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Theoretical Investigation of Thermodiffusion
(Soret Effect) in Multicomponent Mixtures
by
Alireza Abbasi
A thesis submitted in conformity with requirements for the degree of Doctor of
Philosophy Graduate Department of Chemical Engineering and Applied Chemistry
University of Toronto
©Copyright by Alireza Abbasi 2010
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Abstract
Theoretical Investigation of Thermodiffusion (Soret Effect) in Multicomponent
Mixtures
By: Alireza Abbasi
A thesis submitted in conformity with requirements for the degree of Doctor of
Philosophy Graduate Department of Chemical Engineering and Applied Chemistry
University of Toronto 2010
Thermodiffusion is one of the mechanisms in transport phenomena in which molecules
are transported in a multicomponent mixture driven by temperature gradients.
Thermodiffusion in associating mixtures presents a larger degree of complexity than non-
associating mixtures, since the direction of flow in associating mixtures may change with
variations in composition and temperature. In this study a new activation energy model is
proposed for predicting the ratio of evaporation energy to activation energy. The new
model has been implemented for prediction of thermodiffusion for acetone-water,
ethanol-water and isopropanol-water mixtures. In particular, a sign change in the
thermodiffusion factor for associating mixtures has been predicted, which is a major step
forward in modeling of thermodiffusion for associating mixtures.
In addition, a new model for the prediction of thermodiffusion coefficients for linear
chain hydrocarbon binary mixtures is proposed using the theory of irreversible
thermodynamics and a kinetics approach. The model predicts the net amount of heat
transported based on an available volume for each molecule. This model has been found
to be the most reliable and represents a significant improvement over the earlier models.
Also a new approach to predicting the Soret coefficient in binary mixtures of linear chain
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and aromatic hydrocarbons using the thermodynamics of irreversible processes is
presented. This approach is based on a free volume theory which explains the diffusivity
in diffusion-limited systems. The proposed model combined with the Shukla and
Firoozabadi model has been applied to predict the Soret coefficient for binary mixtures of
toluene and n-hexane, and benzene and n-heptane. Comparisons of theoretical results
with experimental data show a good agreement. The proposed model has also been
applied to estimate thermodiffusion coefficients of binary mixtures of n-butane & carbon
dioxide and n-dodecane & carbon dioxide at different temperature. The results have also
been incorporated into CFD software FLUENT for 3-dimensional simulations of
thermodiffusion and convection in porous media. The predictions show the
thermodiffuison phenomenon is dominant at low permeabilities (0.0001 to 0.01), but as
the permeability increases convection plays an important role in establishing a
concentration distribution.
Finally, the activation energy in Eyring’s viscosity theory is examined for associating
mixtures. Several methods are used to estimate the activation energy of pure components
and then extended to mixtures of linear hydrocarbon chains. The activation energy model
based on alternative forms of Eyring’s viscosity theory is implemented to estimate the
thermodiffusion coefficient for hydrocarbon binary mixtures. Comparisons of theoretical
results with the available thermodiffusion coefficient data have shown a good
performance of the activation energy model.
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Acknowledgements
I sincerely appreciate the guidance provided by Professor M. Kawaji and Professor M.Z.
Saghir during my graduate study on thermodiffusion research.
My appreciation goes to the University of Toronto’s School of Graduate Studies and
Ontario Graduate Scholarship Program for providing me with a graduate scholarship
during my study. Furthermore I would like to express my appreciation for the financial
support provided by my thesis supervisors during the entire period of my study.
Lastly, I would like to thank my sisters, my parents and my wife for their encouragement,
love and support.
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To My Family
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Table of contents
Abstract .......................................................................................................................... i
Acknowledgements ....................................................................................................... iv
Table of contents .......................................................................................................... vi
List of Tables ................................................................................................................ ix
List of Figures ............................................................................................................... xi
Nomenclature ............................................................................................................. xiv
1. Introduction ........................................................................................................... 1
1.1. Literature review .............................................................................................. 2
1.2. Basics concepts and equations for diffusion phenomena ................................... 7
1.3. Objectives ...................................................................................................... 11
2. Theoretical models for thermodiffusion calculation .......................................... 14
2.1. Phenomenological and kinetic approaches in deriving the net heat of transport
18
2.2. Mass transfer in multicomponent mixtures ..................................................... 21
2.3. Procedure for the calculation of molecular and thermodiffusion coefficients .. 23
3. Equation of State: PC-SAFT ............................................................................... 24
3.1. Hard chain contribution.................................................................................. 25
3.2. Dispersive contribution .................................................................................. 26
3.3. Association contribution ................................................................................ 28
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4. A new approach to evaluate the thermodiffusion factor for associating mixtures
30
4.1. Ratio of evaporation energy to activation energy in associating mixtures ....... 31
4.2. Results and discussion ................................................................................... 42
4.3. Summary ....................................................................................................... 50
5. A new approach to estimate the thermodiffusion coefficients for linear chain
hydrocarbon binary mixtures ..................................................................................... 51
5.1. Free volume ................................................................................................... 51
5.2. Results and discussion ................................................................................... 54
5.3. Summary ....................................................................................................... 61
6. Theoretical and experimental comparison of the Soret effect for binary
mixtures of toluene & n-hexane and benzene & n-heptane ....................................... 71
6.1. Ratio of evaporation energy to activation energy in non-associating mixtures 71
6.2. Results and discussion ................................................................................... 74
6.3. Summary ....................................................................................................... 81
7. Evaluation of the activation energy of viscous flow for a binary mixture in
order to estimate the thermodiffusion coefficient ...................................................... 82
7.1. Activation energy of viscous flow of a pure component ................................. 82
7.2. Activation energy of viscous flow for a binary mixture .................................. 86
First approach ........................................................................................................ 86
Second approach .................................................................................................... 87
7.3. Results and discussion ................................................................................... 89
Activation energy of a mixture ............................................................................... 89
Thermodiffusion coefficients ................................................................................. 94
7.4. Summary ....................................................................................................... 99
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8. Study of Thermodiffusion of Carbon Dioxide in Binary Mixtures of n-Butane &
Carbon Dioxide and n-Dodecane & Carbon Dioxide in Porous Media................... 101
8.1. Mathematical Model .................................................................................... 102
8.2. Model Description ....................................................................................... 104
8.3. Solution Technique and Mesh Sensitivity ..................................................... 106
8.4. Numerical Results ........................................................................................ 109
Density variation ................................................................................................. 110
Calculation of thermodiffusion coefficient for binary mixtures ............................. 113
Compositional variation ...................................................................................... 122
Separation ratio ................................................................................................... 127
8.5. Summary ..................................................................................................... 128
9. Conclusions and recommendations ................................................................... 130
10. References .......................................................................................................... 135
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List of Tables
Table 4-1: 0i and
i values for acetone-water, ethanol-water and isopropanol-water
mixtures. ....................................................................................................................... 39
Table 4-2: Pure component parameters from PC-SAFT EoS. ........................................ 41
Table 4-3: Binary interaction parameter, ijk , for acetone-water, ethanol-water and
isopropanol-water mixtures. .......................................................................................... 41
Table 5-1: Pure component parameters from PC-SAFT EoS (Gross and Sadowski, 2001).
...................................................................................................................................... 54
Table 5-2: Thermodiffusion coefficients x 1012
(m2/sK) in 50% mole fraction for nCi-C10
(i=5, 6, 7, 15, 16, 17, 18), nCi-C12 (i=5, 6, 7, 8, 9), and nCi-C18 (i=5, 6, 7, 8, 9, 12)
mixtures at 298.15 K. Method 1: New net heat of transport model, Eq. 5.3; Method 2:
Haase model, Eq. 2.14; Method 3: Kempers model, Eq. 2.15; Method 4: Shukla &
Firoozabadi model, Eq. 2.19, considering i = 4. Experimental data are extracted from
Yan et al. 2008 and Blanco et al. 2007, 2008. ................................................................ 57
Table 5-3: Thermodiffusion coefficients x 1012
(m2/sK) in 50% wt fraction for nCi-C12
(i=5, 6, 7, 8, 9) and nCi-C18 (i=5, 6, 7, 8, 9, 12) mixtures at 298.15 K. Method 1: New net
heat of transport model, Eq. 5.3; Method 2: Haase model, Eq. 2.14; Method 3: Kempers
model, Eq. 2.15; Method 4: Shukla & Firoozabadi model, Eq. 2.19, considering k = 4.;
Method 5: Shukla & Firoozabadi model , Eq. 2.19, considering i calculated by Yan et
al. ( 2008). Experimental data are extracted from Yan et al. 2008 and Blanco et al. 2007,
2008 .............................................................................................................................. 58
Table 5-4: The activation energy x10-4
(j/mol) in 50% mole fraction nCi-C10 (i=5, 6, 7,
15, 16, 17, 18), nCi-C12 (i=5, 6, 7, 8, 9), and nCi-C18 (i=5, 6, 7, 8, 9, 12) mixtures at
298.15 K.(Example of larger and smaller molecules in a binary mixture: C12 is the larger
molecule and nCi (i=5, 6, 7, 8, 9) are the smaller molecules in binary mixture of C12-nCi
(i=5, 6, 7, 8, 9)) ............................................................................................................. 59
Table 5-5: The activation energy x10-4
(j/mol) in 50% wt fraction for nCi-C12 (i=5, 6, 7,
8, 9) and nCi-C18 (i=5, 6, 7, 8, 9, 12) mixtures at 298.15 K. (Example of larger and
smaller molecules in a binary mixture: C12 is the larger molecule and nCi (i=5, 6, 7, 8, 9)
are the smaller molecules in binary mixture of C12-nCi (i=5, 6, 7, 8, 9)) ......................... 60
Table 6-1: Pure component parameters from PC-SAFT EoS (Gross and Sadowski, 2001).
...................................................................................................................................... 74
x
Table 6-2: The ratio of evaporation energy to activation energy,i for the binary
mixtures of toluene and n-hexane at different temperatures. .......................................... 78
Table 6-3: The ratio of evaporation energy to activation energy, i for the binary
mixtures of benzene and n-heptane at different temperatures. ........................................ 79
Table 6-4: Soret coefficient x 103 (K
-1) for the binary mixtures of toluene and n-hexane
at different temperatures. Experimental data was extracted from Wittko and Kohler,
2007. ............................................................................................................................. 80
Table 6-5: Soret coefficient x 103 (K
-1) for the binary mixtures of benzene and n-
heptane at different temperatures. Experimental data was extracted from Wittko and
Kohler, 2007. ................................................................................................................ 81
Table 7-1: Activation energy functions for a pure component. ...................................... 85
Table 7-2: Activation energy functions for a hydrocarbon mixture. ............................... 88
Table 7-3: The relative permittivity i of pure materials of nCi (i=5, 6, 7, 8, 9, 10, 12,
15, 16, 17, 18) at 298.15 K obtained by Kotas and Valesova’s approach (1986)............. 88
Table 7-4: The relative permittivity mix of 50% mole fraction mixtures of nCi-C10 (i=5,
6, 7, 15, 16, 17, 18), nCi-C12 (i=5, 6, 7, 8, 9), and nCi-C18 (i=5, 6, 7, 8, 9, 12) at 298.15 K.
mix is obtained by Peon–Iglesias approach (1994). ....................................................... 89
Table 8-1: The physical properties of porous medium. ................................................ 105
Table 8-2: Pure component parameters from PC-SAFT EoS ((Gross and Sadowski,
2001). .......................................................................................................................... 109
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List of Figures
Figure 4-1: Variations of i using Eq. 4.5 .................................................................... 34
Figure 4-2-a: Activation energy calculated with Eq. 4.3 in an acetone-water mixture. .. 36
Figure 4-2-b: Activation energy calculated with Eq. 4.3 in an ethanol-water mixture. ... 37
Figure 4-2-c: Activation energy calculated with Eq. 4.3 in an isopropanol-water mixture.
...................................................................................................................................... 38
Figure 4-3-a: Variations of i calculated using the new model given by Eq. 4.6 in an
acetone-water mixture. .................................................................................................. 43
Figure 4-3-b: Variations of i calculated using the new model given by Eq. 4.6 in an
ethanol-water mixture. ................................................................................................... 44
Figure 4-3-c: Variations of i calculated using the new model given by Eq. 4.6 in an
isopropanol-water mixture. ............................................................................................ 45
Figure 4-4-a: Evaluation of thermodiffusion factor for an acetone-water mixture. ........ 47
Figure 4-4-b: Evaluation of thermodiffusion factor for an ethanol-water mixture......... 48
Figure 4-4-c: Evaluation of thermodiffusion factor for an isopropanol-water mixture. . 49
Figure 4-5: Evaluation of Soret coefficient for an ethanol-water mixture. .................... 50
Figure 5-1-a: Thermodiffusion coefficients in 75% mole fraction of C10 in nCi-C10 (i=5,
6, 7) mixtures. ............................................................................................................... 62
Figure 5-1-b: Thermodiffusion coefficients in 50% mole fraction of C10 in nCi-C10 (i=5,
6, 7) mixtures. ............................................................................................................... 63
Figure 5-1-c: Thermodiffusion coefficients in 25% mole fraction of C10 in nCi-C10 (i=5,
6, 7) mixtures. ............................................................................................................... 64
Figure 5-2-a: Thermodiffusion coefficients in 75% mole fraction of C12 in nCi-C12 (i=5,
6, 7, 8, 9) mixtures......................................................................................................... 65
Figure 5-2-b: Thermodiffusion coefficients in 50% mole fraction of C12 in nCi-C12 (i=5,
6, 7, 8, 9) mixtures......................................................................................................... 66
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Figure 5-2-c: Thermodiffusion coefficients in 25% fraction of C12 in nCi-C12 (i=5, 6, 7, 8,
9) mixtures. ................................................................................................................... 67
Figure 5-3-a: Thermodiffusion coefficients in 25% mole fraction of C18 in nCi-C18 (i=5,
6, 7, 8, 9) mixtures......................................................................................................... 68
Figure 5-3-b: Thermodiffusion coefficients in 50% mole fraction of C18 in nCi-C18 (i=5,
6, 7, 8, 9) mixtures......................................................................................................... 69
Figure 5-3-c: Thermodiffusion coefficients in 25% mole fraction of C18 in nCi-C18 (i=5,
6, 7, 8, 9) mixtures......................................................................................................... 70
Figure 6-1: Pure Activation energy of toluene and n-hexane (j/mol) at different
temperatures. ................................................................................................................. 76
Figure 6-2: Pure Activation energy of benzene and n-heptane (j/mol) at different
temperatures. ................................................................................................................. 77
Figure 7-1: The activation energy of pure hydrocarbon components at 298.15 K. ......... 91
Figure 7-2: Comparison of estimated activation energy of binary hydrocarbon mixtures
of C10-nCi (i=5, 6, 7, 15, 16, 17, 18) at 298.15 K. .......................................................... 92
Figure 7-3: Comparison of estimated activation energy of binary hydrocarbon mixtures
of C12-nCi (i=5, 6, 7, 8, 9) at 298.15 K. .......................................................................... 93
Figure 7-4: Comparison of estimated activation energy of binary hydrocarbon mixtures
of C18-nCi (i=5, 6, 7, 8, 9, 12) at 298.15 K. .................................................................... 93
Figure 7-5: Thermodiffusion coefficients x 10-12
(m2/sK) in 50% mole fraction for nCi-
C10 (i=5, 6, 7, 15, 16, 17, 18), nCi-C12 (i=5, 6, 7, 8, 9), and nCi-C18 (i=5, 6, 7, 8, 9, 12) at
298.15 K. Experimental data are extracted from Yan et al. 2008 and Blanco1 et al. 2007,
2008. ............................................................................................................................. 96
Figure 7-6: Thermodiffusion coefficients x 10-12
(m2/sK) in 50% mass fraction for nCi-
C12 (i=5, 6, 7, 8, 9) and nCi-C18 (i=5, 6, 7, 8, 9, 12) at 298.15 K. Experimental data are
extracted from Yan et al. 2008 and Blanco1 et al. 2007, 2008. ...................................... 97
Figure 7-7: The difference between the predicted thermodiffusion coefficients and the
experimental values for nCi-C10 (i=5, 6, 7, 15, 16, 17, 18), nCi-C12 (i=5, 6, 7, 8, 9), and
nCi-C18 (i=5, 6, 7, 8, 9, 12) at 298.15 K. ........................................................................ 98
Figure 7-8: The difference between the predicted thermodiffusion coefficients and the
experimental values for nCi-C12 (i=5, 6, 7, 8, 9) and nCi-C18 (i=5, 6, 7, 8, 9, 12) at 298.15
K. .................................................................................................................................. 99
Figure 8-1: Schematic of the porous medium .............................................................. 105
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Figure 8-2: Temperature profile in horizontal and vertical directions. ......................... 106
Figure 8-3: Variation of Nusselt number with mesh NxNxN in 3D ............................. 108
Figure 8-4-a: Density variation in horizontal direction for permeability of 0.001md (n-
butane & carbon dioxide mixture). .............................................................................. 111
Figure 8-4-b: Density variation in horizontal direction for permeability of 0.001md (n-
dodecane & carbon dioxide mixture). .......................................................................... 112
Figure 8-4-c: Density variation in vertical direction for permeability of 0.001md. ...... 113
Figure 8-5-a: n-Butane thermodiffusion coefficients as a function of carbon dioxide in n-
butane & carbon dioxide mixtures. (present model) ..................................................... 116
Figure 8-5-b: n-Butane thermodiffusion coefficients as a function of carbon dioxide in
n- butane & carbon dioxide mixtures. (Sukula and Firoozabadi model, 1998) .............. 117
Figure 8-5-c: n-Dodecane thermodiffusion coefficients as a function of carbon dioxide
n-dodecane & carbon dioxide mixtures. (present model)............................................. 118
Figure 8-5-d: n-Dodecane thermodiffusion coefficients as a function of carbon dioxide
n-dodecane & carbon dioxide mixtures. (Sukula and Firoozabadi model, 1998)........... 119
Figure 8-5-e: Carbon dioxide thermodiffusion coefficient for permeability of 0.001md.
.................................................................................................................................... 120
Figure 8-5-f: Carbon dioxide thermodiffusion coefficient for permeability of 0.001md.
.................................................................................................................................... 121
Figure 8-6-a: Carbon dioxide mass fraction in horizontal direction (n-butane & carbon
dioxide mixture). ......................................................................................................... 123
Figure 8-6-b: Carbon dioxide mass fraction in vertical direction (n-butane & carbon
dioxide mixture). ......................................................................................................... 124
Figure 8-6-c: Carbon dioxide mass fraction horizontal direction (n-dodecane & carbon
dioxide mixture). ......................................................................................................... 125
Figure 8-6-d: Carbon dioxide mass fraction in vertical direction (n-dodecane & carbon
dioxide mixtures). ....................................................................................................... 126
Figure 8-7: Separation ratio of carbon dioxide in n-butane & carbon dioxide and n-
dodecane & carbon dioxide mixtures. .......................................................................... 128
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Nomenclature
Symbols
ja The coefficient defined by Eq. 3.14
ja0 The universal coefficients in Eq. 3.14
ja1 The universal coefficients in Eq. 3.14
ja2 The universal coefficients in Eq. 3.14
nba The influence coefficients for the neighboring cells in Eq. 8.7
pa The center coefficient in Eq. 8.7
b The contribution of the constant part of the source term in Eq. 8.7
jb The coefficient defined by Eq. 3.15
jb0 The universal coefficients in Eq. 3.15
jb1 The universal coefficients in Eq. 3.15
jb2 The universal coefficients in Eq. 3.15
B The matrix defined by Eq. 2.22
Bij The components of matrix B defined by Eq. 2.22
c The total molar density, [mol/m3]
ic The molar density of component i, [mol/m3]
C The packing parameter
1C The abbreviation defined by Eq. 3.12
xv
2C The abbreviation defined by Eq. 3.13
minC The minimum concentration of a specific component in the porous
medium, [mol/m3]
maxC The maximum concentration of a specific component in the porous
medium, [mol/m3]
id The temperature dependent segment diameter of component i, [0
A ]
xdT The temperature gradient in x direction, [K/m]
ydT The temperature gradient in y direction, [K/m]
zdT The temperature gradient in z direction, [K/m]
M
D The Fick’s or molecular diffusion coefficient for a binary mixture, [m2/s]
T
D The conventional binary thermodiffusion coefficient, [m2/(s.K)]
*
11D The Fick’s or molecular diffusion coefficient for the molar diffusion flux
with a molar average reference velocity in a porous medium, [m2/s]
MD The Fick’s diffusion coefficient matrix with elements
mol
ikD (i, k = 1, 2, 3...n-1)
TD The thermodiffusion coefficient vector with element s
mol
TiD (i = 1, 2, 3...n-1)
0
ijD The Fick’s or molecular diffusion coefficient of infinitely diluted in a
binary mixture with a molar average reference velocity, [m2/s]
D~
ij The Maxwell-Stephane diffusion coefficient between component i and j,
[m2/s]
mass
ikD The Fick’s or molecular diffusion coefficient for the mass diffusion flux
with a mass average reference velocity, [m2/s]
mol
ikD The Fick’s or molecular diffusion coefficient for the molar diffusion flux
with a molar average reference velocity, [m2/s]
xvi
*
1TD The thermodiffusion coefficient for the molar diffusion flux with a molar
average reference velocity in a porous medium, [m2/(s.K)]
mass
TiD The thermodiffusion coefficient for the mass diffusion flux with a mass
average reference velocity, [m2/(s.K)]
mol
TiD The thermodiffusion coefficient for the molar diffusion flux with a molar
average reference velocity, [m2/(s.K)]
if The molar fugacity of component i, [Pa]
g The gravitational acceleration vector [m/s2]
hs
iig The radial pair distribution function for segments of component i
hs
ijg The function defined by Eq. 3.5
h The Plank constant, [J·s]
kH The partial molar enthalpy of component k, [J/mol]
1I The abbreviation defined by Eq. 3.11
2I The abbreviation defined by Eq. 3.11
ij The mass diffusion flux of component i with respect to a mass average
reference velocity, [kg/(m2.s)]
ij The molar diffusion flux of component i with respect to a molar average
reference velocity, [mol/(m2.s)]
qj The total heat flux, [J/(m2.s)]
*
qj The heat flux with diffusional enthalpy flux subtracted off, [J/(m2.s)]
J The molar diffusion flux vector with elements ij (i = 1, 2, 3...n-1)
k The permeability of a porous medium
ji BA
k The association volume between sites iA and jB
xvii
Bk The Boltzmann constant, [J/K]
fk The fluid thermal conductivity in a porous medium, [J/( s.m.K)]
ijk The binary interaction parameter of components i and j
Pk The porous thermal conductivity, [J/( s.m.K)]
ikL The phenomenological coefficients under the frame of reference moving
with mass average velocity
iqL The phenomenological coefficients under the frame of reference moving
with mass average velocity
qkL The phenomenological coefficients under the frame of reference moving
with mass average velocity
qqL The phenomenological coefficients under the frame of reference moving
with mass average velocity
im The number of segments in a chain of component i
m The mean segment number in the mixture
kM The molecular weight of component k, [kg/mol]
mixM The molecular weight of the mixture, [kg/mol]
n The number of components in mixture
AVN The Avogadro’s number
q The separation ratio in a porous medium
kQ The heat transport of component k defined under the frame of the
reference moving with mass average velocity, [J/Kg]
kQ The net heat transport of component k defined under the frame of the
reference moving with molar average velocity, [J/mol]
xviii
*
kQ The net heat transport of component k defined under the frame of the
reference moving with mass average velocity, [J/Kg]
P The pressure, [Pa]
R The ideal gas constant, 8.314 [J/(mol.K)]
R The pressure-based solver scaled residual defined by Eq. 8.7
TS The Soret coefficient, [1/K]
t The time, [s]
T The temperature, [K]
u The velocity component in x direction, [m/s]
iU The partial internal energy of component i, [J/mol]
mixVapU , The evaporation energy of the mixture, [J/mol]
v The velocity component in y direction,[ m/s]
iv The velocity of component i, [m/s]
mass
avev The mass average velocity, [m/s]
mol
avev The molar average velocity, [m/s]
iV The molar volume of component i, [m3/mol]
mixV The molar volume of the mixture, [m3/mol]
inorV , The molar volume of component i at its normal boiling point, [m3/mol]
xix
V The velocity vector, [m/s]
w The velocity component in z direction, [m/s]
iw The mass fraction of component i
HW The energy needed to detach a molecule from its neighbors, [J/mol]
LW The energy given up when one molecule fills a hole, [J/mol]
x The spatial coordination in x-direction, [m]
ix The mole fraction of component i
iA
X The fraction of A sites on molecule i that do not form associating bonds
with other active sites on molecule j
y The spatial coordination in y-direction, [m]
z The spatial coordination in z-direction, [m]
Z The compressibility factor
dispZ The dispersive part compressibility factor
assocZ The association part compressibility factor
hcZ The hard-chain part compressibility factor
hsZ The hard-chain contribution to
hcZ
Greek Letters
T The thermodiffusion factor
i The coefficient defined by Eq. 4.5
The pressure, temperature, and composition in iteration
xx
i The activity coefficient of component i
The errors between iterations defined by Eq. 8.7
i The depth of the potential well component i, [J]
ij The average depth of potential well of components i and j, [J]
ji BA
The association energy between sites iA and jB
The matrix defined by Eq. 2.22
ij The components of matrix defined by Eq. 2.22
i The temperature independent segment diameter of component i, [0
A ]
*
The entropy generation rate, [J/(K.Kg.s)]
ij The average temperature-independent segment diameter of components i
and j, [0
A ]
jk The Kronecker delta function
n The abbreviation defined by Eq. 3.6
The reduced segment density
32m The abbreviation defined by Eq. 3.9
322 m The abbreviation defined by Eq. 3.9
k The chemical potential of component k, [J/mol]
i The viscosity of component i, [Pa.s]
mix The viscosity of the mixture, [Pa.s]
The porosity of a porous medium
xxi
i The number of molecules which move into the hole left by a molecule i
i The relative permittivity of component i
mix The relative permittivity of the mixture
The total mass density, [kg/m3]
i The mass density of component i, [kg/m3]
n The total number density of molecules
fPC The fluid volumetric heat capacity in a porous medium, [J/(m
3.K)]
PPC The porous volumetric heat capacity, [J/m
3.K]
i The ratio of the energy of evaporation to the activation energy of
component i
0i The i value when the component i is the solute in an infinite dilution
ipure, The ratio of the energy of evaporation to the activation energy of pure
component i
i The i value when the component i is the solvent in an infinite dilution
* The tortuosity in the porous medium
iG
The Gibbs free activation energy of viscous flow of component i, [J/mol]
ji BA
The measure of the association strength between the site A on molecule i
and the site B on molecule j, [0
A ]
iH The viscous flow activation enthalpy of component i, [J/mol]
iS The activation entropy of viscous flow of component i, [J/(K.mol)]
iVapU , The evaporation energy of component i, [J/mol]
xxii
0,viscU The iviscU , value when the component i is the solute in an infinite
dilution
iviscU , The activation energy of component i, [J/mol]
ipureviscU , The activation energy of pure component i, [J/mol]
mixviscU , The activation energy of the mixture, [J/mol]
iviscU , The ivisU , value when the component i is the solvent in an infinite
dilution
T The temperature gradient
kw The mass fraction gradient of component k
kx The mole fraction gradient of component k
1
1. Introduction
Diffusion is one of the major mechanisms of transport phenomena. Molecular diffusion is
the movement of molecules from a higher to lower chemical potential. Thermodiffusion
and pressure diffusion are additional ways in which molecules are transported in a
muticomponent mixture driven by temperature and pressure gradients, respectively. The
thermodiffusion phenomenon was discovered by Ludwig (1856) and Soret (1879), and
named as the Soret effect. The Soret coefficient is the ratio of the thermodiffusion
coefficient to the molecular diffusion coefficient.
Thermodiffusion along with molecular diffusion occurs in many engineering systems and
in nature. Thermodiffusion has a great effect on the concentration distribution in a
muticomponent mixture. The variations of composition and temperature may either
lessen or enhance the separation in mixtures. The thermodiffusion phenomenon also
plays a major role in the hydrodynamic instability analysis in mixtures, investigations of
mineral migration, the mass transport modeling in living matters, and composition
variation studies in hydrocarbon reservoirs.
Furthermore, the study of thermohaline instability with thermodiffusion in a fluid
saturated porous medium is of importance in geophysics, ground water hydrology, soil
science, oil extraction (Parvathy and Patil, 1989). The reason is that the earth’s crust is a
porous medium saturated by a mixture of different types of fluids such as oil, water,
gases and molten form of ores dissolved in fluids. Thermal gradients present between the
2
interior and exterior of the earth’s crust may help convection to set in. The thermal
gradient in crude oil can have a strange effect on the distribution of petroleum
components in an oil deposit. The thermal gradient causes the Soret effect which makes
the larger molecule components to have a tendency to rise, while smaller molecule
components go down to the bottom of the well. However, gravity causes the heavy
components in a fluid to fall and the lighter ones to rise. This means that the distribution
of components in a given well is neither consistent nor readily predictable.
1.1. Literature review
For years, various attempts have been made to generate reliable thermodiffusion
coefficient models for binary mixtures, however, the prediction of diffusion coefficients
for associating and non-associating mixtures is still a new subject. For example, one may
mention the kinetic theory of Prigogine et al. (1950a&b), Haase’s model (1969) based on
the barodiffusion theory, a series of models based on the irreversible thermodynamics and
Erying’s rate theory (Dougherty and Drickamer, 1955a&b; Shukla and Firoozabadi,
1998), Kempers’ model through the maximization of the partition function of two
idealized bulbs (2001), the fluctuation theory by Shapiro (2004), and the lattice model by
Luettmer-Strathmann (2003, 2005). Among these models, the Dougherty and Drickamer
model, Haase model, Kempers model, and Shukla and Firoozabadi model have been
developed using four different approaches to provide relatively reasonable results for
non-associating liquid mixtures.
3
Dougherty and Drickamer (1955a&b) developed a kinetics model for the net heat of
transport in a binary mixture. The net heat of transport for each component in a binary
mixture was suggested to be considered as the energy needed to detach a molecule from
its neighbors and the energy given up when one molecule fills a hole (Denbigh, 1952).
Initially the proposed activation energy by Eyring’s viscosity theory was considered as a
good approximation for calculating the energy needed to detach a molecule from its
neighbors (Glasstone et al., 1941). In Eyring’s theory, the activation energy has to be
supplied for molecular motion. This energy is related to the latent heat of evaporation of a
molecule or a molecular segment making the jump. In 1969, Haase (1969) obtained a
model for the thermodiffuion factor from an analogy between thermodiffusion and
barodiffusion, i.e., pressure diffusion, which is the mass diffusion due to a pressure
gradient.
In a model published by Kempers (1989, 2001), Kempers suggested a thermostatic
strategy based on statistical mechanics for the estimation of a thermodiffuion factor. The
single assumption in Kempers’ model is that a steady state will be achieved by a
maximum number of microstates. The thermodifuion factor has been derived by
maximizing the canonical partition function of a two idealized bulb system with two
constraints. Shukla and Firoozabadi (1998) proposed a model for the net heat of
transport in a binary mixture. Their kinetics model was based on Dougherty and
Drickamer’s kinetics approach.
4
In the Shukla and Firoozabadi model, the activation energy was considered to be one-
fourth of the latent heat of evaporation energy. Yan et al. (2008) modified the Shukla and
Firoozabadi model and presented a new model for the evaluation of the ratio of
evaporation energy to activation energy. Yan et al.’s model has been used for
hydrocarbon binary mixtures taking into consideration the vapor-liquid equilibrium
variation. In their approach, the mixture properties, i.e., the viscosity of the mixture and
the energy of vaporization in the mixture were used for estimating the ratio of the energy
of evaporation to the activation energy for a larger component.
All four different thermodiffuion models (Dougherty and Drickamer, Haase, Kempers,
and Shukla and Firoozabadi) along with Yan et al.’s model demonstrated a good starting
point in calculating the net heat of transport in non-associating mixtures, but they did not
satisfy completely the thermodiffusion estimation (Abbasi et al., 2009a&b). The
performance of the existing models shows that the mechanism of thermodiffusion is still
not understood very well. Therefore, a quantitative study needs to be performed to
provide a reliable thermodiffusion model for non-associating mixtures (Abbasi et al.,
2009a&b).
A sign change in the Soret coefficient with compositional or temperature variation in
associating mixtures is one of the most interesting subjects in thermodiffuison research.
This particular phenomenon is mainly observed in aqueous alkanol solutions where the
hydrogen bonding exists. The sign change was reported by Tichacek et al., 1956; Poty et
al., 1974; Kolodner et al., 1988; Zhang et al., 1999; Platten, 2000; Platten et al., 2002;
5
Dutrieux et al., 2002; Costeseque et al., 2003; and Kita et al., 2004a,b. Only a few
successful theoretical models were developed to investigate this phenomenon. Nieto-
Draghi (2003), Nieto-Draghi et al. (2005), and Rousseau et al. (2004) have successfully
predicted the sign change in the thermodiffusion factor for some water alcohol mixtures
using molecular dynamics (MD) simulation. In addition, Prigogine et al.’s model
(1950a&b) and Luettmer-Strathmann’s model (2003, 2005) were able to quantitatively
show a sign change in the thermodiffusion factor, whereas the other models such as the
Kempers model (1989, 2001), Haase model (1969) and Shukla and Firoozabadi model
(1998) have failed to predict the sign change. Prigogine et al.’s model considered
thermodiffusion to be a stepwise activated process by defining the activation energy to be
the potential energy necessary to break cohesive bonds. In their model, the energy of
evaporation was used to calculate the activation energy for the thermodiffuion factor.
Luettmer-Strathmann’s model (2003, 2005) is based on a thermostatic approach. In their
model, the fluid mixture in a two-chamber system is assumed to be of a cubic lattice
configuration. The expression for the canonical partition function of the system with two
chambers at different temperatures will lead to the probability of finding different species
in the mixtures and an expression for the Soret coefficient. Following Shukla and
Firoozabadi’s model (1998), Pan et al. (2007a) used the PC-SAFT equation of state for
predicting thermodiffusion coefficients for associating binary mixtures. The PC-SAFT
equation of state with two adjustable parameters calculated from experimental data
provided a close agreement with experimental data. Particularly, this approach was
found to be capable of predicting a sign change in thermodiffusion factors for associating
6
mixtures such as methanol-water, ethanol-water and isopropanol-water, which was a
major step forward in thermodiffusion studies for associating mixtures. However, their
model could not predict the second sign change in the thermodiffuison factor for
isopropanol-water (Abbasi et al., 2009c). Their proposed model was not able to predict
precisely the thermodiffuison factor for binary mixtures of acetone-water (Abbasi et al.,
2009c). Therefore, an additional theoretical investigation needs to be carried out in order
to achieve a quantitatively reliable prediction of thermodiffuion in small-molecule
associating mixtures.
A variety of approaches have been applied to study thermodiffusion in porous media
(Leahy-Dios et al., 2005; Haugen et al., 2005; Platten, 2006) and convection has been
found to have an important effect on the accuracy of Soret measurements. Costeseque et
al. (2004) carried out diffusion experiments in both free and porous media Soret cells. It
was found that the molecular diffusion and thermodiffusion coefficients in porous media
were related to those in a clear fluid via the tortuosity. However, the ratio of molecular
diffusion coefficient to thermal diffusion coefficient (i.e., Soret coefficient) for binary
mixtures is the same for both configurations. Riley and Firoozabadi (1998) presented a
model to investigate the effects of natural convection and thremodiffusion along with
molecular and pressure diffusion on a single-phase of a binary hydrocarbon mixture in a
horizontal cross-sectional reservoir in the presence of a prescribed linear temperature
field. The permeability was found to have a great effect on the horizontal compositional
variation. Delware et al. (2004) studied these phenomena for a binary system in a square
cavity. Their results showed that in the lateral heating case the Soret effect was found to
7
be weak, whereas in the bottom heating case the Soret effect was more prominent.
Nasrabadi et al. (2006) presented a numerical simulation of two-phase muticomponent
diffusion and natural convection in a porous medium. Thermodiffusion, pressure
diffusion, and molecular diffusion were included in the diffusion expression from the
thermodynamics of irreversible processes. Results showed that the natural convection has
an important role in the phase distribution in the non-isothermal gas and oil medium.
Jaber et al. (2008) simulated the Soret effect for a ternary mixture in a porous cavity
considering variable viscosity and diffusion coefficients. For permeability below 200 md,
the thermodiffusion phenomenon was found to be governing; but above this permeability,
buoyancy convection would become the dominant mechanism. The variation of viscosity
was also found to have a large effect on molecular and thermodiffusion. In the present
work, all of the above models for investigating the effects of natural convection and
thremodiffusion are examined in two dimensional systems with a lateral heating
condition or horizontal and vertical temperature gradients. The lateral heating condition
was just implemented for thermo-convection model in three dimensional systems.
Therefore, a more theoretical study is needed to obtain a better estimation of thermo-
convection effects in porous media such as an oil reservoir (Abbasi et al., 2010b).
1.2. Basics concepts and equations for diffusion phenomena
Before reviewing and discussing the thermodiffusion research, it is essential to define the
diffusion flux and several basic concepts. Two types of diffusion fluxes are often used in
8
the thermodiffusion research. The first one is the mass diffusion flux with a mass average
velocity defined by the following general equation:
mass
aveiii vvj (1.1)
where ij is the mass diffusion flux,
i is the mass density, and iv is the velocity of
component i. mass
avev is an arbitrary reference velocity (mass average velocity) defined by
n
i
ii
mass
ave vwv (1.2)
where iw is the mass fraction of component i and n is the number of components in a
mixture.
The second diffusion flux is the molar diffusion flux defined by the following general
equation:
mol
aveiii vvcj (1.3)
where ij is the molar diffusion flux and ic is the molar density of component i. mol
avev is
an arbitrary reference velocity (molar average velocity) defined by
n
i
ii
mol
ave vxv (1.4)
where ix is the mole fraction of component i.
9
One of the fundamental properties of diffusion flux valid for both mass and molar
definitions is as follows:
n
i
ij 0 (1.5)
n
i
ij 0 (1.6)
Equations 1.5 and 1.6 imply that there are only n-1 independent diffusion fluxes in an n-
component system. The diffusion flux equations when only the concentration and
temperature gradients are considered can be expressed as follows:
1
1
n
k
mass
Tik
mass
iki TDwDj (1.7)
1
1
n
k
mol
Tik
mol
iki TDxDcj (1.8)
where is the total mass density ( n
i
i ) and c is the total molar density of the
mixture ( n
i
icc ). mass
ikD and mol
ikD are the molecular diffusion coefficient matrices
(Fickian diffusion coefficient) corresponding to ij and ij , respectively. mass
TiD and mol
TiD
are the thermodiffusion coefficient corresponding to ij and ij , respectively. kw is a
mass fraction gradient and kx is a mole fraction gradient of component k. T is the
10
temperature and T is a temperature gradient. According to Eq. 1.7 and Eq. 1.8, the
diffusion flux for a binary system can be written as follows:
TDwDj mass
T
mass 11111 (1.9)
TDxDcj mol
T
mol
i 1111 (1.10)
In addition to the conventional binary diffusion coefficient, another two important
quantities, i.e. the Soret coefficient TS and thermodiffusion factor T , have been
defined for thermodiffusion in binary mixtures,
M
T
TD
DS (1.11)
and
M
T
TTD
DTTS (1.12)
where MD and TD are the molecular diffusion coefficient and the thermodiffusion
coefficient, respectively, defined by the following equations:
molmassM DD 1111 D (1.13)
21
1
21
1
xx
D
ww
D mol
T
mass
TT D (1.14)
11
1.3. Objectives
The overall objective of this work is to theoretically and numerically investigate
thermodiffusion phenomena or the Soret effect. It can be divided into the following
specific objectives:
1. To achieve a better physical understanding of the thermodiffusion process by
studying the thermodiffusion-related theories such as thermodynamics of non-
equilibrium states, transport theory, heat transfer, and fluid mechanics.
2. To improve an existing thermodiffusion model for associating mixtures.
Thermodiffusion behaviors in associating mixtures have an important role in
separation processes in the oil industry. The variations of composition and
temperature may either lessen or enhance the separation of mixtures. A new
model for predicting the thermodiffusion phenomena in associating mixtures will
be developed. The new model will be implemented for prediction of
thermodiffusion in acetone-water, ethanol-water and isopropanol-water mixtures.
3. To develop a thermodiffusion model for binary hydrocarbon mixtures.
Developing a thermodiffusion model for non-associating mixtures is under an
increasing demand from the petroleum industry due to its important role in
separation processes of the oil industry. Here, two new models for predicting the
thermodiffusion phenomena in binary hydrocarbon mixtures of linear
hydrocarbon chains and combinations of aliphatic and aromatic compounds will
12
be provided. The binary mixture of linear chain hydrocarbons will be nCi-C12
(i=5, 6, 7, 8, 9), nCi-C18 (i=5, 6, 7, 8, 9, 12) and nCi-C10 (i=5, 6, 7, 15, 16, 17, 18).
The binary mixtures of linear chain and aromatic hydrocarbon will be toluene and
n-hexane, and benzene and n-heptane as well as binary mixtures of nC4 and
carbon dioxide, and nC12 and carbon dioxide.
4. To study the thermo-convection effect in porous media, numerical solutions of
diffusion and convection equations for binary mixtures of n-butane and carbon
dioxide, and n-dodecane and carbon dioxide will be presented. Based on the
thermodiffuison theory such as the thermodynamics in a hydrocarbon reservoir,
and thermodynamics of equilibrium and non-equilibrium states, transport theory,
heat transfer, and fluid mechanics, a better understanding of thermodiffusion
phenomena in oil reservoirs will be achieved. This theory will be incorporated
into CFD software FLUENT. Molecular diffusion and thermodiffusion
expressions will be included in the diffusion terms in order to study diffusion
effects on hydrocarbon reservoirs. The thermodynamic properties such as density
and viscosity of components will be functions of temperature.
The contents of this thesis will be presented in the following order. The Introduction is
presented in Chapter 1. Theoretical models for thermodiffusion calculations will be
presented in Chapter 2, followed by the PC-SAFT (perturbed chain statistical associating
fluid theory) equation of state in Chapter 3. Chapter 4 will provide a new approach to
evaluate the thermodiffusion factor for associating mixtures. A new approach to the
prediction of thermodiffusion in linear chain hydrocarbon binary mixtures will be given
in Chapter 5. Following them, theoretical and experimental comparisons of the Soret
13
effect for binary mixtures of toluene and n-hexane, and benzene and n-heptane will be
presented in Chapter 6. Chapter 7 will provide the evaluation of the activation energy of
viscous flow for a binary mixture in order to estimate the thermodiffusion coefficient.
Three-dimensional modeling of thermo-solutal convection in porous medium containing
binary mixtures of n-butane and carbon dioxide, and n-dodecane and carbon dioxide, will
be given in Chapter 8. Finally, the conclusions and recommendations for future work will
be provided in Chapter 9.
14
2.Theoretical models for thermodiffusion
calculation
In transport theory, there are four postulates, called Curie’s postulates (Groot and Mazur,
1984):
1. The equilibrium thermodynamics relations apply to systems that are not in
equilibrium, provided that the gradients are not too large (quasi-equilibrium
postulate).
2. All fluxes in the system may be expressed by linear relations involving all the
forces (linear postulate).
3. No coupling of fluxes and forces occurs if the difference in tensorial orders of the
flux and force is an odd number (Curie’s postulate).
4. In the absence of magnetic fields the matrix of the coefficients in the flux-force
relations is symmetric (Onsager’s reciprocal relations).
The entropy generation rate of a system without considering any chemical reaction and
viscous dissipation can be given in the following form (Groot and Mazur, 1984):
)(.11 1
112
*
n
n
k
kT
n
k
k
n
k
k
k
kq
MMj
TTj
M
Hj
T
(2.1)
15
where qj is the total heat flux; T is the absolute temperature;
kH is the partial molar
enthalpy, k is the chemical potential,
kM is the molecular weight, andkj is the
diffusion mass flux of component k .
For simplicity,
n
k
k
k
kqq j
M
Hjj
1
*
(2.2)
The thermodynamic force conjugating to *
qj is the temperature gradient, whereas the
chemical potential under constant temperature is the thermodynamic force forkj . The
difference between *
qj and qj illustrates heat transfer due to mass diffusion. According
to the linearity postulate, each force can be written as a linear function of all the forces
(Donald, 1962). In the diffusion flux which is a vector, the effect from higher or lower
order forces can be omitted. The coupling among the vector forces and vector fluxes is
being considered if order of flux and force is an even number. The phenomenological
equations for the heat flux and mass diffusion fluxes can be written as:
1
12
*
)(.11 n
k n
n
k
kTqkqqq
MML
TT
TLj
(2.3)
1
12
)(.11 N
k N
N
k
kTikiqi
MML
TT
TLj
(2.4)
16
where qqL , qkL , iqL , and ikL are the Onsager coefficients [Onsager 1931]. In addition to
the expressions in Eq. 2.2, a new quantity called the net heat of transport is defined to
construct a connection between the phenomenological coefficients (Denbigh, 1952).
n
k
k
k
kq j
M
Qj
1
with 0T (2.5)
kQ is the heat of transport of component k defined under the frame of the same
reference moving velocity. The net heat of transport is defined as follows :
n
k
k
k
kn
k
k
k
kkn
k
k
k
k
qq jM
Qj
M
HQj
M
Hjj
1
*
11
*
(2.6)
*
kQ is defined as the net heat of transport of component k. Substituting Eq. 2.6 into the
entropy generation rate, Eq. 2.1, and applying the irreversible process theory, one can
have.
1
12
1
1
**
)(.11
)(n
k n
n
k
k
ik
n
k n
n
k
k
ikiMM
TLT
TTM
Q
M
QLj
(2.7)
The net heat of transport term in Eq. 2.7 can be written as follows:
17
1
1
**
)(n
k n
n
k
k
ikiqM
Q
M
QLL (2.8)
By considering relations between chemical potential and fugacity, and the relation
between chemical potential and partial molar volume given by Gibbs-Duhem expression
in the absence of the pressure gradient, molar diffusion flux can be written as (Groot and
Mazur, 1984):
2
,,
1
1
1
1
1
1
ln
T
TL
x
f
M
xMxML
xM
Rj iq
PTx
n
k
n
j
N
l l
j
j
jknnjj
ik
nn
i
l
(2.9)
where R is the gas constant, if is the fugacity of component i , and jk is the delta
function, 1jk when kj , and 0jk when kj . On the other hand, the mass
diffusion flux for a muticomponet system fluid mixture in the absence of a pressure
gradient can be expressed by the following equation:
)( TDxDcJ TM (2.10)
where MD and TD are the molecular and thermodiffusion coefficient matrix and vector ,
respectively, x and T are mole fraction and temperature gradient vectors, c is the
molar density of the mixture, and J is the tensor of all component molar mass, ij . By
comparing Eq. 2.8, 2.9, and 2.10, thermodiffusion and molecular diffusion coefficients
can be given by:
18
PTx
n
k
n
j
N
l l
j
j
ikNNjj
ik
NN
mol
ij
l
x
f
M
xMxML
xcM
RD
,,
1
1
1
1
1
1
ln
(2.11)
1
122
)(1 n
k n
n
k
k
ik
iqmol
TiM
Q
M
QL
cTcT
LD (2.12)
where kQ is the net heat of transport of component k under the frame with the molar
average refrence velocity.
For a binary mixture the themodiffusion factor, T based on Eq. 2.4 can be given by:
pT
Txx
,111
21
/
(2.13)
2.1. Phenomenological and kinetic approaches in deriving
the net heat of transport
Two different phenomenological approaches have been adopted for the derivation of the
net heat of transport. In the first method, Haase’s model (1969) is obtained from an
analogy between thermodiffusion with barodiffusion, i.e., pressure diffusion, which is the
mass diffusion due to a pressure gradient. The model for net heat of transport for binary
mixtures is defined by:
19
2211
2211MxMx
MHxHxHQ k
kk
(2.14)
where kH is the partial molar enthalpy, kM is the molecular mass and
kx is the mole
fraction of component k. The second method of phenomenological approach is Kempers
model (1989, 2001). This model was not originally derived from the non-equilibrium
thermodynamics theory. Instead, it was based on a statistical description of a “two-bulb
system”. However, Faissat et al. (1994) proved that the Hasse and Kempers models also
fall in the scope of the theory of non-equilibrium thermodynamics. One can obtain the
expression for the net heat of transport for binary mixtures based on the mass average
velocity frame as follows:
2211
2211VxVx
VHxHxHQ k
kk
(2.15)
where kV is the partial molar volume of component k.
The other approach in determining the net heat of transport is the kinetics approach.
Denbigh (1952) considered the net heat of transport as the amount of energy which must
be absorbed by the region per mole of the component diffusing out in order to maintain
the constancy in the temperature and pressure of the mixture. He suggested that the net
heat of transport for each component in a binary mixture can be expressed in terms of the
energy needed to detach a molecule from its neighbors, HW , and the energy given up
20
when one molecule fills a hole, LW . In this model, HW depends on the type of the
detached molecule while LW is identical for the mixture. The amount of energy supplied
for a molecule filling the hole is considered as:
2H21H1L WxWxW (2.16)
The net heat of transport for each component is suggested to be given by:
kk
LkHkk
x
WWQ
1 (2.17)
It is reasonable to consider that the average number of molecules that fill the hole
depends on the size of the detached molecule. Dougherty and Drickamer (1955a&b)
proposed k as the number of molecules which would move into the hole left by a
molecule k. The net heat of transport for components one and two are defined as follows:
1
2
111 HH W
V
VWQ
(2.18)
2
1
222 HH W
V
VWQ
On the basis of the frame of non-equilibrium thermodynamics and kinetic theory by
Dougherty & Drickamer (1955a&b), Shukla & Firoozabadi
(1998) proposed the
following expression for the net heat of transport for binary mixtures.
21
22112
22
1
11
VxVx
VUxUxUQ k
k
k
k
(2.19)
where kU is partial molar internal energy and
k is the ratio of evaporation energy to
activation energy of component k. Here the k value may be fixed at 4.0 for hydrocarbon
mixtures, as suggested by Firoozabadi and his coworkers (2000).
2.2. Mass transfer in multicomponent mixtures
For an ideal gas, the molecular diffusion coefficient can be expressed by a theoretical
approach. However, for non-ideal mixtures, empirical expressions are usually used. In
multi-component mixtures, the molecular diffusion coefficient, D~
ij, is calculated based
on the binary coefficients. The expression used here is based on Taylor’s approach
(Taylor and Krishna, 1993):
2/ojk
oik
n
j,ik1k
oji
oijij
kijx
DDx
Dx
DD~
(2.20)
where 0
ijD is the molecular diffusion coefficient of component i infinitely diluted in a
binary mixture. According to the theory of mass transfer, The Fick’s molecular diffusion
MD can be expressed in terms of D~
ij as follows:
22
1BDM (2.21)
where B and are matrices, which are defined formulas follows:
x
f
fx
jiD
x
D
xB
jiDD
xB
j
i
i
iij
in
in
ikk ik
kij
inij
iij
1
,~~
,~1
~1
1
(2.22)
In this procedure, the diffusion coefficients in dilute binary mixtures have to be evaluated
before estimating Fick’s diffusion coefficients in any mixtures. For hydrocarbon mixtures
the following equation given by Hayduk–Minhas (Taylor and Krishna, 1993) is used in
this study.
47.1791.0/2.1071.0
,
80 1103.13 TVDV
jinorij
(2.23)
The diffusion coefficients in dilute water-alcohol binary liquid mixtures can be estimated
by the Tyn–Calus method (Taylor and Krishna, 1993):
jjnorinorij TVVD 269.0
,
433.0
,
80 1093.8 (2.24)
where inorV , is the molar volume of component i at its normal boiling point.
23
2.3. Procedure for the calculation of molecular and
thermodiffusion coefficients
After evaluating the net heat of transport and mass transfer coefficient using a proper
Equation of State such as the Perturbed Chain Statistical Associating Fluid Theory (PC-
SAFT), the thermodiffuion coefficient can be determined according to the following
procedure:
1. Specify a mixture and provide parameters for each component.
2. Specify known parameters such as temperature, pressure, and mole fraction of the
mixture.
3. Calculate the required thermodynamic properties for thermodiffusion models,
including 11 xlnd/lnd , 11 x/ , iU , and
iV .
4. Calculate the molecular diffusion coefficients using 0
ijD , Bij and ij matrices.
5. Calculate the Onsager Coefficients, Lij .
6. Calculate the thermodiffuion coefficients.
24
3.Equation of State: PC-SAFT
Besides having proper thermodiffusion coefficient models, it was found that amongst the
reasons for good agreement is using the appropriate equation of state. In associating
molecules, the PC-SAFT equation of state (Perturbed Chain Statistical Associating Fluid
Theory), using two adjustable parameters calculated from experimental data, provides a
good agreement with experimental data (Pan et al., 2007; Abbasi et al., 2009c). In the
case of non-associating mixtures such as hydrocarbons, the properties of the fluid mixture
at equilibrium states were calculated using the Peng–Robinson (PR) equation of state
(Shukla and Firoozabadi, 1998; Yan et al., 2008) and PC-SAFT equation of state (Abbasi
et al, 2009a&b; Pan et al., 2007b). The PC-SAFT equation of state was found to be more
accurate in calculating the thermodynamic properties than cubic equations of state such
as Peng–Robinson equation of state (Pan et al., 2007b; Bataller et al., 2009). Therefore,
PC-SAFT equation of state has been applied to calculate the thermodynamic properties of
the mixtures of interest. PC-SAFT equation of state was derived and described in detail
by Gross and Sadowski (2002, 2003). The compressibility factor, Z, is the sum of the
ideal gas contribution (id), the hard-chain term (hc), the dispersive part (disp), and the
contribution due to association (assoc). The effect of multipole interactions is not taken
into account in this study.
assocdisphc ZZZZ 1 (3.1)
25
3.1. Hard chain contribution
Chapman et al. (1988, 1990) developed an expression for homonuclear hard-sphere
chains based on thermodynamic perturbation theory of first-order (Wertheim, 1984 &
1986) in which comprising m segments is defined by:
i n
hs
ii
nii
hshc gmxZmZ
ln)1( (3.2)
i
imxm (3.3)
where, im is the number of segments in a chain, ix is the mole fraction and hs
iig is the
radial pair distribution function for segments of component i . n is the total number
density of molecules, m is the mean segment number in the mixture. In PC-SAFT
equation of state, the expressions of Mansoori et al. (1971) and Boublik (1970) for the
hard sphere mixtures are used in Eq. 3.2, which can be written as:
330
3
23
3
2
2
30
21
3
3
1
3
1
3
1
hsZ (3.4)
33
2
2
2
3
2
3 1
2
1
3
1
1
ji
ji
ji
jihs
ijdd
dd
dd
ddg (3.5)
where
i
n
iiinn dmx
6
with 0,1,2,3n (3.6)
26
The effective collision diameter of the chain segments, id , is considered to be a function
of temperature T by the Barker–Henderson approach (1967a&b). The specific equation
proposed by Huang and Radosz (Chiew 1991) is given by:
Tk
3exp12.01Td
B
iii
(3.7)
where i is the temperature independent segment diameter (the effective collision
diameter at absolute zero), i stands for the depth of the potential well, and Bk is the
Boltzmann constant.
3.2. Dispersive contribution
The dispersive contribution is calculated from the second-order perturbation suggested by
Barker and Hendreson (1967a&b). In this model, the total interaction between two chains
of spherical molecules is given by the sum of all individual segments interactions. The
compressibility term of dispersive contribution based on an average segment–segment
radial distribution function can be defined by the following equations (Gross and
Sadowski, 2001) :
322
222
1
3212
mIC
ICmm
IZ dis
(3.8)
where
27
332
ij
i j B
ij
jijiTK
mmxxm
and 3
2
322
ij
i j B
ij
jijiTK
mmxxm
(3.9)
j
j
j jmaI
1
6
0
1
and j
j
j jmbI
1
6
0
2
(3.10)
j
j
j mamI
6
0
1 , and j
j
j mbmI
6
0
2 , (3.11)
Here, is the reduced segment density which is equal to 3 defined in Eq 3.6. 1C and 2C
are abbreviations defined as :
22
432
4
2
1
1
21
21227201
1
2811
mmZ
ZC
hc
hc (3.12)
3
23
5
22
1
1
221
40481221
1
82041
mmC
CC (3.13)
Coefficients ma j and mb j depend on the chain length defined by:
jjjj am
m
m
ma
m
mama 210
211
(3.14)
jjjj bm
m
m
mb
m
mbmb 210
211
(3.15)
where ja0 , ja1 , ja2 , jb0 , jb1 , and jb2 have been given by Cross and Sadowski (2001).
The parameters ij and ij for a pair of unlike segments i and j are defined by the
conventional rules:
ijjiij k 1 (3.16)
28
jiij
2
1 (3.17)
where ijk is the interaction parameter.
3.3. Association contribution
Two pure-component parameters determine the associating interactions between the
association site iA and iB of a pure component i . For many systems, the cross-
association between two different associating substances can be determined from pure-
component association parameters. The compressibility term of association contribution
is given by Chapman et al. (1988, 1990):
i j A
A
cTi
A
ji
assoc
j
j
ik
j
Xc
XcxZ
2
11
,
(3.18)
where jc is molar density of component j , iAX is the fraction of A sites on molecule i
that do not form associating bonds with other active sites on molecule j . This value of
iAX can be obtained by solving the following system of equations:
j
BA
B
B
jAV
A
ji
j
i
i
XcNX
1
1 (3.19)
29
where jB indicates summation over all sites and AVN is the Avogadro’s number. ji BA
is a measure of the association strength between the site A on molecule i and the site B
on molecule j .
1exp
3
Tkkg
B
BABA
iji
BAji
jiji
(3.20)
where ji BAk is the association volume and ji BA
is the association energy between sites iA
and jB that can be defined by the following equations (Gross and Sadowski, 2001).
2
jjii
ji
BABABA
(3.21)
3
2
ji
jiBABABA jjiiji kkk
(3.22)
With pure component parameters i , i , im , ijk , iiBA , and iiBAk , the compressibility
factor Z can be obtained by the approach stated above. Using Z as a starting point, the
required thermodynamic properties for thermodiffusion models, including
11 xlnd/lnd , 11 x/ , iU , and iV can be derived easily.
30
4. A new approach to evaluate the
thermodiffusion factor for associating
mixtures
Diffusion behaviors in associating mixtures present a larger degree of complexity than
those in non-associating mixtures. The direction of flow in associating mixtures may
change with the variation of composition and temperature. In this paper a new activation
energy model is proposed for predicting the ratio of evaporation energy to activation
energy. The new model was implemented for prediction of thermodiffusion for acetone-
water, ethanol-water and isopropanol-water mixtures. In particular, this approach is
implemented to predict the sign changes in the thermodiffusion factor for associating
mixtures, which has been a major step forward in thermodiffusion studies for associating
mixtures. In this work the thermodiffusion coefficient was determined for binary
mixtures of acetone-water, ethanol-water and isopropanol-water at different temperatures
with a newly proposed mixing rule for calculating the ratio of evaporation energy to the
activation energy (second adjustable parameter). As described in this chapter,
comparisons of numerical results with benchmark data have led to a completely revised
theory capable of evaluating the thermodiffusion coefficient for binary associating
mixtures.
31
4.1. Ratio of evaporation energy to activation energy in
associating mixtures
One of the major difficulties in predicting the thermodiffusion factor is finding i , the
ratio of the energy of evaporation to the activation energy, in Shukla and Firoozabadi’s
model given by Eq. 2.19. Under the non-equilibrium thermodynamics framework, the
thermodiffusion phenomenon in associating liquid mixtures has been studied by Pan et
al. (2006, 2007a) and Saghir et al. (2004). In their first attempt, they used PC-SAFT and
Cubic Plus Association (CPA) equations of state which are suitable for associating
mixtures. It has been found that in aqueous alcohol solutions, the choice of i = 4
proposed by Shukla and Firoozabadi (1998) for hydrocarbon mixtures along with
Dougherty and Drickamer’s work (1955a&b) did not give a good agreement with
experimental results. By changing the value of i to 10 or more it was possible to obtain
a good match with the experimental data for low water concentrations, but the model
failed to predict the change in the sign of the thermodiffusion factor. These findings led
Pan et al. (2007a) to suggest that i may be a variable quantity rather than a constant
over the entire concentration range.
By suggesting that 10 and 2 are the values for 1 and 2 , respectively, when 1x1
and, similarly 1 and 20 when 0x1 , new expressions for 1 and 2 as a function of
1x , 10 , 2 , 1 and 20 were proposed. Here 1 and 2 represent the effect of unlike
intermolecular interactions and 10 and 20 represent the effect of alike intermolecular
32
interactions. Pan et al. (2007a) proposed two simple mixing rules for calculating the
energy ratio, i :
Mixing rule 1:
1
1
21202
2
1
1
12101
1
ln
ln1
,
ln
ln1
xd
d
xx
xd
d
xx
(4.1)
Mixing rule 2:
1
1
x
2
x
202
1
1
x
1
x
101
xlnd
lnd1
,
xlnd
lnd1
1221
(4.2)
where 1 is the activity coefficient of component 1. Energy of activation for viscous flow
or so called activation energy or viscous energy is the energy required to jump a molecule
or molecular segment into an existing hole and thereby creating a new hole that is
scattered throughout a liquid matrix. In another word, the activation energy of each
component in the mixture represents the energy required to remove the molecule from its
surroundings. Activation energy for viscous flow tends to be one third to one fourth of
heat of vaporization because the molecules remain in the liquid state and intermolecular
forces are replaced. i is the ratio of the energy of vaporization to the energy of
activation or viscous energy of component i.
Prediction of i values requires calculation of the energy of evaporation as well as the
activation energy. The energy of evaporation, ivapU , , is the energy required to bring a
compound from liquid to an ideal gas state. Using PC-SAFT equation of state, an
33
equilibrium condition for specific temperature and pressure can be achieved. Thus
ivapU , calculation for each mole fraction is manageable. However, the calculation of
the activation energy, iviscU , , as a function of mole fraction does not follow any
specific model. Therefore, we propose a mixing rule for estimating the activation energy
as a function of mole fraction as follows:
2,1,ln
ln1,,,
1
1102
i
xd
dUxUxU iivisciviscivisc
(4.3)
where
2,1,,
,,,
,0
0
0
iU
UU
Ui
ivap
ivisc
i
ivap
ivisc
(4.4)
21
5.0
2
1
2
1
2
1
xx
i
ii
i
i
ii
i
i
ii
i
i
Mx
M
Vx
V
Ux
U
(4.5)
where iM is the molecular weigh of component i. The expression
1
11xlnd
lnd is a
reflection of non-ideal compositional dependence of Fick’s diffusion coefficients (Poling
et al., 2001). i is considered for the effect of intermolecular interaction and molecule
size for each component in the mixture (Figure 4-1). According to the free volume
theory, the transfer kinetics of diffusing molecules depends greatly on molecular size and
shape, with small molecules having higher molecular diffusion coefficients (Sung, 1998).
In addition, the interaction between the molecules as a result of intermolecular interaction
34
such as hydrogen bonding has an effect on diffusivity of the molecules (Abbasi et al.,
2008b). The effect of molecular size can be expressed by the geometric average value of
specific volume and molecular weight fraction and the effect of intermolecular
interaction can be defined by of evaporation energy of each component. The tendency of
change in the evaporation energy depends inversely on molecular size in associating
mixture. This can be seen by increasing water concentration in associating mixtures. As a
result, the evaporation energy of the organic component increases, however, it leads to
decrease the volume and molecular weight fraction of organic component in the mixture.
This trend was used in proposing the above mathematical model fori . Results show
that the values of acetone, ethanol and isopropanol in their mixtures with water are
smaller than 1. However, the values of water in the mixtures are larger than 1.
Figure 4-1: Variations of i using Eq. 4.5
35
The main rational for proposing the above model for activation energy is evident by
examining Figure 4-2. As one can observe, the activation energies calculated using Eq.
4.3 for acetone-water, ethanol-water and isopropanol-water mixtures are displayed in
Figure 4-2-a, 4-2-b and 4-2-c, respectively. In all three systems one can observe a major
change in activation energy with concentration for the organic component. Figure 4-2-a
presents the activation energy of acetone and water in their mixture. As the concentration
of acetone increases, the activation energy of acetone decreases and at high
concentrations of acetone, the activation energy will have a zero slope. Regarding the
water activation energy, its variation for this particular mixture is not high when
compared to acetone. Figure 4-2-b shows the case of an ethanol-water mixture. Here
again a decrease in the activation energy of ethanol is observed as the concentration
increases but a very weak variation is observed for the water. Finally in Figure 4-2-c, for
an isopropanol-water mixture, two different activation energy slopes are observed: the
first at a low concentration and the second at a high concentration of isopropanol.
Thus, the variation of activation energy of the organic component is considered to mainly
control the sign change in the thermodiffusion factor. In these three mixtures, the
activation energy of the organic component decreases sharply with its concentration,
which leads to a sign change in the thermodiffusion factor. The thermodiffusion factor
then increases gradually with a decreasing water concentration in the systems. For the
isopropanol-water mixture, isopropanol activation energy increases high enough to have
the second sign change. However, the activation energies of acetone and ethanol in their
36
mixtures do not increase enough to cause the second sign change in the thermodiffusion
factor.
Figure 4-2-a: Activation energy calculated with Eq. 4.3 in an acetone-water mixture.
37
Figure 4-2-b: Activation energy calculated with Eq. 4.3 in an ethanol-water mixture.
38
Figure 4-2-c: Activation energy calculated with Eq. 4.3 in an isopropanol-water mixture.
39
Estimating the values of iviscU , using Eq. 4.3 and ivapU , leads to the calculation of
i . The calculated i values as a function of mole fraction can therefore be expressed
mathematically as follows:
ivisc
ivap
iU
U
,
,
(4.6)
In order to calculate the proposed activation energy in Eq. 4.3 and 4.4, 1 and 2 were
obtained from extrapolating the thermodiffusion factors from experimental data. The
0i values were obtained by plotting the relationship between natural logarithm of
viscosity and ivapU , /RT for saturated water, acetone, ethanol, and isopropanol in the
temperature range of 280–380 K. Evaluated values of 10 , 20 , 1 and 2 are listed in
Table 4-1. The second sign change ofin themodiffuison factor for an isopropanol-water
mixture requires a high value of 1 . The required thermodynamic properties for
thermodiffusion models, including 11 xlnd/lnd , 11 x/ , iU , and iV can be
derived easily using the PC-SAFT equation of state .
Table 4-1: 0i and i values for acetone-water, ethanol-water and isopropanol-water
mixtures.
mixtures 10 (water) 1 (water) 20 (Organic) 2 ( Organic)
Acetone-water 3.63 5.0 5.00 0.37
Ethanol-water 3.63 8.6 4.43 0.79
Isopropanol-water 3.63 50 3.13 0.61
40
In this work, all associating components, i.e., acetone-water, ethanol-water and
isopropanol-water are assigned two and four association sites respectively (often referred
to as the 4B and 2C) as proposed by Gross and Sadowski (2002). The pure component
parameters, including i (temperature independent segment diameter),
i (depth of the
potential well), im (number of segments in a chain), iiBA (association energy), and ii BA
k
(association volume) are listed in Table 4-2. In addition to the parameter sets in the
literature, the pure component parameters of acetone have also been optimized by fitting
the vapor pressure and saturated liquid density data from Physical and Thermodynamic
Properties of Pure Chemicals: Data Compilation (Hemisphere, New York, 1989).
Interaction parameter coefficients, ijk , that have been optimized through the correlation
of vapor pressure and saturated liquid volumes are listed in Table 4-3. With pure
component parametersi ,
i , im , iiBA , and ii BA
k , the compressibility factor Z can be
obtained. Using Z as a starting point, the thermodynamic properties required for thermal
diffusion models can be predicted easily.
41
Table 4-2: Pure component parameters from PC-SAFT EoS.
Component Ref. Associating
scheme
im
i
(Ǻ)
i / Bk
(K)
iiBAk
iiBA / Bk
(K)
Water Pan et al.,
2007a
4C 3.1043 1.9879 179.30 0.471821 1248.65
Acetone This work 2B 3.3156 3.01145 165.568 1.25676 1224.52
Ethanol Pan et al.,
2007a
2B 2.5802 3.0795 191.85 0.039088 2620.9
Isopropanol Pan et al.,
2007a
2B 3.7043 2.9877 187.61 0.058664 2108.8
Table 4-3: Binary interaction parameter, ijk , for acetone-water, ethanol-water and
isopropanol-water mixtures.
Mixture Ref. Temperature (K)
298.15 313.15
Acetone- water This work -0.005 -0.005
Ethanol-water Ref. 13 0.008 0.013
Isopropanol-water Ref. 13 0.003 NA
42
4.2. Results and discussion
The rational for the newly proposed i model given by Eq. 4.6 is shown in Figure 4-3
where we compare the predicted variations of 1 and 2 with the water mole fraction for
acetone-water, ethanol-water and isopropanol-water mixtures. As expected the activation
energy for viscous flow is smaller than the heat of vaporization but the ratio is much
larger than that in pure limits. This infers that there is less energy required for a molecule
or molecular segment making the jump as the hole size distribution changes. To our
knowledge, measuring the activation energy experimentally is not achievable, so
evaluation of the activation energy is performed by comparing the predicted and
experimental thermodiffusion factors. Results show that the i values calculated with the
modified rule given by Eq. 4.6 have a rounded peak behavior. For each mixture, the
energy ratio reaches a peak at different water concentrations. The 1 value for
isopropanol in isopropanol-water mixtures has one maximum and one minimum peak
which again confirms the double sign changes in their mixture. In order to verify our new
approach, the thermodiffusion factor was evaluated for the three mixtures. Figure 4-4
shows the thermodiffusion factors calculated with our modified rule given by Eq. 4.6
and compared with the predictions obtained with the simple mixing rules proposed by
Pan et al. (2007a) given by Eq. 4.1 along with the available experimental results. The
modified rule predictions are shown to be in good agreement with the experimental
results.
43
Figure 4-3-a: Variations of i calculated using the new model given by Eq. 4.6 in an
acetone-water mixture.
44
Figure 4-3-b: Variations of i calculated using the new model given by Eq. 4.6 in an
ethanol-water mixture.
45
Figure 4-3-c: Variations of i calculated using the new model given by Eq. 4.6 in an
isopropanol-water mixture.
46
As one can observe in Figure 4-4-b, a better agreement for the water-ethanol mixture is
obtained when compared with Pan et al.’s simple mixing rules. It is noticed that a more
accurate slope of the thermodiffusion factor at low ethanol concentrations is obtained.
The mean deviation from the experimental data is found to be 10% compared to Pan et
al.’s simple mixing rules where the mean error was indicated to be 23%. Figure 4-4-c
demonstrates the importance of the rules by showing the thermodiffusion factor as a
function of isopropanol concentration. It is seen that the newly proposed model can
predict the sign change at a low isopropanol concentration and the second change at a
high isopropanol concentration. However, Pan et al.’s mixing rules were not capable of
predicting the two sign changes obtained experimentally. The error between our results
and the experimental data is in the range of 10%. Finally, Figure 4-4-a presents a
comparison between the present model and the experimental data for acetone-water
mixture. Very good agreement is observed with an error in the order of 5%. Next, rules
given by Eq. 4.1 and Eq. 4.6 have been implemented to predict the Soret coefficient for a
ethanol-water solution at 295.65 K and 315.65 K. Figure 4-5 shows the predicted Soret
coefficient for the water-ethanol mixture at 295.65 K and 315.65 K by the newly
proposed rule along with the experimental results of Kolodner et al.(1988). This figure
shows that the predictions of the new model are in good agreement with the experimental
data. Experimentally, the Soret coefficient changed its sign at an ethanol mole fraction of
0.142 for both 295.65 K and 315.65 K that can be predicted by the newly proposed
rule.
47
Figure 4-4-a: Evaluation of thermodiffusion factor for an acetone-water mixture.
48
Figure 4-4-b: Evaluation of thermodiffusion factor for an ethanol-water mixture.
49
Figure 4-4-c: Evaluation of thermodiffusion factor for an isopropanol-water mixture.
50
Figure 4-5: Evaluation of Soret coefficient for an ethanol-water mixture.
4.3. Summary
A new theoretical approach for calculating the activation energy and the ratio of
evaporation energy to activation energy was presented to evaluate the thermodiffusion
factor in associating mixtures, including acetone-water, ethanol-water, and isopropanol-
water. The Firoozabadi model combined with the PC-SAFT equation of state and the
newly proposed rule were used in the calculations. The results showed very good
agreement between the experimental data and the predicted values.
51
5. A new approach to estimate the
thermodiffusion coefficients for linear
chain hydrocarbon binary mixtures
Thermodiffusion behaviors in non-associating mixtures have an important role in
separation processes of oil industry. The variations of composition and temperature may
either lessen or enhance the separation in mixtures. A new model regarding the prediction
of thermodiffusion coefficients for linear chain hydrocarbon binary mixtures using the
thermodynamics of irreversible process is proposed. The model predicts the net amount
of heat transported based on an available volume for each molecule. This newly proposed
model combined with the Perturbed Chain Statistical Associating Fluid Theory (PC-
SAFT) equation of state has been applied to predict thermodiffusion coefficients for
binary hydrocarbon mixtures of C10-nCi (i=5, 6, 7, 15, 16, 17, 18), C12-nCi (i=5, 6, 7, 8,
9), and C18-nCi (i=5, 6, 7, 8, 9, 12) at T=298.15 K and P=1 atm. Comparisons of the
calculated results with the experimental data show a good performance of the proposed
model. In particular, this model based on the kinetics approach has been found to be most
reliable and represents a significant improvement over the previous models.
5.1. Free volume
The activation or viscous energy, iviscU , , is the total energy needed to detach all the
molecules of the specific component. This energy depends on the molecule type and
52
concentration of a specific component in order to satisfy the free volume theory. In
diffusion-limited systems, the free volume is the paramount factor in controlling the
release rate of molecules (Abbasi et al., 2008a&b). The free volume is generally defined
as the volume of a system at a given temperature minus its volume at absolute zero.
Rearrangement of the free volume makes pores or voids through which diffusing species
may pass (Fujita, 1967). According to the free volume theory, the transfer kinetics of
diffusing molecules depends greatly on the molecular size and shape, with small
molecules having higher molecular diffusion coefficients (Sung et al., 1998). Pores that
are larger than the diffusing molecule will permit diffusion with little or no resistance,
whereas diffusing species larger than the pores will encounter resistance against their
flow as they entangle with matrix mesh (Abbasi et al., 2008a&b). As a result, the
required energy for detaching a molecule is directly related to the molecular size and
shape. The effect of the molecular size and average mesh size for specific molecules can
be shown in the molecular weight and specific volume of each component. This
molecular size effect may be defined as the geometric average value of specific volume
and molecular weight fraction of each component (Abbasi et al., 2009c). As result, the
tendency of change in the activation energy for the movement of specific component is
directly related to its specific volume and molecular weight fraction. The physical sense
of the phenomena is that the higher specific volume and molecular weight yields higher
viscosity (Glasstone et al., 1941). This trend was used in proposing a new model for
calculating iviscU , based on the free volume theory as follows:
53
ipureviscii
ivisc UMxMx
M
VxVx
VU ,
5.0
2211
5.0
2211
,
(5.1)
where ipureviscU , is the pure activation energy of component i. The pure activation
energy may be calculated based on Eyring’s viscosity theory as follows (Glasstone et al.,
1941):
2/32/12/16/1
,
3/2
,)2(
lnTMkCRN
UVRTU
iBAV
iiVapi
ipurevisc
(5.2)
where i is the viscosity and iVapU , is energy of vaporization of component i.
AVN is
Avogadro’s number and Bk is Boltzmann’s constant For a cubic lattice packing, 2C .
By modifying the Dougherty and Drickamer model, the newly proposed activation
energy, Eq. 5.1, may be used as a good approximation for HW in the calculation of the
net heat of transport for each component kQ . Therefore Eq. 2.18 can be written as
follows:
1,
2
11,1 viscvisc U
V
VUQ
(5.3)
2,
1
22,2 viscvisc U
V
VUQ
The required thermodynamic properties for thermodiffusion models (Eq. 2.14, Eq. 2.15,
Eq. 2.19, and Eq. 5.3) can be derived easily using Perturbed Chain Statistical Associating
Fluid Theory (PC-SAFT) equation of state. The pure component parameters used in PC
54
SAFT equation of state, including i (temperature independent segment diameter),
i
(depth of the potential well), and im (number of segments in a chain) are listed in Table
5-1. The viscosity is obtained from the NIST database (2007).
Table 5-1: Pure component parameters from PC-SAFT EoS (Gross and Sadowski, 2001).
Component M
(g/mol)
im i
(Ǻ)
i / Bk
(K)
nC5 72.146 2.6896 3.7729 231.20
nC6 86.177 3.0576 3.7983 236.77
nC7 100.203 3.4831 3.8049 238.40
nC8 114.231 3.8176 3.8373 242.78
nC9 128.25 4.2079 3.8448 244.51
nC10 142.285 4.6627 3.8384 243.87
nC12 170.338 5.3060 3.8959 249.21
nC15 212.419 6.2855 3.9531 254.14
nC16 226.446 6.6485 3.9552 254.70
nC17 240.473 6.9809 3.9675 255.65
nC18 254.5 7.3271 3.9668 256.20
5.2. Results and discussion
Lattice theory states that liquids consist of a matrix of molecules and vacancies or holes
scattered throughout. In a viscous liquid flow, the flow unit, which may be a group of
55
molecules, a single molecule or a segment of a molecule would jump into an existing
hole and thereby create a new hole. Some source of energy is required for the molecule to
jump over an energy barrier and into a hole. In a mixture of two different molecules,
smaller molecules would required less energy to be removed from their surroundings, as
their molecular size does not exceed the perimeter of the average pore size within the
matrix. Pores that are larger than the diffusing molecule will permit diffusion with little
or no resistance, whereas diffusing species larger than the pores will encounter resistance
against their movment as they entangle with matrix mesh. Therefore larger molecules
require more energy to remove them from their surroundings. In a binary mixture the
hole size distribution will be altered by increasing the fraction of one component. That
will change the required energy for each molecule to be removed from its surroundings.
Thus viscous energy of larger molecules will increase by decreasing the percentage of
larger molecule inside the mixture. However the viscous energy of smaller molecules
will decrease if their percentage decreases in the binary mixture. This confirms the jump
from one pore in the mixture matrix to another for a given pore size distribution which
will be easier for smaller than larger molecules.
The results of the newly proposed model of net heat of transport are shown in Tables 5-2
and 5.3 where the predicted variations of thermodiffusion coefficients are compared with
the Haase model, Kempers model, Shukla & Firoozabadi model (considering the i =4
and k calculated by Yan et al.( 2008)), and the experimental data. It is evident that the
results of the newly proposed model in predicting thermodiffusion coefficients are in a
very good agreement with the experimental data. Also it is important to mention that the
56
proposed model performance is superior to other theoretical models. By examining
carefully the comparison presented in Table 5-2 for C10-nCi (i=5,6,7,15,16,17,18), C12-
nCi (i=5, 6, 7, 8, 9), and C18-nCi (i=5, 6, 7, 8, 9, 12) mixtures, one can discover that the
new model as well as Shukla & Firoozabadi and Kempers model predicted the sign
change as seen in the experiment. However the Haase model failed to predict the sign
change. Also it is important to notice that the maximum difference between the
experimental data and the new model is around 10% whereas the differences were greater
for the other models. The comparison was repeated for C12-nCi (i=5, 6, 7, 8, 9), and C18-
nCi (i=5, 6, 7, 8, 9, 12) mixtures with different compositions (Table 5-3). It is evident that
the new model gave the best prediction in comparison with the other models.
To better understand the reason for this good agreement with experiment, Tables 5-4 and
5-5 show the comparisons of the predicted activation energy for larger and smaller
molecules. Results show the activation energy of the larger molecule such as C12 in a
binary mixtures of C12-nCi (i=5, 6, 7, 8, 9) increases and that of smaller molecules
decreases by decreasing the percentage of the larger molecule inside the mixture. The
free volume theory states that the transfer kinetics of diffusing molecules depends greatly
on the molecular size and shape as well as pore size. This can be examined by comparing
our calculated thermodiffusion coefficients with the experimental results.
57
Table 5-2: Thermodiffusion coefficients x 1012
(m2/sK) in 50% mole fraction for nCi-C10
(i=5, 6, 7, 15, 16, 17, 18), nCi-C12 (i=5, 6, 7, 8, 9), and nCi-C18 (i=5, 6, 7, 8, 9, 12)
mixtures at 298.15 K. Method 1: New net heat of transport model, Eq. 5.3; Method 2:
Haase model, Eq. 2.14; Method 3: Kempers model, Eq. 2.15; Method 4: Shukla &
Firoozabadi model, Eq. 2.19, considering i = 4. Experimental data are extracted from
Yan et al. 2008 and Blanco et al. 2007, 2008.
Mixture
50% mole
Exp.
Method
1
Diff.
%
Method
2
Diff. % Method
3
Diff. % Method
4
Diff.
%
nC5-nC10 8.78 ±0.439 9.21 -4.85 -0.81 109.21 4.52 48.55 1.42 83.79
nC6-nC10 6.08 ±0.304 6.71 -10.42 -0.56 109.26 2.76 54.63 0.89 85.37
nC7-nC10 3.90 ±0.195 4.26 -9.19 -0.40 110.26 1.67 57.21 0.54 86.19
nC15-nC10 -2.15 ±0.107 -2.10 2.10 0.49 123.01 -0.26 87.89 -0.12 105.70
nC16-nC10 -2.23 ±0.111 -2.11 5.45 0.50 122.34 -0.35 84.12 -0.15 106.66
nC17-nC10 -2.29 ±0.114 -2.16 5.67 0.53 123.06 -0.36 84.48 -0.15 106.61
nC18-nC10 -2.38 ±0.119 -2.19 7.88 0.54 122.85 -0.45 81.12 -0.18 107.38
nC5-nC12 8.81 ±0.440 9.13 -3.66 -1.04 111.78 3.98 54.86 1.29 85.34
nC6-nC12 6.45 ±0.322 7.13 -10.54 -0.83 112.89 2.53 60.83 0.85 86.86
nC7-nC12 4.74 ±0.237 5.07 -6.97 -0.68 114.33 1.61 66.12 0.55 88.44
nC8-nC12 3.23 ±0.161 3.44 -6.46 -0.43 113.46 0.98 69.69 0.34 89.41
nC9-nC12 2.15 ±0.107 2.12 1.37 -0.32 114.75 0.46 78.45 0.18 91.77
nC5-nC18 7.38 ±0.369 7.06 4.34 -1.08 114.58 3.13 57.54 1.04 85.91
nC6-nC18 5.90 ±0.295 5.95 -0.81 -0.96 116.33 2.16 63.42 0.74 87.48
nC7-nC18 5.00 ±0.250 4.70 5.91 -0.86 117.23 1.51 69.88 0.53 89.50
nC8-nC18 3.94 ±0.197 3.67 6.97 -0.68 117.30 1.05 73.24 0.38 90.48
nC9-nC18 3.00 ±0.150 2.78 7.43 -0.58 119.38 0.67 77.60 0.25 91.62
nC12-nC18 1.33 ±0.066 1.20 9.68 -0.31 122.98 0.28 78.82 0.10 92.17
58
Table 5-3: Thermodiffusion coefficients x 1012
(m2/sK) in 50% wt fraction for nCi-C12
(i=5, 6, 7, 8, 9) and nCi-C18 (i=5, 6, 7, 8, 9, 12) mixtures at 298.15 K. Method 1: New net
heat of transport model, Eq. 5.3; Method 2: Haase model, Eq. 2.14; Method 3: Kempers
model, Eq. 2.15; Method 4: Shukla & Firoozabadi model, Eq. 2.19, considering k = 4.;
Method 5: Shukla & Firoozabadi model , Eq. 2.19, considering i calculated by Yan et
al. ( 2008). Experimental data are extracted from Yan et al. 2008 and Blanco et al. 2007,
2008
Mixture
50% wt
Exp.
Method
1
Diff. Method
2
Diff. Method
3
Diff. Method
4
Diff. Method
5
Diff.
nC5-nC12 10.94 ±0.547 12.28 -12.21 -1.27 111.61 4.44 59.43 1.44 86.88 3.25 70.33
nC6-nC12 7.45 ±0.372 8.79 -18.01 -0.97 113.01 2.75 63.13 0.92 87.64 1.96 73.69
nC7-nC12 5.15 ±0.257 5.80 -12.61 -0.75 114.48 1.70 67.01 0.58 88.74 1.71 66.76
nC8-nC12 3.39 ±0.169 3.72 -9.83 -0.46 113.61 1.01 70.17 0.35 89.57 1.09 67.98
nC9-nC12 2.15 ±0.107 2.21 -2.70 -0.33 115.17 0.47 78.12 0.18 91.63 0.78 63.82
nC5-nC18 11.86 ±0.593 13.44 -13.32 -1.68 114.19 3.90 67.11 1.30 89.05 - -
nC6-nC18 8.90 ±0.445 8.90 0.01 -1.40 115.68 2.61 70.70 0.90 89.89 - -
nC7-nC18 6.28 ±0.314 7.08 -12.67 -1.14 118.13 1.75 72.06 0.62 90.18 - -
nC8-nC18 4.69 ±0.234 5.01 -6.82 -0.85 118.06 1.18 74.79 0.42 90.94 - -
nC9-nC18 3.57 ±0.178 3.50 1.90 -0.68 119.03 0.73 79.60 0.28 92.29 - -
nC12-nC18 1.49 ±0.074 1.30 12.48 -0.32 121.64 0.29 80.54 0.11 92.78 0.88 40.60
59
Table 5-4: The activation energy x10-4
(j/mol) in 50% mole fraction nCi-C10 (i=5, 6, 7,
15, 16, 17, 18), nCi-C12 (i=5, 6, 7, 8, 9), and nCi-C18 (i=5, 6, 7, 8, 9, 12) mixtures at
298.15 K.(Example of larger and smaller molecules in a binary mixture: C12 is the larger
molecule and nCi (i=5, 6, 7, 8, 9) are the smaller molecules in binary mixture of C12-nCi
(i=5, 6, 7, 8, 9))
Mixture
50% mole visU
larger
molecule,
Eq. 5.1
visU
smaller
molecule,
Eq. 5.1
purevisU ,
larger
molecule,
Eq. 5.2
purevisU ,
smaller
molecule,
Eq. 5.2
nC5-nC10 1.25 0.34 0.97 0.49
nC6-nC10 1.19 0.45 0.97 0.58
nC7-nC10 1.12 0.58 0.97 0.69
nC15-nC10 1.61 0.79 1.36 0.97
nC16-nC10 1.70 0.76 1.40 0.97
nC17-nC10 1.80 0.74 1.45 0.97
nC18-nC10 1.90 1.70 1.50 0.97
nC5-nC12 1.56 1.80 1.14 0.49
nC6-nC12 1.48 0.40 1.14 0.58
nC7-nC12 1.41 0.52 1.14 0.69
nC8-nC12 1.35 0.65 1.14 0.79
nC9-nC12 1.29 0.77 1.14 0.89
nC5-nC18 2.28 0.23 1.50 0.48
nC6-nC18 2.19 0.31 1.50 0.58
nC7-nC18 2.11 0.41 1.50 0.69
nC8-nC18 2.04 0.51 1.50 0.79
nC9-nC18 1.97 0.61 1.50 0.89
nC12-nC18 1.78 0.93 1.50 1.14
60
Table 5-5: The activation energy x10-4
(j/mol) in 50% wt fraction for nCi-C12 (i=5, 6, 7,
8, 9) and nCi-C18 (i=5, 6, 7, 8, 9, 12) mixtures at 298.15 K. (Example of larger and
smaller molecules in a binary mixture: C12 is the larger molecule and nCi (i=5, 6, 7, 8, 9)
are the smaller molecules in binary mixture of C12-nCi (i=5, 6, 7, 8, 9))
Mixture
50% wt visU
larger
molecule,
Eq. 5.1
visU
smaller
molecule,
Eq. 5.1
purevisU ,
larger
molecule,
Eq. 5.2
purevisU ,
smaller
molecule,
Eq. 5.2
nC5-nC12 1.83 0.36 1.14 0.49
nC6-nC12 1.64 0.45 1.14 0.58
nC7-nC12 1.50 0.56 1.14 0.69
nC8-nC12 1.40 0.67 1.14 0.79
nC9-nC12 1.32 0.79 1.14 0.89
nC5-nC18 3.19 0.33 1.50 0.49
nC6-nC18 2.58 0.37 1.50 0.58
nC7-nC18 2.56 0.50 1.50 0.69
nC8-nC18 2.36 0.59 1.50 0.79
nC9-nC18 2.19 0.68 1.50 0.89
nC12-nC18 1.85 0.96 1.50 1.14
61
The newly proposed model of net heat of transport, Eq. 5.3, was then used for predicting
the thermodiffusion coefficients in binary mixtures of C10-nCi (i=5, 6, 7), C12-nCi (i=5, 6,
7, 8, 9), and C18-nCi (i=5, 6, 7, 8, 9, 12) for different compositions and temperatures.
Figures 5-1, 5-2 and 5-3 show the calculated thermodiffusion coefficients for 25%, 50%
and 75% mole fractions of the larger component in the mixtures of interest at different
average temperatures. Results show that the thermodiffusion coefficient increases as the
temperature increases. On the other hand, adding larger component in a binary mixture
decreases the thermodiffusion coefficient of that component. Decreasing the percentage
of the larger component inside the mixture or increasing the temperature, will decrease
the required energy for detaching the molecule (according to Eq. 5.1 and Eq. 5.2). As the
energy required for a molecule to jump over an energy barrier and into a hole is reduced,
the diffusivity of the molecule increases which confirms the free volume theory.
5.3. Summary
A new theoretical approach based on the free volume theory for calculating the activation
energy was presented to evaluate the thermodiffusion coefficients in linear chain
hydrocarbon binary mixtures. In this model, the size of the molecule that has a significant
effect on thermodiffusivity of the molecule was considered. The new model combined
with the PC-SAFT equation of state provides a significant improvement in the accuracy
of thermodiffusion modeling for the mixtures investigated.
62
Figure 5-1-a: Thermodiffusion coefficients in 75% mole fraction of C10 in nCi-C10 (i=5,
6, 7) mixtures.
63
Figure 5-1-b: Thermodiffusion coefficients in 50% mole fraction of C10 in nCi-C10 (i=5,
6, 7) mixtures.
64
Figure 5-1-c: Thermodiffusion coefficients in 25% mole fraction of C10 in nCi-C10 (i=5, 6,
7) mixtures.
65
Figure 5-2-a: Thermodiffusion coefficients in 75% mole fraction of C12 in nCi-C12 (i=5,
6, 7, 8, 9) mixtures.
66
Figure 5-2-b: Thermodiffusion coefficients in 50% mole fraction of C12 in nCi-C12 (i=5,
6, 7, 8, 9) mixtures.
67
Figure 5-2-c: Thermodiffusion coefficients in 25% fraction of C12 in nCi-C12 (i=5, 6, 7, 8,
9) mixtures.
68
Figure 5-3-a: Thermodiffusion coefficients in 25% mole fraction of C18 in nCi-C18 (i=5,
6, 7, 8, 9) mixtures.
69
Figure 5-3-b: Thermodiffusion coefficients in 50% mole fraction of C18 in nCi-C18 (i=5,
6, 7, 8, 9) mixtures.
70
Figure 5-3-c: Thermodiffusion coefficients in 25% mole fraction of C18 in nCi-C18 (i=5,
6, 7, 8, 9) mixtures.
71
6. Theoretical and experimental
comparison of the Soret effect for
binary mixtures of toluene & n-hexane
and benzene & n-heptane
Thermodiffusion along with molecular diffusion occurs in many engineering systems and
in nature. Thermodiffusion has a great effect on concentration distributions in binary
mixtures. A new approach to predicting the Soret coefficient in binary mixtures of linear
chain and aromatic hydrocarbons using the thermodynamics of irreversible processes is
presented. In particular, this approach is based on the free volume theory which explains
the diffusivity in diffusion-limited systems. Free volume states that the transfer kinetics
of molecules depends greatly on the molecular size and shape. The proposed model
combined with Shukla and Firoozabadi’s model (1998) was applied to predict the Soret
coefficient. The Perturbed Chain Statistical Associating Fluid Theory (PC-SAFT)
equation of state was used to calculate the related thermodynamic properties.
Comparisons of theoretical results with experimental data show a good agreement.
6.1. Ratio of evaporation energy to activation energy in
non-associating mixtures
The net heat of transport is the main challenge in the prediction of thermodiffusion.
Shukla and Firoozabadi (1998) defined the net heat of transport based on a ratio of
72
evaporation energy to activation energy. It has been found that in non-associating
mixtures, the choice of i = 4 proposed by Shukla and Firoozabadi
(1998) for
hydrocarbon mixtures along with Dougherty and Drickamer’s work (1955a&b) did not
give a very good agreement with experimental results. With the choice of i = 4 or less it
was possible to obtain a good match with the experimental data for certain
concentrations, but the model failed to estimate the thermodiffusion coefficient for a
wide range of concentrations. i is the ratio of the energy of vaporization to the energy of
activation or viscous energy of component i. Prediction of i values requires calculation
of the energy of evaporation as well as the activation energy. Using the Perturbed Chain
Statistical Associating Fluid Theory (PC-SAFT) equation of state, an equilibrium
condition for specific temperature and pressure can be calculated. Thus, the calculation of
evaporation energy for each mole fraction is manageable. However, the calculation of the
activation energy, iviscU , , as a function of mole fraction does not follow any specific
model.
Recently, Abbasi et al. (2009a) proposed a model for the activation energy, iviscU , , for
each component in a binary mixture, Eq. 5.1. The modified activation energy model, Eq.
5.1, is based on a combination of the volume fraction and molecular weight that reflects
the average free volume of each component in the mixture. The free volume is generally
defined as the volume of a system at a given temperature minus its volume at absolute
zero. It is essentially the volume of a system not occupied by all components of the
system (Fujita, 1967). Based on the lattice theory, liquids consist of a matrix of molecules
and holes scattered throughout (Totten, 1999). Rearrangements of free volumes make
73
pores through which diffusing species may pass. Some source of energy is required for a
molecule to jump over an energy barrier and into a hole. In a mixture of two different
molecules, smaller molecules require less energy to be removed from their surroundings,
as their molecular size does not exceed the perimeter of the average pore size within the
matrix. The modified ratio of the energy of vaporization to the energy of activation
energy based on the modified activation energy shown in Eq. 5.1 may be presented as
follows:
ipure
i
i
ii
i
i
ii
iM
Mx
V
Vx
5.02
1
2
1
(6.1)
The ratio of the evaporation energy to the activation energy in pure limits of component i,
ipure, , may be calculated based on Eyring’s viscosity theory as follows (Glasstone et al.,
1941):
TRT
U
i
i
ipure
ln2
, (6.2)
where iU is the partial internal energy of component i. The thermodynamic properties
required for thermodiffusion models can be derived easily using PC-SAFT equation of
state. The pure component parameters used in PC-SAFT equation of state, including i
74
(temperature independent segment diameter), i (depth of the potential well), and
im
(number of segments in a chain) are listed in Table 6-1. The viscosity is obtained from
the Handbook of Transport Property Data: Viscosity, Thermal Conductivity, and
Diffusion Coefficients of Liquids and Gases.
Table 6-1: Pure component parameters from PC-SAFT EoS (Gross and Sadowski, 2001).
Component M
(g/mol)
km k
(A0)
k / Bk
(K)
nC6 86.177 3.0576 3.7983 236.77
nC7 100.203 3.4831 3.8049 238.40
Benzene 78.114 2.4653 3.6478 287.35
Toluene 92.141 2.8149 3.7169 285.69
6.2. Results and discussion
According to the Lattice theory, some source of energy is required for a molecule to jump
over an energy barrier and into a hole. The required energy to move the molecules
depends greatly on the molecular size and shape. Among the molecules with the same
shape, the larger molecule has higher activation energy such as linear hydrocarbon
chains. The pure activation energies of the components of interest obtained from Eq. 5.2
are shown in Figures 6-1 and 6-2. Results show the activation energies of toluene and
benzene are greater than those of n-hexane and n-heptanes. However, the molar volume
75
of n-hexane and n-heptanes are larger than those of toluene and benzene. This indicates
that the size of the molecule is not the only condition to have a higher activation energy.
The shape of the molecule as well as the size of the molecule has a direct effect on the
activation energy. At high temperatures, a higher energy is available and more jumps can
be made per unit time, resulting in lower viscosity and lower activation energy. In a
binary mixture the hole size distribution will change with the increasing fraction of one
component. The required energy will also change for each molecule to be removed from
its surrounding. Thus, the activation energy of larger molecules will increase and go far
from the value of its pure limit by decreasing the percentage of larger molecules inside
the mixture; however, the activation energy of smaller molecules will decrease and go
toward the value of its pure limit. This confirms that the jump from one pore in the
mixture matrix to another for a given pore size distribution will be easier for smaller than
for larger molecules. Changing the temperature can also alter the energy barrier for
moving the molecule. The rising temperature increases the free volume for each
molecule, therefore the required energy for a molecule to jump decreases. In other words,
the activation energy increases by quenching temperature. The ratio of the evaporation
energy to activation energy, i , has an inverse behavior. In a binary mixture, the
difference between the values of k and ipure, for larger molecules decreases as the
percentage of larger molecule inside the mixture is increased; however, the difference
between the values of k and ipure, of smaller molecules increases. Tables 6-2 and 6-3
show the i values for the component of interest in their binary mixtures and their pure
limits. The average pure values calculated by Eq. 6.2 for toluene, benzene, n-heptane,
and n-hexane are, respectively, 4.16, 3.2, 4.41, and 4.62 for the range of temperatures of
76
interest. Results show the differences between the values of k and ipure, of the larger
molecules (n-hexane and n-heptanes) decrease and those of smaller molecules (toluene
and benzene) increase with the increasing percentage of the larger molecules inside the
mixture. Rising the temperature decreases the energy required for detaching the
molecules (according to Eq. 5.1 and Eq. 5.2). In contrast with the viscous energy, the
ratio of evaporation energy to activation energy,i increases.
Figure 6-1: Pure Activation energy of toluene and n-hexane (j/mol) at different
temperatures.
77
Figure 6-2: Pure Activation energy of benzene and n-heptane (j/mol) at different
temperatures.
78
Table 6-2: The ratio of evaporation energy to activation energy,i for the binary
mixtures of toluene and n-hexane at different temperatures.
Mixture of
C7H8 - nC6
mole%
T
(°C)
C7H8, Eq. 6.1
nC6, Eq. 6.1
0.05 nC6 5 4.16 4.38
0.25 nC6 5 4.22 4.43
0.50 nC6 5 4.30 4.50
0.75 nC6 5 4.38 4.56
0.95 nC6 5 4.45 4.62
0.05 nC6 15 4.21 4.37
0.25 nC6 15 4.28 4.42
0.50 nC6 15 4.36 4.49
0.75 nC6 15 4.44 4.56
0.95 nC6 15 4.51 4.61
0.05 nC6 25 4.22 4.36
0.25 nC6 25 4.28 4.41
0.50 nC6 25 4.36 4.48
0.75 nC6 25 4.45 4.55
0.95 nC6 25 4.53 4.61
0.05 nC6 35 4.17 4.35
0.25 nC6 35 4.24 4.40
0.50 nC6 35 4.32 4.47
0.75 nC6 35 4.41 4.54
0.95 nC6 35 4.49 4.60
0.05 nC6 45 4.08 4.34
0.25 nC6 45 4.14 4.39
0.50 nC6 45 4.23 4.46
0.75 nC6 45 4.32 4.54
0.95 nC6 45 4.40 4.60
79
Table 6-3: The ratio of evaporation energy to activation energy, i for the binary
mixtures of benzene and n-heptane at different temperatures.
Mixture of
C6H6 – nC7
mole%
T
(°C)
C6H6, Eq. 6.1
nC7, Eq. 6.1
0.05 nC7 20 3.18 3.11
0.25 nC7 20 3.47 3.39
0.50 nC7 20 3.82 3.73
0.75 nC7 20 4.17 4.07
0.05 nC7 30 3.18 3.11
0.25 nC7 30 3.47 3.39
0.50 nC7 30 3.82 3.73
0.75 nC7 30 4.17 4.07
0.05 nC7 40 3.18 3.11
0.25 nC7 40 3.47 3.39
0.50 nC7 40 3.82 3.73
0.75 nC7 40 4.17 4.07
In order to verify our new approach, the calculated Soret coefficient was compared with
experimental data. Tables 6-4 and 6-5 show the Soret coefficient calculated with the
Shukla and Firoozabadi model combined with the modified rule given by Eq. 6.1 along
with the available experimental results. Results from the proposed new model are in good
agreement with the experimental results. The mean deviation from the experimental data
is found to be 4%. However, the difference between the experimental data and calculated
values in the dilute mixtures is high. This can be seen in 95% n-hexane mole fraction for
the binary mixtures of toluene and n-hexane and 75% n-heptane mole fraction for the
binary mixtures of benzene and n-heptane. An increase in the percentage of the larger
compound inside the mixture as well as the rising temperature reduces the required
energy for a molecule to jump over an energy barrier. As a result the diffusivity of the
molecule increases which confirms the free volume theory.
80
Table 6-4: Soret coefficient x 103 (K
-1) for the binary mixtures of toluene and n-hexane at
different temperatures. Experimental data was extracted from Wittko and Kohler, 2007.
Mixture of
C7H8 - nC6
mole%
T
(°C)
Exp.
Present
Work
Diff.
%
0.05 nC6 5 7.20 ±0.360 6.22 13.54
0.25 nC6 5 6.63 ±0.331 6.34 4.40
0.50 nC6 5 5.63 ±0.281 5.98 -6.28
0.75 nC6 5 4.38 ±0.219 5.25 -19.75
0.95 nC6 5 3.33 ±0.166 4.55 -36.67
0.05 nC6 15 6.54 ±0.327 5.54 15.22
0.25 nC6 15 6.10 ±0.305 5.62 7.86
0.50 nC6 15 5.26 ±0.263 5.30 -0.69
0.75 nC6 15 4.19 ±0.209 4.65 -10.98
0.95 nC6 15 3.23 ±0.161 4.05 -25.24
0.05 nC6 25 5.96 ±0.298 5.11 14.28
0.25 nC6 25 5.56 ±0.278 5.16 7.19
0.50 nC6 25 4.92 ±0.246 4.86 1.28
0.75 nC6 25 3.98 ±0.199 4.27 -7.35
0.95 nC6 25 3.17 ±0.158 3.73 -17.56
0.05 nC6 35 5.53 ±0.276 4.88 11.82
0.25 nC6 35 5.20 ±0.260 4.91 5.54
0.50 nC6 35 4.64 ±0.232 4.62 0.42
0.75 nC6 35 3.85±0.192 4.07 -5.79
0.95 nC6 35 3.12 ±0.156 3.56 -14.15
0.05 nC6 45 5.15 ±0.257 4.81 6.58
0.25 nC6 45 4.86 ±0.243 4.84 0.49
0.50 nC6 45 4.41 ±0.220 4.55 -3.17
0.75 nC6 45 3.74 ±0.187 4.02 -7.45
0.95 nC6 45 3.10 ±0.155 3.52 -13.62
81
Table 6-5: Soret coefficient x 103 (K
-1) for the binary mixtures of benzene and n-
heptane at different temperatures. Experimental data was extracted from Wittko and
Kohler, 2007.
Mixture of
C6H6 – nC7
mole%
T
(°C)
Exp.
Present
work
Diff.
%
0.05 nC7 20 8.30 ±0.415 7.62 8.15
0.25 nC7 20 6.95 ±0.347 6.41 7.76
0.50 nC7 20 4.60 ±0.230 4.81 -4.52
0.75 nC7 20 2.90 ±0.145 3.60 -24.06
0.05 nC7 30 7.45 ±0.372 7.06 5.23
0.25 nC7 30 6.20 ±0.310 5.90 4.81
0.50 nC7 30 4.08 ±0.204 4.43 -8.52
0.75 nC7 30 2.80 ±0.140 3.32 -18.73
0.05 nC7 40 6.80 ±0.340 6.56 3.57
0.25 nC7 40 5.80 ±0.290 5.45 6.02
0.50 nC7 40 3.80 ±0.190 4.09 -7.65
0.75 nC7 40 2.60 ±0.130 3.08 -18.48
6.3. Summary
The proposed activation energy model was used to estimate the ratio of evaporation
energy to activation energy. The new approach was then used to evaluate the Soret
coefficient for binary mixtures of linear chain and aromatic hydrocarbons. The new
model of the ratio of evaporation energy to activation energy combined with the Shukla
and Firoozabadi model (1998) along with the PC-SAFT equation of state provides a
significant improvement in the accuracy of thermodiffusion modeling for the mixtures
investigated.
82
7. Evaluation of the activation energy of
viscous flow for a binary mixture in
order to estimate the thermodiffusion
coefficient
The evaluation of the activation energy in Eyring’s viscosity theory is of great
importance in estimating the thermodiffusion coefficient for associating and non-
associating fluid mixtures. Several methods were used to estimate the activation energies
of pure components and then extended to mixtures of linear hydrocarbon chains. Results
show that the recent model of Abbasi et al. (2009a) gives a good outcome in determining
the activation energy of the components in binary mixtures. The activation energy model
for pure components is shown to be useful for obtaining the activation energy of the
mixture. In this chapter, the activation energy model using alternative forms of Eyring’s
viscosity theory is used to estimate the thermodiffusion coefficient values for
hydrocarbon binary mixtures. Comparisons of predicted thermodiffusion coefficients
using different theoretical models with the experimental data show good capability of the
activation energy model.
7.1. Activation energy of viscous flow of a pure component
The activation energy or viscous energy is an energy required to overcome the internal
resistance to flow. The internal friction of a fluid is represented by its viscosity.
Molecules generate friction as they pass each other in the flow (Totten, 1999). Detailed
and comprehensive theories have presented significant understanding of the physical
83
chemistry and molecular movement in a viscous flow. However, the theories are not so
well advanced that accurate predictions of the activation energy of viscous flow can not
yet be obtained for a wide range of temperatures and pressures. As a result, many
investigators have attempted to use the viscosity principle to explain the variations of the
activation energies of different types of molecules with temperature.
Based on the lattice theory, liquids consist of a matrix of molecules and holes scattered
throughout (Totten, 1999). In a viscous flow of liquid, the flow unit is considered to be a
group of molecules, a single molecule or a segment of a molecule, and jump into an
existing hole, thereby creating a new hole. Some amount of energy is required for a
molecule to jump over an energy barrier into a hole. In a binary mixture consisting of two
different components, the energy required to remove a molecule from its surroundings is
represented by the activation energy of that component in the mixture. Smaller molecules
need less energy to be removed from their surroundings, as their molecular size does not
exceed the perimeter of the average pore size within the matrix. At higher temperatures,
more energy is available and more jumps can be made per unit time, resulting in lower
viscous or activation energy.
Eyring (Glasstone et al., 1941) determined the Gibbs free activation energy as a function
of viscosity as follows:
hN
VRTG
Av
iii
ln (7.1)
84
whereiG ,
iV , and i are Gibbs free activation energy of viscous flow, the molar
volume, and the viscosity of component i, respectively. The parameters AVN , h, R, and T
are the Avogadro’s number, the Plank constant, the universal gas constant, and the
temperature respectively. Sinceiii STHG , Eq. 7.1 may be re-written in the
following form.
i
Av
iii ST
hN
VRTH
ln (7.2)
where iH is the viscous flow activation enthalpy and
iS is the activation entropy of
viscous flow of component i. Using Eq. (7.2), four different equations (Glasstone et al.,
1941; Kotas and Valesova, 1986) were suggested for the calculation of the activation
energy of viscous flow. Table 7-1 summarizes the activation energy equations,
ipureviscU , , for a pure component. In the first equation, (see Table 7-1, Eq. 7.3), the
activation entropy of viscous flow and molar volume were assumed to be constant. This
is a fair assumption since the molar volume of a liquid does not vary greatly with
temperature (Glasstone et al., 1941). A relationship of this kind was suggested
empirically by Arrhenius (1916) and by Guzman (1913), and derived theoretically in a
different manner by Andrade (1934).
The second equation, (see Table 7-1, Eq. 7.4), is another form of Eq. 7.3 obtained by
considering the molar volume of a liquid as a variable. The third activation energy
equation, (see Table 7-1, Eq. 5.2), was derived based on the assumption that a liquid is a
pseudocrystalline fluid. The activation energy is then related to a single molecule moving
85
in its free volume. In this method, Eq. 7.2 in Eyring’s theory was further expanded to
express the activation energy as a function of parameters such as molecular weight,
temperature, and molar volume of the material (Glasstone et al., 1941). In the third
activation energy equation, iVapU , and iM are the energy of vaporization and the molar
mass of component i, respectively. The constant Bk is the Boltzmann’s constant, and the
constant C relates the free volume to the incompressible diameter and the volume of each
molecule. For a cubic packing of the lattice, this constant is set equal to 2.
Table 7-1: Activation energy functions for a pure component.
Eq. 7.3
P
i
ipureviscT
RU
/1
ln,,
Eq. 7.4
P
ii
ipureviscT
VRU
/1
ln,,
Eq. 5.2
2/32/12/16/1
,
3/2
,,)2(
lnTMkCRN
UVRTU
iBAv
iiVapi
ipurevisc
Eq. 7.5
1
ln
ln
9
7
2
1
ln
lnln,,
TTRTU iki
ipurevisc
In addition to all of the foregoing expressions for the activation energy, Kotas and
Valesova (1986) presented an equation (see Table 7-1, Eq. 7.5), which is labeled as the
fourth equation to determine the activation energy of viscous flow. In their approach, the
value of the activation energy of viscous flow can be determined from the Eyring
relation, i.e., Eq. 7.1, when the molar volume and the free volume per molecule are
86
expressed as a function of the densityi , and the relative permittivity
i . One of the
advantages of using this equation is that the temperature dependent variables are
expressed in the form of easily measurable physico-chemical parameters of the liquid
which is valid only in the case of non-polar compounds.
7.2. Activation energy of viscous flow for a binary mixture
Two different approaches may be used to calculate the activation energy of viscous flow
for a binary mixture. In the first approach, the activation energy of the mixture is
calculated from those of both components. However, in the second approach, one
considers the whole mixture as a single component.
First approach
The free volume theory states that the transfer kinetics of diffusing molecules depends
greatly on the molecular size and shape. Based on the free volume theory, small
molecules have a higher activation energy and as a result higher viscosity. The diffusivity
of the molecules is directly related to the size of the molecules. Pores that are larger than
the diffusing molecule will permit diffusion with little or no resistance, whereas diffusing
species larger than the pores will be entangled inside the mesh (Abbasi et al.,
2009a&b&c). The effects of the molecular size and average mesh size for specific
molecules on the required energy for detaching a molecule can be shown in the molecular
weight and the specific volume of each component. Abbasi et al. (2009a&b&c) showed
that this molecular size effect may be defined as the geometric average value of the
87
specific volume and the molecular weight fraction of each component. They proposed a
model for calculating iviscU , for each component in a mixture based on the free volume
theory, Eq. 5.1. The pure viscous or activation energy of component i required for Eq. 5.1
can be calculated from Eqs. 5.2 and 7.3 to 7.5 given in Table 7-1. The activation energy
of the mixture as a function of the activation energy of each component is suggested by
Singh and Sinha (1984, 1985) as follows:
2,21,1, viscviscmixvisc UxUxU (7.6)
Second approach
Besides the proposed model of activation energy of a mixture given by Eq. 7.6, the most
noticeable way to find the activation energy of viscous flow for a mixture is by using a
method similar to that for a single component. Table 7-2 shows four different expressions
(Eqs. 7.7-7.10) used to calculate the activation energy of a mixture at a given
temperature. These equations consider that the entire mixture behaves like a single fluid
with one viscosity associated with it. Therefore, only one value for the activation energy
of viscous flow is defined that is associated with the entire mixture and not each single
component.
The thermodynamic properties required for thermodiffusion coefficients and activation
energy models can be derived using the Perturbed Chain Statistical Associating Fluid
Theory (PC-SAFT) equation of state. The pure component parameters used in PC- SAFT
88
equation of state, including the temperature-independent segment diameter, depth of the
potential well, and the number of segments in a chain are listed in Abbasi et al.’s work
(2009a). The relative permittivities i of pure materials and mixtures
mix obtained by
using Peon–Iglesias approach (1994) are listed in Tables 7-3 and 7-4, respectively. The
viscosity values used in these calculations are obtained from the NIST database (2007).
Table 7-2: Activation energy functions for a hydrocarbon mixture.
Eq. 7.7
P
mix
mixviscT
RU
/1
ln,
Eq. 7.8
P
mixmix
mixviscT
VRU
/1
ln,
Eq. 7.9
2/32/12/16/1
,
3/2
,)2(
lnTMkCRN
UVRTU
mixB
mixmixVapmix
mixvisc
Eq. 7.10
1
ln
ln
9
7
2
1
ln
lnln,
TTRTU mixmix
mixvisc
Table 7-3: The relative permittivity i of pure materials of nCi (i=5, 6, 7, 8, 9, 10, 12,
15, 16, 17, 18) at 298.15 K obtained by Kotas and Valesova’s approach (1986).
Component nC5 nC6 nC7 nC8 nC9 nC10 nC12 nC15 nC16 nC17 nC18
i 1.829 1.887 1.917 1.942 1.963 1.985 2.016 2.054 2.067 2.078 2.085
89
Table 7-4: The relative permittivity mix of 50% mole fraction mixtures of nCi-C10 (i=5,
6, 7, 15, 16, 17, 18), nCi-C12 (i=5, 6, 7, 8, 9), and nCi-C18 (i=5, 6, 7, 8, 9, 12) at 298.15 K.
mix is obtained by Peon–Iglesias approach (1994).
Mixture
50%
mole
C10-
C5
C10-
C6
C10-
C7
C15-
C10
C16-
C10
C17-
C10
C18-
C10
C12-
C5
C12-
C6
C12-
C7
C12-
C8
C12-
C9
C18-
C5
C18-
C6
C18-
C7
C18-
C8
C18-
C9
C18-
C12
mix 1.863 1.907 1.930 1.998 2.002 2.004 2.006 1.953 1.969 1.977 1.985 1.973 1.897 1.937 1.958 1.975 1.990 2.030
7.3. Results and discussion
Four expressions were used to calculate the activation energy of a binary mixture and the
results are compared with the available experimental data in this section. Then, these
models were used to determine the thermodiffusion coefficients and the predicted results
are compared with the available experimental data.
Activation energy of a mixture
The main objective for calculating the activation energy of a mixture using different
methods was to investigate the accuracy of the proposed model, Eq. 5.1, for predicting
the activation energy of each component in a mixture. The four methods used to estimate
the activation energy of a mixture were;
Method 1: Comparison of estimated activation energy of a mixture using Eq. 7.3,
Eq. 5.1, and Eq. 7.6 (method 1.1) or Eq. 7.7 (method 1.2).
90
Method 2: Comparison of estimated activation energy of a mixture using Eq. 7.4,
Eq. 5.1, and Eq. 7.6 (method 2.1) or Eq. 7.8 (method 2.2).
Method 3: Comparison of estimated activation energy of a mixture using Eq. 5.2,
Eq. 5.1, and Eq. 7.6 (method 3.1) or Eq. 7.9 (method 3.2).
Method 4: Comparison of estimated activation energy of a mixture using Eq. 7.5,
Eq. 5.1, and Eq. 7.6 (method 4.1) or Eq. 7.10 (method 4.2).
Figure 7-1 shows the activation energy values for pure components estimated by using
Eqs. 5.2 and 7.3 – 7.5 at a temperature of T = 298.15 K and pressure, P = 101,325 Pa.
The results show that the variations among the four activation energy values estimated
using different methods are more than 10%. As one can observe the deviations of the
estimated activation energy values from the available experimental data of Qureshi
(1971) are more than 20%.
91
Figure 7-1: The activation energy of pure hydrocarbon components at 298.15 K.
The activation energy values for mixtures were then estimated using methods 1-4 are
compared in Figures 7-2-7-4. As expected the activation energy for viscous flow of a
mixture using the proposed model given by Eq. 5.1 is in good agreement with the results
of Eq. 7.7 to Eq. 7.10 where the entire mixture was considered to behave like a single
fluid with one viscosity associated with it. This confirms that the free volume theory is
the paramount factor in the required energy for detaching a molecule and as a result
controlling the transfer kinetics of diffusing molecules. In addition, the results obtained
from methods 3.1 and 3.2 are very close. As a matter of fact both methods 3.1 and 3.2 are
based on the free volume theory. The results confirm that using Eq. 5.2 for the estimation
92
of the activation energy of a pure component gives less errors than the other methods.
Using Eq. 5.2, the liquid is considered as a pseudocrystalline fluid. In this model, a liquid
may be treated as if it were composed of individual molecules each moving in a free
volume with an average potential field due to its neighbors. In this approach, the
rotational and vibrational movements of the molecules are considered in the partition
function of a molecule in a liquid mixture.
Figure 7-2: Comparison of estimated activation energy of binary hydrocarbon mixtures
of C10-nCi (i=5, 6, 7, 15, 16, 17, 18) at 298.15 K.
93
Figure 7-3: Comparison of estimated activation energy of binary hydrocarbon mixtures
of C12-nCi (i=5, 6, 7, 8, 9) at 298.15 K.
Figure 7-4: Comparison of estimated activation energy of binary hydrocarbon mixtures
of C18-nCi (i=5, 6, 7, 8, 9, 12) at 298.15 K.
94
Thermodiffusion coefficients
The main reason to estimate the thermodiffusion coefficients is to investigate the effect of
the activation energy of viscous flow equations, Eq. 5.2 and Eq. 7.3 to Eq. 7.5, for each
component on the proposed net heat of transport model. The following four methods
were used to predict the thermodiffusion coefficients for different hydrocarbon binary
mixtures.
Method 1: Equations 7.3 and 5.1 were used to estimate the activation energies of
the components in the mixtures.
Method 2: Equations 7.4 and 5.1 were used to estimate the activation energies of
the components in the mixtures.
Method 3: Equations 5.2 and 5.1 were used to estimate the activation energies of
the components in the mixtures.
Method 4: Equations 7.5 and 5.1 were used to estimate the activation energies of
the components in the mixtures.
The predicted thermodiffusion coefficients using different methods are compared with
available experimental data in Figures 7-5 and 7-6. The difference between the predicted
thermodiffusion coefficients and the experimental values are shown in Figures 7-7 and 7-
8. It is evident that the results of the proposed models for predicting the thermodiffusion
coefficients are in very good agreement with the experimental data. Comparing two
components such as C5 and C6 in their mixtures with C10, C12 and C18, shows that the
95
smaller component, C5, has a larger thermodiffusion coefficient than the bigger
component, C6, in the mixtures (Figures 7-5 and 7-6). This observation can be explained
with the free volume theory, which states that the transfer kinetics of diffusing molecules
depends greatly on the molecular size and shape. Pores that are larger than the diffusing
molecule will permit diffusion with little or no resistance. However, diffusing species
larger than the pores will not flow easily as they become entangled with the matrix mesh
(Abbasi et al., 2009a&b). One can notice that the predicted thermodiffusion coefficients
are not very sensitive to the activation energy values for pure components. This can be
one of the positive aspects compared to the Shukla and Firoozabadi model (1998). It was
shown that the Shukla and Firoozabadi model is very sensitive to the ratio of the
evaporation energy to the activation energy. If the ratio of the evaporation energy to the
activation energy changes slightly, the estimated thermodiffusion coefficient may vary
significantly for some mixtures. Again the results show that using Eq. 5.2 for the
estimation of the activation energy of pure components and thermodiffusion coefficients
are more accurate than the other methods.
96
Figure 7-5: Thermodiffusion coefficients x 10-12
(m2/sK) in 50% mole fraction for nCi-
C10 (i=5, 6, 7, 15, 16, 17, 18), nCi-C12 (i=5, 6, 7, 8, 9), and nCi-C18 (i=5, 6, 7, 8, 9, 12) at
298.15 K. Experimental data are extracted from Yan et al. 2008 and Blanco1 et al. 2007,
2008.
97
Figure 7-6: Thermodiffusion coefficients x 10-12
(m2/sK) in 50% mass fraction for nCi-
C12 (i=5, 6, 7, 8, 9) and nCi-C18 (i=5, 6, 7, 8, 9, 12) at 298.15 K. Experimental data are
extracted from Yan et al. 2008 and Blanco1 et al. 2007, 2008.
98
Figure 7-7: The difference between the predicted thermodiffusion coefficients and the
experimental values for nCi-C10 (i=5, 6, 7, 15, 16, 17, 18), nCi-C12 (i=5, 6, 7, 8, 9), and
nCi-C18 (i=5, 6, 7, 8, 9, 12) at 298.15 K.
99
Figure 7-8: The difference between the predicted thermodiffusion coefficients and the
experimental values for nCi-C12 (i=5, 6, 7, 8, 9) and nCi-C18 (i=5, 6, 7, 8, 9, 12) at 298.15
K.
7.4. Summary
Different methods were used to estimate the activation energies of pure components and
the mixtures of linear hydrocarbon chains. Results show that the present model, Eq. 5.1,
provides good predictions of the activation energy of hydrocarbon mixtures. The
100
predicted activation energies of hydrocarbon mixtures using different approaches to
obtain the activation energy of the pure components are in good agreement supporting the
proposed idea where we consider the mixture is considered to behave like a single
component. The estimated thermodiffuion coefficients using alternative forms of
Eyring’s viscosity theory (i.e., Eqs. 5.2 and 7.3 to 7.5), combined with the proposed
model are in good agreement with the experimental data. Among these activation energy
equations, Eq. 5.2 which considers the liquid as a pseudocrystalline fluid has performed
better with the present model (i.e., with Eq. 5.1). This can be attributed to the
consideration of the free volume theory as the foundation for the development of the
present model and Eyring’s activation energy function presented in Eq. 5.2.
101
8. Study of Thermodiffusion of Carbon
Dioxide in Binary Mixtures of n-Butane
& Carbon Dioxide and n-Dodecane &
Carbon Dioxide in Porous Media
Convection due to a thermodiffusion phenomenon has an important effect on component
separation of hydrocarbon mixtures in a porous medium. A numerical study of carbon
dioxide diffusion in porous medium is investigated in the presence of different fluid
mixtures such as n-butane & carbon dioxide and n-dodecane & carbon dioxide single
phase. In this chapter, all physical properties with an exception of the mixture
conductivity is assumed as varying with temperature and concentration. The fluid is
maintained at a pressure of 150 bar and remains in the liquid state. Constant temperature
gradients in horizontal and vertical directions are applied on the three dimensional porous
domain. Thermodiffusion coefficients applied in simulation were calculated by using
Abbasi et al’s. (2009b) thermodiffusion model. Results reveal that for a certain
concentration of carbon dioxide the thermodiffusion coefficient reaches a maximum
leading to large separation. In the presence of the Soret effect, the vertical density
distribution tends closer to the one without the Soret effect. With an increase in the
permeability, the convection becomes dominant and contributes to a decrease in the
vertical and horizontal component separation considerably.
102
8.1. Mathematical Model
The mathematical description of the flow of a viscous fluid through a three-dimensional
porous medium is based on the Darcy equation for momentum conservation. The porous
medium is assumed to be homogeneous and isotropic. The mass continuity equation may
be written in the following form:
Vc
t
c.
(8.1)
where c represents molar density of the fluid per unit volume, wkji vuV is the
average velocity vector of the mixture, and u , v , w are the velocity components in x, y,
z directions, respectively. The continuity equation for the component i may be given as
follows:
ii
i Jcxt
cx..
V (8.2)
where ix is mole fraction and iJ is the molar diffusion flux of the component i. When
only the contributions of molecular diffusion and thermodiffuison are considered, the
diffusion flux can be described as follows:
TDxDJ Ti *
11
*
11. (8.3)
103
Here, *
11D and *
1TD are the molecular diffusion and thermodiffusion coefficients of the
fluid mixture in the porous medium, respectively, which are related to molecular
diffusion and thermodiffusion coefficients in free liquid as follows:
2
*
1*
12
*
11*
11 ,
mol
TT
mol DD
DD (8.4)
where molD11 and mol
TD 1 are the molecular and thermodiffusion coefficients, which are
functions of the temperature and composition of the fluid mixture. * is the tortuosity for
molecular diffusion and thermodiffusion coefficients in the porous medium. Darcy’s law
for the fluid in a porous medium is expressed as follows:
gPk
mix
V (8.5)
where k and are the permeability and the porosity of the porous medium, respectively,
mix is the dynamic viscosity, is the mass density of the fluid mixture, and g is the
gravitational acceleration vector.
The thermal energy conservation equation is expressed as follows:
TkkTCV
t
TCCPffP
PPfP 21.1
(8.6)
104
where fPC is the fluid volumetric heat capacity,
PPC is the matrix volumetric heat
capacity, fk is the fluid thermal conductivity, and Pk is the matrix or porous medium’s
thermal conductivity.
8.2. Model Description
The porous medium has a horizontal length of 500 m, a width of 500 m, and a height of
500 m as shown in Figure 8-1. Zero mass flux and no-slip condition were set for the
boundaries. The physical properties of porous medium are given in Table 8-1. Similar to
Riley & Firoozabadi (1998) and Nasrabadi et al. (2006), it is assumed that the porous
medium is bounded by the rock that has constant temperature gradients in the horizontal
and vertical directions. Therefore the boundary temperatures of the porous medium are
set as 0TzdTydTxdTT zyx where xdT , ydT , and zdT are the temperature
gradients in x , y and z directions, which were specified to be 1.5K/km, -3K/100m, and
1.5K/km, respectively. 0T is the temperature at
0xx , 0yy , and
0zz (Figure 8-2).
The reference temperature 0T at 0xx ,
0yy , and 0zz was set to 313.15K and the
pressure of the medium was considered to be 150 bar. The reason for the choice of the
values for temperature and pressure values is to have a liquid phase in the porous
medium. Phase diagrams for the mixtures of interest were calculated by NIST database
(2007). Initially, the porous medium has a 50% weight fraction of carbon dioxide. It
should be considered that the carbon dioxide is in supercritical condition at 313.15K;
however, the carbon dioxide mixtures with n-butane and n-dodecane are in liquid phase
at this tempreature.
105
Figure 8-1: Schematic of the porous medium
Table 8-1: The physical properties of porous medium.
Density 3983.60 kg/m3
Heat capacity 786.27 J/kg.K
Thermal conductivity 43 J/s.m.K
Tortuosity 35.2
Permeability 10
-4 md, 10
-3 md , 10
-2 md
10-1
md, 1md
Porosity 0.2
Uniform initial pressure 150 bar
y
x
z
500m
500m 500m
0,0 ViJ
0,0 ViJ
0,0 ViJ
g
106
Figure 8-2: Temperature profile in horizontal and vertical directions.
8.3. Solution Technique and Mesh Sensitivity
Governing equations (8.1-8.6) are solved numerically by using the control volume
method with a rectangular grid system. The second-order centered scheme is used in the
space discretization, and an implicit first-order scheme is used for the temporal
integration. The Gauss-Seidel convergence method with a given convergence criterion is
used to solve the linear system of algebraic equations. The convergence criterion is set
107
for three parameters, the pressure, temperature, and composition, respectively. The
pressure-based solver is used for residuals as follows.
Pcells pp
Pcells nb ppnbnb
a
abaR
(8.7)
where represents the pressure, temperature, and composition, respectively; Here pa is
the center coefficient, nba are the influence coefficients for the neighboring cells, and b is
the contribution of the constant part of the source term. A tight convergence criterion is
applied in the time integration to determine the establishment of the steady state. The
mesh size is chosen based on the convergence limit of the average Nusselt number, Nu,
during the mesh refining process. The average Nusselt number defined at the hot and cold
wall of the cavity is given below:
dydzx
T
T
L
WHNu
wall
1 (8.8)
The average Nusselt number is equivalent to non-dimensional heat flux averaged over the
wall surface of the cavity. Figure 8-3 shows the values of the average Nusselt number
obtained with different types of mesh size for a permeability of 0.001md. The value of
Nu approaches an asymptotic value, 1.16 with an increase in the mesh number. If a 5%
relative error in the average Nusselt number is considered to be accepted, then mesh
numbers from 111111 up to 818181 can be used. In our calculation, 414141
control volume has been adopted.
108
Figure 8-3: Variation of Nusselt number with mesh NxNxN in 3D
The solution procedure begins by assuming initial pressure, temperature, and
concentration values in the mixture. The thermodynamic properties in each cell are
changing. So, the density and viscosity of each component were calculated by NIST
database (2007) as functions of temperature. The required thermodynamic properties for
molecular diffusion and thermodiffusion coefficients were derived using the Perturbed
Chain Statistical Associating Fluid Theory (PC-SAFT) equation of state. The pure
component parameters used in the PC-SAFT equation of state, including i (temperature
independent segment diameter), i (depth of the potential well), and im (number of
109
segments in a chain) are listed in Table 8-2. Molecular diffusion and thermodiffusion
coefficients are functions of the temperature and the species composition.
Table 8-2: Pure component parameters from PC-SAFT EoS ((Gross and Sadowski,
2001).
Component M
(g/mol)
km k
(A0)
k / Bk
(K)
Carbon
Dioxide 44.01 2.0729 2.7852 169.21
n-Butane 58.123 2.3316 3.7086 222.88
n-Dodecane 170.338 5.3060 3.8959 249.21
8.4. Numerical Results
The main objective of this work is to investigate the effect of thermodiffusion
coefficients on compositional variations in the porous medium. Then, the convection
effect on the thermodiffusion process is investigated for different permeability values in
terms of the separation ratio.
110
Density variation
Figures 8-4-a to 8-4-c show the horizontal ( my 250 and mz 250 ) and vertical
( mx 250 and mz 250 ) variations of density for the permeability of 0.001md. In the
vertical direction, a uniform density distribution with or without thermodifusion effect is
observed (Figure 8-4-c). This confirms that the flow is weak and does not significantly
affect the vertical distribution. However, there are changes in the density variation in the
horizontal direction. This density variation comes from the thermodiffusion effect. In the
presence of thermodiffusion, the smaller component (n-butane and n-dodecane) migrates
to the cold spots. The carbon dioxide being the heavy component of the mixture migrates
to the hot spots. As a result, there is a less density variation when thermodiffusion is
present. The average densities in the horizontal direction (Figure 8-4-a and 8-4-b) are
684.8 kg/m3 for n-butane & carbon dioxide mixture and 785 kg/m
3 for n-dodecane &
carbon dioxide mixture. The maximum density variations in n-butane & carbon dioxide
and n-dodecane & carbon dioxide mixtures are 2.01 and 2.28 kg/m3 when the
thermodiffuison is absent and 1.22 and 2.11 kg/m3
when the thermodiffuison is present.
Therefore the density variation decreases from 0.29% to 0.18% for the n-butane & carbon
dioxide mixture and from 0.29% to 0.27% for the n-dodecane & carbon dioxide mixture.
111
Figure 8-4-a: Density variation in horizontal direction for permeability of 0.001md (n-
butane & carbon dioxide mixture).
112
Figure 8-4-b: Density variation in horizontal direction for permeability of 0.001md (n-
dodecane & carbon dioxide mixture).
113
Figure 8-4-c: Density variation in vertical direction for permeability of 0.001md.
Calculation of thermodiffusion coefficient for binary mixtures
Thermodiffusion coefficient estimation based on the free volume theory has been found
to be highly reliable and represents a significant improvement over the earlier
thermodiffusion models (Abbasi et al., 2009a&b) as presented in chapter 6.
Abbasi et al. (2009a&b) have developed a new model for predicting the activation
energy of viscous flow for each component in a mixture based on the free volume theory.
The model combined with the Dougherty and Drickamer model and Shukla and
114
Firoozabadi model were used to predict the net heat of transport for each component in
binary mixtures. As a matter of fact, the size of the molecule that has a significant effect
on the diffusivity of the molecule is considered in predicting the net heat of transport.
Initially, Dougherty and Drickamer’s model was modified for predicting thermodiffusion
coefficients for binary linear chain hydrocarbon mixtures (Abbasi et al. (2009a).
Thermodiffusion coefficients for C10-nCi (i=5, 6, 7, 15, 16, 17, 18), C12-nCi (i=5, 6, 7, 8,
9), and C18-nCi (i=5, 6, 7, 8, 9, 12) were calculated by the new proposed model.
Comparisons of the calculated theoretical results with the experimental data showed a
good performance of the proposed model. Next, the ratio of evaporation energy to
viscous or activation energy used in the Shukla & Firoozabadi model was modified for
predicting thermodiffuion of binary mixtures of aliphatic and aromatic compounds
(Abbasi et al. model 2009b), k calculated by Eq. 6.2. The modified Shukla &
Firoozabadi model was used to calculate Soret coefficient for binary mixtures of toluene
& n-hexane, and benzene & n-heptane. The new model of the ratio of evaporation energy
to viscous or activation energy, Eq. 6.2, showed a significant improvement in the
accuracy of thermodiffusion modeling for the mixtures investigated.
The predicted thermodiffusion coefficients for carbon dioxide in binary mixtures with n-
butane and n-dodecane are shown in Figures 8-5-a to 8-5-d. Results shows
thermoddiffusion coefficients calculated by Shukla & Firoozabadi model (1998), with
k = 4, are less than 10 times of the values obtained from the present model with k
calculated with Eq. 6.2. Variation of temperature has a major effect on the calculated
thermodiffusion coefficients using the Shukla and Firoozabadi model and present model.
115
Comparing the predicted thermodiffusion coefficients of carbon dioxide in binary n-
butane-carbon dioxide mixtures in Figures 8-5-a and 8-5-b shows the rising temperature
has an inverse effect on the Shukla and Firoozabadi model results. It illustrates the
increase temperature would decrease the thermodiffusion coefficients of carbon dioxide
in binary n-butane-carbon dioxide mixtures. This effect is not confirmed by the free
volume theory. However, the temperature variations make a specific explanation for the
results obtained from the present model. The free volume theory states that the transfer
kinetics of diffusing molecules depends greatly on the molecular size and shape. Pores
that are larger than the diffusing molecule will permit diffusion with little or no
resistance. However, diffusing species larger than the pores will not flow easily as they
become entangled with the matrix mesh. As a result, the required energy for detaching a
molecule is directly related to the molecular size and shape. At high temperatures, a more
energy is available and more jumps can be made per unit time, resulting in lower viscous
energy. In another word, a rising temperature reduces the required energy for a molecule
to jump over an energy barrier. As a result the diffusivity of the molecule increases. The
molecular size as well as the matrix size of molecules and holes may vary by changing
the temperature. The present model accounts for the molecular size effect in which the
rising temperature would enlarge the hole sizes and as a result themodiffusion coefficient
increases.
116
Figure 8-5-a: n-Butane thermodiffusion coefficients as a function of carbon dioxide in n-
butane & carbon dioxide mixtures. (present model)
117
Figure 8-5-b: n-Butane thermodiffusion coefficients as a function of carbon dioxide in n-
butane & carbon dioxide mixtures. (Sukula and Firoozabadi model, 1998)
118
Figure 8-5-c: n-Dodecane thermodiffusion coefficients as a function of carbon dioxide n-
dodecane & carbon dioxide mixtures. (present model)
119
Figure 8-5-d: n-Dodecane thermodiffusion coefficients as a function of carbon dioxide
n-dodecane & carbon dioxide mixtures. (Sukula and Firoozabadi model, 1998)
In this work, Abbasi et al.’s model (2009b) was used to calculate the thermodiffusion
coefficients in the porous medium. The thermodiffusion coefficients of carbon dioxide in
n-butane and n-dodecane mixtures for the permeability of 0.001md are shown in Figures
8-5-e and 8-5-f. The predicted thermodiffusion coefficients are all negative. Results
illustrate that the thermodifusion coefficients vary considerably in the vertical direction
( mx 250 and mz 250 ); however the variation of those values in the horizontal
directions is not considerable ( my 250 and mz 250 ). The changes in the
thermodiffuison coefficients indicate a change in the fluid temperature. The temperature
120
gradient in the vertical direction is 20 times more than that in the horizontal direction
which causes a great difference in the thermodiffusion coefficients.
Figure 8-5-e: Carbon dioxide thermodiffusion coefficient for permeability of 0.001md.
121
Figure 8-5-f: Carbon dioxide thermodiffusion coefficient for permeability of 0.001md.
122
Compositional variation
Figures 8-6-a to 8-6-d show the compositional separation of carbon dioxide for
permeability values ranging from 0.0001 to 1md. The figures illustrate the carbon
dioxide distribution along the vertical ( mx 250 and mz 250 ) and the horizontal
( my 250 and mz 250 ) directions. These results show that the permeability has a
strong effect on the carbon dioxide separation. In the vertical direction, the separation of
carbon dioxide decreases continuously as the permeability increases from 0.0001 to 1md.
In terms of the fluid compositional variation, it is found that when the permeability varies
between 0.0001 and 0 .01md, the convection flow is weak and the thermodiffusion effect
is dominant. In this range of the permeability, the transport of carbon dioxide is shown to
be effective in the horizontal and vertical directions. In n-butane & carbon dioxide
mixtures, the largest variation of the concentration is found at 0.0001 and 0.001md
permeabilities where the separation of the carbon dioxide is at its maximum in the
vertical and horizontal directions, respectively. As the value of the permeability increases
from 0.0001 to 1md, the variation of the carbon dioxide concentration will be limited. In
n-dodecane & carbon dioxide mixtures, the maximum separation of the carbon dioxide
occurs for 0.01md permeability in the horizontal direction and 0.0001md permeability in
the vertical direction. In both n-butane & carbon dioxide and n-dodecane & carbon
dioxide mixtures at 1md permeability, a flat carbon dioxide concentration distribution in
both vertical and horizontal directions is observed. It is therefore important to examine
the separation ratio and its relationship to the permeability.
123
Figure 8-6-a: Carbon dioxide mass fraction in horizontal direction (n-butane & carbon
dioxide mixture).
124
Figure 8-6-b: Carbon dioxide mass fraction in vertical direction (n-butane & carbon
dioxide mixture).
125
Figure 8-6-c: Carbon dioxide mass fraction horizontal direction (n-dodecane & carbon
dioxide mixture).
126
Figure 8-6-d: Carbon dioxide mass fraction in vertical direction (n-dodecane & carbon
dioxide mixtures).
127
Separation ratio
In order to better understand the separation of components, the separation ratio is defined
as follows:
minmin
maxmax
1/
1/
CC
CCq
(8.9)
where maxC and minC are the maximum and minimum carbon dioxide concentrations in
the porous medium. As already discussed, the convection effect is critical to the analysis
of the thermodiffusion phenomenon. As the permeability of the porous medium
increases, three different regimes are found based on the analysis of the carbon dioxide
concentration distributions. These three different regimes are in accordance with the three
ranges of permeability. The variation of the separation ratio as a function of the
permeability is shown in Figure 8-7. In butane-carbon dioxide mixtures, for permeability
values below 0.00001md, the separation ratio remains constant at about 1.06. This
separation ratio is due to the contribution of the molecular diffusion and thermodiffusion.
The convection effect is too small. The maximum separation ratio occurs at 0.0001md
permeability. As the permeability becomes greater than 0.001md, it is observed that the
separation ratio decreases rapidly. When the permeability is equal to 1md, the separation
ratio is close to 1.003 as the fluid is mixed and separation does not occur. Obviously for
the permeability values greater than 0.01md, the convection effect becomes dominant and
the thermodiffusion effect is suppressed. In n-dodecane & carbon dioxide mixtures, the
128
peak in the separation ratio is predicted to occur at a permeability of 0.001md. The
separation ratio drops to 1.007 at a permeability of 1md.
Figure 8-7: Separation ratio of carbon dioxide in n-butane & carbon dioxide and n-
dodecane & carbon dioxide mixtures.
8.5. Summary
A model of thermosolutal convection for binary mixtures of n-butane & carbon dioxide
and n-dodecane & carbon dioxide in porous media is presented. The thermodiffuion
model presented in chapter 6 was implemented in the simulation. The diffusion
coefficients, the density, viscosity, and thermal conductivity were calculated with time
129
and space dependent fluid properties and compositions. The effect of permeability on
concentration distributions was investigated. Results show the thermodiffuison
phenomenon is dominant at low permeabilities (0.0001 to 0.01). As the permeability
increases convection plays an important role in the concentration distribution. At 1md
permeability, a flat carbon dioxide concentration distribution in both vertical and
horizontal directions is predicted.
130
9. Conclusions and recommendations
Theoretical analyses of thermodiffuison phenomena have been performed in the
framework of non-equilibrium thermodynamics and free volume theory. A new model
for predicting the thermodiffusion phenomena in non-associating mixtures was developed
and used to predict thermodiffusion in acetone-water, ethanol-water and isopropanol-
water mixtures. Additionally, two new models for predicting the thermodiffusion
phenomena in binary hydrocarbon mixtures of linear hydrocarbon chains and
combinations of aliphatic and aromatic compounds were developed. Finally, thermo-
convection effects in porous media were investigated numerically by solving diffusion
and convection equations for binary mixtures. From the results of these investigations,
the following conclusions can be drawn.
1. A new theoretical approach for calculating the activation energy and the ratio of
evaporation energy to activation energy, , was presented to evaluate the
thermodiffusion factor in associating mixtures, including acetone-water, ethanol-
water, and isopropanol-water. In particular, this approach was implemented to
predict the sign changes in the thermodiffusion factor for associating mixtures,
which has been a major step forward in thermodiffusion studies. The Firoozabadi
model combined with the PC-SAFT equation of state by using one adjustable
parameter calculated from experimental data was used for evaluating
thermodiffusion. The adjustable binary interaction parameter for the mixture of
interest under a range of temperatures has been optimized based on available
131
experimental data in vapor–liquid equilibrium. The results showed very good
agreement between the experimental data and the calculated values.
2. The free volume theory was used to develop a new theoretical approach for
calculating the activation energy in non-associating mixtures. The new model for
activation energy was used to evaluate the thermodiffusion coefficients in linear
chain hydrocarbon binary mixtures of C10-nCi (i=5, 6, 7, 15, 16, 17, 18), C12-nCi
(i=5, 6, 7, 8, 9), and C18-nCi (i=5, 6, 7, 8, 9, 12). In this model, the size of the
molecule that has a significant effect on thermodiffusivity of the molecule was
considered. This molecular size effect was considered through the geometric
average value of specific volume and molecular weight fraction of each
component. The new model combined with the PC-SAFT equation of state has
been applied to predict thermodiffusion coefficients. The new model provides a
significant improvement in the accuracy of thermodiffusion modeling for linear
chain hydrocarbon binary mixtures.
3. The proposed activation energy model for non-associating mixtures was used to
estimate the ratio of evaporation energy to activation energy, . The new
approach was used to evaluate the Soret coefficient in binary mixtures of linear
chain and aromatic hydrocarbons (toluene and n-hexane, and benzene and n-
heptane). The new model of the ratio of evaporation energy to activation energy
was combined with the Shukla and Firoozabadi model and the PC-SAFT equation
132
of state providing a significant improvement in the accuracy of thermodiffusion
prediction for the mixtures investigated.
4. In addition to the new models for associating and non-associating binary
mixtures, different methods were used to estimate the activation energies of pure
components and the mixtures of linear hydrocarbon chains. Results revealed that
the proposed activation energy model in non-associating mixtures could provide
good predictions of the activation energy of hydrocarbon mixtures. The predicted
activation energies of the hydrocarbon mixtures using different approaches to
obtain the activation energy of the pure components were in good agreement with
the data, supporting the present approach to consider the mixture like a single
component. The estimated thermodiffuion coefficients using alternative forms of
Eyring’s viscosity theory combined with the proposed activation energy model
are in good agreement with the experimental data. Among these activation energy
equations, Eyring’s viscosity model which considers the liquid as a
pseudocrystalline fluid has performed better with the present model. This can be
attributed to the free volume theory which is the basis for the development of the
proposed activation energy model and Eyring’s activation energy function.
5. Thermodiffusion coefficients for n-butane and carbon dioxide, and n-dodecane
and carbon dioxide mixtures were estimated at different temperatures. Rising
temperature changes the diffusivity of the component in their mixtures. There is a
significant difference in the calculated thermodiffusion coefficients using the
proposed activation energy model and Shukula and Firoozabadi model. Rising
133
temperature increases the thermodiffusion coefficients estimated by the proposed
activation energy model which is validated by free volume theory. However, the
Shukula and Firoozabadi model predictions for n-butane in binary n-butane and
carbon dioxide mixtures cannot be explained by the free volume theory.
6. A 3-dimensional model for thermosolutal convection in binary mixtures of n-
butane and carbon dioxide, and n-dodecane and carbon dioxide in porous media
was presented. The proposed activation energy model for estimating
thermodiffuion coefficients was implemented in the simulation. The diffusion
coefficients, density, and viscosity were calculated with time and space dependent
fluid properties and compositions under constant temperature gradients in
horizontal and vertical directions applied on the walls. The effect of permeability
on concentration distributions was investigated. Results illustrate the
thermodiffusion phenomenon is dominant at low permeabilities (0.0001 to 0.01
md). As the permeability increases, convection plays an important role in
concentration distribution, and the variation of carbon dioxide concentration will
be limited. At 1 md permeability, a flat carbon dioxide concentration profile was
predicted in both vertical and horizontal directions.
Based on the present work, future work may be carried out to address the following
points.
1. Although the present thermodiffusion models based on non-equilibrium
thermodynamics show a good capability for thermodiffusion estimation in associating
134
and non-associating mixtures, the contribution of free volume theory must be clarified. It
will be interesting to find an optimum equation for the molecular size effect presented in
free volume theory.
2. The proposed approach for estimating thermodiffuion coefficients for binary mixtures
in associating and non-associating mixtures may be extended to muticomponent
mixtures.
3. It will be interesting to perform numerical simulations of multiphase flow in porous
media with multiple permeabilities having thermosolutal convection to see the Soret
effect on concentration distribution.
135
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