Page 275 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

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6. THERMODYNAMIC RELATIONS FOR PROPERTY ESTIMATIONS 255
where x i is mole fraction of component i in the mixture. Both
◦
i
ˆ a i and γ i are dimensionless parameters. With the above defini- where G is the molar Gibbs energy of pure i at T of the
system and pressure of 1 atm (ideal gas state). By replac-
ˆ
¯
tions one may calculatef i from one of the following relations: ing G = y i ˆμ i , and V = y i V i in the above equation and
removing the summation sign we get
ˆ
(6.113) f i = ˆ φ i y i P
ˆ
(6.114) f i = x i γ i f i (6.122) P ¯ V i − RT dP + RT ln(y i P) + G ◦
ˆ μ i =
P i
Although generally ˆ φ i and γ i are defined for any phase, but 0
usually ˆ φ i is used to calculate fugacity of i in a gas mixture Integration of Eq. (6.119) from pure ideal gas at T and P = 1
and γ i is used to calculate fugacity of component i in a liq- atm to real gas at T and P gives
uid or solid solution. However, for liquid mixtures at high
ˆ
pressures, i.e., high pressure VLE calculations,f i is calculated (6.123) ˆ μ i − μ = RT ln f i ˆ
◦
from ˆ φ i through Eq. (6.113). In such calculations as it will be i 1
shown later in this section, for the sake of simplicity and con- where μ is the chemical potential of pure component i at
◦
i
venience, ˆ φ i for both phases are calculated through an equa- T and pressure of 1 atm (ideal gas as a standard state). For
tion of state. Both ˆ φ i and γ i indicate degree of nonideality for a a pure component at the same T and P we have: μ = G .
◦
◦
system. In a gas mixture, ˆ φ i indicates deviation from an ideal Combining Eqs. (6.122) and (6.123) gives i i
gas and in a liquid solution, γ i indicates deviation from an
ideal solution. To formulate phase equilibrium of mixtures a f i ˆ P RT
new parameter called chemical potential must be defined. (6.124) RT ln = RT ln ˆ φ i = ¯ V i − dP
y i P P
∂G
t 0
(6.115) ˆ μ i ≡ = ¯ G i
∂n i where ¯ V i is the partial molar volume of component i in the
T,P,n j =i
mixture. It can be seen that for a pure component ( ¯ V i = V i and
where ˆμ i is the chemical potential of component i in a mixture y i = 1) this equation reduces to Eq. (6.53) previously derived
and ¯ G i is the partial molar Gibbs energy. General definition for calculation of fugacity coefficient of pure components.
of partial molar properties was given by Eq. (6.78). For a pure There are other forms of this equation in which integration
component both partial molar Gibbs energy and molar Gibbs is carried over volume in the following form [21]:
energy are the same: ¯ G i = G i . For a pure ideal gas and an ideal
∞
gas mixture from thermodynamic relations we have ∂P RT
t
(6.125) RT ln ˆ φ i = − t dV − ln Z
(6.116) dG i = RTdln P ∂n i T,V,n j =i V
V t
(6.117) d ¯ G i = RTdln(y i P) where V is the total volume (V = nV). In using these equa-
t
t
tions one should note that nis the sum of n i and is not constant
where in Eq. (6.117) if y i = 1, it reduces to Eq. (6.116) for pure
component systems. For real gases these equations become when derivative with respect to n i is carried. These equations
are the basis of calculation of fugacity of a component in a
(6.118) dG i = dμ i = RTdln f i mixture. Examples of such derivations are available in vari-
(6.119) d ¯ G i = dˆμ i = RTdln f i ˆ ous texts [1, 4, 11, 20–22]. One can use an EOS to obtain ¯ V i
and upon substitution in Eq. (6.124) a relation for calculation
Equation (6.118) is the same as Eq. (6.46) derived for pure of ˆ φ i can be obtained. For the general form of cubic equations
components. Equation (6.119) reduces to Eq. (6.118) at given by Eqs. (5.40)–(5.42), ˆ φ i is given as [11]
y i = 1. Subtracting Eq. (6.118) from Eq. (6.119) and using
ˆ b i A b i
Eq. (6.114) for f i one can derive the following relation for ˆμ i ln ˆ φ i = b (Z − 1) − ln (Z − B) + √ u −u 2 b − δ i
2
B
in a solution: √ 1 2

2
2Z+B u 1 + u −4u 2
1
(6.120) ˆ μ i − μ = RT ln γ i x i (6.126) × ln 2Z+B u 1 − √ u −4u 2
◦

2
i
1
where μ is the pure component chemical potential at T and b i T ci /P ci
◦
i
P of mixture and x i is the mole fraction of i in liquid solu- where b = y j T cj /P cj
tion. For ideal solutions where γ i = 1, Eq. (6.120) reduces to 1/2 j
ˆ μ i − μ = RT ln x i . In fact this is another way to define an ideal 2a i x j a 1/2 (1 − k ij )
◦
i
solution. A solution that is ideal over the entire range of com- and δ i = a i j
position is called perfect solution and follows this relation.
1/2
if all k ij = 0 then δ i = 2 a i
a
6.6.2 Calculation of Fugacity Coefficients All parameters in the above equation for vdW, RK, SRK, and
from Equations of State PR equations of state are defined in Tables 5.1 and 6.1. Pa-
Through thermodynamic relations and definition of G one rameters a and b for the mixture should be calculated from
can derive the following relation for the mixture molar Gibbs Eqs. (5.59)–(5.61). Equation (6.126) can be used for calcula-
free energy [21]. tion of fugacity of i in both liquid and vapor phases provided
appropriate Z values are used as for the case of pure compo-
nent systems that was shown in Example 6.7. For calculation
RT

P
(6.121) G = V − dP + RT y i ln (y i P) + y i G ◦ i of ˆ φ i from PR and SRK equations through the above relation,
P
0 i i use of volume translation is not required.
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