3. Thermodynamic and kinetic modelling
3.1. Modelling of gas hydrate dissociation conditions
3.1.2. Fugacity approach (equality of water fugacity in the neighbouring phases)
The probable equilibrium phases during gas hydrate formation include (Sloan and Koh, 2008):
Gas phase (G): This phase mainly contains gases which can form hydrate crystal (maybe some of them are not able to form crystal hydrate). Depending on the equilibrium conditions, this phase may include a small amount of water vapour.
The liquid phase (Lw): This phase contains mainly water. In addition, other components in the system which have been dissolved in water can exist in this phase.
Liquid phase containing mainly hydrocarbons (LH): This phase consists mainly of hydrocarbons and small amount of water which can be dissolved in this phase.
Solid hydrate phase (H): This phase consists of crystalline hydrates which contain water molecules and gas molecules which can be trapped inside the hydrate crystal lattice. Different types of hydrate crystals may be simultaneously formed in certain conditions (such as crystals, type I and II).
Ice-phase (I): This is the phase consists of water molecules which have been frozen.
Note that depending on the system and the related equilibrium conditions, some of the aforementioned phases may not exist. The existence of the available equilibrium phases can be determined using the Gibbs free energy theory (Sloan and Koh, 2008). The thermodynamic models for the representation of gas hydrate dissociation conditions are based on the solid solution theory of van der Waals and Platteeuw (1959) or vdWP theory (van der Waals and Platteeuw, 1959). The fundamental assumptions in the vdWP theory are summarized as follows (Sloan and Koh, 2008):
1) Each cavity in the crystal lattice comprises at most one molecule of hydrate former.
2) Ideal gas partition function for guest molecule is applicable.
3) A potential function is applied for expressing the intermolecular forces between guest and water molecules in a spherical cavity.
4) Interaction between the encaged guest molecules is negligible.
5) The interaction forces are considered between a surrounding guest molecule and water molecules in the same cavity.
6) The host molecules’ contribution to the free energy is independent of the occupation of the cavity.
CHAPTER 3………....………… THERMODYNAMIC AND KINETIC MODELLING
29 According to the solid solution theory of van der Waals and Platteeuw (vdWP) (1959) (van der Waals and Platteeuw, 1959), at the equilibrium conditions the equality of the fugacity of water in the liquid, vapour and hydrate phases is applied as a criterion for hydrate dissociation conditions as the following equation (Mohammadi et al., 2005, Javanmardi et al., 2012):
) , ( ) , ( ) ,
(PT f PT f PT
fwL wV wH 3-3
where,
f
wL,f
wV andf
wH present the fugacity of water in the liquid, vapour and hydrate phases, respectively. The fugacity of water in the vapour and liquid phases can be determined using an EoS (Equation of State) and an appropriate mixing rule. It is believed that Valderrama’s modification of Patel–Teja equation of state (VPT EoS) (Valderrama, 1990) together with the non-density-dependent (NDD) mixing rules (Avlonitis et al., 1994) are the strong tools for the modelling of the phase equilibria for the systems containing water and polar molecules. In this thesis, the VPT EoS (Valderrama, 1990) and the NDD mixing rules (Avlonitis et al., 1994) along with the flash calculations was used for predicting the fugacity of water in the vapour and liquid phases. The reader is referred to Appendix A for the details of the VPT EoS (Valderrama, 1990) and the NDD mixing rules (Avlonitis et al., 1994). More details regarding the calculation of the fugacity of water in the hydrate phase,f
wH, in Equation 3-3, are given in Appendix B. Figure 3-3 represents the algorithm which was used in this study for calculating the hydrate dissociation temperature in the fugacity approach. Since the fugacity approach is based on the flash calculations, it is useful for high soluble hydrate forming systems such as CO2 hydrates.30 Figure 3-3. The algorithm used in this study based on fugacity approach (Li et al., 2008, Abbott et al., 2001) for calculating the hydrate dissociation temperature (Newton- Raphson method (Ben-Israel, 1966) was used in the flash calculations). The weight fraction value was 0.001 in these calculations.
CHAPTER 3………....………… THERMODYNAMIC AND KINETIC MODELLING
31 3.1.3 Chemical potential approach
Holder et al. (1980) presented a model using van der Waals and Platteeuw (1959), the (vdWP) theory (van der Waals and Platteeuw, 1959), and based on the equality of the chemical potential of water in the hydrate phase (wH) and in liquid/ ice phase (wl/) as a criteria for the phases equilibria (Holder et al., 1980):
wH wl/ 3-4Subtracting the chemical potential of water in the empty hydrate lattice (w) from the both sides of Equation 3-4, the following equation is derived:
wH wl/ 3-5
in which,
wH represents the water potential difference between the hydrate phase and the hypothetical empty hydrate lattice phase. In addition,
wl/ in Equation 3-5 shows the water potential difference between the hypothetical empty hydrate lattice phase and the liquid or ice phase. The details of the parameters in equation 3-5 are given in Appendix B. The final equation for calculating the hydrate dissociation temperature in the activity approach is presented as follows (Holder et al., 1980):
P
P w
w l TT
w l l w
w dP a
T R dT v
T R h T
R T
R / 0 0 / 0 / ln( )
3-6
All the parameters in the above equation are given in Appendix B. The activity of water (aw) in Equation 3-6 is considered to be unity for the gas hydrates in the presence of pure water.
Li et al. (2008) used the fugacity and the activity approach for the hydrate dissociation conditions of the single and mixed gas hydrates (Li et al., 2008). In many cases, they obtained more accurate results using the fugacity approach.