Page 92 - Handbook of Natural Gas Transmission and Processing Principles and Practices

2.3. Natural Gases Phase Behavior Modeling With Cubic
EoS

2.3.1. Some Words About Cubic Equations of State History
Our current industrialized world transports and produces chemicals on an unprecedented scale.
Natural gas and oil are today key raw materials from which are derived the gaseous and liquid
fuels energizing factories, electric power plants, and most modes of transportation as well.
Processes of evaporation and condensation, of mixing and separation, underlie almost any
production method in the chemical industry. These processes can be grandly complex, especially
when they occur at high pressures. Interest for them has mainly started during the industrial
revolution in the 19th century and has unceasingly grown up since then. A huge leap in
understanding of the phase behavior of fluids was accomplished during the second half of this
century by the scientists Van der Waals and Kamerlingh Onnes and more generally by the Dutch
School. It is undisputable that their discoveries were built on many talented anterior works as for
instance,

• the successive attempts by Boyle and Mariotte in the 17th century and Gay-Lussac, in the
19th century, to derive the perfect-gas equation,
• the first observation of critical points by Cagniard de la Tour in 1822,
• the experimental determination of critical points of many substances by Faraday and
Mendeleev throughout the 19th century,
• the measurement of experimental isotherms by Thomas Andrews showing, at the end of
the 19th century, the behavior of a pure fluid around its critical point.

As the major result of the Dutch School, the Van der Waals equation of state (1873)—connecting
variables P (pressure), v (molar volume), and T (temperature) of a fluid—was the first mathematical
model incorporating both the gas–liquid transition and fluid criticality. In addition, its foundation
on, more or less rigorous, molecular concepts (Van der Waals' theory assumes that molecules are
subject to attractive and repulsive forces) affirmed the reality of molecules at a crucial time in
history. Before Van der Waals, some attempts to model the real behavior of gases were made. The
main drawback of the P-v-T relationships presented before was that they did not consider the finite
volume occupied by the molecules, similarly, to the perfect-gas equation. Yet the idea of including
the volume of the molecules into the repulsive term was suggested by Bernoulli at the end of the
18th century and was then ignored for a long time. Following this idea, Dupré and Hirn (in 1863–
64) proposed to replace the molar volume v by ( ), where b is the molar volume that molecules
exclude by their mutual repulsions. This quantity is proportional to the temperature-independent
molecular volume and named covolume by Dupré (sometimes also called excluded volume).
However, none of these contributions were of general use, and none was able to answer the many
questions related to fluid behavior remaining at that time. It was Van der Waals with his celebrated
doctoral thesis on “The Continuity of the Liquid and Gaseous States” (1873) and his famous equation of
state who proposed for the first time a physically coherent description of fluid behavior from low to
high pressures. To derive his equation, he considered the perfect-gas law (i.e., ) and took
into account the fact that molecules occupy space by replacing v by ( ) and the fact that they
exert an attraction on each other by replacing P by (cohesion effect). Therefore owing to
mutual repulsion of molecules, the actual molar volume v has to be greater than b while molecular
attraction forces are incorporated in the model by the coefficient a. Note also that in Van der Waals'
theory the molecules are assumed to have a core in the form of an impenetrable sphere. Physicists
and more particularly thermodynamicists rapidly understood that Van der Waals' theory was a
revolution-upsetting classical conceptions and modernizing approaches used until then to describe
fluids. As a consecration, Van der Waals was awarded the Nobel Prize of physics on December 12,
1910; he can be seen as the father of modern fluid thermodynamics. The equation that Van der
Waals proposed in his thesis (Van der Waals, 1873) writes as

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