Page 234 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

T1: IML
P2: IML/FFX
P1: IML/FFX
QC: IML/FFX
August 16, 2007
17:42
AT029-Manual
AT029-05
AT029-Manual-v7.cls
214 CHARACTERIZATION AND PROPERTIES OF PETROLEUM FRACTIONS
5
2
2
E o /(RT V c ) = 0.00645 − 0.022143ω × exp(−3.8ω)
B o /V c = 0.44369 + 0.115449ω TABLE 5.7—Coefficients for the BWRS EOS—Eq. (5.89) [21]. d/(RT V ) = 0.0732828 + 0.463492ω
c c
c
3
A o /(RT c V c ) = 1.28438 − 0.920731ω α/V = 0.0705233 − 0.044448ω
c
3 2 3 2
C o /(RT V c ) = 0.356306 + 1.7087ω b/(V ) = 0.528629 + 0.349261ω c/(RT V ) = 0.504087 + 1.32245ω
c
c
c c
4
2
2
D o /(RT V c ) = 0.0307452 + 0.179433ω a/(RT c V ) = 0.484011 + 0.75413ω γ /V = 0.544979 − 0.270896ω
c
c
c
predictions are obtained. Equation (5.72) also gives reason- volume that be can occupied by N molecules of diameter σ is
able results but Eq. (5.73) gives a less accurate estimate of B.
The best result is obtained from Eq. (5.76) with Eqs. (5.71) V oN = N V o
and (5.78), which give a deviation of 0.2%. (e) The ideal gas (5.90) N A
1
V
3
law (Z = 1) gives V = 2500.7cm /mol with a deviation of √ σ 3
+22.8%. V o = 2 N A
where N A is the Avogadro’s number and V o is the volume of
5.6.2 Modified Benedict–Webb–Rubin 1 mol (N A molecules) of hard spheres as packed molecules
Equation of State without empty space between the molecules. V oN is the total
volume of packed N molecules. If the molar volume of fluid is
Another important EOS that has industrial application is the V, then a dimensionless reduced density, ξ, is defined in the
Benedict–Webb–Rubin (BWR) EOS [45]. This equation is in following form:
fact an empirical expansion of virial equation. A modification
√
of this equation by Starling [46] has found successful applica- 2 V o
tions in petroleum and natural gas industries for properties (5.91) ξ = 6 π × V
of light hydrocarbons and it is given as
1 C o D o E o 1 Parameter ξ is also known as packing fraction and indicates
P = RT + B o RT − A o − + − fraction of total volume occupied by hard molecules. Substi-
V T 2 T 3 T 4 V 2
tuting V o from Eq. (5.90) into Eq. (5.91) gives the following
d 1 d 1

(5.89) + bRT − a − + α a + relation for packing fraction:
T V 3 T V 6
π N A σ
3
c γ −γ (5.92) ξ = ×
+ 1 + exp 6 V
2
T V 3 V 2 V 2
The Carnahan–Starling EOS is then given as [6]
where the 11 constants A o , B o , ..., a, b, ..., α and γ are given
2
in Table 5.7 in terms of V c , T c , and ω as reported in Ref. [21]. HS PV 1 + ξ + ξ − ξ 3
This equation is known as BWRS EOS and may be used for (5.93) Z = RT = (1 − ξ) 3
calculation of density of light hydrocarbons and reservoir flu- HS
ids. In the original BWR EOS, constants D o , E o , and d were all where Z is the compressibility factor for hard sphere
zero and the other constants were determined for each spe- molecules. For this EOS there is no binary constant and
cific compound separately. Although better volumetric data the only parameter needed is molecular diameter σ for each
can be obtained from BWRS than from cubic-type equations, molecule. It is clear that as V →∞ (P → 0) from Eq. (5.93)
HS
but prediction of phase equilibrium from cubic equations are ζ → 0 and Z → 1, which is in fact identical to the ideal
quite comparable in some cases (depending on the mixing gas law. Carnahan and Starling extended the HS equation to
rules used) or better than this equation in some other cases. fluids whose spherical molecules exert attractive forces and
Another problem with the BWRS equation is large computa- suggested two equations based on two different attractive
tion time and mathematical inconvenience to predict various terms [6]:
physical properties. To find molar volume V from Eq. (5.89), (5.94) Z = Z HS − a
a successive substitutive method is required. However, as it RTV
will be discussed in the next section, this type of equations can and
be used to develop generalized correlations in the graphical HS a −1 −1/2
or tabulated forms for prediction of various thermophysical (5.95) Z = Z − RT (V − b) T
properties.
where Z HS is the hard sphere contribution given by Eq.
(5.93). Obviously Eq. (5.94) is a two-parameter EOS (a, σ)
5.6.3 Carnahan–Starling Equation of State and Eq. (5.95) is a three-parameter EOS (a, b, σ). Both Eqs.
and Its Modifications (5.94) and (5.95) reduce to ideal gas law (Z → Z HS → 1) as
Equations of state are mainly developed based on the un- V →∞ (or P → 0), which satisfies Eq. (5.18). For mixtures,
derstanding of intermolecular forces and potential energy the quadratic mixing rule can be used for parameter a while
functions that certain fluids follow. For example, for hard a linear rule can be applied to parameter b. Application of
sphere fluids where the potential energy function is given by these equations for mixtures has been discussed in recent ref-
Eq. (5.13) it is assumed that there are no attractive forces. For erences [8, 47]. Another modification of CS EOS is through
such fluids, Carnahan and Starling proposed an EOS that has LJ EOS in the following form [48, 49]:
been used extensively by researchers for development of more HS 32εξ
accurate EOS [6]. For hard sphere fluids, the smallest possible (5.96) Z = Z − 3k B T
--`,```,`,``````,`,````,```,,-`-`,,`,,`,`,,`---
Copyright ASTM International
Provided by IHS Markit under license with ASTM Licensee=International Dealers Demo/2222333001, User=Anggiansah, Erick
No reproduction or networking permitted without license from IHS Not for Resale, 08/26/2021 21:56:35 MDT