Page 235 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

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pressure variations from the critical point. By substituting pa-
where ε is the molecular energy parameter and ζ (see Eq. 5.92)
rameter h into the first equation in Eq. (5.99) one can see that
is related to σ the size parameter. ε and σ are two parameters 5. PVT RELATIONS AND EQUATIONS OF STATE 215
in the LJ potential (Eq. 5.11) and k B is the Boltzman constant.
One advanced noncubic EOS, which has received significant (5.101) Z = f (T r , P r )
attention for property calculations specially derived proper- This equation indicates that for all fluids that obey a two-
ties (i.e., heat capacity, sonic velocity, etc.), is that of SAFT parameter EOS, such as RK, the compressibility factor, Z,
originally proposed by Chapman et al. [50] and it is given in is the only function of T r and P r . This means that at the
the following form [47]: critical point where T r = P r = 1, the critical compressibility
factor, Z c , is constant and same for all fluids (0.333 for RK
(5.97) Z SAFT = 1 + Z HS + Z CHAIN + Z DISP + Z ASSOC
EOS). As can be seen from Table 2.1, Z c is constant only for
where HS, CHAIN, DISP, and ASSOC refer to contributions simple fluids such as N 2 ,CH 4 ,O 2 , or Ar, which have Z c of
from hard sphere, chain formation molecule, dispersion, and 0.29, 0.286, 0.288, and 0.289, respectively. For this reason RK
association terms. The relations for Z HS and Z CHAIN are simple EOS is relatively accurate for such fluids. Equation (5.101)
and are given in the following form [47]: is the fundamental of corresponding states principle (CSP) in
classical thermodynamics. A correlation such as Eq. (5.101)
4ξ − 2ξ 2 5ξ − 2ξ 2
Z SAFT = 1 + r + (1 − r) is also called generalized correlation. In this equation only
(1 − ξ) 3 (1 − ξ)(2 − ξ) two parameters (T c and P c ) for a substance are needed to
(5.98) + Z DISP + Z ASSOC determine its PVT relation. These types of relations are usu-
ally called two-parameter corresponding states correlations
where r is a specific parameter characteristic of the substance (CSC). The functionality of function f in Eq. (5.101) can be
of interest. ζ in the above relation is segment packing fraction determined from experimental data on PVT and is usually
and is equal to ζ from Eq. (5.92) multiplied by r. The relations expressed in graphical forms rather than mathematical
for Z DISP and Z ASSOC are more complex and are in terms of equations. The most widely used two-parameter CSC in a
summations with adjusting parameters for the effects of asso- graphical form is the Standing–Katz generalized chart that
ciation. There are other forms of SAFT EOS. A more practical, is developed for natural gases [52]. This chart is shown in
but much more complex, form of SAFT equation is given by Li Fig. 5.12 and is widely used in the petroleum industry [19, 21,
and Englezos [51]. They show application of SAFT EOS to cal- 53, 54]. Obviously this chart is valid for light hydrocarbons
culate phase behavior of systems containing associating fluids whose acentric factor is very small such as methane and
such as alcohol and water. SAFT EOS does not require criti- ethane, which are the main components of natural gases.
cal constants and is particularly useful for complex molecules Hall and Yarborough [55] presented an EOS that was based
--`,```,`,``````,`,````,```,,-`-`,,`,,`,`,,`---
such as very heavy hydrocarbons, complex petroleum fluids, on data obtained from the Standing and Katz Z-factor chart.
water, alcohol, ionic, and polymeric systems. Parameters can The equation was based on the Carnahan-Starling equation
be determined by use of vapor pressure and liquid density (Eq. 5.93), and it is useful only for calculation of Z-factor of
data. Further characteristics and application of these equa- light hydrocarbons and natural gases. The equation is in the
tions are given by Prausnitz et al. [8, 47]. In the next chapter, following form:
the CS EOS will be used to develop an EOS based on the
y

velocity of sound. (5.102) Z = 0.06125P r T r −1 −1 exp −1.2 1 − T r −1 2
where T r and P r are reduced temperature and pressure and y
5.7 CORRESPONDING STATE is a dimensionless parameter similar to parameter ξ defined
CORRELATIONS in Eq. (5.91). Parameter y should be obtained from solution
of the following equation:
One of the simplest forms of an EOS is the two-parameter RK F(y) =− 0.06125P r T −1 exp −1.2 1 − T −1 2

EOS expressed by Eq. (5.38). This equation can be used for 2 3 r 4 r
fluids that obey a two-parameter potential energy relation. In + y + y + y − y − 14.76T −1 −9.76T −2 + 4.58T −3 y 2

fact this equation is quite accurate for simple fluids such as (1 − y) 3 r r r
methane. Rearrangement of Eq. (5.38) through multiplying + 90.7T −1 − 242.2T −2 + 42.4T −3 y ( 2.18+2.82T r ) = 0
−1

both sides of the equation by V/RT and substituting param- r r r
eters a and b from Table 5.1 gives the following relation in (5.103)
terms of dimensionless variables [1]: The above equation can be solved by the Newton–Raphson
1 4.934 h 0.08664P r method. To find y an initial guess is required. An approximate
(5.99) Z = − where h ≡ relation to find the initial guess is obtained at Z = 1 in Eq.
1 − h T r 1.5 1 + h ZT r
(5.102):
where T r and P r are called reduced temperature and reduced (k) −1
pressure and are defined as: (5.104) y = 0.06125P r T r exp −1.2 1 − T r −1 2
(k)
T P Substituting y (k) in Eq. (5.103) gives F , which must be used
(5.100) T r ≡ P r ≡ in the following relation to obtain a new value of y:
T c P c
where T and T c must be in absolute degrees (K), similarly P (5.105) y (k+1) = y (k) F (k)
and P c must be in absolute pressure (bar). Both T r and P r are − dF (k)
dimensionless and can be used to express temperature and dy

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