# Reduced Scale Experimental Modelling of Distributed Thermal Response Tests for the Estimation of the Ground Thermal Conductivity

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}emissions than the condensing gas boiler technology (92–95% efficiency), even when the primary carbon intensity of electricity consumption is taken into account. GCHP applications are composed of a heat pump coupled with the ground through multiple vertical or horizontal ground heat exchangers constituting a closed-loop system. Typically, vertical borehole heat exchangers (BHEs) are the most frequently adopted solution.

^{3}sand tank with known thermal properties), and Beier et al. [20] obtained data from a “sandbox” containing a borehole with a U-tube. Other similar examples are those presented by Gu and O’Neal [21], Eslami-nejad and Bernier [22], Kramer et al. [23], Salim Shirazi and Bernier [24], and more recently by Mazzotti et al. [25].

## 2. Theoretical Background on Thermal Response Test

_{gr}. This measurement procedure (firstly proposed by Mogensen [4]) allows evaluating also the effective borehole thermal resistance, and the undisturbed ground temperature. According to the standard TRT process, a carrier fluid flow rate evolves along the BHE active depth H. The initial period of the experiment (few hours) is characterised by the fluid circulation without any heat transfer rate injected into the fluid. This preliminary part is usually devoted to allowing the carrier fluid to reach the thermal equilibrium with the surrounding ground and the measured fluid temperature value is assumed as the depth-averaged undisturbed ground temperature T

_{gr,∞}. Then, the carrier fluid is constantly heated above the ground by providing a constant heat transfer rate $\dot{Q}$ by means of an electric heater. During the test, the fluid temperatures at the inlet and outlet of the TRT machine are measured and recorded. The heat injection continues up to reach, for the average fluid temperature T

_{f,ave}(the mean of the inlet T

_{f,in}and outlet T

_{f,out}temperature), a linear profile in a semilogarithmic time coordinate (typically about 70 h). Finally, from the analysis of this time-temperature profile, the ground thermal conductivity k

_{gr}and the effective BHE thermal resistance R

_{b}* can be evaluated.

_{1}is employed for providing the temperature field into the ground around the linear source [5]

_{r}based on the radial coordinate is defined as

_{1}(x) can be approximated by formulas or series expansion, as discussed in detail by Fossa [3]:

_{b}* that links the borehole temperature T

_{b}to the average fluid temperature T

_{f,ave}:

_{gr}and R

_{b}*, respectively, and the estimated values can be derived according to the following expressions:

_{gr}value can be derived from the slope m inside an appropriate Fo

_{rb}range (i.e., time range), for which the profile can be assumed as linear. Typically, the suggested range in a TRT analysis starts from Fo

_{rb}≥ 10, as reported by Eskilson [26]. The upper limit of the Fo

_{rb}range (and as a consequence of the time range) is associated with the fact that the assumption of an infinite linear source is fulfilled from a finite BHE only if the heat transfer rate can be assumed as radial. This approximation can be considered satisfied when the effect of the heat source in the ground temperature field reaches a radial distance sufficiently low with respect to the depth H of the BHE itself. Fo

_{rb}= 1000 is frequently considered as a valid limit in this sense. As a consequence, for Fo

_{rb}up to nearly 1000, the ILS model can effectively represent the thermal behaviour of borefield also with complex geometries, as demonstrated by [2,3].

## 3. Scaling the Experiment

^{6}J/m

^{3}K and k = 2.5 W/m·K. The slate domain is a parallelepiped with side L and high H, with adiabatic top and bottom surfaces. These boundary conditions are aimed to reproduce ideally the condition of infinite volume in the axial direction. The imposed initial condition is a uniform temperature distribution in the ground and BHE volumes, T

_{gr,∞}= 20 °C. For the sake of completeness, the Reader is directed to [18] for more details about the model. The robust 3D numerical model by the present research group was validated against real TRT experiments, as reported in [18].

_{b}= 0.02 m, with a geometrical reduction with respect to actual cases at 1/2.5. In the middle of the BHE volume, a hole with radius r

_{h}= 0.002 m and a boundary condition of constant heat transfer rate per unit length ${\dot{Q}}^{\prime}$ = 40 W/m represents the heater.

_{b}, r = 0.5·r

_{b}and r = r

_{b}.

_{gr,∞}. The sensitivity analysis is based on comparing the numerical results carried out by adopting the two different far-field boundary conditions according to the approach reported in different studies by the present research group [18,27,28,32].

_{b}= 0.02 m), the period of τ = 15,000 s corresponds to about Fo

_{rb}= 45, that, if referred to a not-scaled geometry (r

_{b}= 0.05 m) with the same thermophysical properties, corresponds to about τ = 26 h, a reasonable time duration for that type of TRT.

^{2}·K and an external fluid temperature equal to T

_{f}= T

_{gr,∞}= 20 °C. This boundary condition can better represent the actual situation of the scale prototype located in a laboratory. The temperature into BHE domain is now calculated not only as an average along the three vertical ideal lines previously discussed, but also in specifically defined sample points, as sketched in Figure 3.

_{b}, except for the point TK4 that is located at the heater surface, namely for r = 0.1 r

_{b}.

_{b}), almost overlap (with an error with respect to the nearly central point TK3 of less than 0.2 °C) in the previously discussed time range up to τ = 15,000 s (ln(τ) = 9.62), except for the point TK1n. In fact, the temperature of the point TK1n is influenced by the boundary effects in the axial direction and the slope of its profile seems not useful for a proper calculation of the thermal conductivity.

_{gt}= 0.8 W/m·K. Figure 5 shows the corresponding temperature profile for the reference sample point TK3.

## 4. Experimental Apparatus

## 5. Inner Borehole Temperature Evolution

_{rb}) and later the slate one (for higher Fo

_{rb}values). As a consequence, the temperature profile measured by the thermocouples shows different slopes for different Fo

_{rb}ranges, in agreement with the simulated behaviour presented in Figure 5. For this reason, two zones can be recognised on each graph, and only the second one can provide useful information to deduce the thermal conductivity of the slate/ground, namely for 7.5 < ln(τ) < 9.5.

_{i%}, which is defined by the following equation:

## 6. Conclusions

- -
- The analysis was firstly addressed at the correct sizing of the prototype through an accurate preliminary evaluation based on dimensionless analysis and numerical simulations with Comsol Multiphysics. The suitable size was theoretically and numerically assessed in terms of the Fourier number time window of interest.
- -
- Then, the scaled heat exchanger, inserted into a slate block (0.8 × 0.8 × 0.4 m), was created with high-precision additive technology (3D printer), enabling the positioning of a central resistive heating cable that makes the experiment more consistent with ILS-based analyses.
- -
- In a second phase, preliminary experimental measurements were carried out providing electrical power to the central resistive heating cable and recording the temperature values measured by the K-type thermocouples distributed along the central axis.
- -
- The results highlight the possibility to reliably estimate the slate/ground thermal conductivity with an accuracy of about +10% with respect to the reference measured values. Based on the results obtained it can be concluded that the presented experimental apparatus related to the innovative TRT method can reliably estimate the slate/ground thermal conductivity without having to resort to expensive standard TRT methods.

## 7. Patents

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

b | constant (K) |

c | specific heat (J/kg K) |

E_{1} | exponential integral in ILS model (-) |

ε_{i%} | relative percentage error (-) |

Fo | Fourier number (-) |

H | active depth of the BHE (m) |

k | thermal conductivity (W/(m·K)) |

m | slope (K/cycles) |

$\dot{Q}$ | heat transfer rate (W) |

${\dot{Q}}^{\prime}$ | heat transfer rate per unit length (W/m) |

R | thermal resistance (m·K/W) |

r | radial coordinate (m) |

T | temperature (K) |

z | vertical coordinate (m) |

Greek letters | |

α | thermal diffusivity (m^{2}/s) |

γ | Euler constant (-) |

ρ | density (kg/m^{3}) |

π | pi constant (-) |

σ | standard deviation (%) |

τ | time (s) |

Subscripts | |

ave | average, as inlet-outlet fluid temperature outside the ground |

b | Borehole |

f | heat carrier fluid |

gr | of the ground medium, of the ground domain |

innovative | TRT related to the measurement method |

meter | related to the measurement method |

∞ | far field and initial condition |

Superscripts | |

* | Effective |

## References

- IEA. Heat Pumps; IEA: Paris, France, 2020; Available online: https://www.iea.org/reports/heat-pumps (accessed on 3 August 2021).
- Fossa, M.; Rolando, D. Improving the Ashrae method for vertical geothermal borefield design. Energy Build.
**2015**, 93, 315–323. [Google Scholar] [CrossRef] - Fossa, M. Correct design of vertical BHE systems through the improvement of the ASHRAE method. Sci. Technol. Built Environ.
**2017**, 23, 1080–1089. [Google Scholar] [CrossRef] - Mogensen, P. Fluid to duct wall heat transfer in duct system heat storages. Doc.-Swed. Counc. Build. Res.
**1983**, 16, 652–657. [Google Scholar] - Carslaw, H.S.; Jaeger, J.C. Conduction of Heat in Solids; Claremore Press: Oxford, UK, 1947. [Google Scholar]
- Ingersoll, L.R.; Zobel, O.J.; Ingersoll, A.C. Heat conduction with engineering, geological, and other applications. Phys. Today
**1955**. [Google Scholar] [CrossRef] - Eklöf, C.; Gehlin, S. TED—A Mobile Equipment for Thermal Response Test. Master’s Thesis, Lund University, Lund, Sweden, 1996. [Google Scholar]
- Austin, W.A. Development of an in Situ System for Measuring Ground Thermal Properties. Ph.D. Thesis, Oklahoma State University, Stillwater, OK, USA, 1998. [Google Scholar]
- Gehlin, S. Thermal Response Test: Method Development and Evaluation. Ph.D. Thesis, Luleå Tekniska Universitet, Luleå, Sweden, 2002. [Google Scholar]
- 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] - 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] - Fossa, M.; Rolando, D.; Pasquier, P. Pulsated Thermal Response Test experiments and modelling for ground thermal property estimation. IGSHPA Res. Track Stockh.
**2018**. [Google Scholar] [CrossRef] [Green Version] - Acuña, J.; Mogensen, P.; Palm, B. Distributed Thermal Response Test on a U-Pipe Borehole Heat Exchanger. In Proceedings of the Effstock 2009, 11th International Conference on Thermal Energy Storage, Stockholm, Sweden, 14–17 June 2009. [Google Scholar]
- Franco, A.; Conti, P. Clearing a Path for Ground Heat Exchange Systems: A Review on Thermal Response Test (TRT) Methods and a Geotechnical Routine Test for Estimating Soil Thermal Properties. Energies
**2020**, 13, 2965. [Google Scholar] [CrossRef] - Ingersoll, L.; Adler, F.; Plass, H.; Ingersoll, A. Theory of earth heat exchangers for the heat pump. Trans. ASHVE
**1951**, 57, 167–188. [Google Scholar] - Deerman, J.D.; Kavanaugh, S.P. Simulation of vertical U-tube ground-coupled heat pump systems using the cylindrical heat source solution. ASHRAE Trans.
**1991**, 97, 287–295. [Google Scholar] - Blackwell, J.H. A transient-flow method for the determination of thermal constants of insulating materials in bulk. J. Appl. Phys.
**1954**, 25, 137–144. [Google Scholar] [CrossRef] - Fossa, M.; Rolando, D.; Priarone, A.; Vaccaro, J. Numerical evaluation of the Ground Response to a Thermal Response Test experiment, European Geothermal Congress 2013. In Proceedings of the EGC 2013 Pisa, Pisa, Italy, 3 June 2013. [Google Scholar]
- Cimmino, M.; Bernier, M. Experimental determination of the g-functions of a small-scale geothermal borehole. Geothermics
**2015**, 56, 60–71. [Google Scholar] [CrossRef] - Beier, R.A.; Smith, M.D.; Spitler, J.D. Reference data sets for vertical borehole ground heat exchanger models and thermal response test analysis. Geothermics
**2011**, 40, 79–85. [Google Scholar] [CrossRef] - Gu, Y.; O’Neal, D.L. Modelling the effect of backfills on U-tube ground coil performance. ASHRAE Trans.
**1998**, 104, 356–365. [Google Scholar] - Eslami-nejad, P.; Bernier, M. Freezing of geothermal borehole surroundings: A numerical and experimental assessment with applications. Appl. Energy
**2012**, 98, 333–345. [Google Scholar] [CrossRef] - Kramer, C.A.; Ghasemi-Fare, O.; Basu, P. Laboratory thermal performance tests on a model heat exchanger pile in sand. Geotech. Geol. Eng.
**2015**, 33, 253–271. [Google Scholar] [CrossRef] - Salim Shirazi, A.; Bernier, M. A small-scale experimental apparatus to study heat transfer in the vicinity of geothermal boreholes. HVACR Res.
**2014**, 20, 819–827. [Google Scholar] [CrossRef] - Mazzotti, W.; Acuña, J.; Lazzarotto, A.; Palm, B. Deep Boreholes for Ground Source Heat Pump, Final Report Effsys Expand (Energimyndigheten); KTH: Stockholm, Sweden, 2018; pp. 1–77. [Google Scholar]
- Eskilson, P. Thermal Analysis of Heat Extraction Boreholes. Ph.D. Thesis, Lund University, Lund, Sweden, 1987. [Google Scholar]
- Morchio, S.; Fossa, M. Thermal modeling of deep borehole heat exchangers for geothermal applications in densely populated urban areas. Therm. Sci. Eng. Prog.
**2019**, 13, 100363. [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] - Beier, R.A.; Fossa, M.; Morchio, S. Models of thermal response tests on deep coaxial borehole heat exchangers through multiple ground layers. Appl. Therm. Eng.
**2020**, 184, 116241. [Google Scholar] [CrossRef] - Morchio, S.; Fossa, M. Modelling and Validation of a New Hybrid Scheme for Predicting the Performance of U-pipe Borehole Heat Exchangers during Distributed Thermal Response Test Experiments. Appl. Therm. Eng.
**2020**, 186, 116514. [Google Scholar] [CrossRef] - Morchio, S.; Fossa, M.; Beier, R.A. Study on the best heat transfer rate in Thermal Response Test experiments with coaxial and U-pipe Borehole Heat Exchangers. Appl. Therm. Eng.
**2022**, 200, 117621. [Google Scholar] [CrossRef] - Priarone, A.; Fossa, M. Modelling the ground volume for numerically generating single borehole heat exchanger response factors according to the cylindrical source approach. Geothermics
**2015**, 58, 32–38. [Google Scholar] [CrossRef]

**Figure 2.**Simulation data analysis: axial average temperature in the slate volume at different radial distances from the heater. In abscissa the logarithm of time.

**Figure 4.**Simulation data analysis: temperature in the slate volume at different axial positions according to Figure 3. In abscissa the logarithm of time.

**Figure 5.**Simulation data analysis: temperature of the slate volume in sample point TK3 for k

_{gt}= 0.8 W/m·K. In abscissa the logarithm of time.

**Figure 6.**The rock sample (slate) and the cart that is used to keep the block stably raised during the planned experimental activity.

**Figure 8.**3D representation of the reduced scale BHE assembly, including its 3D printer manufactured prototype and its spacer.

**Figure 9.**Dimensions (in mm) of the section of each pipe manufactured using the Cartesian 3D printer.

**Figure 10.**Infrared temperature map close to the top part of the scale prototype. On the right the concentric temperature profiles can be observed.

**Figure 11.**Temperature vs. time profiles for the different sensors inserted into the “all-in-one” heat exchanger.

Material | Measurement Method | Thermal Conductivity (W/m·K) | Heat Capacity (MJ/m^{3·}K) |
---|---|---|---|

Slate block | Applied Precision Isomet 2114 conductivity meter | 2.85 σ = 0.18% | 2 σ = 0.26% |

Cylindrical slate samples | Applied Precision Isomet 2114 conductivity meter | 2.85 σ = 0.12% | 2.12 σ = 0.035% |

Cylindrical slate samples | Unige Steady-state meter | 2.48 σ = 0.023% | - |

Ground | ||
---|---|---|

Thermocouple ID | Slope | k_{gr} (W/m·K) |

TK2 | 0.854 | 3.58 |

TK3 | 1.017 | 3.01 |

TK5 | 0.880 | 3.48 |

TK6 | 1.121 | 2.73 |

Average | 3.20 |

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

© 2021 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**

Morchio, S.; Fossa, M.; Priarone, A.; Boccalatte, A.
Reduced Scale Experimental Modelling of Distributed Thermal Response Tests for the Estimation of the Ground Thermal Conductivity. *Energies* **2021**, *14*, 6955.
https://doi.org/10.3390/en14216955

**AMA Style**

Morchio S, Fossa M, Priarone A, Boccalatte A.
Reduced Scale Experimental Modelling of Distributed Thermal Response Tests for the Estimation of the Ground Thermal Conductivity. *Energies*. 2021; 14(21):6955.
https://doi.org/10.3390/en14216955

**Chicago/Turabian Style**

Morchio, Stefano, Marco Fossa, Antonella Priarone, and Alessia Boccalatte.
2021. "Reduced Scale Experimental Modelling of Distributed Thermal Response Tests for the Estimation of the Ground Thermal Conductivity" *Energies* 14, no. 21: 6955.
https://doi.org/10.3390/en14216955