## 4 Results and Discussion
However, the supply of hydrate former needed to sustain the growth will still occur via diffusion. Assuming a quasi-equilibrium between liquid water and water adsorbed on the hydrate surface, one can obtain a fair estimate of hydrate nucleation rate from CO\({}_{2}\) dissolved in water. Figure 13 presents the critical radius calculated for homogeneous hydrate formation from CO\({}_{2}\) dissolved in water for 280 K and 100 bars as an example. The corresponding nucleation times are plotted in Figure 1.8 We should note that this route to hydrate formation is characterized by almost \"instant\" hydrate nucleation in the macroscopic sense (nucleation time of mere seconds). Figure 1.9 Calculated nucleation times in nanoseconds in the case of homogeneous hydrate formation from liquid water and CO\({}_{2}\) dissolved in water at 280 K and 100 bars. Figure 1.0 Calculated critical radius in the case of homogeneous hydrate formation from liquid water and dissolved CO\({}_{2}\) at 280 K and 100 bars as function of mole fraction of CO\({}_{2}\) in liquid water. Figure 15 presents the growth rates calculated by using the value of \(\rm D_{\rm liq}=10^{-08}\) m\({}^{2}\)/s in Equation (10). The time for visible hydrate film appears to be in a good agreement with the observations of [PERSON] et al. [10], as well as our own observations of CH\({}_{4}\) hydrate film growth [40] that showed visible (Magnetic Resonance Imaging) hydrate on a resolution scale of approximately 300 microns after 100 h.  
It is beyond the scope of this work to provide a more detailed analysis of the induction times for various sets of transport characteristics and thermodynamic conditions. However, the results presented in this work will be applied in our modeling of hydrate dynamics in sediments, as well as in the dynamics of hydrate nucleation during pipeline transport of hydrate formers containing dissolved water. MDIT (multicomponent diffuse interface theory) theory [45] is equally a simple and convenient alternative for practical applications that require quickly solved kinetic models. The heat transport modeling in this work will also find its way into our present level of Phase Field Theory (PFT) models. See, for instance, [24; 25; 26; 38; 39; 40] for representative examples. While the heat transport across aqueous systems is very fast, the opposite is true for hydrate formation from water dissolved in gas even though this process is thermodynamically feasible. Mass transport is slow since about 150 water molecules must assemble and structure themselves into a hydrate nucleus for it to be able to grow steadily. In this case, heat transport will be the rate limiting factor since the enthalpy of creating the hydrate core needs to be transported through a heat insulator such as hydrocarbon gas. If the heat cannot be transported away from the hydrate core fast enough, the hydrogen bonds will break due to the accumulated heat, and the hydrate core will melt. The important point here is that any specific case has to be analyzed in detail in terms of coupled mass transport, heat transport, and thermodynamic control (free energy change effect). Then, the appropriate simplifications and approximations can be performed based on physical arguments. Far too often, including examples mentioned here, kinetic models for hydrate phase transitions are based on physically wrong assumptions. While the kinetic model for hydrate phase transitions presented in this work (CNT) is numerically simple, it still incorporates a newly developed mass transport term based on modern theoretical concepts and results from molecular dynamics simulations. The implicit heat transport model we employ is equally numerically simple but has the advantage of being coupled to a new model for enthalpy calculations completely consistent with the free energies in hydrate and all co-existing phases. Figure 1.1 Calculated CO\({}_{2}\) hydrate film thickness as a function of time for a liquid side diffusivity in Equation (10) equal to \(\rm D_{\rm liq}=10^{-08}\) m\({}^{2}\)/s.