Page 223 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

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5. PVT RELATIONS AND EQUATIONS OF STATE 203
universal gas constant in which its values in different units
are given in Section 1.7.24. The conditions that Eq. (5.14)
can be used depend on the substance and its critical proper-
σ ties. But approximately this equation may be applied to any
Γ
gas under atmospheric or subatmospheric pressures with an
acceptable degree of accuracy. An EOS can be nondimension-
alized through a parameter called compressibility factor, Z,
0 defined as
r
V PV
(5.15)
FIG. 5.7—Hard sphere po- Z ≡ V ig = RT
tential model.
where for an ideal gas Z = 1 and for a real gas it can be greater
or less than unity as will be discussed later in this chapter. Z in
energy we have, as r →∞, → 0. For example, in the fact represents the ratio of volume of real gas to that of ideal
Sutherland model it is assumed that the repulsion force is gas under the same conditions of T and P. As the deviation
∞ but the attraction force is proportional with 1/r , that is of a gas from ideality increases, so does deviation of its Z
n
6
for r >σ, =−D/r , where D is the model parameter [6]. factor from unity. The application of Z is in calculation of
Potential energy models presented in this section do not de- physical properties once it is known for a fluid. For example,
scribe molecular forces for heavy hydrocarbons and polar if Z is known at T and P, volume of gas can be calculated from
compounds. For such molecules, additional parameters must Eq. (5.15). Application of Eq. (5.15) at the critical point gives
be included in the model. For example, dipole moment is a critical compressibility factor, Z c , which was initially defined
parameter that characterizes degree of polarity of molecules by Eq. (2.8).
and its knowledge for very heavy molecules is quite useful for In ideal gases, molecules have mass but no volume and
better property prediction of such compounds. Further dis- they are independent from each other with no interaction.
cussion and other potential energy functions and intermolec- An ideal gas is mathematically defined by Eq. (5.14) with the
ular forces are discussed in various sources [6, 7]. following relation, which indicates that the internal energy is
only a function of temperature.
ig
5.4 EQUATIONS OF STATE (5.16) U = f 4 (T)
Substitution of Eqs. (5.14) and (5.16) into Eq. (5.5) gives
An EOS is a mathematical equation that relates pressure, vol-
ume, and temperature. The simplest form of these equations (5.17) H = f 5 (T)
ig
is the ideal gas law that is only applicable to gases. In 1873,
ig
van der Waals proposed the first cubic EOS that was based where H is the ideal gas enthalpy and it is only a function
on the theory of continuity of liquids and gases. Since then of temperature. Equations (5.14), (5.16), and (5.17) simply
many modifications of cubic equations have been developed define ideal gases.
and have found great industrial application especially in the
--`,```,`,``````,`,````,```,,-`-`,,`,,`,`,,`---
petroleum industry because of their mathematical simplic-
ity. More sophisticated equations are also proposed in re- 5.4.2 Real Gases—Liquids
cent decades that are useful for certain systems [8]. Some of Gases that do not follow ideal gas conditions are called real
these equations particularly useful for petroleum fluids are gases. At a temperature below critical temperature as pressure
reviewed and discussed in this chapter. increases a gas can be converted to a liquid. In real gases, vol-
ume of molecules as well as the force between molecules are
5.4.1 Ideal Gas Law not zero. A comparison among an ideal gas, a real gas, and
a liquid is demonstrated in Fig. 5.8. As pressure increases
As discussed in the previous section the intermolecular forces behavior of real gases approaches those of their liquids. The
depend on the distance between the molecules. With an space between the molecules in liquids is less than real gases
increase in molar volume or a decrease in pressure the and in real gases is less than ideal gases. Therefore, the in-
intermolecular distance increases and the intermolecular termolecular forces in liquids are much stronger than those
forces decrease. Under very low-pressure conditions, the in- in real gases. Similarly the molecular forces in real gases are
termolecular forces are so small that they can be neglected higher than those in ideal gases, which are nearly zero. It is
( = 0). In addition since the empty space between the for this reason that prediction of properties of liquids is more
molecules is so large the volume of molecules may be ne- difficult than properties of gases.
glected in comparison with the gas volume. Under these con- Most gases are actually real and do not obey the ideal gas
ditions any gas is considered as an ideal gas. Properties of law as expressed by Eqs. (5.14) and (5.16). Under limiting
ideal gases can be accurately estimated based on the kinetic conditions of P → 0(T > 0) or at T and V →∞ (finite P)we
theory of gases [9, 10]. The universal form of the EOS for ideal can obtain a set of constraints for any real gas EOS. When
gases is T →∞ translational energy becomes very large and other
ig
(5.14) PV = RT energies are negligible. Any valid EOS for a real gas should
obey the following constraints:
where T is absolute temperature, P is the gas absolute pres-
sure, V ig is the molar volume of an ideal gas, and R is the (5.18) lim (PV) = RT
P→0

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