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1D Dissolution of a gas phase in a liquide phase
1D Dissolution of a gas phase in a liquide phase
This test case presented by [1] describes a flat interface separating a liquid phase from a gas phase, where the dissolution of the gas phase into the liquid phase induces a flow in the surrounding fluid. This flow, caused by the density difference between the phases, modifies the movement of the interface. In other words, if the densities are equal, the flow disappears, but the interface still moves without this effect.
This validation aims to confirm several key aspects: (i) the calculation of interfacial mass transfer, (ii) the volume variation of the gas phase, (iii) the variation of the mass fraction of chemical species during their transfer and its impact on the interface movement, and (iv) the accurate capture of the discontinuity of the concentration field across the interface.
The study focuses here on the diffusion of a single chemical species, denoted \(A\). The second species composing the solution also diffuses, and the magnitude of its flux is equal to that of species \(A\).
The study domain considered is a rectangle of arbitrary height and total length \(L = 10 \times L_{\text{ref}} \), where the characteristic length is set to \(L_{\text{ref}} = 1 \times 10^{-2}\,\mathrm{m}\).
The initial configuration places the interface between the phases at \(l_I(t=0) = 0\), thus defining a spatial distribution in which the gas phase occupies the left part of the domain \(-L/2 \leq x \leq l_I\), while the liquid phase is located in the right part \(l_I \leq x \leq L/2\).
The properties of the system are:
| phases | \(\rho~\left[\text{kg}.\text{m}^{-3}\right]\) | \(\mu~\left[\text{N.s}.\text{m}^{-2}\right]\) |
|---|---|---|
| liquid | 1000 | \(1.05\times 10^{-3}\) |
| gas | 1 | \(1.46\times 10^{-5}\) |
The simulation is performed with a henry's coefficient \(H=0.5\) and for two different values of the diffusion coefficient: \(D_{A,l} = 1 \times 10^{-6}\,\mathrm{m^2\,s^{-1}}\) and \(D_{A,l} = 1 \times 10^{-8}\,\mathrm{m^2\,s^{-1}}\). By introducing the dimensionless Schmidt number \(\mathrm{Sc}\), which compares momentum diffusion to mass diffusion, defined as
\[ \mathrm{Sc} = \frac{\mu_l}{\rho_l D_{A,l}}, \]
these two diffusion coefficients correspond respectively to Schmidt numbers \(\mathrm{Sc} = 1.05\) and \(\mathrm{Sc} = 1.05 \times 10^2\).
The initial concentrations are imposed as \(\rho_{A,g}^0 = 1~\mathrm{kg}.\mathrm{m}^{3}\) in the gas phase and \(\rho_{A,l}^0 = 0~\mathrm{kg}.\mathrm{m}^{3} \) in the liquid phase. These values are also maintained as Dirichlet boundary conditions, respectively on the left and right side of the domain.
A Neumann-type condition (outflow condition) is imposed at the right boundary of the domain at \(x = L/2\), while symmetry conditions are applied on the other boundaries.
The spatial discretization is based on a structured mesh comprising 1600 uniform cells in the \(x\)-direction. The number of cells in the \(y\)-direction remains arbitrary due to the one-dimensional nature of the problem.
The results for both methods unshifted and shifted are the same. Figure 1 presents a comparison between the interface displacement kinetics obtained numerically and those predicted analytically. Figure 2 illustrates the comparison of the concentration profiles for the fastest kinetics (i.e., for \(\mathrm{Sc} = 1.05\)).
The obtained results show an excellent agreement between numerical and analytical solutions, regardless of the value of \(\mathrm{Sc}\).
[1] Fleckenstein, S., & Bothe, D. (2015). A Volume-of-Fluid-based numerical method for multicomponent mass transfer with local volume changes. Journal of Computational Physics, 301, 35-58. https://doi.org/10.1016/j.jcp.2015.08.011
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