Page 66 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

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46 CHARACTERIZATION AND PROPERTIES OF PETROLEUM FRACTIONS
where
2

(2.27) ∂ P

= 0
A ◦ ∂V 2
T c ,P c
rep =
r n
Application of Eqs. (2.26) and (2.27) to any two-parameter
B ◦
att =− EOS would result in relations for calculation of parameters
r m
A and B in terms of T c and P c , as shown in Chapter 5. It should
be noted that EOS parameters are generally designated by
rep = repulsive potential lower case a and b, but here they are shown by A and B. No-
att = attractive potential
A ◦ = parameter characterizing the repulsive force (>0) tation a, b, c, ... are used for correlation parameters in various
B ◦ = parameter characterizing the attractive force (>0) equations in this chapter. Applying Eqs. (2.26) and (2.27) to
r = distance between molecules Eq. (2.25) results into the following three relations for T c , P c ,
n, m = positive numbers, n > m and V c :
The main characteristics of a two-parameter potential energy (2.28) T c = f 4 (A, B)
function is the minimum value of potential energy, min , desig-
nated by ε =− min and the distance between molecules where (2.29) P c = f 5 (A, B)
the potential energy is zero ( = 0) which is designated by σ. (2.30) V c = f 6 (A, B)
London studied the theory of dispersion (attraction) forces
and has shown that m = 6 and it is frequently convenient for Functions f 4 , f 5 , and f 6 are universal functions and are the
mathematical calculations to let n = 12. It can then be shown same for all fluids that obey the potential energy relation ex-
that Eq. (2.20) reduces to the following relation known as pressed by Eq. (2.20) or Eq. (2.21). In fact, if parameters A
Lennard–Jones potential [39]: and B in a two-parameter EOS in terms of T c and P c are rear-
ranged one can obtain relations for T c and P c in terms of these
12 6 two parameters. For example, for van der Waals and Redlich–

σ
σ
(2.21) = 4ε −
r r Kwong EOS the two parameters A and B are given in terms
of T c and P c [21] as shown in Chapter 5. By rearrangement of
In the above relation, ε is a parameter representing molec-
ular energy and σ is a parameter representing molecular the vdW EOS parameters we get
size. Further discussion on intermolecular forces is given in 8 −1 1 −2
Section 5.3. T c = 27R AB P c = 27 AB V c = 3B
According to the principle of statistical thermodynamics
there exists a universal EOS that is valid for all fluids that and for the Redlich–Kwong EOS we have
follow a two-parameter potential energy relation such as
5 2/3
Eq. (2.21) [40]. (0.0867) R 2/3 −2/3
T c = A B
0.4278R
(2.22) Z = f 1 (ε, σ, T, P)
(0.0867) R 2/3 −5/3
5 1/3
P c = A B V c = 3.847B
PV T,P 2
(2.23) Z = (0.4278)
RT
Similar relations can be obtained for the parameters of other
where
Z = dimensionless compressibility factor two-parameter EOS. A generalization can be made for the
V T,P = molar volume at absolute temperature, T, and pres- relations between T c , P c , and V c in terms of EOS parameters
sure, P A and B in the following form:
f 1 = universal function same for all fluids that follow (2.31) [T c , P c , V c ] = aA B c
b
Eq. (2.21).
By combining Eqs. (2.20)–(2.23) we obtain where parameters a, b, and c are the constants which differ
for relations for T c , P c , and V c . However, these constants are
(2.24) V T,P = f 2 (A ◦ , B ◦ , T, P)
the same for each critical property for all fluids that follow
where A ◦ and B ◦ are the two parameters in the potential en- the same two-parameter potential energy relation. In a two-
ergy relation, which differ from one fluid to another. Equation parameter EOS such as vdW or RK, V c is related to only one
(2.24) is called a two-parameter EOS. Earlier EOS such as van parameter B so that V c /B is a constant for all compounds.
der Waals (vdW) and Redlich–Kwong (RK) developed for sim- However, formulation of V c through Eq. (2.30) shows that
ple fluids all have two parameters A and B [4] as discussed in V c must be a function of two parameters A and B. This is
Chapter 5. Therefore, Eq. (2.24) can also be written in terms one of the reasons that two-parameters EOS are not accurate
of these two parameters: near the critical region. Further discussion on EOS is given
in Chapter 5.
(2.25) V T,P = f 3 (A, B, T, P) To find the nature of these two characterizing parameters
The three functions f 1 , f 2 , and f 3 in the above equations vary one should realize that A and B in Eq. (2.31) represent the two
in the form and style. The conditions at the critical point for parameters in the potential energy relation, such as ε and σ in
any PVT relation are [41] Eq. (2.22). These parameters represent energy and size char-
acteristics of molecules. The two parameters that are readily

(2.26) ∂ P

= 0 measurable for hydrocarbon systems are the boiling point,
∂V
T c ,P c T b , and specific gravity, SG; in fact, T b represents the energy

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