Page 226 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

P2: IML/FFX
QC: IML/FFX
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AT029-Manual-v7.cls
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AT029-Manual
August 16, 2007
206 CHARACTERIZATION AND PROPERTIES OF PETROLEUM FRACTIONS
TABLE 5.1—Constants in Eq. (5.40) for four common cubic EOS (with permission from Ref. [15]).
Equation u 1 u 2 T1: IML a c 17:42 α b Z c
2 2
vdW 0 0 27 R T c 1 RT c 0.375
64 P c 8P c
2 2 −1/2
RK 1 0 0.42748R T c T r 0.08664RT c 0.333
P c P c
2
2 2 1/2
SRK 1 0 0.42748R T c 1 + f ω 1 − T r 0.08664RT c 0.333
P c P c
2
f ω = 0.48 + 1.574ω − 0.176ω
2
2 2 1/2
PR 2 −1 0.45724R T c 1 + f ω 1 − T r 0.07780RT c 0.307
P c P c
f ω = 0.37464 + 1.54226ω − 0.2699
when a = b = 0, the equation reduces to ideal gas law, Eq.
(5.14). In addition, all equations satisfy the criteria set by Eqs. Actual Isotherm
(5.18)–(5.20) as well as Eq. (5.9). For example, consider the Predicted by Cubic EOS
PR EOS expressed by Eq. (5.39). To show that criteria set
by Eq. (5.18) are satisfied, the limits of all terms as V →∞ C
(equivalent to P → 0) should be calculated. If both sides of
Eq. (5.39) are multiplied by V/RT and taking the limits of all Pressure, P
terms as V →∞ (or P → 0), the first term in the RHS ap-
proaches unity while the second term approaches zero and
we get Z → 1, which is the EOS for the ideal gases.
Reid et al. [15] have put vdW, RK, SRK, and PR two-
parameter cubic EOS into a practical and unified following
form:
RT a
(5.40) P = −
V − b V + u 1 bV + u 2 b 2 Volume, V
2
where u 1 and u 2 are two integer values specific for each cubic
equation and are given in Table 5.1. Parameter a is in general FIG. 5.9—Prediction of isotherms by a cubic EOS.
temperature-dependent and can be expressed as
(5.41) a = a c α gives three roots for Z. The lowest value of Z corresponds
to saturated liquid, the highest root gives Z for the saturated
where α is a dimensionless temperature-dependent parame- vapor, and the middle root has no physical meaning.
ter and usually is expressed in terms of reduced temperature
(T r = T/T c ) and acentric factor as given in Table 5.1. For both
vdW and RK equations this parameter is unity. Parameters 5.5.2 Solution of Cubic Equations of State
u 1 and u 2 in Eq. (5.40) are the same for both RK and SRK
equations, as can be seen in Table 5.1, but vdW and PR equa- Equation (5.42) can be solved through solution of the follow-
tions have different values for these parameters. Equation ing general cubic equation [16, 17]:
(5.40) can be converted into a cubic form equation similar to (5.43) Z + a 1 Z + a 2 Z + a 3 = 0
2
3
Eq. (5.23) but in term of Z rather than V:
Let’s define parameters Q, L, D, S 1 , and S 2 as
2
Z − (1 + B − u 1 B)Z + (A + u 2 B − u 1 B − u 1 B )Z
2
2
3
3a 2 − a 2 1
3
2
(5.42) − AB − u 2 B − u 2 B = 0 Q = 9
aP bP 9a 1 a 2 − 27a 3 − 2a 3
where A = and B = 1
2
R T 2 RT L = 54
(5.44)
in which parameters A and B as well as all terms in Eq. (5.42) D = Q + L 2
3
are dimensionless. Parameters a and b and Z c have been de- √
termined in a way similar to the methods shown in Example S 1 = (L + D) 1/3
5.1. Z c for both RK and SRK is the same as 1/3 or 0.333 while S 2 = (L − √ D) 1/3
for the PR it is lower and equal to 0.307 for all compounds.
As it will be shown later performance of all these equations The type and number of roots of Eq. (5.43) depends on the
near the critical region is weak and leads to large errors for value of D. In calculation of X 1/3 if X < 0, one may use
calculation of Z c . Prediction of an isotherm by a cubic EOS is X 1/3 =−(−X) 1/3 .
shown in Fig. 5.9. As is seen in this figure, pressure prediction If D > 0 Eq. (5.43) has one real root and two complex con-
for an isotherm by a cubic EOS in the two-phase region is not jugate roots. The real root is given by
reliable. However, isotherms outside the two-phase envelope (5.45) Z 1 = S 1 + S 2 − a 1 /3
may be predicted by a cubic EOS with a reasonable accuracy.
In calculation of Z for saturated liquid and saturated vapor at If D = 0 all roots are real and at least two are equal. The
the same T and P, Eq. (5.42) should be solved at once, which unequal root is given by Eq. (5.45) with S 1 = S 2 = L 1/3 . The

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