Page 225 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

T1: IML
QC: IML/FFX
P1: IML/FFX
P2: IML/FFX
AT029-Manual-v7.cls
AT029-Manual
AT029-05
August 16, 2007
By taking the second terms to the right-hand side in each
Equation (5.37) indicates that value of Z c is the same for all
equation and dividing Eq. (5.24) by Eq. (5.25) we get 17:42 5. PVT RELATIONS AND EQUATIONS OF STATE 205
compounds. Values of Z c given in Table 2.1 varies from 0.28
to 0.21 for most hydrocarbons. Therefore, vdW EOS signifi-
V c
(5.26) b = cantly overpredicts values of Z c (or V c ) and its performance
3 in the critical region is quite weak. Similar approaches can
By substituting Eq. (5.26) into Eq. (5.24) we get be used to determine EOS parameters and Z c for any other
9 EOS.
(5.27) a = T c V c
8
Since T c and P c are usually available, it is common to express Since the introduction of the vdW EOS as the first cubic
parameters a and b in terms of T c and P c rather than T c and equation 130 years ago, dozens of cubic EOSs have been pro-
V c . For this reason V c can be found from Eq. (5.21) in terms posed, many of them developed in recent decades. The math-
of T c and P c and replaced in the above equations. Similar ematical simplicity of a cubic EOS in calculation of thermo-
results can be obtained by a more straightforward approach. dynamic properties has made it the most attractive type of
At the critical point we have V = V c or V − V c = 0, which can EOS. When van der Waals introduced Eq. (5.21) he indicated
be written as follows: that parameter a is temperature-dependent. It was in 1949
when Redlich and Kwong (RK) made the first modification
3
(5.28) (V − V c ) = 0 to vdW EOS as [11]
Application of Eq. (5.23) at T c and P c gives RT a
(5.38) P = −
a ab V − b V(V + b)
3
2
(5.29) V − b + RT c V + V − = 0
P c P c P c where parameter a depends on temperature as a c /T 0.5 in
Expansion of Eq. (5.28) gives which a c is related to T c and P c . Parameters a and b in Eq.
(5.38) are different from those in Eq. (5.22) but they can be
3
3
2
3
2
(5.30) (V − V c ) = V − 3V c V + 3V V − V = 0 obtained in a similar fashion as in Example 5.1 (as shown
c
c
Equations (5.29) and (5.30) are equivalent and the corre- later). The repulsive terms in Eqs. (5.38) and (5.22) are iden-
0
2
1
3
sponding coefficients for V , V , V , and V must be equal tical. Performance of RK EOS is much better than vdW EOS;
in two equations. This gives the following set of equations for however, it is mainly applicable to simple fluids and rare gases
the coefficients: such as Kr, CH 4 ,orO 2 , but for heavier and complex com-
pounds it is not a suitable PVT relation.

RT c 2
(5.31) − b + =−3V c coefficients of V The RK EOS is a source of many modifications that began
P c in 1972 by Soave [12]. The Soave modification of Redlich–
a
(5.32) = 3V c 2 coefficients of V Kwong equation known as SRK EOS is actually a modifica-
P c tion of parameter a in terms of temperature. Soave obtained
ab 3 parameter a in Eq. (5.38) for a number of pure compounds us-
(5.33) − =−V c coefficients of V 0
P c ing saturated liquid density and vapor pressure data. Then he
By dividing Eq. (5.33) by (5.32), Eq. (5.26) can be obtained. By correlated parameter a to reduced temperature and acentric
substituting V c = 3b (Eq. 5.26) to the right-hand side (RHS) factor. Acentric factor, ω, defined by Eq. (2.10) is a parameter
of Eq. (5.31) the following relation for b is found: that characterizes complexity of a molecule. For more com-
plex and heavy compounds value of ω is higher than simple
RT c
(5.34) b = molecules as given in Table 2.1. SRK EOS has been widely
8P c used in the petroleum industry especially by reservoir engi-
Combining Eqs. (5.26) and (5.34) gives neers for phase equilibria calculations and by process engi-
neers for design calculations. While RK EOS requires T c and
3RT c
(5.35) V c = P c to estimate its parameters, SRK EOS requires an additional
8P c
parameter, namely a third parameter, which in this case is ω.
Substituting Eq. (5.35) into Eq. (5.27) gives As it will be seen later that while SRK EOS is well capable of
9 3RT c 27R T c 2 calculating vapor–liquid equilibrium properties, it seriously
2
a = RT c = underestimates liquid densities.
8 8P c 64P c
Another popular EOS for estimation of phase behavior
Therefore, the final relation for parameter a in terms of T c and and properties of reservoir fluids and hydrocarbon systems
P c is as follows: is Peng-Robinson (PR) proposed in the following form [13]:
2
27R T c 2 RT a
(5.36) a = (5.39) P = −
64P c V − b V(V + b) + b(V − b)
In calculation of parameters a and b unit of R should be con- where a and b are the two parameters for PR EOS and are
sistent with the units chosen for T c and P c . Another useful calculated similar to SRK parameters. Parameter a was cor-
result from this analysis is estimation of critical compress- related in terms of temperature and acentric factor and later it
ibility factor through Eq. (5.35). Rearranging this equation was modified for properties of heavy hydrocarbons [14]. The
and using definition of Z c from Eq. (2.8) gives original idea behind development of PR EOS was to improve
3 liquid density predictions. The repulsive term in all four cubic
P c V c
(5.37) Z c = = = 0.375 equations introduced here is the same. In all these equations
RT c 8
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