An Extension of the Stefan-Type Solution Method Applicable to Multi-component, Multi-phase 1D Systems

• Published in: Transport in porous media Vol. 117; no. 3; pp. 415 - 441
• Main Authors: Lewis, K. C., Coakley, Samuel, Miele, Sean
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.04.2017; Springer Nature B.V
• Subjects: Advection-diffusion equation; Brines; Civil Engineering; Classical and Continuum Physics; Computer simulation; Earth and Environmental Science; Earth Sciences; Economic models; Geotechnical Engineering & Applied Earth Sciences; Green's functions; Heat flux; Heat pipes; Hydrogeology; Hydrology/Water Resources; Industrial Chemistry/Chemical Engineering; Mathematical models; Multiphase; Phase separation; Saline water; Seawater; Temperature profiles; Thickness; Hydrothermal systems; Stefan problem; Heat pipe; Analytic solutions
• ISSN: 0169-3913; 1573-1634
• DOI: 10.1007/s11242-017-0840-1

We present an extension of the Stefan-type solution method applicable to multi-component, multi-phase 1D porous flows, and illustrate the method by applying it to phase separation dynamics in an NaCl– H 2 O -saturated hydrothermal heat pipe. For this example, three mathematical models are constructed. The first two models concern the rate of progression of two interfaces, one separating brine from two-phase fluid and another separating two-phase fluid from single-phase liquid at seawater salinity. The brine layer model shows that the layer may reach quasi-steady-state thickness even while the salt content of the layer continues to increase; the two-phase layer model shows how variable heat flux at the top of the layer leads to departure from the linear growth rate predicted by a simpler model. The third model concerns the temperature profile in the entire column. The governing advection–diffusion equation has highly variable coefficients, with no negligible terms in it in the region of parameter space considered. We present a method to solve this type of equation by constructing a propagator and a corresponding Green’s function. Finally, we show how to use the developed framework to test the internal consistency of numerical simulations, again using the 1D heat pipe as an example.