Page 196 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

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176 CHARACTERIZATION AND PROPERTIES OF PETROLEUM FRACTIONS
in which (1 + 1/B) can be evaluated directly from parame-
ter B. This equation was developed empirically for mathemat- where J is just an integration parameter defined in Eq. (4.79).
A SG is the coefficient in Eq. (4.56) when SG is expressed
ical convenience. Values estimated from this equation vary by in terms of cumulative weight fraction, x cw . For most sam-
a maximum of 0.02% (at B = 1) with those from Eq. (4.43). ples evaluated, parameter A SG is between 0.05 and 0.4; how-
Therefore, for simplicity we use Eq. (4.73) for calculation of ever, for no system a value greater than 0.4 was observed.
average values through Eq. (4.72). By combining Eqs. (4.78) and (4.79) with definition of SG by
∗
As mentioned earlier for many systems fixed values of B Eq. (4.56), SG av can be calculated from the following relation:
for different properties may be used. These values are B M = 1
1
for M, B T = 1.5 for T b , and B SG = B I = B d = 3 for SG, I 20 or (4.80) SG av = SG ◦
d 20 . For these values of B, (1 + 1/B) has been evaluated by J
Eq. (4.73) and substituted in Eq. (4.72), which yields the fol- this equation should be used when SG is expressed in terms
lowing simplified relations for calculation of average proper- of x cw by Eq. (4.56). For analytical integration of Eq. (4.78)
ties of whole C 7+ fraction in terms of coefficient A for each see Problem 4.4.
property: In general, once P is determined from Eq. (4.72), P av can
∗
av
be determined from the definition of P by the following
∗
(4.74) ∗ relation:
M = A M
av
(4.75) T ∗ = 0.689A 2/3 ∗
b,av M (4.81) P av = P ◦ (1 + P )
av
(4.76) SG ∗ = 0.619A 1/3 Average properties determined by Eqs. (4.74)–(4.76) can be --`,```,`,``````,`,````,```,,-`-`,,`,,`,`,,`---
av SG
converted to M av , T bav , and SG av by Eq. (4.81). Equation (4.76)
It should be noted that Eq. (4.76) can be used when SG derived for SG ∗ can also be used for refractive index para-
av
is expressed in terms of cumulative volume fraction. Equa- meter I or absolute density (d) when they are expressed in
tion (4.76) is based on Eq. (4.72), which has been derived
terms of x cv . Similarly Eqs. (4.78)–(4.80) can be applied to I 20
from Eq. (4.71). As it was discussed in Chapter 3 (Section 3.4), or d 20 when they are expressed in terms of x cw . The following
for SG, d (absolute density), and I (defined by Eq. 2.36) two example shows application of these equations.
types of mixing rules may be used to calculate mixture prop-
erties. Linear Kay mixing rule in the form of Eq. (3.45) can Example 4.8—For the gas condensate system of Example 4.7
be used if composition of the mixture is expressed in volume calculate mixture molecular weight, boiling point, and spe-
fractions, but when composition if given in terms of weight cific gravity using the coefficients given in Table 4.13. The ex-
fractions, Eq. (4.46) must be used. Both equations give sim- perimental values are M 7+ = 118.9 and SG 7+ = 0.7569 [24].
ilar accuracy; however, for mixtures defined in terms of very Also calculate the boiling point of the residue (component
few compounds that have SG values with great differences, no. 12 in Table 4.11).
Eq. (3.46) is superior to Eq. (3.45). Equation (4.46) can be
applied to SG in a continuous form as follows: Solution—For molecular weight the coefficients of PDF in
terms of x cm for Eq. (4.66) as given in Table 4.13 are: M o = 91,
1 1 dx cw A M = 0.2854, and B M = 0.9429. From Eq. (4.73), (1 + 1/B) =
(4.77) =
∗
SG av SG(x cw ) 1.02733 and from Eq. (4.72), M = 0.2892. Finally M av is cal-
av
0 culated from Eq. (4.81) as 117.3. For this system B M is very
close to unity and we can use the coefficients in Table 4.13 for
where SG av is the average specific gravity of C 7+ and SG(x cw )is
the continuous distribution function for SG in terms of cumu- M o = 89.86, A M = 0.3105, and B M = 1. From Eqs. (4.74) and
lative weight fraction. SG(x cw ) can be expressed by Eq. (4.56). (4.81) we get M av = 89.86 × (1 + 0.3105) = 117.8. Comparing
Equation (4.77) in a dimensionless form in terms of SG ∗ with the experimental value of 118.9, the relative deviation
becomes is −1%.
For specific gravity, the coefficients in terms of x cv are:
∞ SG o = 0.705, A SG = 0.0232, and B SG = 1.811. From Eq. (4.76),
1 dSG ∗
(4.78) = F(SG ) SG ∗ = 0.0801 and from Eq. (4.81), SG av = 0.7615. Compar-
∗
SG + 1 SG + 1 av
∗
∗
av
0 ing with experimental value of 0.7597, the relative devia-
tion is 0.24%. If the coefficients in terms of x cw are used,
In this equation integration is carried on the variable SG ∗ SG o = 0.6661, A SG = 0.0132, and from Eq. (4.79) we get 1/J =
and F(SG ) is the PDF for SG in terms of x cw . Integration 1.1439 using appropriate range for A SG . From Eq. (4.80),
∗
∗
in Eq. (4.78) has been evaluated numerically and has been SG av = 0.7619 which is nearly the same as using cumulative
correlated to parameter A SG in the following form [24]: volume fraction.
For T b the coefficients in terms x cw with fixed value of B T are
∞
dSG T o = 340 K, A T = 0.1875, and B T = 1.5. From Eq. (4.75) and
∗
∗
J = F(SG )
SG + 1 (4.81) we get: T av = 416.7 K. To calculate T b for the residue we
∗
0 use the following relation:
1 = 1.3818 + 0.3503A SG − 0.1932A 2 for A SG > 0.05
N−1
J SG T av − i=1 x wi T bi
1 2 (4.82) T bN = x wN
= 1.25355 + 1.44886A SG − 5.9777A + 0.02951 ln A SG
J SG where T bN is the boiling point of the residue. For this exam-
(4.79) for A SG ≤ 0.05 ple from Table 4.11, N = 12 and x wN = 0.01. Using values
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