Page 227 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

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5. PVT RELATIONS AND EQUATIONS OF STATE 207
From Table 5.1, u 1 = 1, u 2 = 0 and a SRK and b SRK are calcu-
two equal roots are
lated as
(5.46) Z 2 = Z 3 =−L 1/3 − a 1 /3
2
f ω = 0.48 + 1.574 × 0.3996 − 0.176 × (0.3996) = 1.08087
If D < 0 all roots are real and unequal. In this case S 1 and
2
S 2 (Eq. 5.44) cannot be calculated and the computation is 0.42748 × (83.14) × (568.7) 2
simplified by use of trigonometry as a SRK = 24.9
1 × [1 + 1.08087 × (1 − 0.9717 )]
1/2 2
a 1
Z 1 = 2 −Q Cos θ + 120 ◦ −
3 3 = 3.957 × 10 cm /mol .
2
6
7
1

a 1 0.08664 × 83.14 × 568.7
Z 2 = 2 −Q Cos θ + 240 ◦ − 3
3 3 b SRK = = 164.52 cm /mol.
(5.47) 1 24.9
Z 3 = 2 −Q Cos θ − a 1 Parameters A and B are calculated from Eq. (5.42):

3 3
7
3.957 × 10 × 19.9
L A = = 0.373
2
where Cos θ = (83.14) × (552.65) 2
−Q 3
and
where θ is in degrees. To check validity of the solution, the
three roots must satisfy the following relations 164.52 × 19.9
B = = 0.07126
83.14 × 552.65
Z 1 + Z 2 + Z 3 =−a 1
Coefficients a 1 , a 2 , and a 3 are calculated from Eq. (5.49) as
(5.48) Z 1 × Z 2 + Z 2 × Z 3 + Z 3 × Z 1 = a 2
a 1 =−(1 + 0.07126 − 1 × 0.07126) =−1
Z 1 × Z 2 × Z 3 =−a 3
2
a 2 = 0.373 + 0 × 0.07126 − 1 × 0.07126 − 1 × 0.07126 2
A comparison of Eq. (5.42) and (5.43) indicates that the
following relations exist between coefficients a i (s) and EOS = 0.29664
parameters 2 3
a 3 =−0.37305 × 0.07126 − 0 × 0.07126 − 0 × 0.07126
a 1 =−(1 + B − u 1 B)
=−0.026584
2
(5.49) a 2 = A + u 2 B − u 1 B − u 1 B 2
−4
From Eq. (5.44), Q =−0.01223, L = 8.84 × 10 , and D =
2
a 3 =−AB − u 2 B − u 2 B 3 −6
−1.048 × 10 . Since D < 0, the solution is given by Eq. (5.47).
−1

−4
3
◦
For the case that there are three different real roots (D < 0), θ = Cos (8.84 × 10 / −(−0.01223) ) = 492 and the roots
Z liq is equal to the lowest root (Z 1 ) while Z vap is equal to the are Z 1 = 0.17314, Z 2 = 0.28128, and Z 3 = 0.54553. Accept-
highest root (Z 3 ). The middle root (Z 2 ) is disregarded as phys- able results are the lowest and highest roots while the in-
L
V
ically meaningless. Equation (5.42) may also be solved by suc- termediate root is not useful: Z = Z 1 = 0.17314 and Z =
cessive substitution methods; however, appropriate forms of Z 2 = 0.54553. Molar volume, V, can be calculated from Eq.
the equation and initial values are different for vapor and liq- (5.15): V = ZRT/P in which T = 552.65 K, P = 19.9 bar, and
3
L
3
uid cases. For example, for gases the best initial value for Z is R = 83.14 cm · bar/mol · K; therefore, V = 399.9cm /mol
3
V
1 while for liquids a good initial guess is bP/RT [1]. Solution of and V = 1259.6cm /mol. From Table 5.1, Z c = 0.333 and
cubic equations through Eq. (5.42) is shown in the following V c is calculated from Eq. (2.8) as V c = (0.333 × 83.14 ×
L
3
V
example. 568.7)/24.9 = 632.3cm /mol. Errors for V , V , and V c are
31.5, 3.6, and 30%, respectively. It should be noted that Z c
Example 5.2—Estimate molar volume of saturated liquid and can also be found from the solution of cubic equation with
vapor for n-octane at 279.5 C and pressure of 19.9 bar from T = T c and P = P c . However, for this case D > 0 and there is
◦
the RK, SRK, and PR cubic EOS. Values of V and V V ex- only one solution which is obtained by Eq. (5.45) with similar
L
3
tracted from the experimental data are 304 and 1216 cm /mol, answer. As is seen in this example, liquid and critical volumes
respectively [18]. Also estimate the critical volume. are greatly overestimated. A summary of results for RK, SRK,
and PR EOSs are given in Table 5.2.
Solution—To use SRK and PR EOS pure component data
for n-C 8 are taken from Table 2.1 as T c = 295.55 C (568.7 K), 5.5.3 Volume Translation
◦
3
P c = 24.9 bar, ω = 0.3996, and V c = 486.35 cm /mol. When T is
3
in K, P is in bar, and V is in cm /mol, value of R from Section In practice the SRK and PR equations are widely used for VLE
3
1.7.24 is 83.14 cm · bar/mol · K. Sample calculation is shown calculations in industrial applications [19–21]. However, their
here for SRK EOS. T r = (279.5 + 273.15)/568.7 = 0.972. ability to predict volumetric data especially for liquid systems
TABLE 5.2—Prediction of saturated liquid, vapor and critical molar volumes for n-octane in
Example 5.2.
3
3
3
L
V
Equation V ,cm /mol %D V ,cm /mol %D V c ,cm /mol %D
Data ∗ 304.0 . . . 1216.0 . . . 486.3 . . .
RK 465.9 53.2 1319.4 8.5 632.3 30
SRK 399.9 31.5 1259.6 3.6 632.3 30
PR 356.2 17.2 1196.2 −1.6 583.0 19.9
L
V
Source: V and V from Ref. [18]; V c from Table 2.1.
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