Diffusion-kinetic model of gas hydrate film growth along the gas–water interface

Abstract

An analytic model that describes the kinetics of the process of gas hydrate film growth along the gas–water interface is presented. This model is based on the assumption that this process is controlled only by the mass transfer of gas molecules dissolved in water to the moving front of the gas hydrate film. In the presented model, the driving force of the process of gas hydrate film growth along the gas–water interface is the concentration driving force. The calculated data obtained in the framework of the presented model are compared with the available experimental data on the kinetics of methane hydrate film growth on a planar water surface and on the surface of a methane bubble suspended in water. Moreover, the calculated data obtained in the framework of the presented model are compared with the available experimental data on the kinetics of carbon dioxide hydrate film growth on the surface of a carbon dioxide bubble suspended in water. As a result of this comparison, the dependence of the thickness of carbon dioxide hydrate film on the concentration driving force was determined.

Appendices

Appendix 1: Calculation of the concentration driving force

Since the concentration driving force is Δc = c_s − c_eq, then, for its calculation under given thermobaric conditions, it is necessary to calculate the quantities c_s and c_eq. In the case of the aqueous gas solution, the following relations for the quantities c_s and c_eq are valid:

$$ {c}_{\mathrm{s}}=\frac{\rho_{\mathrm{w}}{x}_{\mathrm{s}}}{M_{\mathrm{w}}}, $$

(25)

$$ {c}_{\mathrm{eq}}=\frac{\rho_{\mathrm{w}}{x}_{\mathrm{eq}}}{M_{\mathrm{w}}}, $$

(26)

where x_s and x_eq are the mole fractions of the gas dissolved in water at the liquid water–gas equilibrium and the liquid water–hydrate–gas equilibrium, respectively. In the case of an ideal gas, the quantities x_s and x_eq are determined from Henry’s law:

$$ p=H{x}_{\mathrm{s}}, $$

(27)

$$ {p}_{\mathrm{eq}}=H{x}_{\mathrm{eq}}, $$

(28)

where H is the Henry’s constant. In the case of a real gas, the quantities x_s and x_eq are determined from the following relations:

$$ f=H{x}_{\mathrm{s}}\exp \left[\frac{\upsilon_{\mathrm{g}}^{\infty}\left(p-{p}_{\mathrm{v}}\right)}{RT}\right], $$

(29)

$$ {f}_{\mathrm{eq}}=H{x}_{\mathrm{eq}}\exp \left[\frac{\upsilon_{\mathrm{g}}^{\infty}\left({p}_{\mathrm{eq}}-{p}_{\mathrm{v}}\right)}{RT}\right], $$

(30)

where f ≡ f ( p, T ) and f_eq ≡ f ( p_eq, T ) are the gas fugacities under given thermobaric conditions and under liquid water–hydrate–gas equilibrium conditions, respectively, ${\upsilon}_{\mathrm{g}}^{\infty }$ is the partial molar volume of the gas in water at infinite dilution, p_v is the saturated vapor pressure of water, and R is the gas constant.

In this work, the quantity c_s for methane and carbon dioxide was calculated by Eqs. (25) and (29), and the quantity c_eq for these gases was calculated by Eqs. (26) and (30). After that, the concentration driving force Δc was calculated. The fugacities f and f_eq for methane and carbon dioxide were calculated from the Peng–Robinson equation of state.

Appendix 2: Thermophysical properties of substances

The kinematic viscosity of water ν was calculated using the relation ν = μ/ρ_w, where μ is the dynamic viscosity of water. The values of the mass density of water ρ_w at different temperatures were taken from a handbook [33].

Carbon dioxide:

• ${\upsilon}_{\mathrm{g}}^{\infty }=33.9$ cm³/mol Ref. [34],

• H = exp(13.99194 − 2650.11724/T) MPa Ref. [35],

• D = 0.013942(T/227 − 1)^1.7094 mm²/s Ref. [36].

Carbon dioxide hydrate:

• M_h = 175.52 g/mol (n_h = 7.3), ρ_h = 1.14 g/cm³ Ref. [37],

• p_eq = exp(44.58 − 10246.28/T) kPa Ref. [38].

Methane:

• ${\upsilon}_{\mathrm{g}}^{\infty }=\exp \left(3.541+0.00123\left(T-273.15\right)\right)$ cm³/mol Ref. [39],

• H = 101325 exp(183.786 − 9112.582/T − 25.0405 ln T + 0.00015T) Pa Ref. [40],

• D = 0.01595(T/229.8 − 1)^1.8769 mm²/s Ref. [41].

Methane hydrate:

• M_h = 124.13 g/mol (n_h = 6.0), ρ_h = 0.91 g/cm³ Ref. [38],

• p_eq = exp(38.98 − 8533.8/T) kPa Ref. [38].

Water:

• ${\displaystyle \begin{array}{l}{M}_{\mathrm{w}}=18.015\ \mathrm{g}/\mathrm{mol},\\ {}{p}_{\mathrm{v}}=22.064\exp \left(647.096{T}^{-1}\right(-7.85951783\tau +1.84408259{\tau}^{1.5}\\ {}-11.7866497{\tau}^3+22.6807411{\tau}^{3.5}-15.9618719{\tau}^4\\ {}+1.80122502{\tau}^{7.5}\left)\right)\mathrm{MPa},\mathrm{where}\ \tau =1-T/647.096\end{array}}$ Ref. [42],

• ${\displaystyle \begin{array}{c}\mu =280.68{\tilde{T}}^{-1.9}+511.45{\tilde{T}}^{-7.7}+61.131{\tilde{T}}^{-19.6}\\ {}+0.45903{\tilde{T}}^{-40}\upmu \mathrm{Pa}\ \mathrm{s},\mathrm{where}\ \tilde{T}=T/300\end{array}}$ Ref. [43].