Abstract
In the present work, a network model for the numerical resolution of the heat transport problem in porous media coupled with a water flow is presented. Starting from the governing equations, both for 1D and 2D geometries, an equivalent electrical circuit is obtained after their spatial discretization, so that each term or addend of the differential equation is represented by an electrical device: voltage source, capacitor, resistor or voltage-controlled current source. To make this possible, it is necessary to establish an analogy between the real physical variables of the problem and the electrical ones, that is: temperature of the medium and voltage at the nodes of the network model. The resolution of the electrical circuit, by means of the different circuit resolution codes available today, provides, in a fast, simple and precise way, the exact solution of the temperature field in the medium, which is usually represented by abaci with temperature-depth profiles. At the end of the article, a series of applications allow, on the one hand, to verify the precision of the numerical tool by comparison with existing analytical solutions and, on the other, to show the power of calculation and representation of solutions of the network models presented, both for problems in 1D domains, typical of scenarios with vertical flows, and for 2D scenarios with regional flow.
Highlights
The topic of heat transport coupled to water flow in porous media has aroused great interest among researchers in recent decades [1,2], finding in the literature a great variety of temperature-depth profiles, both for regional [3,4] and exclusively vertical flows [5,6]and for a wide variety of cases in the soil surface temperature condition [7,8,9,10,11]
Koch et al [34] present a very similar network model that includes the possibility of considering different fluid and solid phase temperatures. This situation of thermal non-equilibrium at the local level is typical of phenomena associated with chemical reactions, evaporation or heat/cold injections which do not occur in the groundwater flow scenarios that we address in this work
It is important to note that, since the thermal characteristics of the material are included in the expressions of the electrical devices that make up each elementary cell, with the network method it would be possible to model and simulate non-homogenous porous media, since this technique allows designing circuits with cells of different properties, which would be implemented through simple programming routines
Summary
The topic of heat transport coupled to water flow in porous media has aroused great interest among researchers in recent decades [1,2], finding in the literature a great variety of temperature-depth profiles, both for regional [3,4] and exclusively vertical flows [5,6]. The method has two phases: (i) the elaboration of a network model (electrical circuit equivalent to the process) and (ii) the simulation of the problem (obtaining the results of the network model) by means of a suitable program that allows the resolution of electrical circuits [21,22] This technique has been successfully used in the simulation of a wide variety of problems in different fields of engineering, such as electrochemistry [23,24], elasticity [25], heat transfer [26] and tribology [27], providing accurate solutions in all cases. And after verifying the results obtained with our tool with the existing solutions in the reference literature [7,8,9,10], a series of selected applications (both for regional flow and exclusively vertical upflow) will be simulated in which different functions (dependent on time) will be assumed for the boundary condition of the soil surface temperature, obtaining the temperature-depth profiles once the stationary situation has been reached
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