# Novel Methodology for Sizing a Single U-Tube Ground Heat Exchanger Useful at the Early Design Stage

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review & Problem Statement

## 3. Establishment of a Methodology for Sizing a Single U-Tube Ground Heat Exchanger (GHE)

#### 3.1. Major Concept

#### 3.2. Transient Simulation to Derive the Sizing Method

#### 3.2.1. Transient Simulation Condition

#### 3.2.2. Simulation Results

#### 3.3. Method for Estimating SHER

- Only a single borehole of a single U-tube type was considered. The surrounding ground was assumed to be homogeneous and isotropic.
- The ground had a uniform initial temperature. The heat loss or gain at the ground surface and effects of groundwater were neglected.
- In extracting heating or cooling energy from the GHE, the entering temperature and mass flow rate of the circulating fluid were assumed to be constant.

## 4. Verification

#### 4.1. Design Cases for Verification

#### 4.2. Results of the Ground Region Temperature

#### 4.3. Results of SHER

#### 4.4. Discussion on the Applicability of Design Method

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$q\left(t\right)$ | SHER of the GHE at a certain time | W/m |

$\dot{m}$ | Mass flow rate of the circulating fluid in the GHE | kg/s |

$\dot{{V}_{f}}$ | Volumetric flow rate of the circulating fluid in the GHE | m^{3}/s |

${c}_{p,f}$ | Specific heat of the circulating fluid | J/kgK |

${\rho}_{f}$ | Density of the circulating fluid | kg/m^{3} |

${\lambda}_{gr}$ | Thermal conductivity of the grouting material | W/mK |

$C$ | Equivalent thermal conductance | W/K |

${R}_{b}^{*}$ | Effective borehole thermal resistance | mK/W |

${R}_{b}$ | Borehole thermal resistance | mK/W |

${R}_{a}$ | Internal fluid to fluid thermal resistance | mK/W |

${\theta}_{f,in}$ | Inlet fluid temperature of the GHE | °C |

${\theta}_{f,out}$ | Outlet fluid temperature of the GHE | °C |

${\theta}_{f,avg}$ | Arithmetic average of the inlet and outlet fluid temperatures | °C |

${\theta}_{g}$(d) | Ground region temperature on a day | °C |

$H$ | Borehole length | m |

${a}_{1}$ | Performance coefficient for estimate of the SHER | - |

${b}_{1}$ | Performance constant for estimate of the SHER | W/m |

${a}_{2}$ | Performance coefficient for estimate of the ground region temperature | - |

${b}_{2}$ | Performance constant for estimate of the ground region temperature | °C |

${\alpha}_{1}$ | Dimensionless parameter for calculating borehole thermal resistance | - |

${\alpha}_{2}$ | Dimensionless parameter for calculating borehole thermal resistance | - |

${\alpha}_{3}$ | Dimensionless parameter for calculating borehole thermal resistance | - |

$\beta $ | Dimensionless thermal resistance of one U-tube leg | - |

$\mathsf{\sigma}$ | Thermal conductivity ratio | - |

$t$ | Time in an operating time in a day | s |

$\mathrm{CV}({\mathrm{RMSE}}_{Period})$ | Coefficient of variation of the root mean square error in the period | - |

${\mathrm{RMSE}}_{Period}$ | Root mean square error in the period | - |

${A}_{Period}$ | The mean of data for the period | - |

${N}_{Interval}$ | The number of time intervals in the period | - |

$M$ | The results of the transient simulation during the time interval | - |

$S$ | The results of the design method during the same time interval | - |

## References

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**Figure 1.**Variation in the borehole wall temperature at the end of the operating time of each day ((

**a**): heating season, (

**b**): cooling season).

**Figure 2.**Variation in the SHER during the first five days in the heating and cooling seasons ((

**a**): heating season, (

**b**): cooling season).

Parameters | Unit | Value |
---|---|---|

Borehole length | m | 150 |

Borehole diameter | m | 0.15 |

Pipe outside diameter | m | 0.04 |

Pipe inside diameter | m | 0.032 |

Pipe spacing | m | 0.08 |

Thermal conductivity (Fluid) | W/(mK) | 0.574 |

Density (Fluid) | kg/m^{3} | 1000 |

Specific heat (Fluid) | J/(kgK) | 4211 |

Thermal conductivity (Pipe) | W/(mK) | 0.45 |

Density (Pipe) | kg/m^{3} | 550 |

Specific heat (Pipe) | J/(kgK) | 2250 |

Thermal conductivity (Grout) | W/(mK) | 1.6 |

Density (Grout) | kg/m^{3} | 1500 |

Specific heat (Grout) | J/(kgK) | 960 |

Thermal conductivity (Ground) | W/(mK) | 3.8 |

Density (Ground) | kg/m^{3} | 2640 |

Specific heat (Ground) | J/(kgK) | 880 |

Ground initial temperature | °C | 13 |

Fluid mass flow rate (Heating) | kg/s | 0.25 |

Fluid inlet temperature (Heating) | °C | 1 |

Heating season | - | 1 January–28 February |

- | 1 November–31 December | |

Operating time (Heating) | h | 12 |

Stop time (Heating) | h | 12 |

Fluid mass flow rate (Cooling) | kg/s | 0.25 |

Fluid inlet temperature (Cooling) | °C | 30 |

Cooling season | - | 1 June–30 September |

Operating time (Cooling) | h | 12 |

Stop time (Cooling) | h | 12 |

Simulation period | year | 1 |

Influential Variables | Design Case No. | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |

Borehole length [m] | 150 | 150 | 150 | 150 | 150 | 150 | 75 | 150 | 150 | 150 |

Grout thermal conductivity [W/(mK)] | 1.6 | 1.6 | 1.6 | 0.68 | 2.2 | 1.6 | 1.6 | 1.6 | 1.6 | 1.6 |

Ground thermal conductivity [W/(mK)] | 3.8 | 1.9 | 5.2 | 3.8 | 3.8 | 3.8 | 3.8 | 3.8 | 3.8 | 3.8 |

Initial ground temperature [°C] | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 15 | 13 | 13 |

Fluid mass flow rate [kg/s] | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.15 | 0.25 | 0.25 | 0.25 | 0.25 |

Daily operating time [h] | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 8 | 16 |

Daily stop time [h] | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | 8 |

**Table 3.**Results of ground region temperature at the end of the heating and cooling seasons and CV(RMSE).

Design Case No. | ${\mathit{a}}_{1}$ | ${\mathit{b}}_{1}$ | Performance during Heating Season | Performance during Cooling Season | ||||
---|---|---|---|---|---|---|---|---|

Ground Region Temperature at the End of the Season [°C] | CV(RMSE) [%] | Ground Region Temperature at the End of the Season [°C] | CV(RMSE) [%] | |||||

Design Method | Transient Simulation | Design Method | Transient Simulation | |||||

1 | −422 | 11,150 | 10.28 | 10.44 | 1.11 | 16.14 | 16.38 | 1.46 |

2 | −549 | 11,059 | 9.63 | 9.31 | 3.76 | 17.35 | 18.11 | 3.23 |

3 | −377 | 11,215 | 10.67 | 10.92 | 1.66 | 15.86 | 15.85 | 0.33 |

4 | −560 | 11,175 | 10.46 | 11.22 | 5.28 | 16.08 | 15.70 | 1.93 |

5 | −388 | 11,146 | 10.23 | 9.94 | 3.55 | 16.15 | 17.33 | 6.41 |

6 | −322 | 8562 | 10.52 | 10.78 | 1.59 | 16.04 | 16.19 | 1.26 |

7 | −295 | 7227 | 10.13 | 10.41 | 1.78 | 16.12 | 16.88 | 4.35 |

8 | −495 | 13,072 | 12.14 | 12.30 | 0.90 | 18.06 | 18.12 | 0.34 |

9 | −443 | 11,253 | 10.93 | 10.78 | 1.00 | 15.65 | 15.71 | 0.28 |

10 | −410 | 11,111 | 9.86 | 10.06 | 1.32 | 16.26 | 17.04 | 4.51 |

Design Case No. | ${\mathit{a}}_{2}$ | ${\mathit{b}}_{2}$ | Performance during Heating Season | Performance during Cooling Season | ||||
---|---|---|---|---|---|---|---|---|

Average of the SHER [W/m] | CV(RMSE) [%] | Average of the SHER [W/m] | CV(RMSE) [%] | |||||

Design Method | Transient Simulation | Design Method | Transient Simulation | |||||

1 | −0.56 | 12.53 | 39.99 | 41.14 | 6.97 | −58.97 | −59.23 | 5.98 |

2 | −0.69 | 12.44 | 31.12 | 29.04 | 12.16 | −45.10 | −43.58 | 10.37 |

3 | −0.47 | 12.60 | 44.03 | 46.49 | 6.95 | −63.85 | −66.25 | 5.90 |

4 | −0.51 | 12.56 | 33.03 | 35.41 | 11.88 | −48.12 | −50.52 | 10.26 |

5 | −0.56 | 12.53 | 41.75 | 41.14 | 4.98 | −61.75 | −59.23 | 7.23 |

6 | −0.50 | 12.57 | 31.39 | 32.72 | 6.14 | −45.61 | −46.07 | 5.07 |

7 | −0.59 | 12.52 | 48.97 | 50.08 | 14.97 | −73.20 | −71.51 | 12.44 |

8 | −0.58 | 14.52 | 47.75 | 48.90 | 6.66 | −51.11 | −52.15 | 6.09 |

9 | −0.42 | 12.64 | 42.26 | 43.85 | 8.29 | −60.74 | −63.89 | 8.46 |

10 | −0.65 | 12.49 | 38.62 | 38.96 | 5.32 | −58.57 | −55.72 | 7.46 |

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**MDPI and ACS Style**

Baek, S.-H.; Lee, B.-H.; Yeo, M.-S.
Novel Methodology for Sizing a Single U-Tube Ground Heat Exchanger Useful at the Early Design Stage. *Buildings* **2021**, *11*, 651.
https://doi.org/10.3390/buildings11120651

**AMA Style**

Baek S-H, Lee B-H, Yeo M-S.
Novel Methodology for Sizing a Single U-Tube Ground Heat Exchanger Useful at the Early Design Stage. *Buildings*. 2021; 11(12):651.
https://doi.org/10.3390/buildings11120651

**Chicago/Turabian Style**

Baek, Seung-Hyo, Byung-Hee Lee, and Myoung-Souk Yeo.
2021. "Novel Methodology for Sizing a Single U-Tube Ground Heat Exchanger Useful at the Early Design Stage" *Buildings* 11, no. 12: 651.
https://doi.org/10.3390/buildings11120651