This paper describes a macroscopic continuum mechanics approach to formulate the cylindrical transport equations (continuity, momentum and energy) to study simultaneous heat and mass transfer in unsaturated porous materials when a line heat source is embedded in the medium. The coupled continuity, momentum and energy equations are summarized for the constituent species, namely, liquid water and gaseous phase (mixture of water vapor and air). The derived transport equations are applied to study heat transfer characteristics of a light clay bed coupled to a heat pump and to map the temperature profiles around a hot water pipe. It is assumed that no temperature gradient exists along the length of the pipe and that the three-dimensional equations are reduced to a two-dimensional set. Hydrodynamic laws such as Darcy's law and the Darcy-Buckingham theorem are utilized to simplify the continuity and momentum equations. Migration of liquid due to surface tension effects is modeled in the analysis. The effects of phase change on heat transfer are also included in the energy equation. The simplified two-dimensional equations are solved numerically using an explicit finite difference technique based upon the Runge-Kutta-Fehlberg method. The initial and boundary conditions applicable to the soil bed, along with some soil properties such as the aggregate thermal conductivity, are determined experimentally on site. The effects of various heat transfer processes as well as the motion of fluids on heat transfer in a clay bed coupled to a heat pump are discussed. Heat diffusion into the soil by conduction is shown to be predominant through the early stages of heating, while the liquid water motion contributes to heat transfer during the intermediate times and the motion of gaseous mixture is shown to become significant during later stages of drying. The contribution of the convective transport increases with the temperature and becomes equal to the contribution by conduction at moderately high temperatures. Formulation of these transport equations in a cylindrical coordinate system eliminates inaccuracies incorporated into the Cartesian equations and some accompanying simplifying assumptions reported previously. The results obtained with the current cylindrical governing equations are in closer agreement with the experimental results when compared with the results based upon the Cartesian system of equations. Therefore, the cylindrical modeling of the governing equations seems to improve the accuracy of predicting the temperature profiles around a buried hot water pipe in an unsaturated soil with a line heat source.
展开▼
