
force for a binary CO2 interaction
267
1.0
1.0
u.
0.5
0.5
0.0
0.0
10
5
10
5
15
15
C-C DISTANCE (Angstroms)
C-C DISTANCE (Angstroms)
20
6
15
4
10
0U.
2
5
0
0
6
10
5
15
C-C DISTANCE(Angstroms)
FIGURE
9.14
:
Absolute
10
15
C-C DISTANCE (Angstroms)
value
of van
der Waals,
force for a binary C02 interaction
268
coulombic and
total
uJ
100
100
50
50
0
0
5
10
15
5
C-C DISTANCE(Angstroms)
C-C DISTANCE(Angstroms)
I
II
.
15
10
I
II
60
3
40
I
20
2
1
0
0
_
5
15
5
C-C DISTANCE (Angstroms)
FIGURE 9.15 :
.
,
10
Absolute
.
J
10
15
C-C DISTANCE(Angstroms)
value
of van der Waals, coulombic
force for a binary CO2 interaction
269
and total
0.0 I
'
I'
10
-0.5
Il
0cc
0IL
5
-1.0
-1.5
0
I
5
10
I
15
C-C DISTANCE (Angstroms)
I
II
10
15
I
5
C-C DISTANCE(Angst oms)
P
.
.
!
0
0
!
-1
-1
wu
w
0
-2
a .45
-
450
-2
-3
6 - O0
-4
l.
5
10
15
I
5
!
!
.
10
C-C DISTANCE(Angtrons)
C-C DISTANCE (Agstrons)
FIGURE 9.16 : Coulombic force components for a binary C02 interaction
270
.
15
I
I
l
I
0
0
-1
-1
w
0
isw
0
-2
-3
-2
-3
1
.. I
I
5
10
15
5
C-C DISTANCE(Angstroms)
10
15
C-C DISTANCE(Angstroms)
4
I
1
I
10
2
5
0
-2
I
5
10
15
I
,
5
10
C-C DISTANCE(Angstroms)
C-C DISTANCE(Angstroms)
FIGURE 9.17 : Coulombic force components for a binary CO2 interaction
271
.1
I I
15
10
10
&U
U
g
5
0
5
0
5
10
15
5
C-C DISTANCE(Angstoms)
10
15
C-C DISTANCE(Angstroms)
I
0
I
1
10
-2
g
w
)
C)
5
-4
-6
0
IX
5
10
15
5
C-C DISTANCE (Angstroms)
FIGURE9.18 : Coulombic force
10
C-C DISTANCE(Angstroms)
components for a binary
272
I
|
CO2 interaction
15
1
0
0
-1
-2
-2
-4
5
10
15
5
10
15
C-C DISTANCE (Angstroms)
C-C DISTANCE (Angstrom)
I
IV
1.0
0
YZ
0.5
at. gO
If
Iaa . ore
7 .*o0
-5
_
5
di
"
I
0.0
e'
10
5
15
10
15
C-C DISTANCE
(Angstroms)
C-C DISTANCE(krgrom)
FIGURE9.19 : Lennard-Jones force components for a binary C02 interaction
273
I
i
I
·
-
0.5
0.0
~~~-
·
l
0.5
0.0
-
is
a .900
0
Lu
0.00
. 450
'
-0.5
6 450
5
10
15
5
C-C DISTANCE(Angstroms)
I
~
~
450
6 450
10
15
C-C DISTANCE (Angstroms)
.
200
L
I
I
Ij
0
0
ul
-1
0
-2
6*
450
-200
· 45
a - 45e
-3
I
5
10
15
C-C DISTANCE(Angstromsa)
I
I_
I
5
10
15
C-C DISTANCE (Angstroms)
FIGURE 9.20 : Lennard-Jones force components for a binary CO2 interaction
274
1
0
I
0
I
-50
w.
z
_1
-1
-
a~·90°
-
6 89'54'
-2
-100
5
15
10
5
I
I
.
I
I
I
I
I-Tl
0
a 90
-20
15
C-C DISTANCE (Angstroms)
C-C DISTANCE(Angstroms)
0
10
uJ
,
-40
6 -8954'
-50
-0
-
5
10
l00L -
J
5
15
10
15
C-C DISTANCE(Angstroms)
C-C DISTANCE
(Angstrom)
FIGURE 9.21 : Lennard-Jones force components for a binary C02 interaction
275
10: MOLECULAR DYNAMICS SIMULATIONS:
Unless
specified
otherwise,
correspond to simulations. of
107
C02
and
RESULTS AND DISCUSSION
the results reported
in this Chapter
the motion of 108 molecules, representing
1 benzene molecules,
respectively
Interatomic potential
parameters are listed in Table 9.3, and the molecular geometry is shown
in Figure
10.1:
10.1.
VELOCITY DISTRIBUTIONS
Velocity distributions for the 107 solvent molecules were calculated
by counting the number of molecules within a certain velocity interval
at
different
"snapshots".
instants
("snapshots"), and averaging
over the number of
The velocity range considered was O(<v*<3,where v* is expressed
in units of root mean square velocity (v
rms
)
1/2
v
= v
")
(10.1)
3kT
and T* is the run's nominal temperature, i.e., that temperature towards
which velocities were rescaled during the relaxation run.
The
velocity
range was
divided into 20
intervals.
Molecules with
a velocity falling anywhere within a given interval were assigned a nominal
velocity equal to the mid-point velocity.
of 7.5% in v
This implies an uncertainty
units.
In this way, the incremental number of molecules per unit velocity
interval
(AN/Av)
can be plotted against molecular velocity,
and this
can be compared to the Maxwell-Boltzmann expression (Equation (8.58)).
The results are shown in Figures 10.2 to 10.7.
In each case, the
actual number of solvent molecules (107) was used in the Maxwell-Boltzmann
expression
(Equation (8.58)); this normalizes all curves
that the total area equals 107.
in such a way
The number of "snapshots" is indicated
in each case, as well as the run's
average translational temperature,
i.e.,
< T(tr) > = 2 < KE(tr)
>/3Nk
276
(10.2)
C
O
H
H
H
12
H
H
H
FIGURE10.1:
Geometry of benzene and C02 used in the simulations
277
I
-I
I
I
I-
I
e. 25
<T > - 309.3K
0. Be
0.15
E
Z
'0
0.10
't3
0.00
I
I
0
0-C
L
- I
see
I-
I
1000
v (m/s)
FIGURE 10.2:
L
Theoretical and computed velocity distributions;
<T(tr)> - 309.3K; sample size = 64
278
I
0.3
U.
0.
1
I
I
- 310.3K
Ca
E
Z >crI+0.1
0.0
I
I
I
0
I
___·_
I
--
I
I
eee0
se500
v (m/s)
FIGURE 10.3:
Theoretical and computed velocity distributions;
<T(tr)>
=
310.3K; sample size
279
-
64
U
I-
II
I
I
0.as
<T > - 318.6K
09.20
0.1
E
aI
0.10
1 0>
e.es
0.e0
-
I
I_
0
_
_
I
·
S0
_
,
,,
I
lee.
1000
v (m/s)
FIGURE 10.4:
Theoretical and computed velocity distributions;
<T(tr)>
-
318.6K; sample size
280
-
67
0.2
E
7,
0
'0 >
Z
0.1
0.0
50ee
e
eee I
v (m/s)
FIGURE 10.5:
Theoretical and computed velocity distributions;
<T(tr)> - 321.4 K; sample size - 70
281
I
I
e as
0. 20
Mlmlm-
I
I
A
- 337.4K
0.15
E
V
co
O. 10
Z >
go '0
0.05
0.00
I
0
I
_
-I
I
JI
-
I
leeO
v (m/s)
FIGURE
10.6:
Theoretical and computedvelocity distributions;
<T(tr)> - 337.4 K; sample size - 67
282
,
I
·--··
I
I
E. 25
e.ie
0.115
I
<T > - 342.3K
-
g}_
qP6
.10
!D
'~V
0. 00
w
I--
0
-I--
-b-L
I
see
I
·
I
---
I-
0lee
v (m/s)
FIGURE 10.7:
Theoretical and computed velocity distributions;
<T(tr)> - 342.3 K; sample size = 64
283
where N = 107, tr denotes translation, and KE, kinetic energy.
The abscissae
corresponding to interval midpoints were dimensionalized with the nominal
root
mean
square
(i.e., velocity
velocity
corresponding to the run's
nominal temperature), since this was the actual velocity scale used during
simulation.
the
the Maxwell-Boltzmann curve, on
For
the other hand,
the run average translational root mean square velocity was used.
The
curves represent, then, the theoretical and "experimental" velocity frequency
distributions corresponding to a temperature <T(tr)>.
The agreement with theory is very good in all cases.
As an example,
10.2, the run's nominal temperature was 328.2 K, which, for
in Figure
CO2, corresponds to a root mean square velocity of 431.34 ms
intervals for this figure,
64.70
ms
.
The maximum
therefore, have
difference between
to
molecules
1.5
(3/20) vrms, or
the Maxwell-Boltzmann and
(in dN/dv units) is -
the computed frequency distribution
corresponds
a width of
; the velocity
(i.e.,
(AN/Av)
Av),
or
1.4%
0.024, which
the
of
total
number of molecules.
Non-zero computed dN/dv values span a velocity range which can vary
from
32.5
< v < 1068
In
10.6).
ms -
1
(Figure
10.2)
to
32.5
any given simulation, therefore,
< v < 1262 ms -
1
(Figure
molecular velocities vary
by factors of up to 30 or 40.
In the present case, with an integration step of 10
of
32.5
ms
-1
corresponds to 3.25 x
-4
5
sec, a velocity
0
10
A/step, which represents
correspo5 s to 3.25 x 10
4.4
x 10-5 of the cutoff radius (7.4 A), or 26.8% of a potential tabulation
length unit.
-1
For a velocity of 1262 ms
-2
responding
values
potential
tabulation length units,
can
be
seen
are
that
1.26
x 10
, on the other hand, the cor-
o
-3
A/step,
1.7 x 10
repectively.
integration accuracy
, and 1040%,
or 10.4
Numerically, then,
it
(i.e., energy conservation)
is
favoured by the statistical irrelevance of high energy molecules.
10.2:
RADIAL DISTRIBUTION FUNCTIONS
Given a particle i (molecule) located at
a certain point in space,
the number of particles located within a spherical shell of mean radius
r and width 6r about i is given by
284
47rr2 6r
V
g(r)
6N
N-1
where
N is the total
number
(10.3)
of molecules
in a volume
V, and g(r)
is the
radial distribution function (McQuarrie, 1976), the most important features
of which will now be summarized.
In a structureless
fluid, such as an ideal gas, g(r) is unity
since the particles exert no
throughout
influence upon each other, and hence the
number of particles within any given volume about particle i is independent
of i's presence, and is simply proportional to the volume considered.
Molecules
interacting via van der Waals
forces, on the other hand,
repel each other strongly at short distances and attract each other at
long distances.
fluids
(Widom, 1967; Chandler et al., 1983), whereby each molecule is,
statistically
g(r) >
r
This leads to the establishment of local order in dense
speaking,
surrounded by a
1, mathematically).
"nearest neighbour shell"
Moreover, g(r)
(or
decays abruptly to zero as
0 due to the steeply repulsive part of the intermolecular potential,
and becomes unity at large distances, since the influence of the central
molecule
greater
(i)
than
is then
negligible.
Secondary peaks where
1, is smaller than at
the nearest neighbour peak,
at high densities as a consequence of close packing.
features are shown schematically
function like g(r) is two fold.
g(r), though
in Figure
10.8.
arise
These qualitative
The usefulness of
a
In the first place, for a given temperature
and
intermolecular
potential,
knowledge of g(r)
the
compressibility
(Hirschfelder et al.,
implies knowledge of
1964), since this quantity,
as discussed in Chapter 8 in connection with the virial theorem (Equation
8.72), is a function of the average relative positions and forces between
all possible molecular pairs, and of the temperature; g(r), on the other
hand, is simply the mathematical expression of the distribution of intermolecular
will
separations.
not be used here)
In addition to this
quantitative aspect
(which
g(r) provides a very graphical description
of
structure and local order in the fluid.
Finally,
Order
it must be emphasized that g(r) is a statistical concept.
in a fluid cannot
be directly
by statistical arguments.
285
observed:
it
is always
inferred
I
g(r)
Ideal gas
I
r
._
C
0
I
C'
I
g(r)
Dense fluid
r
I
g(r)
Denser fluid
r
FIGURE 10.8:
Qualitative features of the radial distribution function
286
In the simulations, the radial distribution function of the C02 carbon
centers
was
distance
place)
were
computed
(i.e., half
by
dividing
and
intermolecular
the side of the cube where the simulation takes
into twenty equal intervals.
taken,
the maximum possible
the
Several "snapshots" of the system
positions of all the C02 centers were
recorded.
For every "snapshot", all possible pairs were scanned, and the respective
separations classified into one of the twenty distance intervals. Averaging
over all molecules and all "snapshots", the radial distribution was obtained,
where
<g (r)> = <AN(r)>
V
4irr 2 Ar
N-i
< > denotes averaging
(10.4)
over all molecules and "snapshots", r
is
the mid-point of the distance interval, and Ar is (1/40)th of the cube's
side.
Roughly
55
"snapshots"
per
simulation were
taken, which,
coupled
with the number of solvent molecules considered (107), implies that each
calculated
<g(r)> curve is the average of nearly 6000 "measurements".
The details of the computational procedure used to calculate radial distribution functions with periodic boundary conditions can be found in Appendix
5 (computer program NEUTRAL-2).
The nature of the solute-solvent interaction and its temperature
and
density
approach.
dependence
are
ideally suited to a
distribution function
In this case, one would compute the radial and angular distribution
of C02 centers about the solute molecule's
center.
However,
a smooth
curve cannot be generated from a single solute particle, since the averaging
can only be done over the "snapshots".
These can only be increased by
a limited amount (certainly not 100-fold) without making either the duration
of a simulation or the computer memory requirements unacceptable.
Increasing
the number of solute molecules, on the other hand, is an even more impractical
approach, since the number of solvent molecules must then be dramatically
increased if the simulation is to be done at infinite dilution conditions.
The important conclusion, therefore, is that the study of the equilibrium
aspects
of
solute-solvent
interactions
at
infinite dilution requires
computer speed and memory well beyond those used in the present work.
287
The effect of density upon the radial distribution function is shown
The higher density
in Figure 10.9.
(10.53 mol/lt) curve was obtained
"
"snapshots', whereas the lower
from 58
(7.42 mol/lt) density curve
is
the average of 53 "snapshots".
The run average temperatures were, respec-
tively, 314.9 K and 316.5 K.
Although the densities are moderate, and
the fluid structure is limited almost exclusively to the nearest neighbour
in both cases,
peak
it can be seen that a mild secondary peak exists
at 10.53 mol/lt, whereas no structure beyond the nearest neighbour peak
exists at 7.42 mol/lt.
As
above,
explained
corresponds to (1/40)th of
the interval width
the cell size; since separations are assigned a nominal value equal to
the
mid-point,
interval's
For
the cell size.
uncertainty of
this implies an
the densities considered
(1/80)th of
in Figure 10.9, the cell
size and the corresponding uncertainties are 25.73 A and 0.32 A (10.53
-3
4.4xl10
molec/A
effect
The
10
x
6.34
mol/lt;
3
- 3
0
-3
molec/A 3 ),
0
and
28.91
A and
0
0.36
A
(7.42
mol/lt;
).
in Figure
of temperature at constant density is shown
10.10, corresponding to simulations at 13.87 mol/lt (8.35 x 10- 3 molec/A3),
average
with
The
high
run
temperatures
temperature
of 304.2
K and 329.8 K, respectively.
curve was obtained from 56 "snapshots"; the low
temperature curve, from 53.
The cell size corresponding to this density
0
0
is 23.47 A, and the length uncertainty is therefore 0.29 A.
As was the
case with Figure 10.9, we see a mild secondary peak gradually disappearing,
this time due to temperature.
The densities considered in these simulations are moderate.
As will
be explained below in connection with the calculation of diffusion coefficients, statistical problems arise at high densities, and this constitutes
one of the most severe limitations of the one-molecule approach to the
calculation of transport properties.
For the densities
in
the
considered, therefore, we conclude that structure
fluid phase is primarily limited to a nearest neighbour shell
whose density is between 35% and 45% higher than the bulk density.
average radius of this nearest neighbour shell is roughly 4 A .
outer
shell
The
A mild
(density between 3% and 7% higher than the bulk density)
can be detected, under appropriate conditions; the radius of this outer
shell is roughly
7.5 A.
288
a
1.0
I__Y
-I
Ji
1.0
-
0.5
-
I
00 -
g(r)
0.0
I
I
0.
_
_ _
5.
I
_~~~~~~
10.
C-C Distance( Angstrom)
FIGURE 10.9:
Effect of density upon g(r).
<T> = 314.9 K
sample size
=
Density = 10.53 mol/lt,
58; Density = 7.42 mol/lt,
<T> = 316.5 K, sample size = 53
289
r
1.5
1.0
g(r)
0.5
n
10.
5.
0.
C - C Distance ( Angstrom)
FIGURE10.10:
Effect of temperature upon g(r); Density = 13.87 mol/lt.
<T> - 304.2
K, sample
size = 56
290
size
= 53;
T> = 329.8
K, sample
10.3: DIFFUSION COEFFICIENTS
Diffusion coefficients were calculated by a method based on Einstein's
statistical
treatment
of
diffusion
(Section 8.4).
Specifically
(see
Equation (8.41)) the theory predicts that, for times greater than a characteristic relaxation time, the mean squared displacement of an ensemble of
particles with respect
to an arbitrary
initial configuration
increases
linearly with time, the proportionality constant being 6D (or, more generally,
2dD,
where d
is the dimensionality of space where the diffusion under
study takes place, and D is the particles' diffusion coefficient).
In the present case, with
only one solute particle as the sample,
the equivalence between time and ensemble averaging was invoked to generate
an ensemble as explained in Figure 10.11 where
is a generic property.
In a simulation, M diffusion "experiments" with n time steps each were
conducted.
The position of the single solute particle at the beginning
of each experiment was recorded, and squared displacements at corresponding
time intervals were averaged over the M "experiments", to yield the generic
expression
for
the
i
(i
=
0,
1,
...,
n)
mean
squared
displacement,
shown in Figure 10.11.
The equivalence between time and ensemble averaging is strictly applicable
only if each of the M experiments is statistically independent from the
rest, which, physically, requires that the experiments be non-overlapping.
This
was
not possible in the present case since the simulations would
have
become
too long.
The reason behind
this limitation lies
in the
duration of a single experiment, which, as will be seen from the results
below, must be of the order of 1.5 x 10 3
step of 10
integration steps (with a time
5sec) in order to guarantee that the computed mean squared
displacement versus time relationship covers a time span at least three
times greater than the relaxation time.
Given the way in which the ensemble was generated, whenever the solute
particle underwent an interaction which caused an unusual change in its
configuration (position, velocity) or, in other words, a "violent collision",
this event was "felt" throughout the M experiments, and deviations from
linearity
in the <r 2 >
vs. time curve occured.
291
This, of course, would
I
<4> = ir'
Time"
dt
0
T -- aD
I
/t
-r
ro
0, ,
N
---
--
-
n
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
a
I
!
n
I'
X>.
--
n
.is
-I
L· - __
FO~M. I
I
I
t
Ensemble generation for test-particle "experiments"
292
-~~~~~~~~~~~~~~~~~~~~~~~~~
I
U~~~~~~~~~~~~~~~~~~~~~~~
_
<r2 >
FIGURE 10.11:
-
not have happened with an ensemble consisting of truly independent experiments.
Since such events become more frequent the higher the density,
deviations
of high
from
linearity occured in a considerable proportion
density simulations. -This
(- 50%)
is the reason behind the relatively
moderate densities that were used in the present work.
We conclude that the generation of a time ensemble composed of strictly
independent "experiments" from the dynamic simulation of the motion of
one rigid polyatomic solute molecule and as few as 107 rigid polyatomic
solvent molecules is already a problem which requires at least an order
of magnitude increase in computer speed with respect to the machine used
in the present
of
simulations.
This can be seen by comparing the length
a simulation with 21 overlapping experiments of 1600 steps each
28 CPU hours for 3200
(-
integration steps, at 31 seconds per step) with
the corresponding requirements for 21 successive experiments of the same
length (- 289 CPU hours for 33600 integration steps, at the same speed).
The
above
as
computer
explained
in
speed figure
Chapter
9,
this
(31 seconds per step) is indicative;
number
is
a function
of
the
simulated
density.
As discussed above in connection with the calculation of radial distribution
by
functions,
considering
the alternative approach
more
is to generate an
than one solute molecule, with
increase in solvent molecules.
ensemble
the corresponding
Hoheisel (1983) studied binary diffusion
of benzene in dense CO2 using this approach, by considering 62 benzene
and 1310 CO2 molecules, modelled, respectively, as a one center LennardJones and a one-center Lennard-Jones plus point dipole.
Even with these
highly simplified potentials, the simulations required the use of a Cyber
205 computer.
In the present work, 21 "experiments" were conducted in each simulation,
with
eighty
time steps between successive
"experiments", each of which
-12
lasted roughly 1600 time steps (i.e., 1.6 x 10
step of
in
10- 15
different
and a minimum
Squared
seconds).
simulations
of 1440
The duration
between
(1.44 x 10
displacements
a
of the
maximum
of
sec, with an integration
"experiments" was varied
-12
1920
(1.92
x 10
sec)
-12
sec) steps.
with respect to
recorded every fourty time steps.
the initial positions were
Although the use of periodic boundary
293
conditions
(see
Chapter
9)
implies
a
sudden
shift
in
the
coordinates
of a particle whenever it leaves the cube where the simulations occurs,
this was done, for the solute particle, only for force and torque calculations.
For
the calculation of diffusion coefficients, on the other
hand, the true coordinates were recorded (i.e., the solute molecule was
allowed to have coordinates greater than one and/or smaller than zero).
This
implies two separate book-keeping procedures for the single solute
molecule (see Appendix 5, computer program LINALB).
The temperature dependence of binary diffusion coefficients in supercritical fluids, as was discussed
in Chapter 6 in connection with
the
experimental results, cannot be described by a simple power law relationship,
such as the T
dependence at constant density predicted by hard sphere
theory.
Diffusion
has
observed
been
coefficients exhibit an activated behaviour, which
both experimentally
(Feist and Schneider, 1982) and
through computer simulations (Hoheisel, 1983).
Figure 10.12 is a plot of the mean squared displacement versus time
for two different
-3
x 10
simulations at
the same density
3
molecules/A ) but different temperatures.
(10.53 mol/lt, 6.34
The temperatures shown
in the figure are run average translational temperatures.
Four
diffusion
coefficients
corresponding
to simulations
at
the same
density (10.53 mol/lt) are shown in Table 10.1, and are plotted in Figure
10.13 in Arrhenius fashion; the least-squares regressed activation energy
is 14.8
KJ mole
.
This
number
is to be compared
with
the 5.1 KJ mole
figure obtained by Hoheisel (1983) at 13.64 mol/lt in his molecular dynamics
calculations,
and with
the 10.9 KJ mole
-1
-
and 10 KJ mole
-1
(at 9.55 mol/lt
and 14.45 mol/lt, respectively) calculated from the experimental measurements
of Swaid and Schneider (1979), in all cases for the CO2
- benzene system.
The calculated activation energy is therefore 35.7% higher than the experimental value at a similar density.
The
activation
energy
obtained from the Arrhenius plot should be
interpreted with caution, since temperature is not a variable that can
be directly controlled in a simulation.
Instead, while volume and (ideally)
energy are held constant, pressure and temperature fluctuate during the
course of
a simulation.
In addition, the very concept of temperature
is a statistical one, implying a distribution of velocities (see Section
294
10.0
b-
5.0
V
0.0
FIGURE 10.12:
0.0
0.5
Time (10 -
1.0
12 Sec)
1.5
Temperature dependence of the mean squared displacement
versus time relationship.
295
Density = 10.53 moi/lt
4
3
oEU
2
0
I
3
2.5
3.5
103 T-I ( K'I)
FIGURE 10.13:
Arrhenius plot for four different simulations at a density
of 10.53 mol/it.
296
10.1), whereas there is but one solute molecule in the presently considered
simulations.
If, however, we accept the activation energies calculated from Swaid
and
(although more
experiments
Schneider's
than two temperatures were
considered in that work, constant density data were taken at two different
densities and just two temperatures in each case), we must then conclude
that the presently chosen method of ensemble generation not only predicts
the right trends, but also gives reasonable estimates of actual physical
properties.
The calculated standard deviations in the run average translational
temperature and in the run average translational (plus rotational) temperature
given
are
than
compressibility, etc.) performed by the program
energy,
Even though the relative standard deviations are small,
every ten steps.
represent non-negligible numbers when
they
obtained from more
corresponding to an update of key run indicators
three hundred values,
(temperature,
These numbers were
10.1.
in Table
This
converted to degrees.
aspect of the simulations is independent of energy conservation, accuracy
stability considerations, and is due to the fact that fluctuations
and
of macroscopic properties scale inversely as the square root of the ensemble
Lifshitz,
(Landau and
size
Since the duration of a simulation
1980).
is a quadratic function of sample size, we must have
|=
>/L<>
<A
L /<O>J2
is a generic
where
=
(~
N)
N
-
2
property,
from statistical mechanics,
(10.5) we conclude
Equation
by a
of
factor of two
(10.5)
-1/4
in the simulation,
of molecules
N, the number
(in CPU units).
t, the run's duration
and
t
The first equality follows
the second, from pairwise additivity.
From
that reducing a given relative fluctuation
requires increasing the ensemble size by a factor
sixteen-fold increase in computer time (given
four, and leads to a
an event of fixed duration to be simulated).
The isothermal density dependence of the measured diffusion coefficients
7 .42
is shown in Figure 10.14, for simulations at 310.3K,
-3
x 10
molec/3)
corresponding
and 309.3K, 10.53 mol/lt
diffusion
coefficients are
297
-3
(6.34 x 10
shown in Table
(4.47
mol/lt
3
molec/A ).
10.2.
The
If the
Table 10.1:
<T>(*)
(K)
DIFFUSION COEFFICIENTS CORRESPONDING TO FIGURE 10-13
<AT>/<T>(**)
<AT>/<T> (***)
D
(-)
(cm 2 /s)
(-)
309.3
.0554
.0257
1.396
x 10
318.6
.0368
.0171
1.608
x 104
321.4
.0476
.01 72
1.850 x 10
342.3
.0372
.0227
2.430 x 10-4
(*)
run average translational temperature
(**)
relative standard deviation (translational temperature)
4
(***) relative standard deviation (translational & rotational temperature)
Table 10.2: DIFFUSION COEFFICIENTS CORRESPONDING TO FIGURE 10.14
D
P
(mol/lt)
(cm 2 /s)
7.42
1.649 x 10-4
1.396 x 10
10.53
298
10.0
NM:
5.0
V
0.0
0.0
0.5
1.0
1.5
Time ( 10- 12 Sec)
FIGURE 10.14:
Density dependence of the mean squared displacement versus
time relationship.
<T(tr)>
<T(tr)> - 309.3 K, p - 10.53 mol/lt;
310.3K, p - 7.42 mol/lt
299
linear
part of
the mean
squared displacement versus time relationship
is projected to zero displacement, we obtain an abscissa intercept which
can be used as an estimate of the relaxation time.
Although hydrodynamic
arguments
qualitative
to
the
(see
Chapter
moderate
6)
can
densities
useful in the present context.
be
used
only
in
a
way,
due
(viscosities) involved, they are extremely
The relaxation time for a Brownian particle
(Equation (8.48)) is given by
A
where
= 6wn
m and a
(10.6)
are, respectively, the mass and
radius of
the Brownian
particle, and n is the viscosity of the continuum fluid.
Since the viscosity of a dense fluid increases with density at constant
temperature, the trends shown in Figure 10.14 are consistent with Equation
(10.6).
The
relaxation times are, respectively,
respectively,
-
4.51 x 103
10.53 mol/lt, and 8.78 x 10 13 mol/lt at 7.42 mol/lt.
sponding
CO2
pressures, calculated from
state, are 85.6
3.5,
(10.53 mol/lt)
sec at
At 310K, the corre-
the Peng-Robinson equation of
and 80 bar
(7.42 mol/lt).
From Figure
it can be seen that large changes in viscosity occur precisely in
this region.
The detailed discussion of Chapter 6 shows that these arguments
cannot be pursued further in a quantitative way, i.e., since the behaviour
is not truly hydrodynamic, changes in relaxation times cannot be calculated,
but only explained
in terms of viscosity changes.
case with
the interpretation of
arguments
provide an
the experimental
However, as was the
results, hydrodynamic
extremely useful framework for data analysis and
interpretation even when Stokes-Einstein-basedexpressions constitute
a high viscosity limit rather than a quantitative description of molecular
behaviour.
The sharp increase in relaxation times at lower densities introduces
another practical limitation.
As explained above, the interval between
successive experiments is 8 x 10
- 18% of
14
sec (80 time steps), which represents
the relaxation time at a density of 10.53 mol/lt, and - 10%
at the lower density.
Thus, although each
individual integration step
is faster at low densities (see Chapter 9), the statistical independence
of
the
individual "experiments" becomes
300
progressively worse,
and calls
for
a
greater
in computer
inter-experiment interval, with
time
for the same number
In addition,
statistical
the consequent
increase
of "experiments".
it can be seen from Figure 10.14 that, regardless of
independence
considerations, the
long relaxation times at
low densities will give rise to results which can appear unphysical if
not
adequately
illustrated
interpreted.
in
Figure
discussed in Chapter 8.
displacement
shows
is
a
the very
direct
interesting behaviour
consequence
of
the
principles
At short times (Equation (8.47)), the mean squared
is quadratic
and temperature.
simply
10.14
In fact,
in time, and depends only on molecular mass
Thus, for a given solute and temperature, Figure 10.14
how
the longer relaxation
time dominates the short time
behaviour of the low density experiment.
In spite of the limitations explained in detail in Chapter 6 regarding
the predictive use of hydrodynamic expressions in the supercritical region,
it is interesting to compare Equation (10.6) with the relaxation times
obtained
in
the
present
simulations by extrapolating the linear part
of the mean squared displacement versus time relationship down to zero
displacement.
For these purposes, we use the following property values:
m = 1.295 x 10
25
kg (mass of benzene molecule)
o
a - 2.5 A (center of mass-to-hydrogen distance)
n -
.05 cp (Figure
3.5, 310 K, 90 bar)
to obtain
AT - 5.5 x 10-
13
sec
in excellent qualitative agreement with the results obtained in the simulations.
A very interesting theoretical question is raised by the fact that,
although
in
the
simulations
the mean
squared displacement exhibits a
linear behaviour at long times, the relationship nDT1
Chapter
6)
=
f [size] (see
is only an asymptotic law approached at high viscosities.
This apparent paradox can be explained by noting that the long time relationship between <r2 > and time can be derived (see Chapter 8) without postulating
any explicit form for the hydrodynamic drag.
Alternatively (Chandrasekhaar,
1943, see also Chapter 8), starting from the Langevin equation, the limits
<r2 >
-
t2
(t
0) and
<r2 >
-
t (t +
) can again be obtained without
postulating any form for the drag coefficient,
301
, although the drag term
itself
is,
in
this
approach,
proportional
to
the particle's velocity
(with an as yet undefined proportionality constant, 8).
We conclude,
therefore,
that,
if
is non-linear
8 -
in n (i.e.,
n6
for example), the Stokes-Einstein equation (or, more precisely, its form,
i.e.,
DT
= f
[size]) would not describe physical reality; in spite
of this, though, the short and long time limits of <r2> would, of course,
still be parabolic and linear, respectively, and the fundamental relationship
r 2 > and D at long times would still be valid.
between
The
breakdown
of
hydrodynamic
behaviour
in supercritical fluids,
then, is associated with a "hydrodynamic" drag that can best be explained
in terms of
a power law relationship between
the drag coefficient and
viscosity.
Using the activation energy calculated from the log D vs. T
(Figure 10.13), we
10.53
This
by
mol/lt
number
temperature
plot
can estimate a diffusion coefficient at 313.2 K and
from the value obtained
(1.5 x 10
graphical
1
2
cmn /s) is to
interpolation of
at 309.3K
be compared
and the same density.
with the value
obtained
Swaid and Schneider's data at the same
(2.05 cm 2 /s).
The molecular dynamics prediction is 36.7% lower than the experimental
value.
This is a remarkable result, given the facts that no adjustable
parameters were used in this work, and that the ensemble-generating technique
involved just one solute particle.
Linearity
in the mean squared displacement
versus time relationship
is, by itself, an indication of the correctness of the ensemble generating
procedure.
However, since this is, essentially, a test-particle method
(Herman and Alder, 1972; Alder et al., 1974) the accuracy of the calculated
diffusion
coefficients
should be
considered semiquantitative; ensemble
averaging over different runs would obviously make the predictions quantitative; this, however, contradicts the spirit of test-particle studies.
Although,
even
as
shown above, the method
reproduces the basic
trends and
gives good estimates for the actual properties, the test-particle
approach
is not
to be interpreted as a predictive substitute of large
ensemble methods, but as a convenient way of studying the basic physical
phenomena within the limits imposed by time sharing and an average mainframe
computer.
302
10.4:
SIMULATIONS WITH COULOMBIC INTERACTIONS
As
mentioned
- 39
in Chapter
9,
the large
quadrupole
moment
of CO2
(-1.43
2
x 10
Cm ) implies that electrostatic forces should be taken into account
in any realistic
simulation
of this molecule.
In this work,
the approach
was to superimpose localized partial charges (point monopoles) upon the
van der Waals
interactions, with the resulting potential centered upon
the same site (i.e., the van der Waals and electrostatic sites coincide).
Truncation
was
approach
done
being
as
explained
in
based upon the
Chapter
9,
the
plausibility
of
the
effective short range behaviour of the
intermolecular interactions resulting
from pairwise
additive long-range
interatomic interactions.
This approach introduces, at the outset, problems in the determination
of site-site interaction parameters, due to the fact that the van der Walls
part
of
the elementary
parameters
1979)
(a,
binary
) calculated
(Equation 9.32).
This
interaction contained
length and
from the Slater-Kirkwood
formula
(Suter,
equation represents, essentially,
a self-
consistent method of describing site-site
Lennard-Jones
type potential.
The
interactions
in
energy
terms
of
a
fundamental fact about the Slater-
Kirkwood equation, however, is, that the electrostatic properties of the
site
are
already
taken
into
account
through
the
polarizability
and
the
effective number of outer shell electrons, which are used to calculate
a
and
.
The redundancy of adding point monopoles
is thus evident, at
least in principle.
In the case of CO2, however, the Lennard-Jones parameters calculated
from the Slater-Kirkwood
behaviour,
at
least
below in detail).
inability
of
a
behaviour of a
van
in
One
formula do not reproduce
the supercritical
region
experimental
(this
will
be
P-V-T
discussed
is then forced to try to correct this apparent
der
substance
Waals
potential
to
(C02 ) for which
reproduce
the
thermodynamic
important coulombic
effects
have been experimentally measured by introducing electrostatic contributions,
at the expense of internal consistency.
Two
for the
characteristics
forces
failure of
into
the
of electrostatic
the attempt
simulations.
In
interactions were responsible
to successfully incorporate
the
303
first
place,
as
was
coulombic
discussed
at
length in Chapter 9,
(see Figures 9.10 - 9.21) the effective coulombic
intermolecular force and potential obtained from the elementary pairwise
site
additive
potentials
and
forces are both strongly
dependent upon
the relative orientation of the molecules, a feature which is, generally
second place,
speaking, absent in van der Waals interactions.
In the
at distances which correspond, approximately, to
the nearest neighbour
o
(i.e., -
peak
4 A), although the effective van der Waals potential and
force components are roughly an order of magnitude higher than the corresponding Coulombic interactions, this happens as a result of the cancellation of elementary electrostatic site-site
forces and energies which
are considerably greater, in absolute value, than the corresponding van
der Waals elementary interactions.
This is illustrated in Figure 10.15
and Table 10.3 for a specific case.
geometry, charge distribution
Figure 10.15 shows the particular
and Lennard-Jones
parameters considered;
the resulting elementary energies and forces (the latter exerted on the
1-2-3 molecule along the x-y directions) are shown in Table 10.3.
We therefore, conclude that electrostatic forces introduce stiffness
into the problem.
measured
quadruple
Starting from the charge distribution implied by the
moment
- 39
Cm 2 (Murthy
of CO,
2 (-1.43 x 10
et al., 1981))
and the inter-site separations shown in Figure 10.1, we obtain partial
charges of -0.2956 and + 0.5912 on the oxygen and carbon sites, respectively
(in electronic charge units).
This was
superimposed
originally
upon
the Slater-Kirkwood Lennard-Jones potential (see Table 9.3).
Although automatic rescaling during relaxation runs was not implemented
until later into the project, these trial runs already showed the essential
"pathology" of electrostatic simulations: large fluctuations in temperature
and compressibility (the latter often preventing the
time
long
ranged
limit).
The
temperature excursions were,
(timewise), whereas
the compressibility
comparable to the simulation time.
attainment
of
a
in general, short
fluctuated
over
times
After relaxation runs totalling 5000
integration steps at a density of 16.15 mol/lt, the compressibility factor,
which, at the nominal temperature (310K) should have been .295, was apparently
stabilized
An
at a value of approximately 1.2 for
extensive
the
last
series of ad-hoc modifications of the van
1000
steps.
der Waals and
electrostatic parameters was then tested, with the purpose of reproducing
304
I
I
4A
2
(
X
X
.23A
4
3
%F
6
a/k
O'
(K)
(a)
C-C
50.48
3.21
0-0
85.47
2.80
C-O
61.64
3.03
Z
(e)
C
0
FIGURE 10.15: Geometry, charges and site parameters for solvent interaction
case study
305
Table 10 3:
Pair
VAN DER WAALS AND COULOMBIC INTERACTIONS FOR FIGURE 10.15
U(LJ)
U(coul)
fx(LJ)
fx(coul)
fy(LJ)
fy(coul)
(1020)
(10-20)
(10
(10
(10
(10-12N)
2N)
2N)
N)
1-4
-4.90x10 2 +5.05
+6.369
-126.3
1-5
- 2
-4.38x10
-9.735
+5.009
+226.1
-1.54
-69.53
1-6
-2.03x10
2
+4.302
+2.101
-78.02
-1.292
+47.98
2-4
2-5
-4.38x10
-5.46x10 -
-9.735
+20.2
+5.009
+5.203
+226.1
-505.0
+1.54
0
+69.53
0
2-6
-4.38x10
-9.735
+5.009
+226.1
-1.54
-69.53
3-4
-2.03x10
2
+4.302
+2.101
-78.02
+1.292
-47.98
-4.38x10
-4.90x10
-2
-9.735
+5.05
+5.009
+6.369
+226.1
-126.3
+1.54
0
+69.53
0
-9.24
0
0
3-5
3-6
2
- 2
0
0
-
-3.68x10 - 1
-3,6x10
+42.179
306
These modifications included
the true compressibility factor.
·
increasing the energy parameters (c) by 20% to reduce the compressibility
factor
C
reducing the length parameters (a) by - 5% to reduce the compressibility
factor
·
modifying the quadrupole moment by changing both the charge distribution
and the site separation
*
modifying
all
of
plus
the above
parameters to reproduce the empirical
point quadrupole potential parameters proposed by
three
site
Murthy
et al. (1981), where
the point quadrupole is different from
the experimental one
The
essentially empirical nature of this procedure makes a detailed
account
of the effect of each of the above changes irrelevant for the
present purposes.
The essential points, however, can be summarized as
follows:
·
large
gave
electrostatic
rise
to
charges increased temperature fluctuations and
pressure fluctuations over time scales comparable to
the duration of a relaxation run (- 10 3 time steps)
·
no
single combination of
the compressibility.
parameters was found that could reproduce
The best fit (and dynamic behaviour) were obtained
with an ad-hoc modification of the Murthy et. al. (1981) parameters;
the actual values are listed in Table 10.4.
Since the compressibility
factor fit (see Section 10.5), though improved with respect to the
unmodified
Slater-Kirkwood
prediction, was
still not satisfactory,
this empirical approach was abandoned.
The
forces
interesting feature of C02 simulations with electrostatic
most
was
the observation
that, with
the potential parameters listed
in Table 10.4, temperature excursions were not coupled with poor energy
conservation.
In fact, the simulations which were carried out with the
potential parameters as per Table 10.4 exhibited good energy conservation
coupled with
large temperature excursions
307
(.928% standard deviation for
Table 10.4: EMPIRICAL SITE PARAMETERS FOR CO 2 SIMULATION
C-C
energy parameter (K)
34.8
C-C
length parameter (A)
2.646
0-0
energy parameter (K)
99.72
0-0
length parameter (A)
2.863
C-O
energy parameter (K)
58.92
C-O
length parameter (A)
2.755
C-O
separation (A)
1.16
O
charge
(e)
-.214
C
charge
(e)
+.428
308
the
former,
470C
for
(maximum)
the
latter,
and,
in
a second
run,
.91%
This seems to suggest that, although the algorithm
and 42°C, respectively).
is robust enough to handle a potential with moderate Coulombic components,
the dynamics are sufficiently different as to require a much larger sample
size.
The
interesting
and
important
problem of simulating
the dynamics
taking into account the significant electrostatic forces without
of CO2,
resorting
to
empirical
purely
and
time consuming
fitting techniques,
therefore, remains unsolved.
10.5:
COMPRESSIBILITY FACTORS
Compressibility
the virial theorem.
factors are calculated through
As discussed
The expression was developed in Chapter 8 (Equation 8.72).
in chapter 9, the most common approach in molecular dynamics is to obtain
site or molecular parameters by fitting P-V-T behaviour.
was not
followed here,
This approach
the purpose being to perform a simulation with
no adjustable parameters.
is especially justified
This
the determination of diffusion
coefficients, for which
in the case of
(see Chapter 6)
the fundamental variables are density (and not pressure) and temperature.
Because
of the significant
coulombic contribution
to the effective
intermolecular potential of C02 , the use of the Slater-Kirkwood parameters
gives rise to compressibility factors which are considerably higher than
the true values.
Table
is
the
10.5 lists results
average
translational
corresponding to sixteen simulations;
and
rotational
temperature;
P
is
the
<T>
CO 2
pressure corresponding to <T> and V, as read from a P-V-T diagram derived
from
the
International Thermodynamic Tables of the
Fluid State
(Angus
et al., 1976) (Schmitt, private communication); Z(MD) is the compressibility
factor as calculated from the simulation by averaging the instantaneous
values
printed
every
ten
iterations (this value
coincides, to within
a fraction of one percent, with the long-time limit of the time average
value calculated by the program as the simulation proceeds); P(MD)
therefore, the pressure as calculated by the simulation;
value of the percent error (i.e.,
p
309
- 1
x1OO).
is,
is the absolute
Values in parenthesis
x
E -
m LnU N CO
N
N1 (
J
n
CO Ln
O
M. OM
J
M m
cO
L
*
W
n
U
a'
-
**
0'
.O
D
0
N 1 l'D ON J
%P
*
1
C
O
3
x
CO
M
m
' C0
tI
N
t-
M
M
0
Cj
m
CZ
ao '.I
N
*
m m
J
N
* N
N13
t
-
C' s-
o
·
_'.'F- J' a'* NOO
x
O
n
I
'0
C
_
_
L
2:
0
N
- 'N
1
J
C\J
_ 3
3
N
U)
.0)
CD u1D J.0
'D '
_
0
0
0
a
00
W
ON
CO M
-
Nm M
COO
M*0
-N·
oJ- o~
o~ cO
3
-
LJ
: .cOV*
o~ oJ cO
t 0CY
N
CM
* 0. *
cO oJ
ma
_
t-
t-
- * r 0*.. m C N*
Oa
o~ \O ~-- c- ~-- o1 o
I-'
CZ
0cO.
m:
!
0!
L
O
Ln0N-
Ln
kL
L0
Lo
I o
0- o
CD
N-
LI
O
O-
mCO a' aCmO
0
'--
N -
W
14
-
.I
tN
a.
W
J
=-
LUn LC
N N N
a' C~ pc
1N Ca
J
CO m
Ym
0=
N
Cm
m
N
N
N1CO
N
1: CNCO
0
L
0
0
xD
N
CO
A
'
N
CO CO
a-
a' N 3 3 0 % 0
m m m_- 0m m M
m m mt
CO t-
Ln
I
x
r33
E
N
N
3
CJ
-I
_:
M
J
N
mU
CY)
0
a
'0\
.N
p~0
Z
Q)OC
.0
x
N_Y')
0\8~
t:r
CZ
0
M
1-
-
o.
I
0
xX
'·
CO
OO
It'
m
v-J
_
W.
0
LN 1-
0
-
f
t-
00 a'
(V
M
0
M
=
UN
C
G)
0
G)
Wz
W~
11
310
1
in the temperature column indicate standard deviations expressed as percent
of <T>.
Elec #1 and Elec #2 denote two simulations performed with the
parameters listed in Table 10.4.
The last column, ATmax, is the maximum
temperature difference that occured during the simulation.
It should be noted
behaviour attained
that the last two runs correspond to the best
in the simulations with electrostatic charges.
The
average ATmax for non-electrostatic simulations was 32.6 K, the corresponding
value for the best electrostatic simulations was 44.3 K.
This confirms
the previous discussion on the temperature behaviour of simulations with
Coulombic
forces.
In addition,
it must
be noted that high values for
non-electrostatic runs coincide with low temperature simulations (Tc=304.2K),
whereas
the
electrostatic simulations were done at
temperature levels
where ATmax is substantially lower for the purely van der Waals simulations.
From Table 10.5 it must be concluded that the use of Slater-Kirkwood
parameters
is not
properties
of CO2 .
satisfactory for the modelling of the thermodynamic
The
improvement obtained by
introducing localized
electrostatic charges indicates that potential improvement efforts should
be oriented along these lines, but the problem of temperature (and pressure)
fluctuations remains a significant challenge.
10.6:
SUMMARY
The test particle approach predicts diffusion coefficients to within
- 35% of experimental values, and, more importantly, reproduces the main
trends, without adjustable parameters.
This allows the semiquantitative
study of infinite dilution diffusion processes without recourse to supercomputers.
Results can be obtained, in the test particle method, within a narrow
density
range.
At high densities,
frequent deviations
from linearity
in the mean squared displacement versus time relationship occur, probably
as a result of the insufficient statistical independence of the diffusion
"experiments" in the light of the increased frequency of strong interactions.
At low densities, the limitations result from the large relaxation times.
In
the
present
work, diffusion exhibited Arrhenius behaviour; the
resulting activation energy is within - 35% of the value calculated from
311
experiments at two different temperatures.
Velocity distributions are in excellent agreement with the theoretical
(Maxwell-Boltzmann) prediction.
Radial distribution functions have been
The trends exhibited
generated for the carbon centers of the CO2 molecules.
provide qualitative
temperature
information on
variation.
fluid structure and
Quantitative
its density and
information can only be obtained
from radial distributions by generating an ensemble
of curves with an
extremely narrow distance grid, and averaging over these histograms to
eliminate the resulting noise.
The
for
must
the
use
of
unmodified
modelling
Slater-Kirkwood parameters
of P-V-T properties of C0 2 ;
is inappropriate
coulombic
interactions
be taken into account, but this introduces stiffness which, even
in cases where energy
conservation is acceptable
(which could only be
attained by reducing the electrostatic forces), gives rise to large temperature excursions.
312
11
CONCLUSION
Supercritical
fluids have
As a consequence,
mass
transfer
exceptionally low
the inevitable density
kinematic viscosities.
gradients which characterize
give rise, in the presence of a gravitational field,
to
buoyancy-driven flows which, for a given Reynolds number, are more than
two
orders
of magnitude higher
than
in ordinary liquids
(as measured
by the ratio of characteristic buoyant to inertial forces).
Whenever
the
controlling resistance
to mass transfer lies
in the
supercritical phase, significant buoyancy-driven mass transfer enhancements
result.
This
Future work
has
been
verified
should address this
experimentally in the present work.
interesting aspect
in a quantitative
way.
Hydrodynamic
high viscosity
behaviour at
the molecular
(low fluidity) limit.
level is approached as a
Although constancy of nDT- 1 within
a given system can be assumed for data extrapolation provided n
O.004cp,
the real need is for a fundamental theoretical understanding of the way
in which the hydrodynamic limit is approached.
The
concept of an infinite dilution fugacity coefficient, as well
as a simple
and accurate expression for
the composition dependence of
the solute fugacity coefficient in a binary mixture, from infinite dilution
to saturation, have resulted from an analysis of diffusion in the light
of
irrreversible
thermodynamics.
These preliminary ideas merit more
detailed consideration, and may have interesting thermodynamic implications.
A
test-particle molecular dynamics
study of binary
diffusion in a
rigid polyatomic ensemble with solute and solvent having different symmetry
properties has been done.
The results are encouraging, contain all the
basic physics, and yield semiquantitative predictions for binary diffusion
coefficients.
The Einstein plots of mean
squared displacement versus
time constitute a very convenient representation of the relaxation behaviour
and of the transition from a deterministic to a stochastic regime.
The
basic kinetic and thermodynamic characteristics of an equilibrium system
(Maxwellian
velocity distribution, pair
obtained with
distribution functions) can be
as little as 108 molecules, due to
boundary conditions.
313
the use
of periodic
The ensemble generating technique allows the study of infinite dilution
interactions without recourse to supercomputers.
The rigorous
(i.e., using a-priori site
site potential parameters)
dynamic simulation of CO2 , taking into account the electrostatic properties
of this particular molecule represents a challenging problem.
The orien-
tation-sensitivity of the coulombic interactions make the problem stiff,
and multiple time steps methods must be implemented.
314
APPENDIX 1
CUBIC EQUATIONS OF STATE
In the present context, an equation of state is a mathematical relationship between T,P,V, and N, that is, the absolute temperature, pressure, volume and number of moles of a single phase system, which can have
one or more components (throughout this Appendix, V denotes total (extensive)
whereas V denotes molar volume). As an
volume,
example,
the
equation
of state of an ideal gas is
PV
Equation
=
(Al-1)
NRT
(Al-1) is an example of an
analytic
equation
of state.
Its use, however, is limited to the low pressure, high temperature region
where the system behaves effectively as an ideal gas.
A variety of equa-
tions of state are commonly used;to describe the behaviour of real gases,
dense fluids and liquids.
Cubic equations of state originate from the work of van der Waals,
proposed the equation that' bears his name
who
in his
doctoral thesis
(van der Waals, 1873)
a
( P + ,-
).( V - b ) = RT
(A1-2)
).( V - Nb ) = NRT
(A1-3)
2
V
or, in extensive form,
2
aN
( P + --2
V
where
a
and
b
are parameters whose
determination is discussed below.
All existing cubic equations of state are modifications of the van der
Waals equation.
nomial
Their usefulness stems from the fact that a cubic poly-
in V is the simplest analytic relation
describe vapour-liquid equilibrium.
315
that can qualitatively
For a single component, this means
aP
two criticality conditions
av
(A1-4)
= 0
)
(
T=Tc
2
a P
-
(
)
(A1-5)
=0
2
av
T=Tc
plus the existence of three distinct real roots in some range of temperaand
ture
pressure,
(critical) temperature and an
bounded by an upper
upper (critical) pressure.
Although
a cubic equation of state can
behaviour of real
be taken
become
important limitations should
gases and liquids, two
into consideration.
In the first place, density fluctuations
the critical point
unbounded close to
describe qualitatively the
( Stanley,
1971
)9 and,
as a consequence, no analytic equation of state is accurate in this re(A1-5) are still
gion, even though Equations (A1-4) and
true.
In the
second place, an Equation such as (A1-2) cannot possibly describe phase
equilibrium below the triple point of a pure substance, where the solid
phase
must
be
taken
An additional parameter and
into consideration.
an equation of higher order in V are needed.
A
cubic equation is characterized by
its form and its parameters.
The former can be written, in a general way (Schmidt and Wenzel, 1980)
RT
a
P =
(A1-6)
2
V
V-b
where u and w are integers.
2
+ uVb + wb
Values of u and w for several cubic equations
of state are shown in Table Al-1.
Parameters a and b are determined by means of methods which, in general, are modifications of the original van der Waals approach.
the
criticality
(A1-2),
we obtain,
conditions
(Equations (A1-4) and
with T=T c and P=Pc,
316
Applying
(A1-5)) to Equation
2
27
a =
(RTC )
-
--
64
(A1-7)
Pc
RTc
b
-8
(A1-8)
Pc
or, in other words, temperature-independent parameters.
Equations
(A1-7) and
(A1-8) can be
considered particular
cases of
a more general type of functionality,
2
(RTc)
a = a
a
(A1-9)
PC
RTc
b
nb
=
(A1-10)
Pc
where
, B,
a and
b vary according to the particular equation of state
selected, and are introduced to improve agreement with experimental data.
In Equations
(Al-9) and
dependent whereas
a and
(A1-10), a
and B are, in general, temperature
b are constants.
Values of a, B,
a and
b
are shown in Table Al-1 for several different equations of state.
Mixture parameters a and b are obtained from pure component parameters by means of suitable mixing and combining rules, of which the most
commonly used are
a = xi
xj
aij
b = xi bi
aij = (ai aj)
[12- kij(
(Al-11)
-
ij)]
where kij, the binary interaction coefficient, is the single adjustable
parameter once an equation of state is selected, and
317
6
ij is Kronecker's
delta.
318
CC
-I
i
.4.
.
'.
ELz
cs
r.
0
m
m
I
I
Z
0
cO
cO
-
~C.
_N
CQ
u-
0
cc
_
llN
t
"I
LN1i
N
Lle
0
0,3
~ ~3
0n
I _INC: _
4
co
o
C
0
oc0
C,b+
-
.:c.0
N
O
¢
c-.
,N
319
3
'0
APPENDIX 2
A2.1
EQUIPMENT DESIGN; EXPERIMENT DESIGN AND CALCULATIONS
FLAT PLATE DESIGN
The
flat
section's
plate
width
the high
schematically
(2.54 cm) was determined
pressure steel tube
assembly
shown
inches)
is shown
in Figure
4.2
in nominal external
in Figure 4.2.
by practical considerations:
(C2 in Figure 4.1)
is introduced
The coated
had
to
diameter; otherwise,
into which the whole
be
less
than
5 cm
(2
the high pressure end
connections would have become prohibitively expensive and much more complex
than the threaded connections used in this work.
The
entrance
section
(Figure 4.1) was designed
the development of a steady velocity profile.
in order
to allow
The hydrodynamic entrance
length (Lhy) is defined (Shah and London, 1978) as "the duct length required
to
achieve
a maximum
duct
section
velocity
of 99%
of that
for developed
flow when the entering fluid velocity profile is uniform", and is given,
in dimensionless
+
form, by
Lhy
hy hy DhRe
where
In
Dh is the duct hydraulic diameter, and Re, the Reynolds number.
the
than
(A.2-1)
present
the
case,
since the aluminium hemi-cylinders were
steel tube, the approximately 5 cm
shorter
long empty inlet section
was packed with glass wool to guarantee the flatness of the profile at
the duct's entrance.
For a rectangular duct of height 2b, width 2a, and aspect ratio a
=
2b/2a (see Chapter 5), the hydraulic radius is given by
rh
Dh
4
Cross section
Perimeter
b
1 + a
(A.2-2)
Furthermore, the Reynolds number can be written as
Re =
4b
1+a
G
(A.2-3)
n
320
where G is the mass flow rate per unit area, which, in terms of the solvent
flow rate at ambient conditions becomes
it
1
= 4.16666
where V0
which,
(vFMab)
Mmin]
1
L
'see'
7;
V ( tx
G(~-ln)
= F
in)
60 xse
1
3__)
1 2)
mol) x 4ab(m
x 10-6
(A.2-4)
is the molar volume of the solvent gas at ambient conditions,
for the present purposes, can be taken as 22.4 lt/mol, and M is
the solvent's molecular weight.
The hydrodynamic entry length becomes,
therefore,
Equation
F, M, V,
(A.2-5)
x 4.1666 x 10- 6
(1)
n
Lhy = Lhy(l4b )2 ( FM
+ a
V o ab
hy
Lhy
is dimensional,
and will
yield Lhy
(A.2-5)
in meters if
a and b have the units indicated in Equation (A.2-4), and n
is expressed in kg/ms.
monograph,
Values of Ly
are tabulated in Shah and London's
as a function of a; widely differing values are given, the
table in question being a compilation of results from different investigators.
In the present case, the most conservative (i.e., highest) value
was selected,
Li
hy
=
0.08
(A.2-6)
-2
For a typical flow rate of 2 liters per minute, a = 1.27 x 10
m,
b = 1.587 x 10-
3
m, a = 0.125, V
- 22.4
It , n -
5 x 10 - 5 kg/ms,
(a
mol
conservatively low value), Lhy is then given by
-4
Lhy = 9.4 x 10
M
(m)
(A.2-7)
or, in other words, a minimum required hydrodynamic entry length of 41
mm for CO2 , and 137 mm for SF6 .
for CO2
experiments,
Two different flat plates were used:
the entry length was 12.70 cm, and 15.24 cm for
the SF 6 experiments.
321
The coated length was determined by the requirement that the relative
saturation at
sensitivity
the
flat plate's outlet should be neither too high
analysis,
section 6.4)
nor too low
(see
(see minimum weighing
requirements in section 6.4).
For an aspect ratio of 1/8, a 20% relative
saturation implies LD/<v> b2
-
-
cm/s, D
- 7 x 10
0.2
<v> -
0.1
obtain L = 7.2
cm.
(see Figure 5.3).
2
cm /s, and b = 0.15875
cm, we
With
The actual value used was 7.62 cm.
Typical
section,
<v> values are obtained
from the flow rate, F, duct cross
4ab, and solvent molar density under
experimental conditions.
2
For F - 0.09 mol/min, and 4ab = 0.80645 cm , and a molar density of 15
mol/lt, we obtain <v> = 0.12 cm/s, or a mean residence time of - 1 minute.
Before every run, a period of at least 15 minutes was allowed for after
opening the flow control valve, during which solvent gas was vented (see
Figure 4.1) while steady conditions were gradually attained.
A2.2
SAMPLE EQUILIBRIUM AND DIFFUSION CALCULATIONS
As
an
example
of
typical
equilibrium and diffusion calculations,
the determination of the solubility and diffusion coefficient of benzoic
acid
in SF 6 at 65 bar (Pr = 1.73) and 328.2 K (Tr = 1.03) will be explained
in detail.
Two
duration
equilibrium experiments were
was
40 minutes.
conducted simultaneously.
The following numbers were obtained
The run
(values
in parenthesis refer to the second experiment)
(a)
Mass of benzoic acid collected (g)
= .03781 (.03042)
(b)
Moles of benzoic acid collected
= 3.0992 x 10 4 (2.4934 x 10
(c)
DTM final minus initial reading (t)
= 62.703 (52.013)
(d)
Atmospheric
= 759.8
(e)
Ambient
(f)
Temperature-corrected ambient pressure (mm Hg) = 756.88
pressure
temperature
(mm Hg)
= 23.6
(C)
(value read from tables supplied by barometer manufacturer)
(g)
Average temperature of DTM outlet (C)
= 24.2 (24 .7)
(obtained by averaging initial and final readings)
(h)
Average overpressure at DTM outlet (in) = 0.4
322
(0.4)
)
(i)
DTM outlet conditions (mm Hg) =
Ambient pressure
2.54 x (h) X 826 x 760 x 9
101325
.88
756.88)
x 100
(where 826 Kg/m3 is the density of the outlet U-manometer fluid)
(j)
Solvent moles through DTM =
(1) 1
273.2
7
(ig
22.4179 273.2 + (1) 760
[
(c)
(c)
273.2
X
2.558 (2.119)
(where 22.4179 is the ideal gas molar volume, in moles/it, at 1
atmosphere and 0 ° C)
(k)
Solute
mole
fraction
=
C(b
)
=
x 10
1.211
(1.1766 x 10
These two results differ by 2.91%, and the average value can be
taken,
(1)
Average mole fraction -
1.194 x 10-4
For the diffusion run, we write
(m)
Mass of benzoic acid collected (g)
= 0.00853
(n)
DTM final minus initial reading (t)
= 77.670
(o)
Atmospheric pressure (mm Hg)
= 774.60
(p)
Ambient
(q)
Temperature-corrected ambient pressure (mm Hg) = 772.206
(r)
Average temperature at DTM outlet (°C) = 21.6
(s)
Average
(t)
Ambient pressure
_[(q)
temperature
overpressure
+2.54
= 19
(C)
at DTM outlet
(in) = 0.6
DTM outlet conditions (mm Hg) -
x (s) x 826 x 760 x 9.8]
101325 x 100
323
=773.131
(u)
Solvent moles through DTM
(n)
=
1
Ct)
273.2
[ 22.4179
x 273.2 + (r) x760
= 3.26626
x 105-
(v)
Solute
(w)
Relative
(x)
Xo (from mathematical solution; Chapter 5) = .188615
(y)
Run duration (s) - 3600.82
mole fraction
(u) + (m)/122
saturation
> b2
(z)
= [(v)/(Q)]
9.26
=
x 10 -
5
2.14057
= .17928
cm2 /s
(where <v> has been calculated as follows,
(O)
()
mol
s
0.80645
0.80645 cm
em2
mol
= 0.1484
cm/s;
fluid
density
from the Peng-Robinson equation of state)
A2.3
EXPERIMENTAL ERRORS
The sensitivity of the calculated diffusion experiments with respect
to experimental errors in the determination of r, the relative saturation,
was analyzed in Section 6.4.
for
Ar/rl will
In the present section, a numerical value
be estimated.
In the first place, since r is a ratio of two mole fractions (i.e.,
the solute mole fraction at the exit of the test section in equilibrium
and diffusion experiments), we can write, for error estimation,
jAr=
r-
where
that
x
x
Ax,
I Xlldiff
Ax.
I
is the solute mole
is determined
(A2-8)
I Xi equil
fraction.
by weighing
324
From Section A2.2,
it follows
the solid that precipitates in the
U-tubes
upon
decompression
and measuring
the corresponding amount
of
gas that flows through the system.
Weighings were found to be reproducible to within 0.001 g, and the
amount of solute collected varied from a minimum of 0.0085 g (low pressure
benzoic acid - SF6 experiments) to values above 0.2g (high pressure benzoic
acid
-
CO 2 experiments)
for the diffusion experiments
(i.e., weighing
errors ranged from 0.5 to 11.8%).
Weighing
errors
in equilibrium experiments were always
lower
(for
a given system, temperature and pressure) than in the corresponding diffusion
experiment,
since the mss
the former case.
of
of
solute collected was always greater
in
This fact will be used below in the actual evaluation
tAr/ri.
The determination of the amount of gas flowing through the dry test
meter involved reading the instrument (accurate to + .005 lt), and calculating
the number of moles as per Section A2.2.
the
temperature
at
This, in turn required reading
the test meter's outlet
(thermocouple accurate
to
+ 0.2 OC), the pressure differential across the outlet line (U-tube manometer
accurate
to
0.1
inch),
atmospheric pressure
the
atmospheric
(± 0.1 mm Hg).
temperature
(± 0.5
OC) and
the
From Section A2.2 (item (u)), it
follows that, for error analysis, we can write
nI
jn
IDTMI
DTM
=
I+
AP
Po
I T0
T,
(A2-9)
where n is the number of solvent moles, DTM is the dry test meter reading,
and P
and T,, the
respectively.
nation
of
temperature and pressure at DTM outlet conditions,
The ambient temperature contributed to errors in the determi-
the
ambient pressure since
the latter was always corrected
by means of a manufacturer-supplied double-entry table, the independent
variables of which were temperature and pressure.
Pressure errors, therefore,
can be estimated as follows,
AAP.
0.1 (temperature) + 0.1 (manometer) + 0.1 (U-tube)
-
-
PI o
-
3.95 x 10
(21760
(A2-10)
325
For temperature and instrument reading errors, on the other hand,
ATo
TO
0.5
-3
298 = 1.678 x 103
ADTM ~
0'005
I
AD
where
a
-005
=
3.125
conservatively
x 10
(A2-11)
-4
(A2-1 2)
low value of solvent
throughput (i.e., 16
lt)
has been used (see Section A2.2 for typical values).
From Equations (A2-9) to (A2-12), we obtain, finally,
|8n| - 0.00241
The
important
(A2-13)
conclusion here
is that weighing errors are by far
the most important in terms of their contribution to
Ar/rl, for which
we can now write
r
2 [0.0024
+
o.04
]
=
8.48 x 10
(A2-14)
where a typical diffusion weighing error of
4% was used
(corresponding
to a collected solute amount of 0.025 g), and diffusion errors were conservatively equated to equilibrium errors.
We
can
therefore summarize by saying
can be determined to within
except
for
low
pressure
that the relative
saturation
+ 8.5%, this being a conservative estimate
benzoic acid-SF6
experiments, where weighing
errors of up to 11% can exist in diffusion experiments due to the small
amount of solute collected.
A2.4:
DENSITY PROFILES
In Chapter 6, it was shown that, for dilute solutions, density decreases
monotomically away from the solute-fluid interface if
M > LV
M2
V2
Where M
while
(A2-15)
and M2 denote solute and solvent molecular weight, respectively,
V
and V 2
denote solute partial molar
326
volume and
solvent molar
volume, respectively.
In this section, it will be shown that inequality
(A2-15) was indeed satisfied for all of the conditions and systems tested,
a necessary condition for the elimination of buoyant effects in the hydrodynamic experiemtns (Chapter 6).
A simple, sufficient condition will first be developed that enables
to
test
(A2-15) is satisfied with minimum
whether
computations.
isothermal pressure dependence of the equilibrium solubility
The
of solute
1 in fluid 2 is given by
= (V,.
(alnx)
) /
1+
lnx) TP]
(A2-16)
which simply states the fact that, if the solubility increases with pressure,
the molar volume of the solid solute is larger than its partial molar
volume in the fluid phase (V1 s > V,).
This provides a quick, conservative
test of (A2-15) (i.e., a sufficient but not necessary condition for (A2-15)
to be true),
M2
M,
> V2
M2
V,
M2
>
(A2-17)
V,
but
M2 > V-
i>
MM>
2
L>
V2
(A2-18)
In other words, in cases where
the solubility increases with pressure,
consideration of the pure solute and solvent densities provides a sufficient
condition for monotomically decreasing densities away from the interface
which can be easily checked.
In the general case, we write
V = x, V
+ (1-x,) V 2
or, equivalently,
_
V.L
V2
where
V
--
(1 -
V?
1
--V-
)
-
Xl
V2 = V2 has been used
(1
V2
(A2-19)
x1 )
(A2-20)
X2
(this is only valid in dilute solutions).
327
Using the Peng-Robinson equation of state
(see Appendix 1) with binary
interaction coefficients (kij) regressed by minimizing the sum of absolute
values of log [x1 (eq)/x,(eos)], where eos denotes the equation of state
prediction and x(eq),
with the
the measured solubility, Table A2-1 was generated,
second form of the right hand side of Equation
(A2-20) used
in the computations.
From
for
all
Table A2-1 we
of
conclude that
the conditions and
inequality (A2-15) was
satisfied
systems tested; the flow was
therefore
stabilized by gravity, in all cases.
A2.5
CHEMICALS USED
The purity and suppliers of the chemicals used in this work are listed
below:
Benzoic acid
Baker
99.9+ %
Naphthalene
Baker
99.9+ %
2- Naphthol
Aldrich
99%
C02
Matheson
99.8%
SF 6
Matheson
99.8%
328
TABLE A2.1:
BUOYANT STABILITY CALCULATIONS
System
SF 6 -benzoic acid
SF 6-naphtha lene
CO2 -benzoic acid
CO2 -2 Naphthol
T
P
x, (eq)
kij
(K)
(bar)
328
65
1 .194x10
328
80
1 .491x10
328
120
1 .825 x10
338
65
338
80
1 .646x10- 0
2.076xl
338
120
2.803xl0-
318
65
1 .978x10-
318
80
2.152x10- 3
318
120
2.445x10-
328
65
3.184x10- 3
328
80
3.513x10 3
328
120
3.914x10
318
160
2.34x10- 3
318
200
3.580xl 0 -3
328
1 60
2.495x10
328
200
3.864x1 0- 3
308
150
4.460x10
308
200
5.408x10 1 4
308
250
5.910xl 0
318
1 65
5.662x10- 4
318
250
8.655x10 1
41
.128
V1/V 2
.8356
-2.651
4
-0.790
-7.448
.116
329
-4.789
-14
-3.901
-1 .237
3
.184
.8767
-0.325
3
0.515
-2.663
1 72
-1.157
3
0.159
.004
3
4
-1 .084
2.7727
-14.774
-2.330
-6.916
-.004
-3.472
.076
3.2727
-2.557
-0.468
0.538
.078
-3. 425
-0.109
APPENDIX 3
The
Ci,j COEFFICIENTS FOR EQUATION (5.27)
following
Appendix
contains the Cit.j coefficients defined in
Equation (5.27), for i up to 80, and j up to 26.
Coefficients corresponding
to each i value are printed under the hesding "Coefficient a i ",
j should be read by lines.
330
and
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APPENDIX 4
A4.1
COMPUTER PROGRAMS
COMPUTER : TECHNICAL DETAILS
All the computer simulations were run at the Massachusetts Institute
of Technology's Chemical Engineering Department Computer Facility.
The computer is a Data General "Eclipse" MV 4000 32 bit data processing
system.
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APPENDIX 5: THE INTEGRAL ENTROPY BALANCE IN IRREVERSIBLE THERMODYNAMICS
The following derivations follow closely the treatment of the subject
by Landau and Lifshitz (1982).
MASS BALANCE; CONSERVATION EQUATIONS
A5.1:
For an arbitrary volume V, fixed in space, conservation of species
1 can be written
as
I
at
Pw, dV
(pwl v
=
n) df -
*
first term on the right hand
The
j1
*
n df
(A.5-1)
side is the convective transport
of species 1; the second term corresponds to diffusive transport.
The
of a differential equation for the conservation of species
derivation
first
1
is
the
a
relationship
in our
step
we
analysis:
between the diffusive
are
ultimately
interested
in
flux and the appropriate driving
force(s).
Using Green's theorem, we rewrite Equation (A.5-1),
f
at
paw dV
V
-
=
*
+ J
(pv
dV
(A.5-2)
Therefore,
J
(Pl
+
[al
v + jL)] dV
=
O
(A.5-3)
this is independent of our choice of V; we must therefore have
But
am
+
at
V
(P
v + J,) = o
(A.5-4)
Invoking continuity for the fluid as a whole,
P + V
at
pv =
(A5-5)
The required equation follows from combining the last two expressions
P
(a
at
vv
VW)
V
*
= 0
406
(A.5-6)
or, in a more concise form,
PDw
+ V
Dt
-
· Jl
=
(A.5-7)
-(A
5-7)
where D/Dt is the material derivative operator.
A5.2:
THERMODYNAMIC DEFINITIONS
When diffusion is analyzed from the perspective of irreversible thermodynamics,
it is convenient
to introduce a mixture chemical
potential.
We start from the fundamental equation for a binary system,
dU = TdS - PdV + p, dN 1 +
p2
dN2
(A.5-8)
If we impose the constraint of constant mass, we have
dN2 =
(M)
dN
(A.5-9)
M2
and Equation (A.5-8) referred to unit mass of fluid, now reads
du = Tds - Pdv + pdw,
(A.5-10)
with
Ml
M
1
M2
M2
(A.5-11)
A5.3: ENERGY AND ENTROPY RELATIONSHIPS
Diffusion
is one
of
the dissipative
entropy generation in a fluid.
We must therefore obtain an expression
for the rate of entropy generation.
balance,
motion
we
will
to obtain
use
processes that contribute to
Starting from a differential energy
continuity, thermodynamics and the equations
a differential entropy balance.
The
of
integrated form
of this equation gives the rate of entropy generation as a result of
dissipative processes.
We first derive the differential balances.
Consider a volume element fixed in space.
t
[
( 2 + u)] = - V
[pv
- (
407
Energy conservation yields
+ h) - v *
+ ]
(A.5-12)
The right hand side is the sum of reversible convective transport, irreversible
rewrite
viscous
dissipation,
the left hand
and irreversible heat
transfer.
We now
side,
The
1st~e~pv
and
+
a-t
[p=2p-t
(p 4t
+ u)]
=a
2
2
+
aP3at+
(Aat5-13)
au(A5-13)
av
at
and use continuity to transform the 1st and 4th terms in the right hand
side, thermodynamics for the 3rd term, and the equations of motion for
the 2nd term.
Starting with the latter,
av
pv ·
The
av i
-t
-x vi
1st and
aa
x
ppv
i
[(v.V)vi]
(A.5-14)
4th terms in Equation (A.5-13) can be rewritten, using
continuity, as follows,
2- t+
2
u
at
ax
+
at
Vi
2
pv V)
(A.5-15)
Because of Equation (A.5-10) we can put (3rd term of Equation (A.5-13)),
as
au
a=
P
P at
t
(A.5-1
a,
at
and substit ite Equations (A.5-14), (A.5-15) and (A.5-16), and the thermodynamic relation
=
+-+
--
ax.
1
.-
T
x.
ax.
1
i
(A.5-17)
1
into Equation (A.5-13), to obtain
- [
(
at
2
u)]
(
p v + pT as +
V)
aw-
p
at
+
I
ax.
+
t
2
as
ax
at
aw
ax.
]
a CT
ik
i
ax k
Pv i
[(v.V_) vi]
(5-18
whic] can be simplified with:-thehelp of Equation (A.5-6) and the identity
p v [(v
v
vi
=
v
V
(
408
V2)
(A 5-19)
to obtain
at p(u +
[ pv (h + V2
= -
2V
pT
[as
Vs]
+ pT [a
+v
*
at
-
Adding and subtracting V_
i
Vs]+ vi
°vDik
(A.5-20)
D_
q, and noting that
aak
v
+
V · J
-
avi
ik = V
-ik
)
(v
ax
ax k
ikaX
(A.5-21)
we obtain the important relationship
at [P (u +
)] = - V
*
[pv (h +
V
)] - v
*
a
+
q] +
av.
+ pT (as + v
at
Vs) -
p V
-
·
J
V
+
*
q-
aik
ik axk
(A.5-22)
Comparing Equations (A.5-12) and (A.5-22),
av.
pT Ds =ik
Dt
x
-
(q
k
ik axk
(A.5-23)
Vp
This is the required differential entropy balance.
A5.4:
THE INTEGRAL ENTROPY BLAANCE
The rate of change of entropy in a volume fixed
in space is given
by
dt
The
ps dV
integrand, using Equation
s a
[ (P at+
at (ps)dV
dV
(A.5-23) and continuity, can be written
as follows,
as
at
av.
p
at
=
[1
[k
[ V
axk..
q-
409
p,)-j
-
I
V]-
V
psv
(A.5-24)
Therefore,
d Iv.
dt
psdV
=
-
(V
psv) dV +
aX
[a
- V
(A.5-25)
hand side of
The left
(A.5-25) (and hence the right hand
Equation
side) is positive or zero for any isolated system.
Restricting our attention
to this case,
IV
*
ps v dV
=
(ps v
(A.5-26)
n)df - 0
since there can be no flow across the boundaries of an isolated system.
Similarly,
V * (q- pj ) dV=
[(
T
)
*
n] df +
V. (T
J(
T2
dV +
)
f (T ·
· VT dV
I
)
VTdV
(T2
)
*
VT dV
(A.5-27)
where isolation has again been invoked to eliminate the surface integral.
The integral entropy balance for an isolated binary system then becomes
Is.
p dV
dV
dt
-
Closed,
isolated,
dV
T22
T
·
macroscopic
)
V TdV
ITdO
~
dV
dV
(A-5-28)
(-A.5-28)
systems approach stable equilibrium
in an irreversible way, the mechanisms involved being viscous dissipation,
heat flow and diffusion (first, second and third integral, respectively,
in the right hand side of Equation (A.5-28)).
410
NOTATION
A
= roughness factor (Chapter 2), dimensionless
A
= z-dependent coefficient (Chapter 5), dimensionless
A
= defined
A
= reduced attractive parameter in a cubic equation of state (Chapter 7)
dimensionless
A
= inertial-principal transformation matrix (Chapter 8), dimensionless
An
= expansion coefficient defined in Equation (5.72), dimensionless
a
= rectangular
a
= radius of a Brownian sphere (Equation
Chapter 6; Chapter 8), L
a
= attractive parameter for cubic equations of state (Chapter 3;
Chapter 7; Appendix 1), ML 5 /t2 mol 2
ai
= ith coefficient of series expansion (Equation (5.19)), dimensionless
a' i
= scaled
a'.
= scaled ith coefficient evaluated with the nth eigenvalue (Equation
(5.24)), dimensionless
a..
lJ
= parameter
ML 14/t 2
B
= empirical linear coefficient in a density-explicit virial expansion
(Chapter 2), L 3 /M
B
= defined
B
= reduced repulsive parameter in cubic equation of state (Chapter 7),
dimensionless
Bn
= expansion coefficient (Equation (5.73)), dimensionless
b
= rectangular
b
=
bo
- molar second virial coefficient, L 3 /mole
in Equation
(6.26),
dimensionless
L
duct half-width,
ith coefficient
for a 12-6
in Equation
(ai/a o)
type
1.17; Section
(Chapter
interaction
(6.27),
duct half-height,
1.5; Chapter
5), dimensionless
energy
(Equation
(9.31)),
dimensionless
L
repulsive parameter for cubic equations of state
Chapter 7; Appendix 1), L 3 /mole
411
2;
(Chapter 3;
C
= empirical
quadratic coefficient
expansion (Chapter 2), L 6 /M2
Cn
=
expansion coefficient (Equation (5.29)), dimensionless
Ci j
=
expalisioncoefficient (Equation (5.24)), dimensionless
C..
= parameter
for a 12-6 type interaction energy (Equation (9.31)),
ML8 /t 2
c
= solute molar concentration, moles/L3
c
= total molar concentration (Section 1.4; Chapter 7), moles/L3
Ci
=
c+
= 1 - c/c
Co
= inlet solute molar concentration, moles/L3
D
= duct diameter
D
= diffusion
1J
in a density-explicit virial
interface solute molar concentration, moles/L3
i
[ or (c-ci)/(co-ci);
(Section
but c
= 0 throughout],
1.1; Chapter
dimensionless
3), L
L 2 /t
coefficient,
= diffusion coefficient, L 2 /t
Dh
= hydraulic
diameter,
L
d
= dimensionality of space, dimensionless
ei
= ith (i = 0,1,2,3)
F
= force (Section 9.1; Section 9.4), ML/t2
Fm
=
f
= force,
f
=
f
= distribution
f
=
area element (Appendix 5), L2
Gn
=
expansion coefficient, defined in Equation (5.61), dimensionless
Gr
-
Grashof number [(2R) 3 g
g
-
radial distribution function, dimensionless
g
-
acceleration due to gravity (Section 1.1; Chapter 3), L/t2
Cayley-Klein
parameter,
dimensionless
modified force for shifted force potential (Chapter 9), ML/t2
ML/t 2
fugacity (Section 1.4; Chapter 7), M/Lt2
function
(Section
(Ap/p)/V2],
412
8.5),
t 3/L 6
dimensionless
F'
= unit vector, parallel to gravity, dimensionless
H
=
H
= Hamiltonian (Chapter 8), ML 2 /t2
Hn
= scaled "radial" expansion, defined in Equation (5.32), dimensionless
h
= relative height of constant hydrostatic head plane, L
h
= enthalpy per unit mass (Appendix 5), L 2 /t2
I
= moment of inertia, ML 2
J
=
j
= mass flux, M/L2 t
K
= torque, ML 2 /t2
K
= proportionality constant for semi-empirical hydrodynamic expressions
for the diffusion coefficient (Table 6.18), variously defined
K
= exponential decay factor for solute fugacity coefficient (Section
1.4; Chapter 7), dimensionless
KT
= isothermal compressibility, Lt2 /M
KT
= reduced isothermal compressibility, dimensionless
k
= Boltzmann's constant, ML 2 /t2 K
k
= mass transfer coefficient (Section 1.2; Chapter 5), L/t
k
= thermal conductivity (Table 2.1), ML/t 3 K
kij
= binary interaction coefficient, dimensionless
L
= coated length in duct flow, L
L
=
L
= generalized transport coefficient (Chapter 7), variously defined
1
- unit vector, parallel to linear molecule, dimensionless
M
- molecular weight, M/mole
m
=
"radial" part of two-dimensional
diffusion problem, dimensionless
generalized
flux
variously defined
solution to rectangular duct
in irreversible thermodynamics
(Chapter 7),
Avogadro's number (Chapter 2; Section 6.2), mole -1
mass, M
413
m
= velocity profile exponent (Section 1.2; Chapter 5), dimensionless
N
-
# of molecules in a simulation
N
-
# of moles (Chapter 7; Appendix 1; Appendix 5)
N'
=
molar flux, moles/L2 t
n
= velocity profile exponent, dimensionless
n
= number density (Chapter 2; Chapter 6), 1/L 3
n
= unit normal vector (Appendix 5), dimensionless
P
= pressure,
p
= linear momentum, ML/t
Pe
= Peclet
q
= heat flux, M/t
q
=
generalized coordinate (Chapter 8), L
q
=
wave number (Chapter 2), 1/L
R
=
gas constant, ML 2 /t2 mole K
R
= duct radius (Section 1.1; Section 3.1; Section 3.3), L
Re
=
Reynolds number (2R <v>/v), dimensionless
Ra
=
Raleigh number ((2R)3 g Ap/p /v D), dimensionless
r
=
position, displacement, relative position (variously defined), L
r
=
relative saturation (c/ci) (Section 1.1; Section 1.2; Chapter 3;
Chapter 5; Chapter 6; Appendix 2), dimensionless
rc
=
cutoff radius, L
rh
=
hydraulic radius, L
Sc
=
Schmidt number (v/D), dimensionless
Sh
-
Sherwood number (4 rh k/D), dimensionless
s
- entropy per unit mass, L2 /t 2K
s
=
hard sphere solute/solvent ratio (Chapter 6), dimensionless
T
=
absolute temeprature
M/Lt 2
number
(<v. L/D), dimensionless
3
41 4
T+
= reduced temperature (kT/c), dimensionless
t
=
time, t
U
=
interaction energyv ML 2 /t2
U
= internal energy (Appendix 5), ML 2 /t 2
U
= numerical constant for cubic equation of state, dimensionless
u
= internal energy per unit mass, (Appendix 5), L 2 /t2
V
= molar volume, L 3 /mole
V
= total volume (Equation (7.1); Equation (7.25); Chapter 8; Appendix
5), L 3
V,
=
close-packed molar volume for a hard-sphere fluid, L 3 /mole
v
=
velocity, L/t
<v>
=
cross section-average velocity, L/t
v+
= reduced velocity (v/<v>), dimensionless
v
= molecular
v+
= reduced molecular volume (v/a3 ) (Section 2.3), dimensionless
v
=
w
= angular velocity, 1/t
w
= numerical constant for a cubic equation
Chapter 7; Appendix 1), dimensionless
X
=
X
= generalized force (Chapter 7), variously defined
Xo
- modified inverse Graetz number (xD/<v>b2 ), dimensionless
x
- duct axial coordinate, L
x+
= dimensionless axial coordinate (x/b)
xi
= species
x
- inertial coordinate (Section 1.5; Chapter 8), L
volume
(Section
2.3),
L3
molar volume (Appendix 5), L 3/mole
of state
(Chapter 3;
axial part of two-dimensional solution to rectangular duct diffusion
problem, dimensionless
i mole
dimensionless
fraction
(Chapter 6; Section 1.4, Chapter 7),
415
x'
= principal
y
=
duct "radial" coordinate, L
y+
=
dimensionless "radial" coordinate (y/b)
y
=
inertial coordinate (Chapter 8), L
y'
=- principal
z
=
+
coordinate,
L
coordinate, L
duct transverse coordinate, L
=-dimensionless duct transverse coordinate (z/b)
z
= inertial
'
coordinate
(Chapter 8), L
=- principal coordinate, L
z
- compressibility factor (RV/RT) (Chapter 2; Chapter 3; Chapter 7;
Section 8.6), dimensionless
Greek Symbols
a
= aspect ratio for rectangular duct (b/a), dimensionless
= transport coefficient, (Section 1.4; Chapter 7), Mt/L3
a
=
flat plate inclination with respect to horizontal position (Figure
1.10; Figure 6.15), degrees
ai
-
polarizability of site i (Equation 9.32), L 3
aO
-
angle used in specifying the relative orientation of two linear
molecules (Figures 9.9 to 9.21), degrees
aX
=
coefficient
of the attractive parameter
of state (Appendix 1), dimensionless
B
= aspect ratio for rectangular duct (L/2a), dimensionless
Bm
=
mass coefficient of volume expansion (Section 3.2), L 3 /mole
0
-
transport coefficient (Chapter 7), M/Lt K
B
-
frictional time constant (Chapter 8), t
B
-angle
=
B
= coefficient of the size parameter
in a cubic equation
used in specifying the relative orientation of two linear
molecules (Figures 9.9 to 9.21), degrees
(Appendix 1), dimensionless
416
in a cubic equation of state
r
= scattered light autocorrelation decay rate, t1
rn
= defined
_Yn
=
Y
= transport coefficient (Chapter 7), ML/t3 K
Y
= angle used in specifying the relative orientation of two linear
molecules (Figures 9.9 to 9.21), degrees
6
= transport coefficient (Chapter 7), M/Lt
6
= displacement
6
= angle used in specifying the relative orientation of two linear
molecules (Figures 9.9 to 9.21), degrees
in Equation
nth eigenvalue
(8.61),
ML 2 /t
(Chapter 5), dimensionless
(Chapter
8), L
= energy parameter for interaction potential, ML 2 /t2
in
e
=
viscosity,
M/Lt
= Euler angle, dimensionless
= chemical potential per unit mass, L2 /t2
species i chemical potential, ML 2 /t2 mole
v
- kinematic viscosity, L 2 /t
= transformed
5+
=
coordinate
for duct flow
(b-y), L
dimensionless transformed coordinate (1-y+)
- defined in text (Chapter 2)
=
distance measured away from a constant composition solute source
plance
(Chapter
6), L
+ pgh), M/Lt2
In
- modified pressure (P
n+
= dimensionless modified pressure [n/po <v>2]
p
=
Ap
- interface-bulk density difference, M/L3
a
- size parameter for a binary interaction, L
a..
- ijth component of stress tensor (Chapter 7; Appendix 5), M/Lt2
Xr
- relaxation
density, M/L 3
time,
t
417
fugacity coefficient, dimensionless
*
=
On
= cross section-average integral (Equation (5.55)), dimensionless
4)
= association factor
dimensionless
4)
= Euler angle (Chapter 8), dimensionless
Oi
= species i probability distribution function for one-dimensional
displacements
X
in the Wilke-Chang
(Chapter
expression
(Chapter 6),
8), L
= Enskog frequency factor, dimensionless
= composition-dependence function for the thermodynamic transport
coefficient, a, such that D 2 is composition-independent (Chapters
1 and 7), dimensionless
'Pn
= cross section-average integral defined in Equation (5.75), dimensionless
=
Euler angle (Chapter 8), dimensionless
d
=
collision integral for diffusion, dimensionless
X
=
acentric factor, dimensionless
Xi
= species i weight fraction (Appendix 5), dimensionless
Subscripts
1
=
solute
2
= solvent
A
=
B
= solvent (Tables 6.18 and 6.19)
c
= critical property
I
-
i
- interface (Section 1.2; Chapter 3)
i
=
i
- ith principal direction (Equation 1.39; Equation 8.17)
J
= molecule J
solute (Tables 6.18 and 6.19)
molecule
I
ith site (Section 1.5)
4
18
n
=
nth eigenvalue
o
=
solute inlet conditions
r
= reduced
quantity
Superscripts
+
= dimensionless quantity
O
=
T
= transposed
dilute limit
Overbars
partial molar quantity
-
= denotes value of a property for a specie in a mixture
= time derivative
Underbars
=
vector
-=
matrix, tensor
;419
REFERENCES
,B.J. Alder, W.E. Alley and J.H. Dymond, "Studies in molecular dynamics. XIV.
Mass and size dependence of the binary diffusion coeffecient", J. Chem.
Phys., 61(4), 1415-1420, (1974).
B.J. Alder, W.G. Hoover and D.A. Young, "Studies in molecular dynamics.
V. High-density equation of state and entropy for hard disks and spheres",
J. Chem. Phys., 49(8), 3688-3696, (1968).
B.J. Alder and T.E. Wainwright, "Studies in molecular dynamics. I. General
method ", J. Chem. Phys., 31(2), 459-466, (1959).
B.J.
Alder
and
Phys. Rev.,
T.E. Wainwright,
127(2),
359-361,
"Phase transition
in elastic disks",
(1962).
B.J. Alder,and T.E. Wainwright, "Velocity autocorrelations for hard spheres",
Phys. Rev. Lett., 18(23), 988-990, (1967).
B.J. Alder and T.E. Wainwright, "Decay of the velocity autocorrelation
function", Phys. Rev. A, 1(1), 18-21, (1970).
G. Allen, J.G. Watkinson, and K.H. Webb, "An infra-red study of the association
of
benzoic
acid
in
the vapour
phase,
and
in dilute
non-polar solvents", Spechtroch. Acta, 22, 807-814,
solution
in
(1966).
E.N. da C. Andrade, "The Viscosity of Liquids", Nature, 125, 309-310 (1930).
S. Angus, B. Armstrong and K.M. de Reuck, "Carbon Dioxide-International
Thermodynamic Tables of the Fluid State-3", IUPAC, Division of Physical
Chemistry, Commission on Thermodynamics and Thermochemistry, Thermodynamic
Tables Project, Pergamon Press, Oxford, 1976.
V.S. Arpaci, "Conduction Heat Transfer", Addison-Wesley, Reading, Mass.,
1966.
Z. Balenovic,
M. Myers and J.C. Giddings, "Binary diffusion in dense
gases to 1360 atm by the chromatographic peak-broadening method", J. Chem.
Phys., 52(2), 915-922, (1970).
V.J. Berry, Jr., and R.C. Koeller, "Diffusion in compressed binary systems",
AIChE
J., 6(2),
S.J. Bertucci
diffusion
274-280,
(1960).
and W.H. Flygare,
in binary
"Rough hard sphere
liquid mixtures",
J. Chem. Phys.,
R.B. Bird, W.E. Stewart and E.N. Lightfoot,
Wiley and Sons, New York, 1960.
treatment of mutual
63(1), 1-9, (1975).
"Transport Phenomena", John
G. Brunner, "Mass transfer from solid material in gas extraction", Ber.
Bunsenges. Phys. Chem., 88, 887-891, (1984).
420
A.D. Buckingham and R.L. Disch, "The quadruple moment of the carbon dioxide
(1963).
molecule", Proc. Roy. Soc. A, 273, 275-289,
H.C. Burstyn and J.V. Sengers, "Decay rate of critical
concentration
fluctuations in a binary liquid", Phys. Rev. A, 25(1), 448-465, (1982).
D.M. Ceperley, "The relative performances of several scientific computers
for a liquid molecular dynamics simulation", Supercomputers in Chemistry,
P. Lykos and I. Shavitt, eds., ACS Symposium Series #173,
134,
Ch. 9, 125-
(1981).
"Rough hard sphere theory of the self-diffusion constant
D. Chandler,
for molecular liquids", J. Chem. Phys., 62(4), 1358-1363, (1975).
D. Chandler, J. D. Weeks and H.C. Andersen, "van der Waals picture of
liquids,
solids and phase transformations", Science, 220, 787-794,
(1983).
S. Chandrasekhar, "Stochastic problems
Mod. Phys., 15, 1-89, (1943).
in physics and astronomy", Rev.
S. Chapman and T.G. Cowling, "The Mathematical Theory of Non-Uniform
Gases", 3rd edition, Cambridge University Press, Cambridge, 1970.
S. Chen, "A rough-hard-sphere theory for diffusion in supercritical carbon
655-660, (1983).
dioxide", Chem. Eng. Sci., 38(4),
P.S.Y. Cheung and J.G. Powles, "The properties of liquid nitrogen. IV.
A computer simulation", Mol. Phys., 30(3), 921-949, (1975).
K.J. Czworniak, H.C. Andersen, and R. Pecora, "Light scattering measurement
interpretation of mutual diffusion coefficients in
and theoretical
binary liquid mixtures", Chem. Phys., 11, 451-473, (1975).
J. DeZwaan and J. Jonas, "Experimental evidence for the rough hard sphere
model for liquids by high pressure NMR", J. Chem. Phys., 62(10), 4036-4040,
(1975).
Enskog theory and the transport coefficients
J.H. Dymond, "Corrected
of liquids", J. Chem. Phys., 60(3), 969-973, (1974).
J.H. Dymond and B.J. Alder, "van der Waals theory of transport in dense
fluids", J. Chem. Phys., 45(6), 2061-2068, (1966).
A. Einstein, "On the movement of small particles suspended in a stationary
liquid demanded by the molecular-kinetic theory of heat", Ann. d. Phys.,
17, 549, (1905).
J.F. Ely and J.K. Baker, "A review of
NBS Technical Note 1070, (1983).
supercritical fluid extraction",
D. Enskog, K. Svensk. Vet. - Akad. Handl., 63(4), (1921).
421
C.P. Enz (ed.), "Dynamical Critical Phenomena and Related Topics", Proceedings
of the International conference on Dynamic Critical Phenomena, Geneva,
1979; Lecture Notes in Physics, No. 104; Springer-Verlag, Berlin, (1979).
R. Feist and G.M. Schneider, "Determination of binary diffusion coefficients
of benzene, phenol, naphthalene and caffeine in supercritical CO2 between
308 and 333 K in the pressure range 80 to 160 bar with supercritical
fluid chromatography (SFC)", Sep. Sci. and Tech., 17(1), 261-270, (1982).
M. Fury, G. Munie and,J. Jonas, "Transport processes in compressed liquid
pyridine", J. Chem. Phys., 70(3), 1260-1265, (1979).
H. Goldstein, "Classical Mechanics", 2nd edition, Addison-Wesley, Reading,
Mass., 1981.
K.E. Gubbins, K.S. Shing and W.B. Streett, "Fluid phase equilibria: experiment, computer simulation and theory", J. Phys. Chem., 87, 4573-4585,
(1983).
A.S. Gupta and G. Thodos, "Mass and heat transfer in the flow of fluids
through fixed and fluidized beds of spherical particles", AIChE J.,
8(5), 608-610,
(1962).
J.M. Haile, "Molecular dynamics simulations of simple fluids with threebody interactions
included", Computer Modeling of Matter, P. Lykos,
ed., ACS Symposium Series #86, Ch. 15, 172-190, (1978).
W.B. Hall, "Forced convective heat transfer to supercritical pressure
fluids", Arch. Thermod. i Spalania, 6(3),
341-352, (1975).
H.J.M. Hanley, R.D. McCarty and E.G.D. Cohen, "Analysis of the transport
coefficients
for simple dense fluids: applications of the modified
Enskog theory", Physica, 60, 322-356, (1972).
H.J.M. Hanley and E.G.D.Cohen, "Analysis of the transport coefficients
for simple dense fluids: the diffusion and bulk viscosity coefficients",
Physica, 83A, 215-232, (1975).
G.S. Harrison and A. Watson, "Similarity and the formation of correlations
for forced convection to supercritical fluids", Cryogenics, 147-151,
March 1976.
E.G. Hauptmann and A. Malhotra, "Axial development of unusual velocity
profiles due to heat transfer in variable density fluids", J. Heat
Transf., 102, 71-74, (1980).
W. Hayduk
solvent
635-646,
and S.C. Cheng, "Review of relation between diffusivity and
viscosity in dilute liquid solutions", Chem. Eng. Sci., 26,
(1971).
422
P.T. Herman and B.J. Alder, "Studies in molecular dynamics. XI. Correlation
functions of a hard-sphere test-particle", J. Chem. Phys., 56(2), 987-991,
(1972).
J.O. Hirschfelder, C.F. Curtiss and R.B. Bird, "Molecular
and Liquids", John Wiley and Sons, New York, 1964.
Theory
of
Gases
C. Hoheisel, "The binary diffusion coefficient of benzene at high dilution
in dense gas C02 ", Mol. Phys., 45(2), 371-377, (1983).
Y. I'Haya and T. Shibuya, "The dimerization of benzoic acid in carbon
tetrachloride and chloroform", Bull. Chem. Soc. Japan, 38(7), 1144-1147,
(1965).
K. Huang, "Statistical Mechanics", John Wiley and Sons, New York, 1963.
M.B. Iomtev and Y.V. Tsekhanskaya, "Diffusion of naphthalene in compressed
ethylene
and carbon dioxide", Russ. J. Phys. Chem., 38(4), 485-487,
(1964).
J.A. Jossi, L.I. Stiel and G. Thodos, "The viscosity of pure substances
in the dense gaseous and liquid phases", AIChE J., 8(1), 59-63, (1963).
C.R. Kakarala and L.C. Thomas, "Turbulent combined forced and free convection
heat transfer in vertical tube flow of supercritical fluids", Int. J. Heat
and Fluid Flow, 2(3), 115-120, (1980).
A.J. Karabelas, T.H. Wegner and T. J. Hanratty, "Use of asymptotic relations
to correlate mass transfer data in packed beds", Chem. Eng. Sci., 26,
1581-1589, (1971).
R.T. Kurnik, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge,
MA, 1981.
L.D. Landau and E.M Lifshitz, "Statistical Physics", 3rd edition, part 1;
volume 5 of the Course of Theoretical Physics, revised and enlarged
by E.M. Lifshitz and L.P. Pitaevskii, Pergamon Press, Oxford, 1980.
L.D. Landau and E.M. Lifshitz, "Mechanics", 3rd edition; volume
the Course of Theoretical Physics, Pergamon Press, Oxford, 1982.
1 of
L.D. Landau and E.M. Lifshitz, "Fluid Mechanics", volume 6 of the Course
of Theoretical Physics, Pergamon Press, Oxford, 1982.
G. Le Bas, "The Molecular Volumes of Liquid Chemical Compounds", Longmans,
Green, New York, 1915.
M.A. Lusis and G.A. Ratcliff, "Diffusion in binary
infinite dilution", Can. J. Chem. Eng., 46, 385-587,
liquid mixtures at
(1968).
S. Ma, "Modern Theory of Critical Phenomena", Benjamin/Cummings Publishing
Company, Inc., Reading, MA, 1976.
423
G.C. Maitland, M. Rigby, E.B.
Smith and W.A. Wakeham, "Intermolecular
Forces, Their Origin and Determination", Clarendon Press, Oxford, 1981.
V.R. Maynard and E. Grushka, "Measurement of diffusion coefficients by
gas-chromatography broadening techniques: a review", Adv. Chromatogr.,
12, 99-140, (1975).
D.A. McQuarrie, "Statistical Mechanics", Harper and Row, New York, 1976.
B. Metais and E.R.G. Eckert, "Forced, mixed and free convection regimes",
J. Heat Transfer, 86, 295-296, (1964).
N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, and E. Teller,
"Equations of state calculations by fast computing machines", J. Chem.
Phys.,
21, 1087-1092,
(1953).
M. Modell and R.C. Reid, "Thermodynamics and its Applications",
2nd edition,
Prentice-Hall, Englewood Cliffs, NJ, 1983.
of diffusion coefficients
V.S. Morozov
and E.G. Vinkler, "a'surement
of vapours of solids in compressed gases. I. Dynamic method for measurement
coefficients",
Russ. J. Phys. Chem., 49(9), 1404-1405,
of diffusion
(1975).
V.S. Morozov
and E.G. Vinkler, "Measurement of diffusion coefficients
of vapours of solids in compressed gases. II. Diffusion coefficients
of naphthalene in nitorgen and in carbon dioxide, Russ. J. Phys. Chem.,
49(9), 1405-1406, (1975).
S. Murad and K.E. Gubbins, "Corresponding states correlation for thermal
conductivity of dense fluids", Chem. Eng. Sci., 32, 499-505, (1977).
S. Murad and K.E. Gubbins, "Molecular dynamics simulation of methane
algorithm", Computer Modelling of Matter,
using a singularity-free
P. Lykos, ed., ACS Symposium Series #86, ch. 5, 62-71, 1978.
S. Murad and K.E. Gubbins, "Prediction of thermal conductivity for dense
fluids and fluid mixtures", AIChE J., 27(5), 864-866, (1981).
C.S. Murthy, K. Singer, and I.R. McDonald, "Interaction site models for
carbon dioxide", Mol. Phys., 44(1), 135-143, (1981).
K. Nishikawa and T. Ito, "An analysis of free-convection heat transfer
form an isothermal vertical plate to supercritical fluids", J. Heat
Mass Transf., 12, 1449-1463, (1969).
K. Nishikawa, T. Ito, and H. Yamashita, "Free convective heat transfer
to a supercritical fluid", J. Heat Transf., 187-191, May, (1973).
S. Nose and M.L. Klein, "A study of solid and liquid carbon tetrafluoride
using the constant pressure molecular dynamics technique", J. Chem. Phys.,
78(11), 6928-6939, (1983).
424
H.A. O'Hern and J.J. Martin, "Diffusion in carbon dioxide at elevated
pressures", Ind. Eng. Chem., 47, 2081-2087, (1955).
L. Onsager, "Reciprocal Relations in Irreversible Processes. I", Phys. Rev.,
37, 405-426, (1931)
L. Onsager, "Reciprocal Relations in Irreversible Processes. II", Phys. Rev.
38, 2265-2279, (1931).
A.C. Ouano, "Diffusion in liquid systems. I. A simple and fast method
of measuring
diffusion
coefficients", Ind. Eng. Chem. Fund., 11(2),
268-271, (1972).
M.E. Paulaitis,
V.J. Krukonis, R.T. Kurnik and R.C. Reid, "Supercritical
fluid extraction",
W. Pauli,
C.P. Enz,
Rev. in Chem. Eng.,
1(2),
179-250,
(1983).
"Pauli Lectures on Physics", volume 4, Statistical Mechanics,
ed., MIT Press,
1981.
D.Y. Peng and D.B. Robinson, "A new two-constant equation of state",
Ind. Eng. Chem. Fund., 15, 59, (1976).
R.H. Perry and C.H. Chilton, eds., "Chemical Engineers' Handbook",
edition, McGraw-Hill, New York, 1973.
5th
R.H. Perry and D.H. Green, eds., "Perry's Chemical Engineers' Handbook",
6th edition, McGraw-Hill, New York, 1984.
P. Pfeuty and G. Toulouse, "Introduction to the Renormalization Group
and to Critical Phenomena", John Wiley and Sons, Chichester, 1978.
K. Pitzer, D.Z. Lippmann, R.F. Curl, Jr., C.M. Huggins and D.E. Petersen,
"The volumetric and thermodynamic properties of fluids. II. Compressibility
factor, vapour rpessure and entropy of vaporization", J. Am. Chem. Soc.,
77(13), 3433-3440, (1955).
K. Pitzer, "Inter-and intramolecular forces and molecular polarizability",
Adv. Chem. Phys., 2, 59-83, (1959).
A. Rahman, "Correlations in the motion of atoms in liquid argon", Phys. Rev.,
136(2A), A405-A411, (1964).
K. Reddy and L.K. Doraiswamy, "Estimating liquid diffusivity", Ind. Eng.
Chem.
Fund.,
6(1),
77-79,
(1967).
O. Redlich and J.N.S. Kwong, "On the thermodynamics of solutions. V. An
equation of state. Fugacities of gaseous solutions", Chem. Rev., 44(1),
233-244, (1949).
R.C. Reid, J.M. Prausnitz and T.K. Sherwood, "The Properties of Gases
and Liquids", 3rd edition, McGraw-Hill, New York, 1977.
425
P.J. Rossky and M. Karplus, "Solvation. A molecular dynamics study of
a dipeptide in water", J. Am. Chem. Soc., 101(8), 1913-1937, (1979).
H. Saad and E. Gulari, "Diffusion of liquid hydrocarbons in supercritical
CO 2 by photon correlation spectroscopy", Ber. Bunsenges. Phys. Chem.,
88, 834-837, (1984).
E.G. Scheibel, "Liquid diffusivities", Ind. Eng. Chem., 46(9), 2007-2008,
(1954).
G. Schmidt and H. Wenzel, "A modified van der Waals type equation of
state", Chem. Eng. Sci., 35, 1503-1512, (1980).
W.M. Schmitt, private communication, 1984.
R.K.
Shah and A.L. London, "Laminar Flow Forced Convection
Academic Press, New York, 1978.
"Heat transfer
W. Shitsman,
in Ducts",
helium, carbon dioxide,
to supercritical
analysis of thermodynamic and transport properties and experiand water:
mental data", Cryogenics, 77-83, February 1974.
K. Singer, A. Taylor and J.V.L. Singer, "Thermod.,namic and structural
properties of liquids modelled by '2 Lennard-Jones centres' pair potenMol. Phys.,
33(6), 1757-1795, (1977).
tials",
J.C. Slattery and R.B. Bird, "Calculation of the diffusion coefficient
of dilute gases and the self-diffusion coefficient of dense gases",
AIChEJ.,
4(2), 137-142, (1958).
Redlich-Kwong equation
constants from a modified
G. Soave, "Equilibrium
of state",
Chem. Eng. Sci., 27, 1197-1203, 1972.
H.E. Stanley,
Oxford University
Press,
K. Stefan and K. Lucas,
New York, 1979.
F.H. Stillinger
Phenomena",
and Critical
to Phase Transitions
"Introduction
New York, 1971.
"The Viscosity
of Dense Fluids",
and A. Rahman, "Improved simulation
Plenum Press,
of liquid
water
by
molecular dynamics", J. Chem. Phys., 60(4), 1545-1557, (1974).
W.B. Streett, D.J. Tildesley
and an improved potential
of molecular liquids",
and G. Saville, "Multiple time step methods
function for molecular dynamics simulations
Computer Modelling of Matter, P. Lykos, ed.,
ACS Symposium Series #86, ch. 13, 144-158, 1978.
U.W. Suter, "Conformational characteristics of poly(methylvinyl ketone)s
and of simple model ketones", J. Am. Chem Soc., 101, 6481-6496, (1979).
426
I. Swaid and G.M. Schneider, "Determination of binary diffusion coefficients
of benzene and some alkylbenzenes in supercritical C02 between 308
and 328 K in the pressure range 80 to 160 bar with supercritical fluid
chromatography", Ber. Bunsenges. Phys. Chem., 83, 969-974, (1979).
G.I. Taylor,
"Dispersion of soluble matter in solvent flowing slowly
through a tube", Proc. Roy. Soc. (London), A219, 186-203, (1953).
G.I. Taylor, "Diffusion and mass
(London), B67, 857-869, (1954).
transport in tubes", Proc. Phys. Soc.
L.S. Tee, S. Gotoh and W.E. Stewart, "Molecular Parameters for normal
fluids", Ind. Eng. Chem., 5(3), 356-363, (1966).
A.S. Teja, "The correlation
and prediction of diffusion coefficients
in liquids using a generalized corresponding states principle", AIChE
·annual meeting, Los Angeles, CA, 11-1982.
D.N. Theodorou and U.W. Suter, "Geometrical considerations in model systems
with periodic boundary conditions", J. Chem. Phys., in press.
Y.V. Tsekhanskaya, "Diffusion in the system p-nitrophenol-water
critical region", Russ. J. Phys. Chem., 42(4), 532-533, (1966).
in the
Y.V. Tsekhanskaya, "Diffusion of naphthalene in carbon dioxide near the
liquid-gas critical point", Russ. J. Phys. Chem., 45(5), 744, (1971).
Jo van der Waals, Ph.D. Thesis, University of Leiden, 1873.
N.G. van Kampen, "Stochastic Processes in Physics and Chemistry", NorthHolland, Amsterdam, 1983.
L. Verlet, "Computer 'experiments' on classical fluids. I. Thermodynamical
properties
of Lennard-Jones molecules", Phys. Rev., 159(1), 98-103,
(1967).
T. Wainwright and B.J. Alder, "Molecular dynamics computations for the
hard sphere system", Nuovo Cimento, Supplemento al vol. IX, Ser IX,
No. 1, 30 Trimestre, 116-132, (1958).
B. Widom, "Intermolecular forces and the nature
Science, 157, 375-382, (1967).
of the liquid state",
C.R. Wilke and P. Chang, "Correlations of diffusion coefficients in dilute
solutions", AIChE J., 1(2), 264-270, (1955).
J.E. Williamson,
K.E. Bazaire and C.J. Geankoplis, "Liquid-phase mass
transfer at low Reynolds numbers", Ind. Eng. Chem. Fund., 2, 126-129,
(1963).
E.J. Wilson and C.J. Geankoplis, "Liquid mass transfer at very low Reynolds
numbers in packed beds", Ind. Eng. Chem. Fund., 5, 9-14, (1966).
427