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Property Scaling Relations for Nonpolar Hydrocarbons
Sai R. Panuganti1, Francisco M. Vargas1, 2, Walter G. Chapman1
1 Chemical and Biomolecular Engineering Department, Rice University, Houston, USA
2 Department of Chemical Engineering, The Petroleum Institute, Abu Dhabi, UAE
February, 2013
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Outline
• One-Third Rule
• Electronic polarizability
• Dielectric constant
• Critical temperature and pressure
• Surface tension
• Conclusion
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One-Third Rule• Specific Refractivity: independent of the temperature and pressuren, refractive index and ρ, mass density (g/cc)
• For nonpolar hydrocarbons and their mixtures
1
2
12
2
n
n Constant
3
11
2
12
2
D
D
n
n
2
12
2
n
n True volume of the molecules in unit volume
2
12
2
n
n
True density of the molecules
• But strictly speaking, it is a function of the mass density and can be expressed as 2
2
2
2314.03951.05054.01
2
1
n
nL-L Expansion
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One-Third Rule
Increase
Temperature
V increases, ρ decreases
n increases
Volume occupied by molecules without considering space
between molecules
3
11
2
12
2
D
D
n
n
Vargas FM, Chapman WG; Fluid Phase Equilibria, 2010; 290:103-108
For nonpolar hydrocarbons
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Electronic Polarizability Lorentz–Lorenz equation:
where, N – Number of molecules per unit volume α – Polarizability
Refractive index and Polarizability are independent of the amount of material
where, Na – Avogadro number v – Molar Volume (v = MW/ρ)
With the help of One-Third Rule the above expression simplifies as
The above equation is independent of the state of the substance and its polarizability can be computed by just knowing the molecular weight
N
n
n
3
4
2
12
2
v
N
n
n a
3
4
2
12
2
aN
MW
4
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Predicted Polarizability
0 5 10 15 20 25 30 35 40 450
5
10
15
20
25
30
35
40
45 Mean Electronic Polarizability (cm3 x 10-24)
Experiment
Pred
icte
d fr
om O
ne-T
hird
Rul
e + 4 % De-viation
• Data shown is for 80 different nonpolar hydrocarbons belonging to different homologues series
0 5 10 15 20 25 30 35 40 450
5
10
15
20
25
30
35
40
45 Mean Electronic Polarizability (cm3 x 10-24)
X=Y
ExperimentPr
edic
ted
from
L-L
Exp
ansi
on + 2.5 %
Deviation
• Using One-Third Rule• Average absolute deviation is 4.16 %
• Using Lorentz-Lorenz Expansion• Average absolute deviation is 2.72 %
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Dielectric Constant It is well established that for weakly magnetic materials
εr , relative permitivity
For low-loss materials like nonpolar hydrocarbons,
k, dielectric constant
Substituting dielectric constant in the One-Third Rule and solving for dielectric constant
The dielectric constant expression can handle operational variations in temperature and pressure
It is independent of the knowledge of individual constituents of a mixture or the composition allowing the use for complex fluids
such as crude oils and polydisperse polymers
rn
krr )0()(
3
32k
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Predicted Dielectric Constant
1.4 1.6 1.8 2 2.2 2.4 2.6 2.81.4
1.6
1.8
2
2.2
2.4
2.6
2.8Dielectric Constant
Se...
Experiment
Pred
icte
d fr
om O
ne-T
hird
Rul
e
+ 2 % Deviation
X=Y
• Data shown is for 260 nonpolar hydrocarbons, including polymers, mixtures with varying temperatures and pressures
Panuganti SR, Vargas FM, Chapman WG; IEEE Transactions on Dielectrics and Electrical Insulation, 2013; Submitted
1.4 1.6 1.8 2 2.2 2.4 2.6 2.81.4
1.6
1.8
2
2.2
2.4
2.6
2.8Dielectric Constant
Series11
ExperimentPr
edic
ted
from
L-L
Exp
ansi
on
+ 1 % Deviation
X=Y
• Using One-Third Rule• Average absolute deviation is 1.98 %
• Using Lorentz-Lorenz Expansion• Average absolute deviation is 1.0 %
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Critical Temperature and Pressure From literature we have,
Thus, the following expression holds good
Applying One-Third Rule
also
904.22
104.52 2
25.0
D
D
n
n
v
a
Hildebrand and Scott Buckley et al.
),( 202/1MWfunction
P
T
C
C
2020
2
22/1 904.2
2
1042.52
MWMW
n
na
D
D
),( 20
2/1
MWfunctionP
TT
C
BC
2020 1674.0),(
MW
MWMWf
Let,
Hildebrand JH, Scott RL; The Solubility of Nonelectrolytes, 1950 Buckley et al; Petroleum Science and Technology, 1998; 16:251-285
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Critical Temperature and Pressure
0 50 100 150 200 250 300 350 400 450 5000
50
100
150
200
250
300
350
f(x) = 0.612962186296466 x + 24.8584279807217R² = 0.997304082632775
f(MW,ρ20)
Tc/P
c0.5
{K/
atm
0.5}
0 50 100 150 200 250 300 350 400 450 5000
50
100
150
200
250
300
f(x) = 0.57775397294106 x + 11.1206773684791R² = 0.99839713029832
f(MW,ρ20)
(Tb*
Tc/P
c)0.
5 {K
/atm
0.5}
85.24),(613.0 202/1 MWf
P
T
C
C 12.11),(577.0 20
2/1
MWf
P
TT
C
BC
Panuganti SR, Vargas FM, Chapman WG; Industrial and Engineering Chemistry Research, 2013; Accepted
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Predicting Critical Properties
100 300 500 700 900 1100100
300
500
700
900
1100Critical Temperature (K)
X = Y
Experiment
Pred
icte
d
0 10 20 30 40 50 60 700
10
20
30
40
50
60
70 Critical Pressure (atm)
X = Y
Experiment
Pred
icte
dAverage absolute deviation
is 2.2 %Average absolute deviation
is 4.5 %
• Data shown is for 80 different nonpolar hydrocarbons belonging to different homologues series. The applicability to mixtures is limited to nonpolar hydrocarbons
composed of similar sized molecules
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Surface Tension from Hole Theory
12
Volume of hole = Volume of liquid - Volume of solid
Heat of fusion = Energy required for the formation of all the holes
Solving the Schrodinger wave equation for a hole in a liquid,
Using the correlation of a/v2 from the previous section, at a given temperature we have
For example at 20oC we have
2
2
1
22223
22
)(4)(
3
4
m
P
m
PPPrpprEEE rzyx
oPq
509.7)(39.34 2020 h
21 )( ChC 8/1141674.0
)(
h
7/1
7/2
27/8
4.2 h
V
a
where,
Furth R; Proc. Phys. Soc., 1940; 52:768-769 Auluck FC, Rai RN; Journal of Chemical Physics, 1944; 12:321-322
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Predicted Surface Tension
Average absolute deviation is 1.8 %
The practical application of equation can improved further by incorporating the temperature variation of surface tension
With reference temperature as 20°C, surface tension at any other temperature can be calculated as
)( hTTC
)(
)(
293509.7)(39.34
2020
h
h
T
TTh T
c
cT
The parameter of critical temperature can be eliminated using the equation
obtained in the critical properties section.
0 10 20 30 400
10
20
30
40 n-Xylene
Ethylbenzene
Methylcy-clohexane
Cyclopentane
n-Hexane
Experiment
Pred
icte
d
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ConclusionInput Parameters
Property Density MW Boiling Point Function of Temperature
Mixtures
Critical Temperature Y Y Y - Y
Critical Pressure Y Y Y - YSurface Tension Y Y Y Y N
Electronic Polarizability N Y N - -
Dielectric Constant Y N N Y Y
• Polarizability of an asphaltene molecule of molecular weight 750 g/mol will be 99.16x10-24 cc
• Polydispere asphaltene system with density between 1.1 to 1.2 g/cc at ambient conditions will have a dielectric constant
between 2.737 and 3
Panuganti SR, Vargas FM, Chapman WG; IEEE Transactions on Dielectrics and Electrical Insulation, 2013; Submitted
Panuganti SR, Vargas FM, Chapman WG; Industrial and Engineering Chemistry Research, 2013; Accepted