Diffusion model of gas hydrate formation from ice

Abstract

Assuming that gas hydrate formation from ice occurs through a diffusion mechanism, a system of equations was developed that describes the kinetics of gas hydrate formation from a spherical ice particle. This model takes into account the porous structure of the formed gas hydrate as well as the increases in particle volume due to the formation of the gas hydrate. In addition, the intrinsic kinetics of gas hydrate formation from ice is considered. In the framework of a quasi-stationary approximation, a simplified solution for the system of equations was obtained. From the comparison between the calculated and available experimental data, the temperature dependence was determined for the diffusion coefficient of methane in methane hydrate.

Appendices

Appendix 1: Interrelation between radii R(t) and ξ(t)

During the process of gas hydrate formation from ice, the following relations are satisfied:

$$m_{\text{h}} (t) = m^{\prime}_{\text{i}} (t) + m^{\prime}_{\text{g}} (t), \quad t \ge 0,$$

(30)

$$n^{\prime}_{\text{g}} = \frac{{m^{\prime}_{\text{g}} (t)}}{{M_{\text{g}} }},$$

(31)

$$n^{\prime}_{\text{w}} = nn^{\prime}_{\text{g}} = \frac{{m^{\prime}_{\text{i}} (t)}}{{M_{\text{w}} }},$$

(32)

where $m_{\text{h}} (t)$ is the mass of the gas hydrate, $m^{\prime}_{\text{g}} (t)$ is the mass of the gas passed into the structure of the gas hydrate, $n^{\prime}_{\text{g}}$ is the moles of the gas passed into the structure of the gas hydrate and $n^{\prime}_{\text{w}}$ is the moles of water passed into the structure of the gas hydrate. From Eqs. (30)–(32), it follows that

$$m^{\prime}_{\text{i}} (t) = \frac{{m_{\text{h}} (t)}}{{1 + \frac{{M_{\text{g}} }}{{nM_{\text{w}} }}}}, \quad t \ge 0.$$

(33)

Substituting Eq. (33) into Eq. (26) yields

$$\eta (t) = \frac{{m_{\text{h}} (t)}}{{m_{{{\text{i}}0}} \left( {1 + \frac{{M_{\text{g}} }}{{nM_{\text{w}} }}} \right)}}, \quad t \ge 0.$$

(34)

The geometry of the problem allows us to represent the quantities $m_{\text{h}} (t)$ and $m_{{{\text{i}}0}}$ in the form

$$m_{\text{h}} (t) = \rho^{\prime}_{\text{h}} \frac{4}{3}\pi \left( {R^{3} (t) - \xi^{3} (t)} \right) = \rho_{\text{h}} \left( {1 - \varepsilon_{\text{h}} } \right)\frac{4}{3}\pi \left( {R^{3} (t) - \xi^{3} (t)} \right), \quad t \ge 0,$$

(35)

$$m_{{{\text{i}}0}} = \rho_{\text{i}} \frac{ 4}{3}\pi R_{0}^{3} ,$$

(36)

where $\rho^{\prime}_{\text{h}}$ is the apparent mass density of the gas hydrate and $\rho_{\text{h}}$ and $\rho_{\text{i}}$ are the true mass density of the gas hydrate and of ice, respectively. Substituting Eqs. (35) and (36) into Eq. (34), we can obtain the relation

$$R(t) = \sqrt[3]{{\frac{{\rho_{\text{i}} }}{{\rho_{\text{h}} \left( {1 - \varepsilon_{\text{h}} } \right)}}\left( {1 + \frac{{M_{\text{g}} }}{{nM_{\text{w}} }}} \right)R_{0}^{3} \eta (t) + \xi^{3} (t)}}, \quad t \ge 0.$$

(37)

Taking into account Eq. (27) and given that ${ \rho}_{\text{i}} = \omega M_{\text{w}}$ and $\rho_{\text{h}} = \chi M_{\text{h}}$, Eq. (37) goes into Eq. (12).

Appendix 2: Equation of motion for the front of reaction (1)

Near the surface Γ, the following molar balance ratio for gas hydrate is satisfied:

$$r_{\text{h}} = \left. {\chi \left( {1 - \varepsilon_{\text{h}} } \right)\frac{d\Sigma (t)}{dt}} \right|_{\Gamma } , \quad t > 0.$$

(38)

From Eq. (38), taking into account the geometry of the problem, we can obtain the equation of motion for the front reaction (1) in the form

$$\frac{d\xi (t)}{dt} = \frac{{ - r_{\text{h}} }}{{\chi \left( {1 - \varepsilon_{\text{h}} } \right)}}, \quad t > 0.$$

(39)

With allowance for Eq. (7), Eq. (39) goes into Eq. (13).

Near the surface Γ, the following molar balance ratio for water is also satisfied:

$$\left| {r_{\text{w}} } \right| = \left. {\omega \frac{d\Sigma (t)}{dt}} \right|_{\Gamma } , \quad t > 0.$$

(40)

Taking into account the geometry of the problem and given Eq. (6), from Eq. (40) the equation of motion for the front reaction (1) can be obtained in the form

$$\frac{d\xi (t)}{dt} = - nk\omega^{n - 1} \left( {\left. {c(r, \, t)} \right|_{r = \xi (t)} - \frac{{p_{\text{eq}} }}{{{Z}_{\text{eq}} RT}}} \right), \quad t > 0.$$

(41)

Thus, the equation of motion for the front reaction (1) is written ambiguously. However, analysis shows that this has little effect on the results of simulation for the kinetics of gas hydrate formation from ice in the framework developed by the diffusion model.