Modeling Gas Hydrate Formation from Ice Powders Based on Diffusion Theory

Abstract

Based on the diffusion theory, a gas–solid reaction model for gas hydrate formation from monosize ice powders was constructed and an effective diffusion coefficient through the gas hydrate shell was obtained by solving the diffusion equation. This model can be applied for a gas–solid reaction process with gas pressure drop during hydrate formation. Two cases termed as M₁ and M₂ were discussed and compared with each other. In the M₁ case, the spherical particle geometry was assumed to be squeezed by adjacent particles. In the M₂ case, the effect of adjacent particles on the object particle geometry was neglected. The results showed that the gas effective diffusion coefficient was a critical parameter for accurately simulating the reaction process, and it was found to vary during hydrate formation. In this model, the time-dependent gas effective diffusion coefficient was calculated by means of the measured gas pressure during hydrate formation, and the degrees of hydrate formation were obtained. In addition, the geometrical changes of the hydrate shell and ice powders were obtained during hydrate formation.

APPENDIX

The rate of the forward reaction of gas hydrate formation can be represented as

$${{R}_{{{\text{form}}}}} = \;{{k}_{{\text{R}}}}{\rho }_{{{\text{mi}}}}^{n}{{\left. {c\left( {r,t} \right)} \right|}_{{r = {{R}_{{\text{i}}}}\left( t \right)}}},\,\,\,t > 0.$$

((A.1))

In the state of the thermodynamic equilibrium of the ice–gas hydrate–gas system, the following condition is satisfied:

$${{R}_{{{\text{form}}}}} = \;{{R}_{{{\text{dis}}}}} = \;{{k}_{{\text{R}}}}{\rho }_{{{\text{mi}}}}^{n}\frac{{{{p}_{{{\text{eq}}}}}}}{{{{Z}_{{{\text{eq}}}}}RT}}.$$

((A.2))

The rate of decrease in the amount of the gas on the surface is

$$\begin{gathered} {{r}_{{\text{g}}}} = {{R}_{{{\text{form}}}}} - {{R}_{{{\text{dis}}}}} \\ = \,\,\,{{k}_{{\text{R}}}}{\rho }_{{{\text{mi}}}}^{n}\left( {{{{\left. {c\left( {r,t} \right)} \right|}}_{{r = {{R}_{{\text{i}}}}\left( t \right)}}} - \frac{{{{p}_{{{\text{eq}}}}}}}{{{{Z}_{{{\text{eq}}}}}RT}}} \right)\; ,\,\,\,t > 0. \\ \end{gathered} $$

((A.3))

For the diffused gas was fully involved in the generating reaction, the internal boundary condition is

$$\begin{gathered} {{D}_{{{\text{eff}}}}}{{\left. {\frac{{\partial c\left( {r,t} \right)}}{{\partial r}}} \right|}_{{r = {{R}_{{\text{i}}}}\left( t \right)}}} = {{r}_{{\text{g}}}} \\ = \,\,{{k}_{{\text{R}}}}{\rho }_{{{\text{mi}}}}^{n}\left( {{{{\left. {c\left( {r,t} \right)} \right|}}_{{r = {{R}_{{\text{i}}}}\left( t \right)}}} - \frac{{{{p}_{{{\text{eq}}}}}}}{{{{Z}_{{{\text{eq}}}}}RT}}} \right),\,\,\,t > 0. \\ \end{gathered} $$

((A.4))

As it has been assumed, the diffusion is pseudo-steady where it is not in the external boundary; obviously, the following equation is reasonable:

$$\begin{gathered} \frac{{\partial c\left( {r,t} \right)}}{{\partial t}} = \;{{D}_{{{\text{eff}}}}}\left( {\frac{{{{\partial }^{2}}c\left( {r,t} \right)}}{{\partial {{r}^{2}}}} + \;\frac{2}{r}\frac{{\partial c\left( {r,t} \right)}}{{\partial r}}} \right) = 0,~\,\, \\ {{R}_{{\text{i}}}}\left( t \right)~\, \leqslant r~ < ~{{R}_{{\text{h}}}}\left( t \right). \\ \end{gathered} $$

((A.5))

The solution of the above equation is $c\left( {r,t} \right) = \frac{{a\left( t \right)}}{r} + b\left( t \right)$.

Due to the thermodynamic conditions of the external boundary are varied, we have

$$\frac{{\partial c\left( {r,t} \right)}}{{\partial t}} = \frac{{\partial \left( {\frac{p}{{ZRT}}} \right)}}{{\partial t}}~,\,\,\,~r = ~{{R}_{{\text{h}}}}\left( t \right).$$

The improved solution is

$$c\left( {r,t} \right) = \frac{{a\left( t \right)}}{r} + B\left( t \right){{r}^{2}} + b\left( t \right),~\,\,\,r = ~{{R}_{{\text{h}}}}\left( t \right),$$

$$\begin{gathered} {{\left. {B\left( t \right)} \right|}_{{r = {{R}_{{\text{h}}}}\left( t \right)}}} = \frac{1}{{6{{D}_{{{\text{eff}}}}}}}\frac{{\partial \left( {\frac{p}{{ZRT}}} \right)}}{{\partial t}}, \\ {\text{when\;}}p,\,\,T = {\text{const}}, {{\left. {B\left( t \right)} \right|}_{{r = {{R}_{{\text{h}}}}\left( t \right)}}} = 0. \\ \end{gathered} $$

((A.6))

For $B\left( t \right) = 0,~\,\,\,{{R}_{{\text{i}}}}\left( t \right)~\, \leqslant r~ < ~{{R}_{{\text{h}}}}\left( t \right)$, we will obtain a unified solution.

In the external boundary condition,

$$\begin{gathered} {{\left. {c\left( {r,t} \right)} \right|}_{{r = {{R}_{{\text{h}}}}\left( t \right)}}} = \frac{p}{{Z\left( {p,T} \right)RT}} \\ = \,\,~\frac{{a\left( t \right)}}{{{{R}_{{\text{h}}}}}} + B\left( t \right)R_{{\text{h}}}^{2} + b\left( t \right),\,\,\,t > 0. \\ \end{gathered} $$

((A.7))

In the internal boundary condition,

$${{D}_{{{\text{eff}}}}}\left( { - \frac{{a\left( t \right)}}{{R_{{\text{i}}}^{2}}}} \right) = ~\,{{k}_{{\text{R}}}}{\rho }_{{{\text{mi}}}}^{n}\left( {\frac{{a\left( t \right)}}{{{{R}_{{\text{i}}}}}} + b\left( t \right) - \;\frac{{{{p}_{{{\text{eq}}}}}}}{{{{Z}_{{{\text{eq}}}}}RT}}} \right).$$

((A.8))

By (A.7) and (A.8), we will obtain $a\left( t \right)$, $b\left( t \right)$. At the external boundary of hydrate shell, B(t) is obtained by the real gas state equation. At internal boundary of the hydrate shell, while the discrete time is sufficiently small [17], the process is regarded as pseudo-steady state, and B(t) is equal to 0. Then, we obtain ${{\left. {B\left( t \right)} \right|}_{{r = {{R}_{{\text{h}}}}\left( t \right)}}} = \frac{1}{{6{{D}_{{{\text{eff}}}}}}}\frac{{\partial \left( {\frac{p}{{ZRT}}} \right)}}{{\partial t}},$ when $p,T = {\text{const}},$

$${{\left. {B\left( t \right)} \right|}_{{r = {{R}_{{\text{h}}}}\left( t \right)}}} = \;0,\,\,\,{{\left. {B\left( t \right)} \right|}_{{r = {{R}_{{\text{i}}}}\left( t \right)}}} = \;0.$$

((A.9))

$$\begin{gathered} a\left( t \right) \\ = - \frac{{\frac{p}{{ZRT}}\; - \;\frac{{{{p}_{{{\text{eq}}}}}}}{{{{Z}_{{{\text{eq}}}}}RT\;}}{\kern 1pt} \; - {\kern 1pt} \;\frac{1}{{6{{D}_{{{\text{eff}}}}}}}\frac{{\partial \left( {\frac{p}{{ZRT}}} \right)}}{{\partial t}}{{R}_{{\text{h}}}}{{{\left( t \right)}}^{2}}}}{{\frac{{{{D}_{{{\text{eff}}}}}}}{{{{k}_{{\text{R}}}}{\rho }_{{{\text{mi}}}}^{n}{{R}_{{\text{i}}}}{{{\left( t \right)}}^{2}}}} + \;\left( {\frac{1}{{{{R}_{{\text{i}}}}\left( t \right)}} - \;\frac{1}{{{{R}_{{\text{h}}}}\left( t \right)}}} \right)}}, \\ \end{gathered} $$

((A.10))

$$b\left( t \right) = \frac{p}{{ZRT}} - \frac{{a\left( t \right)}}{{{{R}_{{\text{h}}}}\left( t \right)}}\; - \frac{1}{{6{{D}_{{{\text{eff}}}}}}}\frac{{\partial \left( {\frac{p}{{ZRT}}} \right)}}{{\partial t}}{{R}_{{\text{h}}}}{{\left( t \right)}^{2}}.$$

((A.11))

The molar flux through the gas hydrate layer is

$$\begin{gathered} Q = \frac{{4{\pi }{{D}_{{{\text{eff}}}}}\left( {\frac{p}{{ZRT}}\; - {{{\left. {c\left( {r,t} \right)} \right|}}_{{r = {{R}_{{\text{i}}}}\left( t \right)}}}} \right)}}{{\frac{1}{{{{R}_{{\text{i}}}}\left( t \right)}} - \;\frac{1}{{{{R}_{{\text{h}}}}\left( t \right)}}}} \\ = \,\,{{k}_{{\text{R}}}}{\rho }_{{{\text{mi}}}}^{n}\left( {{{{\left. {c\left( {r,t} \right)} \right|}}_{{r = {{R}_{{\text{i}}}}\left( t \right)}}} - \;\frac{{{{p}_{{{\text{eq}}}}}}}{{{{Z}_{{{\text{eq}}}}}RT}}} \right) = {{r}_{{\text{g}}}}. \\ \end{gathered} $$

((A.12))

Because of molar conservation, we have the following formula:

$$ - \frac{{d\left( {{{{\rho }}_{{{\text{mi}}}}}\frac{4}{3}{\pi }{{R}_{{\text{i}}}}{{{\left( t \right)}}^{3}}} \right)}}{{dt}} = n{{r}_{{\text{g}}}}\;.$$

((A.13))

According to (A.12), we have

$${{\left. {c\left( {r,t} \right)} \right|}_{{r = {{R}_{{\text{i}}}}\left( t \right)}}} = \frac{{{{r}_{{\text{g}}}}}}{{{{k}_{{\text{R}}}}{\rho }_{{{\text{mi}}}}^{n}}} + \frac{{{{p}_{{{\text{eq}}}}}}}{{{{Z}_{{{\text{eq}}}}}RT}}.$$

((A.14))

Then, we obtain

$$Q = \;{{r}_{{\text{g}}}} = \frac{{\frac{p}{{ZRT}} - \frac{{{{p}_{{{\text{eq}}}}}}}{{{{Z}_{{{\text{eq}}}}}RT\;}}}}{{\;\frac{{\frac{1}{{{{R}_{{\text{i}}}}\left( t \right)}} - \frac{1}{{{{R}_{{\text{h}}}}\left( t \right)}}}}{{4{\pi }{{D}_{{{\text{eff}}}}}}} + \frac{1}{{{{k}_{{\text{R}}}}{\rho }_{{{\text{mi}}}}^{n}}}}}.$$

((A.15))

The number of ice particles in the sample is as follows:

$$N = \;\frac{{{{m}_{{\text{i}}}}}}{{{{{\rho }}_{{\text{i}}}}\frac{4}{3}{\pi }R_{0}^{3}}}.$$

((A.16))

The variation in the molar flow rate of the gas around the individual hydrate particle is approximately equal to the molar flux through the hydrate layer:

$$\begin{gathered} Q = - \frac{{\partial \left( {\frac{{p\left( t \right)}}{{ZRT}}} \right)}}{{\partial t}}\left( {\frac{V}{N} - \frac{4}{3}{\pi }{{R}_{{\text{h}}}}{{{\left( t \right)}}^{3}}} \right) \\ \approx \,\, - \frac{{{{{\left. {\Delta c} \right|}}_{{r = {{R}_{{\text{h}}}}\left( t \right)}}}}}{{\Delta t}}\left( {\frac{V}{N} - \frac{4}{3}{\pi }{{R}_{{\text{h}}}}{{{\left( t \right)}}^{3}}} \right). \\ \end{gathered} $$

((A.17))

According to molar conservation, we have

$$\begin{gathered} \frac{{4n{{{\rho }}_{{{\text{mh}}}}}}}{3}{\pi }\left( {{{R}_{{\text{h}}}}{{{\left( t \right)}}^{3}} - \;{{R}_{{\text{i}}}}{{{\left( t \right)}}^{3}}} \right) \\ = \,\,\frac{{4{{{\rho }}_{{{\text{mi}}}}}}}{3}{\pi }\left( {{{R}_{{\text{i}}}}{{{\left( t \right)}}^{3}} - \;R_{0}^{3}} \right). \\ \end{gathered} $$

((A.18))

Combining formulas (A.13) and (A.15), we have formula (8). According to (A.18), we obtain formula (9).

NOTATION

C z	slope of the random density function
c(r, t)	gas concentration in the layer of the gas hydrate, mol/m3
Δc	gas concentration variation, mol/m3
D eff	effective diffusion coefficient of the gas in the gas hydrate, m2/s
k R	reaction rate constant for the formation of gas hydrate on the surface, m3n + 1/moln
m i	total mass of ice, kg
N	number of ice particles in the sample
n	hydrate number
p	gas pressure outside the particle, Pa
p(t)	function of pressure over time, Pa
p eq	pressure of the ice–gas hydrate–gas equilibrium, Pa
Q	molar flux of the gas in the gas hydrate layer, mol/s
R	gas constant, J/(mol s)
R 0	initial radius of the ice particle, m
Rh(t)	outside radius of the gas hydrate layer, m
Rh(t)*	outside radius of the gas hydrate layer, m
Ri(t)	inside radius of the gas hydrate layer, m
R form	rate of the reaction of gas hydrate formation on the surface, mol/s
R dis	rate of the reaction of gas hydrate dissociation on the surface, mol/s
r g	rate of decrease in the amount of gas on the surface during a chemical reaction, mol/s; mol/s
T	temperature, K
t	time, s
Δt	discrete time, s
V	volume, m3
Z	compressibility factor
Z 0	current coordination number
Z eq	compressibility factor for a gas that is under equilibrium conditions
δr	thickness of the hydrate shell at the origin time, m
δt	time for hydrate shell formation, s
η(t)	degree of hydrate formation
ρi	density of ice, kg/m3
ρmi	molar density of ice, mol/m3
ρmh	molar density of the gas hydrate, mol/m3