Search Results (12)

A transformation method was applied to the biconfluent (BHE), doubly confluent (DHE), and triconfluent (THE) Heun equations to generate and classify exactly solvable quantum mechanical potentials derived from them. With this, the range of potentials solvable in terms of the confluent hypergeometric function can be extended. The resulting potentials contained five independently tunable terms and two terms originating from the Schwartzian derivative that depended only on the parameters of the $z\left(x\right)$ transformation function. The polynomial solutions of these potentials contain expansion coefficients obtained from three-term (BHE and DHE) and four-term (THE) recurrence relations. For the simplest $z\left(x\right)$ transformation functions, the Lemieux–Bose potentials have been recovered for the BHE and DHE. The coupling parameters of these potentials and also of five potentials derived from the THE have been expressed in terms of the parameters of the respective differential equations. The present scheme offers a general framework into which a number of earlier results can be integrated in a systematic way. These include special cases of potentials obtained from less general versions of the Heun-type equations and individual solvable potentials obtained from various methods that do not necessarily refer to the Heun-type equations considered here. Several potentials derived here were found to coincide with or reduce to potentials found earlier from the quasi-exactly solvable (QES) formalism. Based on their mathematical form, their physically relevant features (domain of definition, asymptotic behaviour, single- or multi-well structure) were discussed, and possible fields of applications were pointed out. Full article

The increasingly severe energy crisis and associated environmental issues pose new challenges for the efficient and rational utilization of renewable energy. The solar-assisted ground-source heat pump (SAGSHP) system is a novel heating system that effectively combines the advantages of both solar and geothermal energy. In this study, an SAGSHP system was established through TRNSYS simulation software to provide winter heating and year-round domestic hot water for a residential building. By varying the area of solar collectors (A) and the number (n) and the depth (H) of the borehole heat exchangers (BHEs), the system operational performance, including the system energy consumption, ground temperature attenuation, and heat pump efficiency, was investigated. A comparison with a single ground-source heat pump (GSHP) system was also conducted. After 20 years of operation, the parameter optimization resulted in a reduction of approximately 60 MWh and 70 MWh in system energy consumption, equivalent to saving 7.37 t and 8.60 t of standard coal, respectively. At the same time, the total costs over 20 years can be reduced by 48.20% and 33.77%, respectively. The proposed design method and simulation results can serve as the reference for designing and analyzing the performance of the SAGSHP system. Full article

The storage of thermal energy within the ground serves as a method to balance irregular energy consumption for heating throughout the year. This principle revolves around the accumulation of thermal energy during the summer months, allowing for its utilization for heating buildings during the winter months. This paper focuses on the technique of storing heat energy in the ground, known as borehole thermal energy storage (BTES), via borehole heat exchangers (BHE), which are designed to harness shallow geothermal energy for heating and cooling purposes. The model of regenerating heat in rocks, after subcooling of the ground in winter months, could be conducted by storing solar energy using a panel collector. The method of solar heat regeneration on a real building with a high number of BHEs was analyzed, with special attention on certain restrictions. In climates such as northern Croatia with cold winters and warm to hot summers, where besides heating loads there are certain cooling loads present, the implementation of this ground temperature regeneration method on the cooling and heating efficiency of heat pumps was studied. This paper presents research on the possibility of using this field as a BTES system coupled with solar collectors in a climate with both heating and cooling loads present. Full article

In the process of development and utilization of a large-scale borehole heat exchanger (BHE) array system, the phenomenon of load shifting within BHE array can be observed. In this paper, OpenGeoSys software coupled with TESPy toolkit is used to establish a comprehensive numerical model of BHE system (without depicting the heat pump part), and the behaviors of load shifting between BHEs with different design parameters are studied. The results show that the outlet temperature of single BHE and BHE array is generally rising, and the soil temperature around the BHE has accumulated unbalanced heat. The soil temperature near the BHEs array fluctuates more obviously than the single BHE system, and the distribution is uneven. At the end of the 15th year, the soil temperature near the center BHE increased by 2 °C compared with the initial soil temperature, which was more favorable in winter, but was not conducive to the performance improvement in summer. Further analysis by changing the inter-borehole spacing shows that with the increase of the inter-borehole spacing, the load shifting behaviors are gradually weakened, and the maximum shifted load of the central BHE is linear with the change of the inter-borehole spacing. After changing the layout methods, we observe that the more intensive the layout is, the more load shifting behavior is, and the unbalanced rate of soil temperature distribution around the linear layout is lower than other layouts. With the increase in the number of BHEs, the load shifting behaviors are further enhanced. By analyzing the proportion of shifted load amount relative to the average value, it is found that the system will take a longer time to reach heat balance with the increase of BHEs’ number. A shutdown of part of BHEs for a certain period of time will help to improve the long-term operational efficiency of the large-scale shallow ground source heat pump (GSHP) system. Full article

Geothermal heat pump systems (GHP) are the spearhead of geothermal achievement and development, and one of the fastest growing applications of renewable energy technologies worldwide. When Swiss GHP activities started in the late 1970s, market introduction and penetration needed science-based proof of reliable, stable, long-term GHP operation. A special, extended project, realized in a field-laboratory setting, provided this proof. Detailed measurements, as well as numerical model simulations, proved the sustainable operation of the installed GHP system. The measurement setup, the recording of the various time series, and their interpretation are presented. Furthermore, basic perceptions were elaborated concerning geothermal resources behavior in production and regeneration. The Swiss GHP was developed from nothing. Early GHP installation costs halved within 20 years; GHP growth was nearly exponential from 1980 to 2020 (8.5% annually). Drilled borehole heat exchanger (BHE) meters are today around 300,000 m per year; heat delivery of GHPs in Switzerland amounted to 3280 GWh 2020—over 85% of Swiss geothermal direct uses (among others like thermal spas, district heating). Large installations with hundreds of BHEs are now common, and are also used for heating and cooling. The international ranking of Swiss GHP realizations is excellent in terms of annual energy use (TJ/yr/area), and is number one worldwide. Switzerland is a global GHP leader, and the Swiss success story is well documented. Full article

With the increasing engineering applications of geothermal borehole heat exchangers (BHEs), accurate and reliable mathematical models can help advance their thermal design and operations. In this study, an analytical model with a time-dependent heat flux boundary condition on the borehole wall is developed, capable of predicting the thermal performance of single, double, and multiple closed-loop BHEs, with an emphasis on solar- and waste-heat-assisted geothermal borehole systems (S-GBS and W-GBS) for energy storage. This analytical framework begins with a one-dimensional transient heat conduction problem subjected to a time-dependent heat flux for a single borehole. The single borehole scenario is then extended to multiple boreholes by exploiting lines of symmetry (or thermal superposition). A final expression of the temperature distribution along the center line is attained for single, double, and multiple boreholes, which is verified with a two-dimensional finite-element numerical model (less than 0.7% mean absolute deviation) and uses much lesser computational power and time. The analytical solution is also validated against a field-scale experiment from the literature regarding the borehole and ground temperatures at different time frames, with an absolute error below 6.3%. Further, the thermal performance of S-GBS and W-GBS is compared for a 3-by-3 borehole configuration using the analytical model to ensure its versatility in thermal energy storage. It is concluded that our proposed analytical framework can rapidly evaluate closed-loop geothermal BHEs, regardless of the numbers of boreholes and the type of the heat flux on the borehole wall. Full article

The objective of this study is to assess the suitability of the analytical infinite moving line source (MLS) model in determining the temperature of vertical grouted borehole heat exchangers (BHEs) for steady-state conditions when horizontal groundwater advection is present. Therefore, a numerical model of a grouted borehole is used as a virtual reality for further analysis. As a result of the first analysis, it has been discovered that established analytical methods to determine the borehole thermal resistance as a mean value over the borehole radius can also be applied to BHEs with groundwater advection. Furthermore, the deviation between a finite MLS and the infinite MLS is found to be only less than 5% for BHEs of a depth of 30 m or more, and Péclet numbers greater than 0.05. Finally, the accuracy of the temperature change calculated with the infinite MLS model at the radius of the borehole wall compared to the temperature change at a numerically simulated grouted borehole is addressed. A discrepancy of the g-functions resulting in a poor dimensioning of BHEs by the infinite MLS model is revealed, which is ascribed to the impermeable grouting material of the numerical model. A correction function has been developed and applied to the infinite MLS model for steady-state conditions to overcome this discrepancy and to avoid poor dimensioning of BHEs. Full article

Krakow has an extensive district heating network, which is approximately 900 km long. It is the second largest city in terms of the number of inhabitants in Poland, resulting in a high demand for energy—for both heating and cooling. The district heating of the city is based on coal. The paper presents the conception of using the available renewable sources to integrate them into the city’s heating system, increasing the flexibility of the system and its decentralization. An innovative solution of the use of hybrid, modular heat pumps with power dependent on the needs of customers in a given location and combining them with geothermal waters and photovoltaics is presented. The potential of deep geothermal waters is based on two reservoirs built of carbonate rocks, namely Devonian and Upper Jurassic, which mainly consist of dolomite and limestone. The theoretical potential of water intake equal to the nominal heating capacity of a geothermal installation is estimated at 3.3 and 2.0 MW, respectively. Shallow geothermal energy potential varies within the city, reflecting the complex geological structure of the city. Apart from typical borehole heat exchangers (BHEs), the shallower water levels may represent a significant potential source for both heating and cooling by means of water heat pumps. For the heating network, it has been proposed to use modular heat pumps with hybrid sources, which will allow for the flexible development of the network in places previously unavailable or unprofitable. In the case of balancing production and demand, a photovoltaic installation can be an effective and sufficient source of electricity that will cover the annual electricity demand generated by the heat pump installation, when it is used for both heating and cooling. The alternating demand of facilities for heating and cooling energy, caused by changes in the seasons, suggests potential for using seasonal cold and heat storage. Full article

This study determined the required lengths of borehole heat exchangers (BHEs) in ground-source heat pump systems for heating/cooling a building (with 300 m² of floor area) across Japan’s four main islands through a simulation approach. Hourly thermal loads were estimated in 10 km gridded cells based on the outside temperature and humidity. Three-dimensional estimates of ground thermal conductivity from our previous study at the depths of the BHEs were used. A 5-year system operation was simulated in a total of 4059 cells with 81 combinations of individual lengths and total numbers of BHEs to determine the shortest total length required to achieve sustainable use and targeted performance. The optimal combination of individual length and total number varied regionally due to climate conditions and locally among adjacent cells due to geological conditions. The total required lengths ranged widely from 78 to 1782 m. However, the lengths were less than 400 m in 85% of the cells. Additionally, cost-effectiveness in 69% of the cells was shown by reducing the total lengths to half or less of those in the practical method. The reduction could potentially increase the feasibility of heat pump system use in Japan. The total lengths were dependent on the heating/cooling loads approximately as secondary-polynomial functions, but the relations with the ground thermal conductivity were not clear. Full article

For reducing the initial GSHP investment, the heat transfer efficiency of the borehole heat exchange (BHE) system can be enhanced to reduce the number or depth of drilling. This paper proposes a novel and simple BHE design by changing the cross-sectional shape of the U-tube to increase the heat transfer efficiency of BHEs. Specifically, in this study, we (1) verified the reliability of the three-dimensional numerical model based on the thermal response test (TRT) and (2) compared the inlet and outlet temperatures of the different U-tubes at 48 h under the premise of constant leg distance and fluid area. Referent to the circular tube, the increases in the heat exchange efficiencies of the curved oval tube, flat oval tube, semicircle tube, and sector tube were 13.0%, 19.1%, 9.4%, and 14.8%, respectively. (3) The heat flux heterogeneity of the tubes on the inlet and outlet sides of the BHE, in decreasing order, is flat oval, semicircle, curved oval, sector, and circle shapes. (4) The temperature heterogeneity of the borehole wall in the BHE in decreasing order is circle, sector, curved oval, flat oval, and semicircle shapes. (5) Under the premise of maximum leg distance, referent to the heat resistance of the tube with a circle shape at 48 h, the heat exchange efficiency of the curved oval, flat oval, semicircle, and sector tubes increased 12.6%, 17.7%, 10.3%, and 7.8%, respectively. (6) We found that the adjustments of the leg distance and the tube shape affect the heat resistance by about 25% and 12%, respectively. (7) The flat-oval-shaped tube at the maximum leg distance was found to be the best tube design for BHEs. Full article

This study proposes a simple ground heat exchanger design capacity that is applicable in the schematic-design stage for several configurations used for borehole heat exchangers (BHEs). Three configurations—single, compact, and irregular types—were selected, and the heat transfer rate per unit BHE was calculated considering heat interference. In a case study with a typical configuration and general range of ground thermal conductivity, the BHE heat transfer rate of the compact configuration decreased owing to heat interference as the number of BHEs increased. However, with respect to the irregular configuration, the heat transfer rate increased as the same number increased. This was attributed to the relatively large increment rate of the distance between the boreholes in the irregular configurations, making the heat recovery factor more dominant than the heat interference. The results show that the average heat transfer rate values per BHE applicable to each configuration type in the schematic-design stage were 12.1 kW for the single configuration, 5.8 kW for the compact type, and 10.3 kW for the irregular configuration. However, owing to the large range of results for each case study, the error needs to be reduced by maximally utilizing the information available at the schematic-design stage. Full article

This work concerns borehole heat exchangers and their testing using apparatus for thermal response tests. In the theoretical part of the article, an equation was derived from the known equation of heat flow, on which the interpretation of the thermal response test was based. The practical part presents the results of several measurements taken in the AGH Laboratory of Geoenergetics. They were aimed at examining the potential heat exchange capacity between the heat carrier and rock mass. Measurement results in the form of graphs are shown in relation to the examined, briefly described wells. Result analysis made it possible to draw conclusions regarding the interpretation of the thermal response test. The method of averaging the measurement results was subjected to further study. The measuring apparatus recorded data at a frequency of one second, however such accuracy was too large to be analyzed efficiently. Therefore, an average of every 1 min, every 10 min, and every 60 min was proposed. The conclusions stemming from the differences in the values of effective thermal conductivity in the borehole heat exchanger, resulting from different data averaging, were described. In the case of three borehole heat exchangers, ground properties were identical. The effective thermal conductivity λ_eff was shown to depend on various borehole heat exchanger (BHE) designs, heat carrier flow geometry, and grout parameters. It is important to consider the position of the pipes relative to each other. As shown in the charts, the best (the highest) effective thermal conductivity λ_eff occurred in BHE-1 with a coaxial construction. At the same time, this value was closest to the theoretical value of thermal conductivity of rocks λ, determined on the basis of literature. The standard deviation and the coefficient of variation confirmed that the effective thermal conductivity λ_eff, calculated for different time intervals, showed little variation in value. The values of effective thermal conductivity λ_eff for each time interval for the same borehole exchanger were similar in value. The lowest values of effective thermal conductivity λ_eff most often appeared for analysis with averaging every 60 min, and the highest—for analysis with averaging every 1 min. For safety reasons, when designing (number of BHEs), safer values should be taken for analysis, i.e., lower, averaging every 60 min. Full article