# Heat Transfer Modeling on High-Temperature Charging and Discharging of Deep Borehole Heat Exchanger with Transient Strong Heat Flux

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## Abstract

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## 1. Introduction

^{3}/h. The simulation settings resulted in weak heat flux along the borehole depth and minor heat extraction output; thus, it is relatively not so challenging for simulation since extreme heat transfer would not occur in this scenario. Kristian and Bastian also investigated the heat storage performance of DBHE [9,10], but the dynamic charging and discharging process of a deep BTES system has not been fully illuminated, especially the transient evolution details of heat flux along the borehole depth at the initial unsteady charging or discharging stage. Large heat exchange temperature differences emerge during the high-temperature charging and subsequent discharging of actual deep BTES systems. Such a large heat exchange temperature difference undoubtedly leads to transient strong heat flux released into or extracted from the rock mass near the borehole. Therefore, the thermal performance of DBHE would evolve drastically throughout the whole operation. Despite the great efforts devoted to the heat transfer modeling of deep BTES systems [4,9,10], these models can only simulate DBHEs under near-constant heat extraction or a release rate with relatively weak heat flux. Existing methods can hardly handle this problem and are subject to temperature oscillation or even simulation failure [11]. It is imperative to develop heat transfer models that are applicable for actual deep BTES systems with strong heat flux.

## 2. Materials and Methods

#### 2.1. Transient Heat Transfer Model for DBHE

#### 2.1.1. Model Assumptions

#### 2.1.2. Unsteady Heat Transfer Outside the Borehole

#### 2.1.3. Quasi-Steady State Heat Transfer Inside the Borehole

- Heat storage mode for charging phase

- Heat extraction mode for discharging phase

#### 2.2. High-Temperature Solar Collector Model

^{2}; ${A}_{c}$ stands for the collector area, m

^{2}; ${F}_{R}$is the heat migration factor; ${\left(\tau \alpha \right)}_{e}$represents the product of effective transmission ratio and absorption ratio; ${U}_{l}$is the heat loss coefficient, W/(m

^{2}∙°C); and ${t}_{ci}$ and${t}_{a}$are the inlet water temperature and outdoor air temperature, respectively, °C.

## 3. Results

#### 3.1. Model Validation Based on a Typical High-Temperature Heat Storage Case from the Literature

^{3}/h. It was determined based on a combined optimization of the heat exchange capacity of deep BTES and energy consumption of the circulating pumps to achieve efficient heat storage and supply.

#### 3.1.1. Simulation Accuracy of the Model

#### 3.1.2. Simulation Efficiency of the Model

#### 3.2. Model Validation against In Situ Field Test of a Demonstration Project

#### 3.2.1. Project Overview

^{2}with the heating demand of 50 W/m

^{2}, and the total heating load sums up to be 1200 kW. During the operation for heating, it is expected to provide a maximum water supply temperature of 50 °C for the building terminal. According to the geothermal exploration data (including geothermal temperature gradient, rock layers and fracture development underground, etc.) in Qingdao area, it shows that the average geothermal temperature gradient of the borehole is 2.8 °C/100 m, which is lower than the expected 3 °C/100 m. In view of the geothermal condition, the coaxial heat exchanger is installed in the deep borehole with a total depth of 2600 m and a borehole diameter of 216 mm. The temperature of the surrounding rock layer is about 85 °C at the bottom of the borehole. The deep borehole heat exchanger takes water as a heat carrier, and its detailed design parameters are summarized in Table 5, Table 6 and Table 7.

#### 3.2.2. In Situ Field Test of the Project under Operation

#### 3.2.3. Model Validation during Charging

#### 3.2.4. Model Validation during Discharging

#### 3.2.5. Simulation Efficiency of the Model

## 4. Discussion

#### 4.1. Advantages of the Heat Transfer Model for Deep BTES

- Generic applicability: The heat transfer model was developed to simulate both heat storage and heat extraction conditions of deep BTES. On one hand, this model couples unsteady heat conduction in the rock and quasi-steady heat transfer inside the deep borehole. The transient charging and discharging of DBHE could be simulated by minorly altering the flow circulation direction inside the borehole, as depicted in Figure 3, while heat conduction in the rock zone remains unchanged. Since the formulations including heat front propagation and heat flux density distribution along the depth are universal, therefore, they could be described in a generic way under charging or discharging of the deep BTES system. On the other hand, this modeling approach performs a heat transfer analysis of the DBHE underground and solar collector in Section 2.1 and Section 2.2 separately. The two major components represent storage and source in a typical energy system, and the corresponding mathematical formulations can be generalized for arbitrary deep BTES simulation with further improvement.
- Good accuracy: Through cross validation of the model in the typical high-temperature heat storage case, as well as the pilot demonstration project, large deviations are shown to mainly exist at the unsteady stage during the charging or discharging of DBHE, as shown in Table 3 and Table 10, while the simulation results are in excellent agreement with the OGS solution and field test data at the steady stage, according to Figure 8, Figure 9 and Figure 10. The prediction errors during both the charging and discharging phases are all around 5%, and they satisfy the accuracy requirement of engineering applications. Moreover, this model succeeded in depicting the dynamic evolution of extreme heat flux density and outlet water temperature in the pilot demonstration project, which was validated well against the OGS solution, as observed in Figure 11 and Figure 12. It could be seen that the large prediction error of the heat transfer model only existed in very short operation days during the initial unsteady stages of charging and discharging. Both relative errors under charging and discharging phases are within 5% during the steady state period. Hence, this model can simulate the challenging heat transfer problem with good accuracy.
- Easy implementation: No complex computations are involved in the overall simulation, as summarized in the flow chart of Figure 7. All the formulations are concise, without time-consuming iterations, which can be easily implemented by manual coding. In addition, this model could also be integrated into a HVAC system simulator such as TRNSYS or Energy-plus toolkits.
- High efficiency: Transient strong heat flux would appear during high-temperature charging and the subsequent discharging of DBHE. Extremely refined spatiotemporal resolution is required to capture the intense heat transfer, especially at the initial start-up. All the key physical laws of heat conduction in rock are formulated based on analytical solutions. Therefore, this model is featured with high computation efficiency in essence. Table 4 and Table 13 compare the simulation cost of both the charging and discharging phases of the deep BTES system, and demonstrate that our efficient heat transfer model achieves an acceleration ratio of 30 times, approximately relative to the fully numerical method.

#### 4.2. Limitations of the Heat Transfer Model for Deep BTES

## 5. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Nomenclature | |

BHE | Borehole heat exchanger |

DBHE | Deep borehole heat exchanger |

BTES | Borehole thermal energy storage |

OGS | Open GeoSys |

notation | |

$\mathit{z}$ | Depth of borehole |

$\mathit{\tau}$ | Operation time of deep borehole heat exchanger |

${\mathit{a}}_{\mathit{s}}$ | Thermal diffusion coefficient |

${r}_{b}$ | Radius of borehole |

$\mathit{H}$ | Drilling depth of borehole |

${\mathit{r}}_{\infty}$ | Thermal affecting radius |

${\mathit{T}}_{\mathit{i}\mathit{n}\mathit{i}\mathit{t}}$ | Initial rock temperature distribution |

${\mathit{T}}_{\mathit{s}}$ | Rock temperature field |

$\mathit{f}\mathit{l}\mathit{u}{\mathit{x}}_{\mathit{r}\mathit{a}\mathit{d}\mathit{i}}$ | Radial heat flux density along depth of borehole |

$\mathit{F}\mathit{l}\mathit{u}\mathit{x}$ | Cumulative heat released into or extracted from the rock segment |

$\dot{\mathit{m}}$ | Circulating flow rate |

${\mathit{t}}_{\mathit{i}\mathit{n}}$ | Inlet temperature of circulating water |

${\mathit{t}}_{\mathit{o}\mathit{u}\mathit{t}}$ | Outlet temperature of circulating water |

${\mathit{T}}_{\mathit{f}1}$ | Flow temperature distribution along annulus of deep borehole heat exchanger |

${\mathit{T}}_{\mathit{f}2}$ | Flow temperature distribution along inner pipe of deep borehole heat exchanger |

${\mathit{Q}}_{\mathit{s}}$ | Total heat storage amount during charging of deep borehole heat exchanger |

${\mathit{Q}}_{\mathit{e}}$ | Total heat extraction output during discharging of deep borehole heat exchanger |

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**Figure 8.**Comparison of high-temperature heat storage performance of DBHE based on the proposed model, OGS simulation results and literature data: (

**a**) evolution of outlet temperature; (

**b**) evolution of heat exchange output.

**Figure 9.**Evolution of inlet and outlet temperature for the deep BTES system during charging phase from in situ field test and simulation results of the model.

**Figure 10.**Evolution of inlet and outlet temperature for the deep BTES system during discharging phase from in situ field test and simulation results of the model.

**Figure 11.**Comparison of simulation results of heat flux density and circulating water temperature distribution along borehole depth during the charging period: (

**a**) 5 days; (

**b**) 10 days; (

**c**) 15 days; (

**d**) 20 days; (

**e**) 25 days; (

**f**) 30 days.

**Figure 12.**Comparison of simulation results of heat flux density and circulating water temperature distribution along borehole depth during the discharging period: (

**a**) 1 days; (

**b**) 5 days; (

**c**) 10 days; (

**d**) 20 days; (

**e**) 30 days; (

**f**) 40 days; (

**g**) 50 days; (

**h**) 120 days.

Geological Parameters | Borehole Heat Exchanger Parameters | ||
---|---|---|---|

Parameter | Value | Parameter | Value |

Thermal conductivity of rock | 2.6 W/m/K | Borehole radius | 152.2 mm |

Bulk heat capacity of rock | 2.3 MJ/m^{3}/K | Inner diameter of annulus | 127 mm |

Thermal conductivity of geofluid | 0.65 W/m/K | Annulus wall thickness | 5.6 mm |

Bulk heat capacity of geofluid | 4.2 MJ/m^{3}/K | Inner diameter of inner pipe | 75 mm |

Porosity of rock | 0.01 | Inner pipe wall thickness | 6.8 mm |

Subsurface temperature | 10 °C | Thermal conductivity of annulus (steel pipe) | 54 W/m/K |

Geothermal temperature gradient | 30 °C/km | Thermal conductivity of inner pipe (PE pipe) | 0.4 W/m/K |

Permeability coefficient | 10^{−8} m/s | Thermal conductivity of backfill | 2 W/m/K |

Hydraulic gradient in the rock | 0 | Bulk heat capacity of heat carrier (water) | 4.12 MJ/m^{3}/K |

Model length | 400 m | Thermal conductivity of heat carrier (water) | 0.65 W/m/K |

Model width | 400 m | Dynamic viscosity coefficient of heat carrier (water) | 5.04 × 10^{−4} kg/m/s |

Model depth | 2000 m | Density of heat carrier (water) | 977 kg/m^{3} |

Literature Data | OGS Solution | Result of This Model | |
---|---|---|---|

Total heat storage (MWh) | 665.68 | 653.16 | 701.48 |

Total heat extraction (MWh) | 103.33 | 112.88 | 78.17 |

Average heat storage rate (kW) | 152.4 | 149.53 | 160.6 |

Average heat extraction rate (kW) | 23.65 | 25.84 | 17.89 |

Average heat flux density during charging (W/m) | 76.2 | 74.77 | 80.3 |

Average heat flux density during discharging (W/m) | 11.83 | 12.92 | 8.95 |

Heat storage efficiency | 15.52% | 17.28% | 11.14% |

**Table 3.**Comparison of outlet temperature during the unsteady stage of charging and discharging phases for the typical high-temperature storage case.

Literature Data | OGS Solution | Result of This Model | |
---|---|---|---|

Average outlet temperature during charging (°C) | 71.03 | 71.85 | 68.7 |

Average outlet temperature during discharging (°C) | 44.45 | 45.31 | 41.99 |

Relative error during charging | - | 1.15% | 3.28% |

Relative error during discharging | - | 1.93% | 5.53% |

Simulation Cost | OGS Software | This Model |
---|---|---|

Simulation for charging phase | 7.51 h | 0.25 h |

Simulation for discharging phase | 7.7 h | 0.26 h |

Parameter | Value |
---|---|

Surface temperature (°C) | 15 |

Geothermal temperature gradient (°C/km) | 28 |

Lithological stratifications | 5 |

Groundwater conditions | Without ground water advection |

Rock Layers | Rock Type | Depth (m) | Thermal Conductivity (W/m/K) | Specific Heat Capacity (J/kg/K) | Density (kg/m ^{3}) |
---|---|---|---|---|---|

Layer 1 | Soil | 0~100 | 1.16 | 840 | 1500 |

Layer 2 | Basalt | 100~1985 | 2.78 | 925 | 2800 |

Layer 3 | Sandstone | 1985~2410 | 2.8 | 920 | 2800 |

Layer 4 | Limestone | 2410~2470 | 2.75 | 900 | 2780 |

Layer 5 | Sandstone | 2470~2600 | 2.8 | 920 | 2800 |

Design Parameters | Value |
---|---|

Drilling depth of DBHE (m) | 2600 |

Inner pipe diameter of DBHE (mm) | 90/110 |

Outer pipe diameter of DBHE (mm) | 159/178 |

Borehole diameter of DBHE (mm) | 216 |

Solar collector area (m^{2}) | 25 |

Operational Parameters | Value |
---|---|

Charging time (days) | 30 |

Discharging time (days) | 120 |

Flow rate during charging (m^{3}/h) | 20 |

Flow rate during discharging (m^{3}/h) | 50 |

Solar radiation intensity (W/m^{2}) | 550~850 |

Operation hours per day (hour) | 24 |

Charging Time (day) | OGS Solution (W/m) | Result of This Model (W/m) | Relative Error | |
---|---|---|---|---|

Average Error | Maximum Error | |||

5 | 242.16 | 253.94 | 4.61% | 13.72% |

10 | 182.02 | 188.49 | 3.65% | 8.71% |

15 | 179.19 | 183.32 | 2.72% | 4.13% |

20 | 135.04 | 136.55 | 1.81% | 4.36% |

25 | 120.18 | 121.56 | 1.8% | 4.35% |

30 | 111.0 | 113.47 | 2.75% | 4.78% |

**Table 10.**Comparison of outlet temperature during the unsteady stage of charging and discharging phases.

Field Test | OGS Solution | Result of This Model | |
---|---|---|---|

Average outlet temperature during charging (°C) | 69.29 | 74.38 | 73.72 |

Average outlet temperature during discharging (°C) | 48.35 | 41.35 | 39.64 |

Relative error during charging | - | 7.34% | 6.39% |

Relative error during discharging | - | 14.47% | 18.01% |

**Table 11.**Comparison of outlet temperature during the steady stage of charging and discharging phases.

Field Test | OGS Solution | Result of This Model | |
---|---|---|---|

Average outlet temperature during charging (°C) | 79.31 | 84.51 | 84.07 |

Average outlet temperature during discharging (°C) | 10.75 | 10.77 | 10.70 |

Relative error during charging | - | 6.55% | 6% |

Relative error during discharging | - | 0.18% | 0.46% |

Discharging Time (day) | OGS Solution (W/m) | Result of This Model (W/m) | Relative Error | |
---|---|---|---|---|

Average Error | Maximum Error | |||

1 | 1549.45 | 1476.25 | 4.38% | 12.07% |

5 | 1163.24 | 1115.03 | 4.13% | 12.06% |

10 | 720.26 | 689.81 | 4.38% | 12% |

20 | 511.98 | 490.5 | 4.38% | 12.02% |

30 | 226.34 | 217.08 | 4.05% | 9.78% |

40 | 169.73 | 162.89 | 3.73% | 7.68% |

50 | 149.46 | 142.63 | 4.21% | 8.23% |

120 | 116.55 | 118.36 | 1.88% | 3.52% |

Simulation Cost | OGS Software | This Model |
---|---|---|

Simulation for charging phase | 1.25 h | 0.042 h |

Simulation for discharging phase | 5.02 h | 0.17 h |

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## Share and Cite

**MDPI and ACS Style**

Zhao, Y.; Qin, X.; Shi, X.
Heat Transfer Modeling on High-Temperature Charging and Discharging of Deep Borehole Heat Exchanger with Transient Strong Heat Flux. *Sustainability* **2022**, *14*, 9702.
https://doi.org/10.3390/su14159702

**AMA Style**

Zhao Y, Qin X, Shi X.
Heat Transfer Modeling on High-Temperature Charging and Discharging of Deep Borehole Heat Exchanger with Transient Strong Heat Flux. *Sustainability*. 2022; 14(15):9702.
https://doi.org/10.3390/su14159702

**Chicago/Turabian Style**

Zhao, Yazhou, Xiangxi Qin, and Xiangyu Shi.
2022. "Heat Transfer Modeling on High-Temperature Charging and Discharging of Deep Borehole Heat Exchanger with Transient Strong Heat Flux" *Sustainability* 14, no. 15: 9702.
https://doi.org/10.3390/su14159702