A model of local thermal non-equilibrium during infiltration

Abstract

A realistic temperature estimation is crucial for many earth-science applications, ranging from hydro-thermal systems to plant physiology. The most common approach to calculate the temperature in multi-phase systems assumes immediate local thermal equilibrium (LTE) between the phases. However, local thermal equilibrium between the phases is not applicable in various scenarios like during the infiltration of rain or melt water in frozen soil, limiting the applicability of the approach and inhibiting the implementation of separate initial and boundary conditions for non-equilibrium situations. In local thermal non-equilibrium (LTNE) models, phase temperatures are described separately to the cost of additional differential equations and an explicitly formulated heat transfer between the phases. Especially a cumbersome parameterization of the explicit heat transfer restricts the use of the LTNE models in multi-phase conditions so far. In this work, we derive a general local thermal non-equilibrium model for dynamic, partly saturated porous media. Heat transfer between the phases is described explicitly using well-known semi-empirical parameterization accounting for velocity changes of the mobile phases. The change in volume fraction introduces an additional term in the heat equation, causing a coupling with the hydraulic model. We validate our model with a numerical simulation of historic experimental data from soil infiltration experiments of warm and cold water into drained soil, posing a perfect example of local thermal non-equilibrium conditions between the phases. Experimentally obtained mixture temperatures are reproduced within experimental accuracy. We further show the benefits of our model by applying it to rainwater infiltration into cold soil. Besides a consistent formulation of initial and boundary conditions, the derived model allows physically based conclusions about the thermal state of the separate phases.