A New Estimate of Sand and Grout Thermal Properties in the Sandbox Experiment for Accurate Validations of Borehole Simulation Codes
Abstract
:1. Introduction
2. The Sandbox Experiment
3. Estimation Method and Results
- The measurements of temperature and of the thermal power released to the BHE are precise. This assumption is based on the accuracies declared in Reference [35] and on the nearly perfect agreement between the value of thermal power released to the BHE determined by measurements of voltage and current and that determined by the energy balance equation in the fluid, in quasi-stationary working conditions. The discrepancy between the mean values of these quantities, in the time period between 1200 min and the end of the TRT, is lower than 0.5%.
- The most probable positions of the U-tube pipes are those declared by the authors [35], with axes of the U-tube pipes and axis of the aluminum tube laying in the same horizontal plane, and a shank spacing 5.3 cm. Indeed, the BHE was only 18.32 m long, and spacings were inserted to ensure that the distance between the pipes and the distance between the outer surface of the pipes and the BHE wall were uniform [35].
- Since the U-tube is very short, the bulk fluid temperature along the flow can be considered as a linear function of the distance from the inlet section. Therefore, the mean fluid temperature in the whole U-tube, Tfm, that in the inlet tube, T1, and that in the outlet tube, T2, can be evaluated as
- There may be a significant discrepancy between the value of the grout thermal conductivity, kg, measured in Reference [35] and the real value, because the measurement was performed in a sample of the grout placed in a container and allowed to cure. The BHE grout, initially prepared in the same way, may have absorbed water from the surrounding wet sand, and an increase in water content may have produced an increase in kg with respect to the measured value.
3.1. Preliminary Estimate of kg, with ks = 2.82 W/(mK)
3.2. Preliminary Estimate of the Thermal Diffusivity of the Sand, αs
3.3. Preliminary Estimate of ks
3.4. Estimate of kg, with ks = 3.18 W/(mK)
3.5. Estimate of the Time Evolution of the Temperature Distribution along the BHE Surface
3.6. Estimate of αs with Non-Uniform Temperature Rise of the BHE Surface
3.7. Estimate of ks with the Values of kg and αs Determined Previously
3.8. Estimate of (ρ c)g
4. Validation of the Simulation Codes
5. Validation of the Estimate by 3D Simulations
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Nomenclature
BHE | borehole heat exchanger |
C | heat capacity (J K–1) |
cp | specific heat capacity at constant pressure (J kg–1K–1) |
FLS | finite line-source |
GCHP | ground-coupled heat pump |
GSHP | ground-source heat pump |
GWHP | ground-water heat pump |
h | convection coefficient (W m–2K–1) |
ICMCS | infinite composite-medium cylindrical source |
ICMLS | infinite composite-medium line-source |
ILS | infinite line-source |
k | thermal conductivity (W m–1K–1) |
mass flow rate (kg s–1) | |
Nu | Nusselt number |
OMEC | one-material equivalent cylinder |
Pr | Prandtl number |
thermal power (W) | |
qb | heat flux per unit length supplied to the borehole (W m–1) |
qg | heat generation term (W m–3) |
r | radius (m) |
r0 | radius of the heat generating surface (m) |
Rb | borehole thermal resistance (m K W–1) |
Rb1 | modified borehole thermal resistance defined in Equation (2) (m K W–1) |
Re | Reynolds number |
RMSD | root mean square deviation |
S1, …, S5 | temperature sensors in the sand |
SWHP | surface-water heat pump |
t | time (s) |
T | temperature (K) or (°C) |
TRCM | thermal resistance capacity model |
TRT | thermal response test |
volume flow rate (m3 s–1) | |
x | horizontal coordinate (m) |
Greek symbols | |
α | thermal diffusivity (m2 s−1) |
(ρ c) | specific heat capacity per unit volume (J m–3 K–1) |
φ | angular coordinate in Equation (7) (rad) |
ψ | angle along the upper part of the BHE surface (degrees) |
Subscripts | |
1 | of the inlet pipe |
2 | of the outlet pipe |
30.5 | for r = 30.5 cm |
100 | for t = 100 min |
approx | approximate |
b | of borehole, of the borehole surface |
eff | effective |
fin | at the final instant |
fm | mean of the fluid |
g | of grout |
in | inlet |
out | outlet |
p | of polyethylene |
s | of sand |
s0 | at the top of the borehole external surface |
S1, …, S5 | of sensor S1, …, S5 |
w | of water |
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Flow Rate [L/s] | Re | Pr | Nu | h [W/(m2K)] |
---|---|---|---|---|
0.197 | 11,460 | 5.4147 | 86.533 | 1948.8 |
Mesh | Elements | t = 1500 min | t = 3106 min | ||||
---|---|---|---|---|---|---|---|
θs2 °C | θs3 °C | θs4 °C | θs2 °C | θs3 °C | θs4 °C | ||
1 | 1130 | 1.4529 | 0.5096 | 0.1592 | 2.4186 | 1.2300 | 0.6676 |
2 | 4520 | 1.4542 | 0.5105 | 0.1592 | 2.4198 | 1.2308 | 0.6676 |
3 | 18,080 | 1.4541 | 0.5104 | 0.1592 | 2.4198 | 1.2308 | 0.6676 |
4 | 72,320 | 1.4541 | 0.5104 | 0.1592 | 2.4198 | 1.2307 | 0.6676 |
Adopted | 28,320 | 1.4541 | 0.5105 | 0.1593 | 2.4198 | 1.2308 | 0.6676 |
Mesh | Elements | t = 1500 min | t = 3106 min | ||||
---|---|---|---|---|---|---|---|
θ1 °C | θ2 °C | θs1 °C | θ1 °C | θ2 °C | θs1 °C | ||
1 | 5472 | 15.7905 | 15.2773 | 5.6623 | 16.9054 | 16.3921 | 6.7453 |
2 | 21,888 | 15.7912 | 15.2780 | 5.6626 | 16.9061 | 16.3928 | 6.7456 |
3 | 87,552 | 15.7912 | 15.2780 | 5.6626 | 16.9062 | 16.3929 | 6.7456 |
Adopted | 52,152 | 15.7912 | 15.2780 | 5.6626 | 16.9062 | 16.3929 | 6.7456 |
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Naldi, C.; Jahanbin, A.; Zanchini, E. A New Estimate of Sand and Grout Thermal Properties in the Sandbox Experiment for Accurate Validations of Borehole Simulation Codes. Energies 2021, 14, 1149. https://doi.org/10.3390/en14041149
Naldi C, Jahanbin A, Zanchini E. A New Estimate of Sand and Grout Thermal Properties in the Sandbox Experiment for Accurate Validations of Borehole Simulation Codes. Energies. 2021; 14(4):1149. https://doi.org/10.3390/en14041149
Chicago/Turabian StyleNaldi, Claudia, Aminhossein Jahanbin, and Enzo Zanchini. 2021. "A New Estimate of Sand and Grout Thermal Properties in the Sandbox Experiment for Accurate Validations of Borehole Simulation Codes" Energies 14, no. 4: 1149. https://doi.org/10.3390/en14041149