Page 224 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

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204 CHARACTERIZATION AND PROPERTIES OF PETROLEUM FRACTIONS
(a) August 16, 2007 17:42 (b) (c)
1 Mole of Ideal Gas 1 Mole of Real Gas 1 Mole of Liquid
(atmospheric pressure) P >P P >P sat
P a b a c
23
1 mole of a fluid consist of 6.02x10 molecules. b = volume of 1 mole of hard molecules.
V=volume of 1 mole of fluid
ig ig g g l l
V =V space V =V space + b V =V space + b
ig g l g ig
b << V space b<V V space << V space < V space
FIG. 5.8—Difference between an ideal gas, a real gas, and a liquid.

∂V R To find V from T and P, the above equation may be rearranged

(5.19) lim =
T→∞ ∂T P as
P
∂ V
2
(5.20) lim 2 = 0 (5.23) 3 RT 2 a ab = 0
T→∞ ∂T P V − b + P V + P V − P
In general for any gas as P → 0 (or V →∞) it becomes an
ideal gas; however, as T →∞ it is usually assumed that gas where it is a cubic equation in terms of V. For this reason the
behavior approaches those of hard sphere gases. Constraints vdW EOS, Eq. (5.22), is known as a cubic EOS. As a matter
set by the above equations as well as Eq. (5.9) may be used to of fact any EOS that can be converted into a cubic form is
examine validity of an EOS for real fluids. called a cubic EOS. In Eq. (5.22), parameters a and b have
physical meanings. Parameter b also called co-volume or re-
pulsive parameter represents volume of 1 mol of hard cores
of molecules and has the same unit as the molar volume (V).
5.5 CUBIC EQUATIONS OF STATE
Parameter a is also referred to as attraction parameter and
2
2
6
has the same unit as that of PV (i.e., bar · cm /mol ). In Eq.
The ideal gas law expressed by Eq. (5.14) is neither applicable (5.22), the term RT/(V − b) represents the repulsive term of a
to real gases (high pressure) nor to liquids where the volume molecule, while a/V represents attractive term and accounts
2
of molecules cannot be ignored in comparison with the vol- for nonideal behavior of gas. V − bis in fact the space between
ume of gas (see Fig. 5.8). Cubic EOS are designed to overcome molecules (Figs. 5.8b and 5.8c). When parameters a and b are
these two shortcomings of ideal gas law with mathematical zero Eq. (5.22) reduces to ideal gas law. Mathematically it can
convenience. Several commonly used equations, their solu- be shown from Eq. (5.22) that as P →∞, V → b and the free
tion, and characteristics are discussed in this section.
volume between molecules disappears.
Since Eq. (5.21) has only two parameters it is also known
5.5.1 Four Common Cubic Equations as a two-parameter EOS. Parameters a and b in the vdW EOS
(vdW, RK, SRK, and PR) can be best determined from experimental data on PVT. How-
ever, mathematically these constants can be determined by
The behavior of high-pressure gases approaches the behavior imposing Eq. (5.9) as shown in the following example.
of liquids until the critical point where both gas and liquid
behavior become identical. van der Waal (vdW) proposed the
idea of continuity of gases and liquids and suggested that a Example 5.1—Obtain vdW parameters in terms of T c and P c
single equation may represent the PVT behavior of both gases using Eq. (5.9) and (5.21). Also determine Z c for fluids that
and liquids. He modified Eq. (5.14) by replacing P and V with obey vdW EOS.
appropriate modifications to consider real gas effects in the
following form [1]: 2 2
Solution—∂ P/∂V and ∂ P/∂V are calculated from Eq. (5.22)
a by keeping T constant and set equal to zero at T = T c , P = P c ,
(5.21) P + (V − b) = RT
V 2 and V = V c as
where a and b are two constants specific for each substance
RT c 2a
∂ P
but independent of T and P. The above equation is usually (5.24) =− 2 + 3 = 0
written as ∂V T c (V c − b) V c
2
RT a ∂ P 2RT c 6a
(5.22) P = − (5.25) = 3 − 4 = 0
V − b V 2 ∂V 2 T c (V c − b) V c

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