Page 228 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

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208 CHARACTERIZATION AND PROPERTIES OF PETROLEUM FRACTIONS
is weak. Usually the SRK equation predicts densities more ac-
curately for compounds with low acentric values, while the 5.5.4 Other Types of Cubic Equations of State
PR predicts better densities for compounds with acentric fac- In 1972 Saove for the first time correlated parameter a in a
tors near 0.33 [21]. For this reason a correction term, known cubic EOS to both T r and ω as given in Table 5.1. Since then
as volume translation, has been proposed for improving vol- this approach has been used by many researchers who tried
umetric prediction of these equations [8, 15, 19, 22]: to improve performance of cubic equations. Many modifica-
tions have been made on the form of f ω for either SRK or
(5.50) V = V EOS − c PR equations. Graboski and Daubert modified the constants
in the f ω relation for the SRK to improve prediction of va-
where c is the volume translation parameter and has the same por pressure of hydrocarbons [24]. Robinson and Peng [14]
unit as the molar volume. Equation (5.50) can be applied to also proposed a modification to their f ω equation given in
both vapor and liquid volumes. Parameter c mainly improves Table 5.1 to improve performance of their equation for heav-
liquid volume predictions and it has no effect on vapor pres- ier compounds. They suggested that for the PR EOS and for
sure and VLE calculations. Its effect on vapor volume is negli- compounds with ω> 0.49 the following relation should be
V
gible since V is very large in comparison with c, but it greatly used to calculate f ω :
improves prediction of liquid phase molar volumes. Values of 2 3
c have been determined for a number of pure components up (5.53) f ω = 0.3796 + 1.485ω − 0.1644ω + 0.01667ω
to C 10 for both SRK and PR equations and have been included Some other modifications give different functions for param-
in references in the petroleum industry [19]. Peneloux et al. eter α in Eq. (5.41). For example, Twu et al. [25] developed
[22] originally obtained values of c for some compounds for the following relation for the PR equation.
use with the SRK equation. They also suggested the following −0.171813 1.77634
correlation for estimation of c for SRK equation: α = T r exp 0.125283 1 − T r

(5.54) + ω T −0.607352 exp 0.511614 1 − T 2.20517

RT c r r
(5.51) c = 0.40768 (0.29441 − Z RA) −0.171813 1.77634
P c − T r exp 0.125283 1 − T r
where Z RA is the Rackett parameter, which will be discussed Other modifications of cubic equations have been derived by
in Section 5.8.1. Similarly Jhaveri and Yougren [23] obtained suggesting different integer values for parameters u 1 and u 2 in
parameter c for a number of pure substances for use with PR Eq. (5.40). One can imagine that by changing values of u 1 and
EOS and for hydrocarbon systems have been correlated to u 2 in Eq. (5.40) various forms of cubic equations can be ob-
molecular weight for different families as follows: tained. For example, most recently a modified two-parameter
cubic equation has been proposed by Moshfeghian that cor-
c P = b PR 1 − 2.258M P −0.1823 responds to u 1 = 2 and u 2 =−2 and considers both parame-

(5.52) c N = b PR 1 − 3.004M −0.2324 ters a and b as temperature-dependent [26]. Poling et al. [8]

N

c A = b PR 1 − 2.516M −0.2008 have summarized more than two dozens types of cubic equa-
A tions into a generalized equation similar to Eq. (5.42). Some
where b PR refers to parameter b for the PR equation as given of these modifications have been proposed for special sys-
in Table 5.1. Subscripts P, N, and A refer to paraffinic, naph- tems. However, for hydrocarbon systems the original forms
thenic, and aromatic hydrocarbon groups. The ratio of c/b of SRK and PR are still being used in the petroleum industry.
is also called shift parameter. The following example shows The most successful modification was proposed by Zudke-
application of this method. vitch and Joffe [27] to improve volumetric prediction of RK
EOS without sacrificing VLE capabilities. They suggested that
parameter b in the RK EOS may be modified similar to Eq.
Example 5.3—For the system of Example 5.2, estimate V L (5.41) for parameter a as following:
V
and V for the PR EOS using the volume translation method.
(5.55) b = b RK β
Solution—For n-C 8 , from Table 2.1, M = 114 and b PR are where β is a dimensionless correction factor for parameter
3
calculated from Table 5.1 as 147.73 cm /mol. Since the hy- b and it is a function of temperature. Later Joffe et al. [28]
drocarbon is paraffinic Eq. (5.51) for c P should be used, determined parameter α in Eq. (5.41) and β in Eq. (5.55) by
3
c = 7.1cm /mol. From Table 5.2, V L(PR) = 356.2 and V V(PR) = matching saturated liquid density and vapor pressure data
3
1196.2cm /mol. From Eq. (5.50) the corrected molar volumes over a range of temperature for various pure compounds. In
L
are V = 356.2 − 7.1 = 349.1 and V = 1196.2 − 7.1 = 1189.1 this approach for every case parameters α and β should be
V
cm /mol. By the volume translation correction, error for V L determined and a single dataset is not suitable for use in all
3
decreases from 17.2 to 14.8% while for V it has lesser effect cases. SRK and ZJRK are perhaps the most widespread cu-
V
and it increases error from –1.6% to –2.2%. bic equations being used in the petroleum industry, especially
for phase behavior studies of reservoir fluids [19]. Other re-
As is seen in this example improvement of liquid volume searchers have also tried to correlate parameters α and β in
by volume translation method is limited. Moreover, estima- Eqs. (5.41) and (5.55) with temperature. Feyzi et al. [29] cor-
tion of c by Eq. (5.51) is limited to those compounds whose related α 1/2 and β 1/2 for PR EOS in terms T r and ω for heavy
Z RA is known. With this modification at least four parameters reservoir fluids and near the critical region. Their correla-
namely T c , P c , ω, and c must be known for a compound to tions particularly improve liquid density prediction in com-
determine its volumetric properties. parison with SRK and PR equations while it has similar VLE

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