Un-stationary thermal analysis of the vertical ground heat exchanger within unsaturated soils
Introduction
Modelling heat transfer in lower heat source is one of the most important skills in predicting the period for which the heat source can be used, its efficiency or duration of individual cycles. An accurate calculation model is necessary in order to be able to correctly determine the individual loss components The quality of the calculation model is influenced by many factors, including: whether it presents the image of the desired parameters in a one-dimensional or three-dimensional manner [1], whether it describes the surrounding soil as a solid body or porous medium … Phase changes occur inside the ground, there is groundwater flow or the possibility of diffusion of gases, compound fluids, or an electromagnetic field exists [2].
When analysing the available models, those in which an additional mechanism, apart from the classic heat conduction in the ground, was diffusion movements caused by water migration should be emphasized [3]. The migration of groundwater was driven by the forces of the gravity or by the pressure difference. The applied model, due to the partial nature, did not include the transport of vapours and phase transitions of water permeating the soil, which had its reference in the curves verifying the model.
Considering the porous medium model with the multi-component fluid flows, it was found that the increase of the soil saturation, the assumed GHE loading time, and the mass flow rate of the working fluid affect the soil temperature changes significantly [4]. Additionally, it was noticed that regeneration of the low temperature heat source using the extremely enlarged heat flux leads to drying of the moist soil, and thus to deterioration of heat conduction properties.
Negligence of moisture changes may lead to an underestimation of the thermal capacity of the soil [5]. Elevation of the groundwater table had a beneficial effect on the operation of the GSHP, connected with the heat exchanger. They also emphasise that considering the transfer of the heat energy along with the groundwater flow, it is important to take into account the groundwater surface elevation. Its fluctuations can cause differences in the temperature indication of 3–4%. These values do not differ from those presented by the authors in this paper.
Advanced models [[6], [7], [8]] include also the convectional flow of the gas, the water vapour and the surface water. It was shown that approximately 83.3% of stored energy is stored in the medium until the end of the 6th day, and 10% of stored energy is being wasted as a result of a convective heat flow until the end of 10th day.
In the current paper the ground heat exchangers mounted in the unsaturated soils are being considered. The proposed calculation model is based on the finite volume method. It provides a three-dimensional description of the surrounding soil, treating it as a porous medium inside which there occur a diffusive movement of the gaseous and liquid components. There is a groundwater flow, conditioned by natural geological processes in the analysed plot of land- Jabłonna near Warsaw.
To solve the problem, a model describing Darcy flow was used with effects of freezing disregarded [[9], [10], [11]]. The air flow was modelled as an ideal gas.
For a more accurate representation of the temperature ragged nature measured by the sensors in the borehole, a new definition of total temperature was proposed. It was completed with an additional term describing both heat and mass diffusion in the element of the backfill material and the surrounding soil.
Section snippets
Physical model
The physical model presents a classic U-tube exchanger, installed in a hole drilled in the soil. It has two round fluid flow ducts, walls separating the channel from backfill material and soil forming the backfill. The detailed dimensions are presented in Fig. 1.
The backfill material and the liquid inside the ducts have constant physical properties, while the soil changes, depending on the depth at which the section of the exchanger was considered. The total height of the model is 80 m, and the
The temperature distribution of the GHE at the centre of the cross-section in the radial direction
Fig. 7 shows the temperature field read in the centre of the section of the considered soil fragment at the depth of 75 m. The temperature field gradually increases during loading of the exchanger, and then decreases during cooling.
After 8.5 h of charging - Fig. 7c - the area with the highest temperature does not exceed half of the considered soil area. It is also visible that the isotherms are clearly directed towards the right. A more pronounced migration of the temperature field is visible
Conclusions
The paper analyses the model for a heat exchanger installed in unsaturated soils. The ground porosity was modelled geometrically and mathematically, also taking the flow of groundwater in liquid and gaseous form into account. I suggested a definition of the total temperature, which takes into account the influence of diffusive mass transfer on the amount of heat accumulated. A 24-hours work time period of the ground heat exchanger was analysed numerically. The temperature value obtained for
CRediT author statement
Daniel Sławiński: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data Curation, Writing – Original Draft, Writing – Review and Editing, Visualization, Supervision, Project administration, Funding acquisition.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Nomenclature
Abbreviations
- CFD
- Computational Fluid Dynamics
- C–F-L
- Courant-Friedrich-Lewy condition
- FEM
- Finite Elements Method
- FVM
- Finite Volume Method
- RES
- Renewable Energy Source
- GSHP
- Ground-Source Heat Pumps
- TRT
- Thermal Response Test
- GHE
- Ground Heat Exchanger
- HP
- Heat Pump
- LTHB
- Low-Temperature Heat Buffer
Parameters
- T
- temperature (K)
- total temperature (K)
- cp
- specific heat at constant pressure (J/kg K)
- k
- thermal conductivity (W/m K)
- velocity fields [m/s]
- vector normal to surface
- diffusion flux for laminar flow (kg/m2)
- Pr
- Prandtl number [−]
- Re
- Reynolds
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