A formulation for dissolution in inhomogeneous temperature field
05/10/2014 14:54
We propose equations governing the dissolution in inhomogeneous temperature field in terms of the variational principle. The derived equations clarify that the interface energy between solute and solvent has a significant effect on the process of the dissolution. The interface energy restrains the dissolution, and the moving interface involves the heat of dissolution.
Dissolution is an important process and frequently discussed in industry as well as in science. For example, the dissolution of supercritical CO2 in interstitial water is one of the most crucial research topics in CO2 capture and storage (CCS), which stores the CO2 into the deep underground in a geological rock formation. The CO2 can be trapped in the micro-pore space as droplets surrounded by water. At this small scale, the contribution of the interface energy to the total energy is consequential, and thus comes into play.
Various phase field models based on free energies are often used to study the dynamics on the assumption of constant temperature and no heat transfer[1, 2]. There can be phenomena involved in the inhomogeneous temperature and the heat transfer. The diffusion flux can be induced by a temperature gradient, which is known as the Soret effect or thermal diffusion[3]. The heat transfer during dissolution across the interface is also considered important for the dynamics of the gasCO2 at near the critical point because of its very large thermal conductivity[4]. Then in previous works, heuristic methods have been proposed to combine the thermodynamics with those phase field models above.
In this study, we propose a completely different method based on the variational principle to derive the governing equations for the dissolution in the inhomogeneous temperature field. In this proposed method, we combine the kinematics and thermodynamics in the variational calculus by using the equation of the entropy. Note that If we don’t know the exact form of this equation, we can obtain it by considering the second law of thermodynamics, and the symmetries associated with each corresponding conservation laws. In this way, we obtain all the equations describing the whole dynamics of the two-component fluid. We clarify that the interface energy plays the important role in thermodynamics and dissolution. Our proposed method can be applied to various more complicated fluids, and yields the governing equations consistent with the conservation laws and thermodynamics.
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[4] Leslie A Guildner. The thermal conductivity of carbon dioxide in the region of the critical point. Proceedings of the National Academy of Sciences of the United States of America, 44(11):1149, 1958.