Geothermal energy known as a clean, renewable energy resource is widely available and reliable. Ground heat exchangers (GHEs) can assist the development of geothermal energy by reducing the capital cost and greenhouse gas emission. In this paper, a novel semianalytical method was developed to study the thermal performance of multiborehole ground heat exchangers (GHEs) with arbitrary configurations. By assuming a uniform inlet fluid temperature (UIFT), instead of uniform heat flux (UHF), the effects of thermal interference and the thermal performance difference between different boreholes can be examined. Simulation results indicate that the monthly average outlet fluid temperatures of GHEs will increase gradually while the annual cooling load of the GHEs is greater than the annual heating load. Besides, two mechanisms, the thermal dissipation and the heat storage effect, will determine the heat transfer underground, which can be further divided into four stages. Moreover, some boreholes will be malfunctioned; that is, boreholes can absorb heat from ground when the GHEs are under the cooling mode. However, as indicated by further investigations, this malfunction can be avoided by increasing borehole spacing.
Geothermal energy is attractive due to its enormous potential, renewability, availability, and low gas emission. With improvements in drilling, completion, and energy conversion systems, geothermal energy is becoming an economically viable alternative. Nonetheless, many challenges remain. The development of geothermal resource may be impeded by high capital requirements. Produced groundwater needs to be reinjected rather than disposing to surface waters to avoid environmental impacts, which raises the operational difficulties and maintaining cost. The ground heat exchangers (GHEs) such as geothermal heat pumps can reduce the cost effectively, and the close-loop system will eliminate the necessary of any reinjections (no groundwater produced). The ground can offer a steady and large heat storage medium as a heat source/sink and for thermal energy utilization, such as geothermal heat pumps. As one of the main geothermal heat pump technologies, the ground source heat pump (GSHP) has been widely used as a viable and economical alternative to traditional air conditioning systems owing to its high-efficient performance in the world [
Many methods, such as analytical/semianalytical, numerical method, and fractal methods [
For the heat transfer inside the borehole, a steady-state process is usually approximated using 1D, 2D, or quasi-3D models [
Although the numerical simulations are flexible, they require significantly more computational time, which is not efficient for practical applications. In contrast, the semianalytical methods are more convenient and have been widely used in practice [
However, an unreasonable assumption with uniform heat flux (UHF) was adopted for these semianalytical methods mentioned above [
Schematic for multiborehole GHEs.
Based on these findings, the main objective of this paper is to develop a semianalytical solution to simulate the underground heat transfer of multiborehole GHEs with the UIFT assumption. The new method can be used for simulating the performance of the GHEs with arbitrary configurations. Furthermore, the effects of the heat capacity of each borehole and the thermal interference between different boreholes on the thermal performance of GHEs are also examined. Overall, this paper is organized as follows: in Section
In this section, the basic thermal response equation for single borehole will be presented followed by the development of heat transfer model for multiple boreholes as well as the corresponding semianalytical solution.
The following assumptions are made.
With the above assumptions, the thermal response at time
Combining (
Substituting (
It is assumed that the GHEs consist of
The inlet temperature
By applying the superposition principle to (
At time
There are
It is worth mentioning that (
Furthermore, (
Then, (
In particular, at time
In this section, we first validate the new semianalytical method by comparing with the results from a numerical method [
In [
In order to examine the effects of multiple boreholes, Zhang [
Figure
Simulation input data [
Borehole spacing | 4.572 m |
Borehole diameter | 0.1524 m |
Ground soil density | 2691.36 kg/m3 |
Borehole wall temperature | 37.78°C |
Undisturbed ground temperature | 22.22°C |
Ground thermal conductivity | 2.422 W/(m·K) |
Comparison results of thermal effectiveness for a 4 × 4 borehole field [
The second comparisons are based on an experimental system at Oklahoma State University [
Experimental data [
Borehole depth | 75 m |
Borehole spacing | 9 m |
Borehole diameter | 114 mm |
U-tube inner diameter | 21.8 mm |
U-tube outer diameter | 26.7 mm |
Undisturbed ground temperature | 17.3°C |
Ground thermal conductivity | 2.3 W/m-K |
Ground volume specific heat | 2012 kJ/m3-K |
Borehole thermal resistance | 0.1622 m-K/W |
Fluid flow rate | 0.631 kg/s |
(a) Monthly average heat extraction/rejection rates; (b) comparison results of monthly average outlet fluid temperature [
In this section, the new method will be used to study the thermal performance of 9-borehole GHEs with different configurations under varying heat fluxes. In particular, the effects of thermal interference and heat storage underground are discussed in detail. The monthly average heat fluxes used in simulation are displayed in Table
Monthly average heat extraction/rejection rates.
Month | Heat rate (kW) |
---|---|
Jan | −45 |
Feb | −34.2 |
Mar | −9 |
Apr | 4.5 |
May | 16.2 |
Jun | 27 |
Jul | 36.9 |
Aug | 63.9 |
Sep | 27.9 |
Oct | 2.7 |
Nov | −13.5 |
Dec | −32.4 |
Simulation input data.
Borehole depth | 100 m |
Borehole spacing | 5 m |
Borehole diameter | 126 mm |
Fluid mass flow rate | 2.651 kg/s |
Undisturbed ground temperature | 18°C |
Ground thermal conductivity | 2 W/m-K |
Ground heat capacity | 2000 kJ/m3-K |
Borehole thermal resistance | 0.12 m-K/W |
We first investigate the thermal performance of 3 × 3 square array GHEs (Figure
Configuration of GHEs with 3 × 3 square borehole array.
Monthly average outlet fluid temperature.
There are two important aspects for the heat transfer of GHEs underground. One is that GHEs will reject heat into ground or extract heat from ground, whose efficiency is determined by the thermal dissipation underground. The other is the extracted or rejected heat can be stored underground, which can be utilized by the GHEs later. Figure
Monthly average heat fluxes for different boreholes (spacing = 5 m).
From the above discussion, we can conclude that while the thermal dissipation dominates, the heat flux of the borehole with stronger thermal interference is smaller. In contrast, while the heat storage effect plays a major role, the boreholes being strongly interfered will have a greater heat flux. This is because the hindrance of thermal dissipation around boreholes due to thermal interference provides a better condition while the ground acts as a heat source/sink for the GHEs working under the opposite mode.
Furthermore, we should note that, in October (circle in Figure
Monthly average heat fluxes for different boreholes (spacing = 4 m).
Monthly average heat fluxes for different boreholes (spacing = 7 m).
In this section, we investigate thermal behaviors of 9-borehole GHEs arranged in “<”-Shape with different angles,
Schematic of 9-borehole GHEs in “<”-Shape (a)
Figure
Average outlet fluid temperatures in 10 years.
Case |
|
|
---|---|---|
3 × 3 square array | 46.3°C | 5.68°C |
|
45.9°C | 4.65°C |
|
45.4°C | 4.18°C |
|
45.3°C | 3.97°C |
|
45.2°C | 3.92°C |
Monthly average GHEs outlet fluid temperature (
Figure
Monthly average heat fluxes for different boreholes (
Monthly average heat fluxes for different boreholes (
In this paper, a novel semianalytical method to study the thermal performance of (GHEs) is developed with uniform inlet fluid temperature (UIFT) assumptions. This new method is verified against numerical and experimental results. The thermal performances of 9-borehole GHEs with different configurations are also analyzed by this new method. Based on this work, several important conclusions are obtained as follows.
Matrix coefficient
Vector coefficient
Specific heat
Specific heat of fluid
Thermal effectiveness
Exponential integral function
Depth of borehole, m
Mass flow rate of fluid
Time
Counter
Total time step
Number of boreholes
Thermal load/heat fluxes
Is the heat flow produced by single isolated borehole, W/m
Heat rate of GHEs
The space between the
Borehole radius, mm
Thermal resistance in borehole
Coefficient, =
Initial ground temperature, °C
The average fluid temperature
Thermal response under unity heat flux
The inlet fluid temperature
Outlet fluid temperature
Location of borehole
Volume specific heat
Heat flux vector.
Coefficient
Thermal conductivity, W/(m-K)
Density, kg/m3
Angle.
The authors declare that they have no conflicts of interest.
This work was supported by the National Natural Science Foundation of China no. 51674227 and the Fundamental Research Funds for the Central Universities.