72
3 Gas Hydrates
3.2.3 Phase Equilibrium Computation of Clathrate Hydrates
The phase diagrams of clathrate hydrates have been determined experimentally for
major guests. However, it is impossible to experimentally determine phase diagrams
of all possible guests because there are literally infinite numbers of compositions
of gas mixtures. So a computational method that can accurately determine a phase
diagram of clathrate hydrate is highly desirable.
We saw in (Sect. 1.1) that, for a pure component, the Clausius–Clapeyron equation
can be used to find the relationship as to how the equilibrium pressure varies with
the equilibrium temperature along a phase boundary. As the first approximation, the
slope of a triple-phase line that extends from a quadruple point may be estimated if
the change in the enthalpy, ΔH, and that of the volume, ΔV, across the triple-phase
line are known at the quadruple point temperature. The applicability of the Clapeyron
equation to the phase equilibria of clathrate hydrate systems is discussed in references
[1, 20, 64, 65]. As we saw in the previous section, new clathrate hydrate phases
are being discovered at very high pressures (>GPa), and the Clausius–Clapeyron
equation cannot tell where the triple-phase curve that extends from a quadruple
point should end.
A more accurate method that can be applied to gas mixtures is a statistical thermodynamic approach [1]. Here, the physical state of a given system is determined so as
to minimize the Gibbs free energy of the system. For a given set of pressure, temperature, and composition, the Gibbs free energy of the system can be calculated for
each case of when the system consists of one phase, two phases, etc., and compared.
To do so, some assumptions and models are required that enable calculations of the
chemical potential of each component involved.
As we saw in Chap. 1, a liquid–gas phase boundary can in principle be determined
by solving an appropriate cubic equation of state. To solve a cubic equation of state,
the coefficients in the cubic equation of state (e.g., a and b for the van der Waals
equation of state) that are specific to each system need to be determined. These
coefficients can be determined by fitting the computed output of the cubic equation
of state to relevant available experimental data. Once the coefficients are determined,
the cubic equation of state can be solved for a system of interest.
In typical flash calculations, the input parameters are the overall composition, the
total amount of the substance, the system pressure, and the temperature. The outputs
are the number of phases and the composition and the amount of each phase. The
Gibbs free energy of the whole system is minimized to find the answer. Alternatively,
computationally simpler fugacity can be used in the place of chemical potential as
a parameter because the fugacity is balanced at a phase boundary where two phases
coexist in equilibrium [66].
For a phase boundary that involves a solid (crystalline) phase, a good model for
the crystal in question is generally required. For clathrate hydrates, a good model
for water is generally required; however, there is no consensus as to what the best
model for water should be. A great deal of work has been carried out over the years
for this effort, which has been summarized in [1].
3 Gas Hydrates
3.2.3 Phase Equilibrium Computation of Clathrate Hydrates
The phase diagrams of clathrate hydrates have been determined experimentally for
major guests. However, it is impossible to experimentally determine phase diagrams
of all possible guests because there are literally infinite numbers of compositions
of gas mixtures. So a computational method that can accurately determine a phase
diagram of clathrate hydrate is highly desirable.
We saw in (Sect. 1.1) that, for a pure component, the Clausius–Clapeyron equation
can be used to find the relationship as to how the equilibrium pressure varies with
the equilibrium temperature along a phase boundary. As the first approximation, the
slope of a triple-phase line that extends from a quadruple point may be estimated if
the change in the enthalpy, ΔH, and that of the volume, ΔV, across the triple-phase
line are known at the quadruple point temperature. The applicability of the Clapeyron
equation to the phase equilibria of clathrate hydrate systems is discussed in references
[1, 20, 64, 65]. As we saw in the previous section, new clathrate hydrate phases
are being discovered at very high pressures (>GPa), and the Clausius–Clapeyron
equation cannot tell where the triple-phase curve that extends from a quadruple
point should end.
A more accurate method that can be applied to gas mixtures is a statistical thermodynamic approach [1]. Here, the physical state of a given system is determined so as
to minimize the Gibbs free energy of the system. For a given set of pressure, temperature, and composition, the Gibbs free energy of the system can be calculated for
each case of when the system consists of one phase, two phases, etc., and compared.
To do so, some assumptions and models are required that enable calculations of the
chemical potential of each component involved.
As we saw in Chap. 1, a liquid–gas phase boundary can in principle be determined
by solving an appropriate cubic equation of state. To solve a cubic equation of state,
the coefficients in the cubic equation of state (e.g., a and b for the van der Waals
equation of state) that are specific to each system need to be determined. These
coefficients can be determined by fitting the computed output of the cubic equation
of state to relevant available experimental data. Once the coefficients are determined,
the cubic equation of state can be solved for a system of interest.
In typical flash calculations, the input parameters are the overall composition, the
total amount of the substance, the system pressure, and the temperature. The outputs
are the number of phases and the composition and the amount of each phase. The
Gibbs free energy of the whole system is minimized to find the answer. Alternatively,
computationally simpler fugacity can be used in the place of chemical potential as
a parameter because the fugacity is balanced at a phase boundary where two phases
coexist in equilibrium [66].
For a phase boundary that involves a solid (crystalline) phase, a good model for
the crystal in question is generally required. For clathrate hydrates, a good model
for water is generally required; however, there is no consensus as to what the best
model for water should be. A great deal of work has been carried out over the years
for this effort, which has been summarized in [1].