Page 277 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

P2: KVU/KXT
QC: —/—
T1: IML
P1: KVU/KXT
20:46
June 22, 2007
AT029-Manual
AT029-Manual-v7.cls
AT029-06
6. THERMODYNAMIC RELATIONS FOR PROPERTY ESTIMATIONS 257
is the saturation pressure or vapor pressure of pure i at T
P
sat
and methods of its calculation are discussed in the next chap- According to the definition of γ i when x i = 1 (pure i) then
γ i = 1. Generally for binary systems when a relation for ac-
ter. φ i sat is the vapor phase fugacity coefficient of pure compo- tivity coefficient of one component is known the relation for
nent i at P i sat and can be calculated from methods discussed activity coefficient of other components can be determined
in Section 6.2. The exponential term in the above equation is from the following relation:
called Poynting correction and is calculated from liquid molar
L
volume. Since variation of V with pressure is small, usually it dln γ 1 dln γ 2
i
is assumed constant versus pressure and the Poynting factor (6.141) x 1 dx 1 = x 2 dx 2
L
L
is simplified as exp[V (P − P i sat )/RT]. In such cases V may
i
i
be taken as molar volume of saturated liquid at temperature which is derived from Gibbs–Duhem equation. One can ob-
T and it may be calculated from Racket equation (Section tain γ 2 from γ 1 by applying the above equation with use of
5.8). At very low pressures or when (P − P sat ) is very small, x 2 = 1 − x 1 and dx 2 =−dx 1 . Constant A in Eq. (6.140) can be
i
the Poynting factor approaches unity and it could be removed obtained from data on the activity coefficient at infinite di-
from Eq. (6.136). In addition, when P i sat is very small (∼ 1 atm lution (γ ), which is defined as lim x i →0 (γ i ). This will result
∞
i
L
∞
or less), φ i sat may be considered as unity and f is simply equal in A = RT ln γ 1 ∞ = RT ln γ . This simple model applies well
2
i
to P sat . Obviously this simplification can be used only in spe- to simple mixtures such as benzene–cyclohexane; however,
i
cial situations when the above assumptions can be justified. for more complex mixtures other activity coefficient models
3
L
For calculation of Poynting factor when V is in cm /mol, P must be used. A more general form of activity coefficients for
i
E
in bar, and T in kelvin, then the value of R is 83.14. binary systems that follow Redlich–Kister model for G are
given as
6.6.5 Calculation of Activity Coefficients 2 3 4 5
RT ln γ 1 = a 1 x + a 2 x + a 3 x + a 4 x + ···
2
2
2
2
(6.142)
Activity coefficient γ i is needed in calculation of fugacity of i 2 3 4 5
RT ln γ 2 = b 1 x + b 2 x + b 3 x + b 4 x + ···
1
1
1
1
in a liquid mixture through Eq. (6.114). Activity coefficients
E
are related to excess molar Gibbs energy, G , through ther- If in Eq. (6.139) coefficient C and higher order coefficients
modynamic relations as [21] are zero then resulting activity coefficients correspond to only
the first two terms of the above equation. This model is called
E

∂ nG
(6.137) RT ln γ i = ¯ G i = four-suffix Margules equation. Since data on γ i ∞ are useful
∂n i in obtaining the constants for an activity coefficient model,
T,P,n j =i
many researchers have measured such data for various sys-
where ¯ G i is the partial molar excess Gibbs energy as defined tems. Figure 6.8 shows values of γ ∞ for n-C 4 and n-C 8 in var-
i
by Eq. (6.78) and may be calculated by Eq. (6.82). This equa- ious n-alkane solvents from C 15 to C 40 at 100 C based on data
◦
tion leads to another equally important relation for the activ- available from C 20 to C 36 [21]. As can be seen from this fig-
ity coefficient in terms of excess Gibbs energy, G : ure, as the size of solvent molecule increases the deviation of
E
activity coefficients from unity also increases.
E

(6.138) G = RT x i ln γ i
Another popular model for activity coefficient of binary
i
systems is the van Laar model proposed by van Laar dur-
where this equation is obtained by substitution of Eq. (6.137) ing 1910–1913. This model is particularly useful for binaries
E
into Eq. (6.79). Therefore, once the relation for G is known whose molecular sizes vary significantly. Van Laar model is
it can be used to determine γ i . Similarly, when γ i is known
E
G can be calculated. Various models have been proposed
E
E
for G of binary systems. Any model for G must satisfy the 1
E
condition that when x 1 = 0or1(x 2 = 0), G must be equal to
C 4 : Y=1.337−0.0269X+0.000241X 2
zero; therefore, it must be a factor of x 1 x 2 . One general model C 8 : Y=1.391−0.0354X+0.000384X 2
E
for G of binary systems is called Redlich–Kister expansion 0.9
and is given by the following power series form [1, 21]:
G E
2
(6.139) = x 1 x 2 [A + B(x 1 − x 2 ) + C(x 1 − x 2 ) + ···] 0.8
RT Activity Coefficient at Infinite Dilution
where A, B,...are empirical temperature-dependent coeffi- 0.7
cients. If all these coefficients are zero then the solution is
ideal. The simplest nonideal solution is when only coeffi-
cient A is not zero but all other coefficients are zero. This 0.6 n-Butane
is known as two-suffix Margules equation and upon applica- n-Octane
tion of Eq. (6.137) the following relations can be obtained for
γ 1 and γ 2 : 0.5
10 20 30 40 50
--`,```,`,``````,`,````,```,,-`-`,,`,,`,`,,`---
A 2
ln γ 1 = x 2 Carbon Number of n-Alkane Solvent
(6.140) RT
A 2 FIG. 6.8—Values of γγ ∞∞ for n-butane and n-octane in
ii
ln γ 2 = x 1 n-paraffin solvents at 100 C.
◦
RT
Copyright ASTM International
Provided by IHS Markit under license with ASTM Licensee=International Dealers Demo/2222333001, User=Anggiansah, Erick
No reproduction or networking permitted without license from IHS Not for Resale, 08/26/2021 21:56:35 MDT