1

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

2

Outline

• One-Third Rule

• Electronic polarizability

• Dielectric constant

• Critical temperature and pressure

• Surface tension

• Conclusion

3

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

4

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

5

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

6

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 %

7

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

8

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 %

9

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

10

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

11

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

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

13

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

14

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