# Estimation of Ground Thermal Properties of Shallow Coaxial Borehole Heat Exchanger Using an Improved Parameter Estimation Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. 3D Heat Transfer Model of SCBHE

_{i}and T

_{a}are the temperatures of internal and annular fluids respectively, °C; t is the time, s; z is the vertical coordinate, m; Q

_{in}is the heat input rate of SCBHE, W; V is the fluid flow rate, m

^{3}·s

^{−1}; (ρc)

_{f}is the fluid thermal capacity, J·m

^{−3}·K

^{−1}; L is the SCBHE length, m.

## 3. Effects of Ground Thermal Properties on the Slope

_{f}is the mean fluid temperature of SCBHE, °C; b is the intercept of the linear Equation (3); λ

_{gr}is the ground thermal conductivity, W·m

^{−1}·K

^{−1}. In practical TRT, inlet and outlet fluid temperatures of SCBHE are generally measured, therefore, the direct method normally regards the average of inlet and outlet fluid temperatures as the mean fluid temperature, and calculates k by linear fitting of Equation (3), and then uses Equation (4) to calculate the ground thermal conductivity. It should be mentioned that the slope k in this paper means the slope of Equation (3), in which T

_{f}is regarded as the average of inlet and outlet fluid temperatures of SCBHE, and that the early-time (normally the first 10 h) TRT data are ignored when using linear fitting of Equation (3).

^{−1}·K

^{−1}, meaning that ground thermal conductivity has a great effect on the slope, however, the relationship between the slope and ground thermal conductivity probably does not satisfy Equation (4). As shown in Figure 4, it is only when (ρc)

_{gr}equals to about 3 × 10

^{6}J·m

^{−3}·K

^{−1}that the relationship between the slope and ground thermal conductivity satisfies Equation (4) well, and there is larger difference between the relationship and Equation (4) for smaller ground thermal conductivity. For smaller ground thermal conductivity or thermal capacity, the mean fluid temperature increases more quickly with time and the effect of measurement error is smaller, but the effect of borehole thermal capacity is larger, which means that the error of ILS model is larger and ground thermal conductivity estimated by Equation (4) has larger error, so the difference between the relationship and Equation (4) is influenced by the two aspects which are mainly determined by ground thermal conductivity and thermal capacity.

^{6}to 4.0 × 10

^{6}J·m

^{−3}·K

^{−1}, and that ground thermal capacity has a larger effect on the slope for smaller ground thermal conductivity, meaning that the ground thermal capacity only has a small effect on the slope.

## 4. Improved Parameter Estimation Method

_{e}is the slope of Equation (3) about the average of inlet and outlet fluid temperatures of TRT, and k

_{c}is the slope of Equation (3) about the average of inlet and outlet fluid temperatures calculated by heat transfer model.

_{ci,i}and T

_{ei,i}are the calculated and experimental inlet fluid temperatures at the i-th test time respectively, °C; T

_{co,i}and T

_{eo,i}are the calculated and experimental outlet fluid temperatures at the i-th test time respectively, °C.

_{gr}(ground thermal conductivity) based on Equations (3) and (4).

_{gr}(ground thermal capacity) is estimated by minimizing S

_{(ρc)}based on Monte Carlo method [20]. It should be mentioned that S

_{(ρc)}is the objective function about the inlet and outlet fluid temperatures estimated by the heat transfer model [25] and those of TRT data.

_{gr}is estimated by minimizing S

_{λ}based on Monte Carlo method. It should be mentioned that S

_{λ}is the objective function about k

_{e}and k

_{c}, and that k

_{c}is calculated based on the heat transfer model in Ref. [25].

_{gr}directly, while the improved PEM adopts an iterative algorithm (i.e., Monte Carlo method) to estimate λ

_{gr}by minimizing the difference between k

_{e}and k

_{c}; (2) the direct method assumes that the mean fluid temperature equals to the average of inlet and outlet fluid temperatures, while the improved PEM has no such assumption; (3) the direct method is not able to estimate (ρc)

_{gr}, while the improved PEM can also estimate (ρc)

_{gr}.

## 5. Results and Discussion

_{gr}accurately when (ρc)

_{gr}equals to about 3.0 × 10

^{6}J·m

^{−3}·K

^{−1}, and would overestimate λ

_{gr}when both λ

_{gr}and (ρc)

_{gr}are large enough, and would underestimate λ

_{gr}when λ

_{gr}or (ρc)

_{gr}is small enough. The main reasons for the errors of direct method are as follows: there exist certain errors when using the ILS model for analyzing the heat transfer in the SCBHE; the assumption of using the average of inlet and outlet fluid temperatures to approximate the mean fluid temperature would also lead to some errors. The precision of the direct method is very high for (ρc)

_{gr}= 3.0 × 10

^{6}J·m

^{−3}·K

^{−1}, which is because the relationship between the slope and ground thermal conductivity satisfies Equation (4) well for the given conditions, shown in Figure 4.

_{gr}and (ρc)

_{gr}accurately for all the situations presented in Table 2, and the errors of the estimated λ

_{gr}and (ρc)

_{gr}are within 0.7% and within 1.4% respectively, indicating that the improved PEM has high precision, which is explained as follows: the heat transfer model used in the improved PEM considers the borehole thermal capacity, and has high accuracy [25,26]; there is no assumption about the mean fluid temperature; because of the strong effect of ground thermal conductivity on the slope, the objective function about the difference between the experimental slope and calculated slope can be used for estimating the ground thermal conductivity accurately; because the ground thermal capacity has a larger effect on the early-time fluid temperatures than on the later-time fluid temperatures, the objective function in Eq. (6) is used to estimate an accurate ground thermal capacity.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Introduction to the Heat Transfer Model Used in the Paper

_{i}and T

_{a}are the temperatures of internal and annular fluids respectively, ℃; r

_{ii}and r

_{ie}are the internal and external radii of internal pipe respectively, m; r

_{ei}and r

_{e}

_{e}are the internal and external radii of external pipe respectively, m; (ρc)

_{f}is the fluid thermal capacity, J·m

^{−3}·K

^{−1}; (ρc)

_{ip}and (ρc)

_{ep}are the thermal capacities of internal and external pipes, J·m

^{−3}·K

^{−1}; t is the time, s; z is the vertical coordinate, m; V is the fluid flow rate, m

^{3}·s

^{−1}; R

_{ia}is the thermal resistance between the internal and annular fluids, m·K·W

^{−1}; q is the heat flow from the annular fluid to the grout:

_{ae}is the thermal resistance between the annular fluid and external surface of external pipe, m·K·W

^{−1}; T

_{eo}is the temperature of external surface of external pipe:

_{g}is the grout thermal conductivity, W·m

^{−1}·K

^{−1}; G(t) is the temperature response function, which is used to analyze the heat transfer in the grout and ground.

_{in}is the heat input rate of SCBHE, W; L is the SCBHE length, m; T

_{0}is the initial ground temperature, °C.

## References

- Lund, J.W.; Toth, A.N. Direct utilization of geothermal energy 2020 worldwide review. Geothermics
**2021**, 90, 101915. [Google Scholar] [CrossRef] - Eswiasi, A.; Mukhopadhyaya, P. Performance of conventional and innovative single U-tube pipe configuration in vertical ground heat exchanger (VGHE). Sustainability
**2021**, 13, 6384. [Google Scholar] [CrossRef] - Spitler, J.D.; Gehlin, S.E.A. Thermal response testing for ground source heat pump systems—An historical review. Renew. Sustain. Energy Rev.
**2015**, 50, 1125–1137. [Google Scholar] [CrossRef] - Javadi, H.; Ajarostaghi, S.S.M.; Rosen, M.A.; Pourfallah, M. Performance of ground heat exchangers: A comprehensive review of recent advances. Energy
**2019**, 178, 207–233. [Google Scholar] [CrossRef] - Zhang, C.; Guo, Z.; Liu, Y.; Cong, X.; Peng, D. A review on thermal response test of ground-coupled heat pump systems. Renew. Sustain. Energy Rev.
**2014**, 40, 851–867. [Google Scholar] [CrossRef] - Zhang, L.; Zhang, Q.; Huang, G.; Du, Y. A p(t)-linear average method to estimate the thermal parameters of the borehole heat exchangers for in situ thermal response test. Appl. Energy
**2014**, 131, 211–221. [Google Scholar] [CrossRef] - Nikolaev, I.V.; Leong, W.H.; Rosen, M.A. Experimental investigation of soil thermal conductivity over a wide temperature range. Int. J. Thermophys.
**2013**, 34, 1110–1129. [Google Scholar] [CrossRef] - Raymond, J. Colloquium 2016: Assessment of subsurface thermal conductivity for geothermal applications. Can. Geotech. J.
**2018**, 55, 1209–1229. [Google Scholar] [CrossRef] [Green Version] - Beier, R.A. Insights into parameter estimation for thermal response tests on borehole heat exchangers. Sci. Technol. Built Environ.
**2019**, 25, 947–962. [Google Scholar] [CrossRef] - Sapińska-Śliwa, A.; Sliwa, T.; Twardowski, K.; Szymski, K.; Gonet, A.; Żuk, P. Method of averaging the effective thermal conductivity based on thermal response tests of borehole heat exchangers. Energies
**2020**, 13, 3737. [Google Scholar] [CrossRef] - Yang, H.; Cui, P.; Fang, Z. Vertical-borehole ground-coupled heat pumps: A review of models and systems. Appl. Energy
**2010**, 87, 16–27. [Google Scholar] [CrossRef] - Li, M.; Lai, A.C.K. Review of analytical models for heat transfer by vertical ground heat exchangers (GHEs): A perspective of time and space scales. Appl. Energy
**2015**, 151, 178–191. [Google Scholar] [CrossRef] - Nian, Y.L.; Cheng, W.L. Insights into geothermal utilization of abandoned oil and gas wells. Renew. Sustain. Energy Rev.
**2018**, 87, 44–60. [Google Scholar] [CrossRef] - Beier, R.A. Use of temperature derivative to analyze thermal response tests on borehole heat exchangers. Appl. Therm. Eng.
**2018**, 134, 298–309. [Google Scholar] [CrossRef] - Pasquier, P.; Zarrella, A.; Marcotte, D. A multi-objective optimization strategy to reduce correlation and uncertainty for thermal response test analysis. Geothermics
**2019**, 79, 176–187. [Google Scholar] [CrossRef] - Zhang, C.; Xu, H.; Fan, J.; Sun, P.; Sun, S.; Kong, X. The coupled two-step parameter estimation procedure for borehole thermal resistance in thermal response test. Renew. Energy
**2020**, 154, 672–683. [Google Scholar] [CrossRef] - Bozzoli, F.; Pagliarini, G.; Rainieri, S.; Schiavi, L. Estimation of soil and grout thermal properties through a TSPEP (two-step parameter estimation procedure) applied to TRT (thermal response test) data. Energy
**2011**, 36, 839–846. [Google Scholar] [CrossRef] - Li, M.; Zhang, L.W.; Liu, G. Step-wise algorithm for estimating multi-parameter of the ground and geothermal heat exchangers from thermal response tests. Renew. Energy
**2020**, 150, 435–442. [Google Scholar] [CrossRef] - Nian, Y.L.; Wang, X.Y.; Xie, K.; Cheng, W.L. Estimation of ground thermal properties for coaxial BHE through distributed thermal response test. Renew. Energy
**2020**, 152, 1209–1219. [Google Scholar] [CrossRef] - Wang, C.; Fang, H.; Lu, J.; Sun, Y.; Zhang, P.; Wang, X. A two-step parameter estimation method for estimating soil thermal properties of coaxial ground heat exchangers. Geothermics
**2021**, 96, 102229. [Google Scholar] [CrossRef] - Beier, R.A.; Acuña, J.; Mogensen, P.; Palm, B. Borehole resistance and vertical temperature profiles in coaxial borehole heat exchangers. Appl. Energy
**2013**, 102, 665–675. [Google Scholar] [CrossRef] - Beier, R.A.; Acuña, J.; Mogensen, P.; Palm, B. Transient heat transfer in a coaxial borehole heat exchanger. Geothermics
**2014**, 51, 470–482. [Google Scholar] [CrossRef] - Morchio, S.; Fossa, M. On the ground thermal conductivity estimation with coaxial borehole heat exchangers according to different undisturbed ground temperature profiles. Appl. Therm. Eng.
**2020**, 173, 115198. [Google Scholar] [CrossRef] - Acuña, J.; Palm, B. Distributed thermal response tests on pipe-in-pipe borehole heat exchangers. Appl. Energy
**2013**, 109, 312–320. [Google Scholar] [CrossRef] - Wang, C.; Lu, Y.; Chen, L.; Huang, Z.; Fang, H. A semi-analytical model for heat transfer in coaxial borehole heat exchangers. Geothermics
**2021**, 89, 101952. [Google Scholar] [CrossRef] - Wang, C.; Fang, H.; Wang, X.; Lu, J.; Sun, Y. Study on the influence of the borehole heat capacity for deep coaxial borehole heat exchanger. Sustainability
**2022**, 14, 2043. [Google Scholar] [CrossRef] - Wang, C.; Wang, X.; Lu, J.; Lu, Y.; Sun, Y.; Zhang, P. A semi-analytical heat transfer model for deep borehole heat exchanger considering groundwater seepage. Int. J. Therm. Sci.
**2022**, 175, 107465. [Google Scholar] [CrossRef]

**Figure 3.**Comparison of the time-varying fluid temperatures simulated by the 3D heat transfer model with the TRT results.

**Figure 4.**Effect of ground thermal conductivity on the slope at different ground thermal capacities.

**Figure 5.**Effect of ground thermal capacity on the slope at different ground thermal conductivities.

**Figure 7.**Comparison of the data in Table 2.

**Table 1.**Detailed parameters of SCBHE [24].

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

SCBHE length L (m) | 168 |

Borehole radius r_{b} (m) | 0.0575 |

Internal radius of internal pipe r_{ii} (m) | 0.0176 |

External radius of internal pipe r_{ie} (m) | 0.020 |

Internal radius of external pipe r_{ei} (m) | 0.0566 |

External radius of external pipe r_{ee} (m) | 0.057 |

Thermal conductivities of internal and external pipes λ_{ip}, λ_{ep} (W·m^{−1}·K^{−1}) | 0.4 |

Thermal capacities of internal and external pipes (ρc)_{ip}, (ρc)_{ep} (J·m^{−3}·K^{−1}) | 1.8 × 10^{6} |

Grout thermal conductivity λ_{g} (W·m^{−1}·K^{−1}) | 0.59 |

Grout thermal capacity (ρc)_{g} (J·m^{−3}·K^{−1}) | 4.19 × 10^{6} |

Ground thermal conductivity λ_{gr} (W·m^{−1}·K^{−1}) | 3.28 |

Ground thermal capacity (ρc)_{gr} (J·m^{−3}·K^{−1}) | 2.24 × 10^{6} |

Fluid thermal conductivity λ_{f} (W·m^{−1}·K^{−1}) | 0.59 |

Fluid thermal capacity (ρc)_{f} (J·m^{−3}·K^{−1}) | 4.19 × 10^{6} |

Fluid Prandtl number Pr | 8.09 |

Initial ground temperature T_{0} (℃) | 8.4 |

Fluid flow rate V (m^{3}·s^{−1}) | 0.58 × 10^{−3} |

Heat input rate of SCBHE Q_{in} (W) | 6360 |

**Table 2.**Comparison of the reference values of ground thermal properties with the results calculated by the improved PEM and direct method.

Reference Values | Calculated Results of Direct Method | Calculated Results of the Improved PEM | |||||
---|---|---|---|---|---|---|---|

(ρc)_{gr} (×10^{6} J·m^{−3}·K^{−1}) | λ_{gr}(W·m ^{−1}·K^{−1}) | λ_{gr}(W·m ^{−1}·K^{−1}) | Error of λ_{gr} | λ_{gr}(W·m ^{−1}·K^{−1}) | Error of λ_{gr} | (ρc)_{gr} (×10^{6} J·m^{−3}·K^{−1}) | Error of (ρc)_{gr} |

1.0 | 1.5 | 1.306 | 12.9% | 1.490 | 0.7% | 1.001 | 0.1% |

1.0 | 2.0 | 1.777 | 11.2% | 1.987 | 0.7% | 1.008 | 0.8% |

1.0 | 2.5 | 2.258 | 9.7% | 2.489 | 0.4% | 1.005 | 0.5% |

1.0 | 3.0 | 2.746 | 8.5% | 2.985 | 0.5% | 1.002 | 0.2% |

1.0 | 3.5 | 3.239 | 7.5% | 3.485 | 0.4% | 1.005 | 0.5% |

1.0 | 4.0 | 3.737 | 6.6% | 3.985 | 0.4% | 1.006 | 0.6% |

1.0 | 4.5 | 4.239 | 5.8% | 4.482 | 0.4% | 1.011 | 1.1% |

2.0 | 1.5 | 1.419 | 5.4% | 1.491 | 0.6% | 2.013 | 0.7% |

2.0 | 2.0 | 1.903 | 4.9% | 1.990 | 0.5% | 2.002 | 0.1% |

2.0 | 2.5 | 2.392 | 4.3% | 2.490 | 0.4% | 1.999 | 0.1% |

2.0 | 3.0 | 2.887 | 3.8% | 2.987 | 0.4% | 2.007 | 0.4% |

2.0 | 3.5 | 3.388 | 3.2% | 3.485 | 0.4% | 2.019 | 1.0% |

2.0 | 4.0 | 3.893 | 2.7% | 3.980 | 0.5% | 2.021 | 1.1% |

2.0 | 4.5 | 4.401 | 2.2% | 4.480 | 0.4% | 2.026 | 1.3% |

3.0 | 1.5 | 1.499 | 0.1% | 1.495 | 0.3% | 3.005 | 0.2% |

3.0 | 2.0 | 1.991 | 0.5% | 1.992 | 0.4% | 3.007 | 0.2% |

3.0 | 2.5 | 2.488 | 0.5% | 2.490 | 0.4% | 3.010 | 0.3% |

3.0 | 3.0 | 2.990 | 0.3% | 2.990 | 0.3% | 3.004 | 0.1% |

3.0 | 3.5 | 3.496 | 0.1% | 3.490 | 0.3% | 3.012 | 0.4% |

3.0 | 4.0 | 4.007 | 0.2% | 3.990 | 0.3% | 3.016 | 0.5% |

3.0 | 4.5 | 4.522 | 0.5% | 4.481 | 0.4% | 3.038 | 1.3% |

4.0 | 1.5 | 1.563 | 4.2% | 1.502 | 0.1% | 3.971 | 0.7% |

4.0 | 2.0 | 2.063 | 3.1% | 1.998 | 0.1% | 3.979 | 0.5% |

4.0 | 2.5 | 2.567 | 2.7% | 2.496 | 0.2% | 3.980 | 0.5% |

4.0 | 3.0 | 3.075 | 2.5% | 2.995 | 0.2% | 3.996 | 0.1% |

4.0 | 3.5 | 3.587 | 2.5% | 3.492 | 0.2% | 4.009 | 0.2% |

4.0 | 4.0 | 4.103 | 2.6% | 3.988 | 0.3% | 4.027 | 0.7% |

4.0 | 4.5 | 4.623 | 2.7% | 4.480 | 0.4% | 4.056 | 1.4% |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, C.; Fu, Q.; Fang, H.; Lu, J.
Estimation of Ground Thermal Properties of Shallow Coaxial Borehole Heat Exchanger Using an Improved Parameter Estimation Method. *Sustainability* **2022**, *14*, 7356.
https://doi.org/10.3390/su14127356

**AMA Style**

Wang C, Fu Q, Fang H, Lu J.
Estimation of Ground Thermal Properties of Shallow Coaxial Borehole Heat Exchanger Using an Improved Parameter Estimation Method. *Sustainability*. 2022; 14(12):7356.
https://doi.org/10.3390/su14127356

**Chicago/Turabian Style**

Wang, Changlong, Qiang Fu, Han Fang, and Jinli Lu.
2022. "Estimation of Ground Thermal Properties of Shallow Coaxial Borehole Heat Exchanger Using an Improved Parameter Estimation Method" *Sustainability* 14, no. 12: 7356.
https://doi.org/10.3390/su14127356