theoretical investigation of thermodiffusion (soret …...theoretical investigation of...

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

ii

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

iii

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.

iv

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.

v

To My Family

vi

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

vii

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


5.3. 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


6.3. 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


Activation energy of a mixture ............................................................................... 89

Thermodiffusion coefficients ................................................................................. 94

7.4. Summary ....................................................................................................... 99

viii

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

ix

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


(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


(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


...................................................................................................................................... 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


-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

xi

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


6, 7, 8, 9) mixtures......................................................................................................... 65


6, 7, 8, 9) mixtures......................................................................................................... 66

xii

Figure 5-2-c: Thermodiffusion coefficients in 25% fraction of C12 in nCi-C12 (i=5, 6, 7, 8,

9) mixtures. ................................................................................................................... 67


6, 7, 8, 9) mixtures......................................................................................................... 68


6, 7, 8, 9) mixtures......................................................................................................... 69


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


of C12-nCi (i=5, 6, 7, 8, 9) at 298.15 K. .......................................................................... 93


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


(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


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

xiii

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

xiv

Nomenclature

Symbols

ja The coefficient defined by Eq. 3.14

ja0 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



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

http://en.wikipedia.org/wiki/Joule

http://en.wikipedia.org/wiki/Second

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


qkL The phenomenological coefficients under the frame of reference moving


qqL The phenomenological coefficients under the frame of reference moving


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

QQ

,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).


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


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


(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


(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


(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


(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


6, 7) mixtures.

63


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


6, 7, 8, 9) mixtures.

66


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


6, 7, 8, 9) mixtures.

69


6, 7, 8, 9) mixtures.

70


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.


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


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


-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


-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


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







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.


of C10-nCi (i=5, 6, 7, 15, 16, 17, 18) at 298.15 K.

93


of C12-nCi (i=5, 6, 7, 8, 9) at 298.15 K.


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.







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


(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


(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


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


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.

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Figure 8-5-f: Carbon dioxide thermodiffusion coefficient for permeability of 0.001md.

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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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