## 3 Determination of the Hydrate Formation Model  
The difference between the chemical potential of the previous and the current phase being evaluated is called the supersaturation, which we refer to as \(\Delta\mu\). We calculate the supersaturation as: [29]  
\[\Delta\mu=kT\ ln\left(\frac{(P_{r},\ T)P_{r}}{(P_{e},T)P_{e}}\right)+\Delta v_{e }(P_{r}-P_{e}), \tag{1}\]  
where \(k\) is the Boltzmann constant [J\(\cdot\)K\({}^{-1}\)], \(T\) is the temperature [K], \(P_{r}\) is the required reactor pressure [MPa], \(P_{e}\) is the equilibrium pressure 2.6 [MPa]. and \(\Delta v_{e}\) is the volume difference \(-4.4\times 10^{-2}\) [nm\({}^{3}\)]. The value of supersaturation \(\Delta\mu\) after calculation is \(3.6\times 10^{-21}\) Joule [29]. The process of formation of methane hydrates in the system, water splashing in the gaseous environment is subject to a heterogeneous nucleation process. The kinetic parameter indicates the size of the heterogeneous nucleation particles and is calculated by:  
\[A_{I}=Z\epsilon(4\pi c)^{\frac{1}{2}}\ v_{h}^{\frac{1}{2}}\ DC_{e}ns^{\frac{1}{3}} \frac{A_{p}C_{p}}{a_{w}}, \tag{2}\]  
where \(Z\) is the Zeldovich factor 0.01 [-], \(\epsilon\) is the coefficient of adhesion 1 [-], \(c\) is the numerical shape factor 4.8 [-], \(v_{h}\) is the unit volume of hydrate 0.216 [nm\({}^{3}\)], \(ns\) is the growth exponent for a stabilization period 50 [-], \(A_{p}\) is the particle surface area 10\({}^{4}\) [nm\({}^{2}\)], \(C_{p}\) is the concentration of active particles 10\({}^{15}\) [m\({}^{-3}\)] and \(a_{w}\). is the area occupied by the water molecule on the surface of the particle 1.2 \(\times\) 10\({}^{-7}\) [nm\({}^{2}\)]. \(C_{e}\) is the concentration of dissolved gas in water at equilibrium conditions 0.03 [nm\({}^{-3}\)]. The kinetic parameter \(A_{I}\) describes the mechanism of binding of hydrate units to the hydrate nucleus and after quantification is \(4.31\times 10^{26}\) [m\({}^{-3}\).s\({}^{-1}\)] [30]. Figure 2: Visualization of the experimental device in 3D. The reference lines are described in Table 1.  
When water and gas particles mix, diffusion of gas through the surface layer of the liquid occurs. The growth constant that defines the growth of crystals during injection is given by the relationship:  
\[G_{I}=(\varepsilon v_{h}DC_{e}/\delta)\bigg{(}\exp\frac{\Delta\mu}{kT}-1\bigg{)}, \tag{3}\]  
where \(D\) is the diffusion coefficient 1000 [\(\mu\)m\({}^{2}\cdot\)s\({}^{-1}\)] and \(\delta\) is the thickness of the immobile layer on the crystal surface 10 [\(\mu\)m]. After quantification, the growth constant has the value \(G_{I}=1.28\times 10^{-5}\) m\(\cdot\)s\({}^{-1}\)[31]. The growth of hydrate crystals is a self-organizing process that leads to the formation of a new thermodynamic phase. The formula for calculating the rate of continuous nucleation of hydrate crystals is given by:  
\[J_{I}=A_{I}\exp\bigg{(}\frac{\Delta\mu}{kT}\bigg{)}\exp\bigg{(}\frac{-4 \epsilon^{3}v_{h}^{2}v_{ef}^{3}}{27 kT\Delta\mu^{2}}\bigg{)}, \tag{4}\]  
where \(\sigma_{ef}\) is specific surface energy [J\(\cdot\)m\({}^{-2}\)] and nucleation rate is \(J_{1}=3.30\times 10^{12}\) [m\({}^{-3}\)s\({}^{-1}\)] [32]. The rate constant of hydrate formation \(ke_{1}(t)\) during the period of water injection into the gaseous environment of the pressure vessel is calculated using Equations (5) and (6). We first derive Equation (5) with respect to time, where all terms of the equation take known values [33]. \[\alpha_{I}(t)=\frac{bG_{I}^{3m}J_{I}}{(1+3m)(2+3m)}\;t^{1+3m} \tag{5}\]  
where \(b\) is the crystal shape factor of 4.05 (-) and \(m\) is the growth rate exponent that characterizes the period of water injection into the pressure vessel and has a value of 1.6 [\(-\)] [34].