# Distributed Thermal Response Tests Using a Heating Cable and Fiber Optic Temperature Sensing

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## Abstract

**:**

## 1. Introduction

^{−1}[13]) to heat the water circulating in the GHE and generate a temperature difference of 3–7 °C at the inlet and outlet of the GHE, reproducing the operation of a geothermal system. The bulk thermal conductivity is estimated over the length intercepted by the borehole [14]. Depending on the site properties, the estimated bulk thermal conductivity can be affected by the ground heterogeneity [15] and groundwater flow [16,17]. The use of fiber optic distributed temperature sensing (FO-DTS), a technology to monitor temperature at high frequency and spatial resolution [18], was proposed by Fujii et al. [15,19] to measure the vertical temperature profile during TRT as an alternative method to assess the vertical distribution of the ground thermal conductivity and associated ground heterogeneity. Subsequently, fiber optic cables have been used to validate analytical models of the vertical temperature evolution [20] and to measure borehole wall temperatures [21] during TRTs. FO-DTS and a heat trace cable was additionally used by Freifeld et al. [22] to estimate the ground thermal conductivity as a function of depth and to infer ground surface temperature history.

^{−1}[24,23] and the required power of less than 1 kW was supplied by connecting the TRT unit to the electrical grid [24]. Tests with a continuous heating cable wete achieved in boreholes having a depth of approximately 30 m, while boreholes with more than 100 m in length have been the subject of tests with heating sections to keep a low power requirement [26].

^{−1}while, at the other site, a second test was achieved in a borehole of equivalent length with a continuous heating cable and a heat injection rate of less than ~10 W·m

^{−1}. This paper describes first heating cable TRTs combined with FO-DTS and a first continuous heating cable TRT in a long borehole with a low heat injection rate. Conventional TRT with water circulation was previously conducted at both sites and the results were used for comparison purposes. The Rayleigh number stability criterion, a methodology to evaluate convective heat transfer in vertical groundwater wells proposed by Love et al. [28], was applied to assess convective heat transfer during the tests. FO-DTS technology used to monitor temperature in boreholes at high frequency and spatial resolution along a fiber-optic cable [18] provided the information needed to evaluate the Rayleigh number stability criterion. Thermal conductivity profiles were estimated using the infinite and finite heat source solution depending on the heating cable used and an uncertainty analysis of each test was performed.

## 2. Field Test Methodology

^{−1}, after which the thermal recovery was monitored. The double U-pipe heat exchanger in Orléans was equipped with a FO-DTS system that allows measuring the temperature in two of the pipes of the GHE. The test was implemented with a continuous heating cable of 95 m. The submersible temperature sensors were placed every 6 m, from 5.50 to 83.60 m depth, and the FO-DTS system was programed to collect data each meter, every 10 min. Due to the small pipe diameter, the heating cable and the fiber optic cables could not be placed in the same pipe. Thus, it was installed in other pipes located at 0.06 and 0.82 m from the pipes containing the fiber optic. The four pipes in this GHE are separated with spacers. The test started with the measurement of the undisturbed subsurface temperatures followed by heat injection at a rate of 9.90 W·m

^{−1}and, finally, thermal recovery monitoring. A test with a heat injection rate below 10 W·m

^{−1}was possible due to the improvement of temperature sensor resolution accomplished in recent years. Sensors with a resolution less than 0.05 °C are able to measure small temperature changes allowing to perform the tests with a low heat injection rate, which was difficult to achieve with previously available technologies.

## 3. Test Analysis

#### 3.1. Thermal Conductivity Assessment

_{0}), defined as the increase of temperature with respect to the initial temperature T

_{0}, were reproduced with the analytical solutions using the temporal superposition principle to consider the recovery period where heat injection is zero. Numerical simulation results described by Raymond et al. [10] indicate that the temperature inside a slice of a borehole varies significantly according to the horizontal measurement location during the heat injection, making it necessary to consider the horizontal position of the temperature sensor to analyze the test. However, during the recovery period, the temperature becomes uniform and the position of the sensor does not influence the measurement. The horizontal position of the temperature sensor is difficult to determine; therefore, the thermal conductivity was estimated using only the late recovery temperatures at each depth. A linear heat source solution of finite length with a dimensionless g-function (Equation (1), [24,25]), which depends on the Fourier number (Fo), was used to analyze the test with heating sections:

^{−1}) is the heat injection rate per unit length; ${\alpha}_{s}$ (m

^{2}s

^{−1}) and λ

_{s}(W·m

^{−1}·K

^{−1}) are the thermal diffusivity and conductivity of the subsurface, respectively; r (m) is the radial distance from the heat source; h (m) is the active heating length; and t

_{off}is the time when heat injection was stopped. The heat capacity of the subsurface was assumed based on the rock type identified with drill cuttings and it was also used to determine the thermal diffusivity (${\alpha}_{s}$) and to compute Fourier’s number (Equations (2) and (3)). The heat injection rate q (W·m

^{−1}) for each heating section was assumed similar and determined according to Joule’s and Ohm’s laws:

_{off}to simulate the recovery period [34] and temperatures are plotted as a function of a normalized time t/(t − t

_{off}) to find thermal conductivity with the slope method:

#### 3.2. Uncertainty Analysis

_{tot}(Equation (10)), the electrical resistivity of the heating section δR

_{h}(Equation (11)) and the heat injection rate δq (Equation (12)) were determined using the accuracy of the individual parameters.

^{−1}(Equation (15)), allowing for determination of the error associated with the analytical solution. The uncertainty δZ was determined over 95% confidence (Equation (16); [27]) by the generalization of the error estimation of non-linear regression (Equation (17)). From Equation (1), we have

_{z}was determined similarly to the standard deviation of a slope (Equation (17)), considering the standard deviation of the calculated temperature increases (ΔT

_{c}) and the measured temperature increases (ΔT

_{m}; Equation (18)).

_{xg}, Equation (19)). Finally, the uncertainty of the thermal conductivity estimation δλ was calculated considering the error of the heat injection rate and the error of Z (Equation (20)).

_{xt}; Equation (24)). Finally, the uncertainty of the thermal conductivity δλ is calculated as a function of the error of the heat injection rate and the slope (Equation (25)).

#### 3.3. Free Convection Assessment Inside GHE Pipe

_{cr}identified with Equation (27).

^{−2}) is the gravitational acceleration, β (K

^{−}

^{1}) is the thermal expansion coefficient of the fluid, H (m) is the height of the water column, ΔT (°C) is the difference in temperature between the top and the bottom of the system, $\nu $ (m

^{2}·s

^{−1}) is the fluid cinematic viscosity and α

_{w}(m

^{2}·s

^{−1}) is the fluid thermal diffusivity. The critical Raleigh number computed for comparison is determined according to the cylinder geometry:

_{t}(m) is the height of the water column. The critical Rayleigh number is therefore a function of the well aspect ratio r

_{a}(-) only and an increment in the well aspect ratio results in a reduction of the Ra

_{cr}value [28].

_{t}was defined as a critical length along the heating cable, rather than the well depth considered for groundwater well with a geothermal gradient. Nevertheless, defining a representative value for this variable is complicated because the critical Rayleigh number depends on the well aspect ratio. Increasing the well aspect ratio (larger radius and shorter height) results in a reduction of the critical Rayleigh number. Consequently, three different values of H

_{t}were defined to evaluate its influence on the results of Rayleigh number stability criterion assessment.

_{t}of 3.00, 6.00 and 9.00 m. These values represent the mean height of the water column used to calculate actual Rayleigh numbers, the length of the maximum temperature difference between heating and non-heating sections and the total length of a system of a single heating and a non-heating section, respectively.

## 4. Results

^{−1}, a lower rate compared to the first test. This rate was defined for the 100 m heating cable to cope with the potential difference provided by the electrical grid (120 V) and respecting the limits of the voltage regulator (140 V and 10 A). The total energy consumption during the test was 127.60 kWh and the average potential difference and electrical current intensity were 104 V and 8.99 A, respectively.

#### 4.1. Thermal Conductivity Estimation

^{−3}·K

^{−1}associated with shales was assumed to compute the temperature increments and reproduce observations recorded for the first test in Québec City. The first 30 h of the recovery period was neglected and the analysis considered only the last 40 h to avoid the influence of the sensor location in the temperature measurement.

^{−1}·K

^{−1}, with an average value of 2.02 W·m

^{−1}·K

^{−1}(Figure 9a). Minimal and maximal thermal conductivities of 1.71 and 2.03 W·m

^{−1}·K

^{−1}were estimated in the conventional TRT with water circulation using the slope method and considering a constant heat injection rate. The maximal value was estimated during a TRT with a constant flow rate, and the minimal value during a second TRT in which the flow rate was reduced after 30 h of heat injection. A thermal conductivity of 1.75 W·m

^{−1}·K

^{−1}was further obtained with analysis of the conventional TRT using the line source model with the superposition principle to consider variations in the heat injection rate [29,38]. The thermal conductivity points estimated with the heating cable section methodology differ between −25.30% and 1.70% from the bulk value estimated during the conventional TRT, using the line source model with a variable heat injection rate, while it is closer to the upper range of the slope analysis assuming constant heat injection rate.

^{−1}·K

^{−1}[39], while thermal conductivity points identified with the heating cable methodology varied from 1.08 to 2.18 W·m

^{−1}·K

^{−1}with a mean value of 1.47 W·m

^{−1}·K

^{−1}. Differences varying from −48.30% to 26.53% were found when comparing the thermal conductivity profile with the bulk result of the conventional test. The maximal difference was found at 71.60 m depth, corresponding to the maximal thermal conductivity (Figure 9b).

#### 4.2. Uncertainty Analysis

^{−1}·K

^{−1}was determined for the test with heating cable sections (Table 4) with a mean value of 0.31 W·m

^{−1}·K

^{−1}(15.18%). An uncertainty varying from 0.02 to 0.05 W·m

^{−1}·K

^{−1}was obtained for the continuous heating cable test (Table 5), with a mean value of 0.03 W·m

^{−1}·K

^{−1}(2.14%).

#### 4.3. Free Convection Assessment

_{t}= 3 m, the Rayleigh

_{actual}> Rayleigh

_{critical}for all the cases considered (28.5, 95, and 121 h), suggesting the presence of significant heat transfer by free convection in all the water column. Nevertheless, the recorded temperature between 41 and 43 m depth did not have an important variation and free convection at this depth was not expected. The choice of H

_{t}= 3 m can be too small for the studied system.

_{t}= 6 m, Rayleigh

_{actual}at the beginning of the heat injection (28.5 h) and during the recovery period (121 h) did not exceed the critical value. However, Rayleigh

_{actual}> Rayleigh

_{critical}at the end of the heating period (95 h) for a portion of the heating section and up to 1.50 m above, suggesting the presence of free convection. The third case, considering H

_{t}= 9 m, shows a critical Raleigh number that is always larger the than actual Rayleigh number, suggesting the absence of free convection. Defining H

_{t}= 9 m can, however, overestimate Rayleigh’s critical value, since temperature increases far above from the heating section is a strong evidence of heat transfer due to free convection.

## 5. Discussion

#### 5.1. Thermal Conductivity Estimation

^{−1}·K

^{−1}was estimated from the test with heating cable sections conducted in Quebec City, Canada. This value is 15.45% higher than the thermal conductivity evaluated during a conventional TRT made on the same GHE with water circulation analyzed with the infinite line source model and considering variations in the heat injection rate [29,38]. Analysis of a conventional TRT can, however, depends on the analysis method itself. The mean thermal conductivity estimated with the heating section TRT carried out in Quebec City differs by only 0.47% when compared to the maximum bulk thermal conductivity evaluated with the slope method for a conventional TRT performed in the same GHE [29,38]. A previous field test with heating cable sections reported by Raymond et al. [24] indicates a mean thermal conductivity 12% higher than that estimated with a conventional TRT. These results suggest that TRT with heating cable sections can slightly overestimate the ground thermal conductivity when compared to a conventional TRT. Free convection generated above the heating cable section can explain this difference since TRT analysis is made considering conductive heat transfer only. Free convection in a TRT using heating cable sections was identified by Raymond and Lamarche [23], through numerical simulation of the borehole temperature evolution during a TRT. The use of perforated plastic disks placed at the extremities of the heating section had been suggested to reduce the possible free convection. Nevertheless, the effect of natural convection was still identified during the full-scale test with heating cable sections, indicating that the four perforated plastic disks are not sufficient to totally control free convection during the heat injection period. A test with a reduced heat injection rate could be performed to evaluate if natural convection induced by the heating sections decreases.

^{−1}·K

^{−1}was estimated for the test with a continuous heating cable carried out in the GHE made of a double U-pipe drilled in limestone in Orléans, France. This value corresponds exactly to the bulk thermal conductivity estimated with the conventional TRT [39]. Previous comparison between TRT using continuous heating cable and a conventional TRT has not been made before. A constant initial temperature increase with depth, influenced by the natural geothermal gradient was observed before the TRT and seems to remain during the heat injection period with the continuous heating cable. Free convective heat transfer due to the geothermal gradient or caused by anthropogenic heat sources is possible since it has been observed in groundwater wells [28] and in the water column of open boreholes [41,42]. However, this weak convection mechanism can be neglected for TRT analysis, where heat conduction triggered upon heat injection is assumed more important than any natural convection. The actual Rayleigh number for the test performs with the continuous heating cable did not exceed the critical Rayleigh number evaluated between 18 m and 42 m depth, indicating that convective heat transfer can be neglected in this case. In future works, the thermal conductivity could be evaluated using the FO-DTS to allow an increase in the spatial resolution currently limited by the number of temperature sensors and to provide an additional independent estimation of the thermal conductivity.

#### 5.2. Uncertainty Analysis

^{−1}). The estimate of the electric resistance of the heating sections is the parameter with the most important uncertainly when calculating the heat injection rate. The uncertainty of the heat injection rate corresponds to 76–96% of the total test uncertainty. The remaining proportion of uncertainty corresponds to the error associated with the analytical model (δZ). The final test uncertainty ranged 14.6–16.4%, showing a reduction of approximately 10% compared to the uncertainly estimated during the laboratory tests performed in three 10 m wells [27]. The power injected underground for the laboratory tests was divided into three heating cable sections using a power switching supplies because three single heating section tests were carried out at the same time. The efficiency of this switch influenced the uncertainty of the heat injection rate, increasing the total uncertainly of the test. The electric current transmitted to the cable inducing the thermal power injected underground for the full-scale test presented in this study was controlled by a voltage regulator and measured using a power meter at the entry of the cable assembly, allowing to reduce the uncertainty associated to the evaluation of the heat injection rate. To adapt error calculation for conventional TRT to continuous heating cable TRT, both of which are using the slope analysis method, some modifications were introduced to the approach proposed by Witte [35]. The error of the heat injection rate and the slope of the regression of the temperature increments were calculated as a combination of errors for the continuous heating cable TRT. It was not necessary to calculate the average temperature, as in a conventional TRT, because the analysis was based on the temperature increments measured at the depth of each temperature sensor. The error associated with the heat injection rate only depended on the electric current intensity and the electric potential difference that are measured with accuracy in the field. Thus, the accuracy of the temperature sensors only influences the error attributed to the slope when plotting the temperature response.

^{−1}, which corresponds to 99% of the total uncertainty. The remaining 1% corresponds to the uncertainty of the slope. The final uncertainty of the thermal conductivity profile was 2.14%, a lower value than the 5.10% estimated for a conventional TRT by Witte [35]. The accuracy of the heat injection rate appears to be the key factor, in any TRT method, that directly affects the uncertainty of the thermal conductivity estimate. Achieving an accurate assessment of the heat injection rate for TRTs with flowing water based on flow rate and temperature measurements tend to be more difficult than evaluating the heat injection rate transmitted through a continuous heating cable with potential difference and current intensity measurements. However, these measurements become complex for heating sections TRT, increasing the uncertainly associated with the analysis performed with the finite line source solution. This last analysis method with the finite line source solution requires assuming the heat capacity of the geological formation [23], adding an additional uncertainly to the analysis.

#### 5.3. Comparison of TRT Methods

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Fo | Fourier’s number (-) |

G | gravitational acceleration (L t^{-2}) |

g | finite heat source function (-) |

h | length of heat source (L) |

H | height of the water column (L) |

I | electric current intensity (I) |

m | slope (-) |

n | number of observations (-) |

q | heat injection rate per unit length (M L t^{−3}) |

R | electric resistance (M L^{2} t^{-3} I^{−2}) |

r | radius (L) |

Ra | Rayleigh number (-) |

S | error of the estimate |

t | time (t) |

T | temperature (T) |

U | electric potential difference (M L^{2} t^{-3} I^{−1}) |

Greek symbols | |

α | thermal diffusivity (L^{2} t^{−1}) |

λ | thermal conductivity (M L T^{−1} t^{−3}) |

δ | uncertainty |

σ | standard deviation |

ν | cinematic viscosity (L^{2} t^{−1}) |

subscripts | |

a | aspect-ratio (-) |

c | calculated |

cr | critical |

h | heating |

nh | non-heating |

m | measured |

off | end of heat injection |

s | subsurface |

t | critical height |

tot | total |

0 | initial condition |

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**Figure 2.**Apparatus for TRT with: (

**a**) heating cable sections; and (

**b**) a continuous heating cable (modified from Raymond [26]).

**Figure 4.**Temperature evolution during the three steps of the TRT in Quebec City with heating cable sections.

**Figure 5.**Observed temperature at the middle of a non-heating section (40 m) and at the top (45 m) and the middle (45.5 m) of a heating section.

**Figure 6.**Temperature evolution during the three steps of the TRT in Orléans with a continuous heating cable.

**Figure 7.**Observed and computed temperature increments (Equation (1)) during the recovery period for the submersible sensor located at 45 m depth for the test with heating cable sections.

**Figure 8.**Observed temperatures during the recovery period for the submersible sensor located at 53.6 m depth for the test with a continuous heating cable.

**Figure 9.**Point subsurface thermal conductivity (crosses) obtained with the (

**a**) heating cable sections in Québec City and (

**b**) the continuous heating cable in Orléans. The results are compared to the bulk values provided by the conventional TRT with water circulation (green and red dashed lines).

**Figure 10.**Temperature distribution during the TRT around a typical heating section at 45–46 m depth.

**Figure 11.**Verification of the Rayleigh number stability criterion considering different critical lengths H

_{t}to calculate the critical Rayleigh number (dashed lines) for the test with heating cable sections.

**Figure 12.**Verification of the Rayleigh number stability criterion for the test with a continuous heating cable.

Parameter | Quebec City | Orléans |
---|---|---|

Heating cable type | Heating section | Continuous heating cable |

Test analysis method | Finite line source | Infinite line source |

Borehole configuration | Single U-pipe | Double U-pipe |

TRT start date | 31 August 2016 | 3 July 2017 |

TRT start time | 11:20:00 | 13:00:00 |

TRT end date | 7 September 2016 | 2 August 2017 |

TRT end time | 12:11:00 | 10:00:00 |

Borehole depth (m) | 150 | 100 |

Heating cable length (m) | 100 | 95 |

Borehole diameter (mm) | 114 | 180 |

Pipe diameter (mm) | 32.00 | 26.20 |

Duration of undisturbed temperature monitoring (h) | 26.00 | 100.00 |

Heat injection period (h) | 73.00 | 135.70 |

Thermal recovery period (h) | 70.00 | 481.30 |

Total test duration (h) | 169.00 | 717.00 |

Initial temperature (°C) | 8.20 | 13.10 |

Heat injection rate (W·m^{−1}) | 42.50 | 9.90 |

Temperature sensors vertical distance (m) | 10 | 6 |

Parameter Measured Repeatedly during the Test | Absolute Error | Relative Error (%) | Reference Value-Quebec City | Reference Value-Orléans |
---|---|---|---|---|

Temperature T (°C) | 0.10 | |||

Electric power P (W) | 0.02 | 0.002 | 985.13 | 940.45 |

Electric current intensity I (A) | 0.02 | 0.17 | 8.71 | 8.99 |

Electric potential difference U (V) | 0.02 | 0.01 | 113.12 | 104.70 |

Parameter measured once and separately | ||||

Length of the heating cable H (m) | 0.01 | 0.01 | - | 95.00 |

Length of the heating section L_{h} (m) | 0.01 | 0.81 | 1.24 | - |

Electrical resistance per unit length of the non-heating section R_{nh} (Ω·m^{−1}) | 0.0002 | 0.50 | 0.04 | - |

Continuous Heating Cable | Absolute Error | Relative Error (%) | Reference Value |

Heat injection rate q (W·m^{−1}) | 0.21 | 2.12 | 9.91 |

Slope m′ (-) | 0.001 | 0.23 | 0.55 |

Heating Cable Section | Absolute Error | Relative Error (%) | Reference Value |

Total electrical resistance of the cable assembly R_{tot} (Ω) | 0.12 | 0.94 | 12.99 |

Electrical resistance of the heating cable sections R_{h} (Ω) | 0.13 | 14.26 | 0.90 |

Heat injection rate q (W·m^{−1}) | 6.07 | 14.29 | 42.50 |

Analytical model Z (-) | 1.06 | 5.00 | 21.14 |

**Table 4.**Uncertainly of the point thermal conductivity assessment for the heating cable sections test in Québec City.

Depth (m) | Thermal Conductivity (W·m^{−1}·K^{−1}) | Absolute Error δk (W·m^{−1}·K^{−1}) | Relative error (%) |
---|---|---|---|

5.3 | 1.72 | 0.27 | 15.61 |

15.5 | 1.86 | 0.31 | 16.38 |

25.4 | 2.03 | 0.30 | 14.85 |

35.6 | 1.91 | 0.29 | 15.02 |

45.5 | 2.07 | 0.30 | 14.62 |

55.7 | 2.19 | 0.33 | 14.95 |

65.6 | 2.08 | 0.32 | 15.44 |

75.5 | 2.12 | 0.33 | 15.31 |

85.4 | 2.12 | 0.32 | 14.88 |

95.6 | 2.06 | 0.30 | 14.73 |

Average | 2.02 | 0.31 | 15.18 |

**Table 5.**Uncertainly of the point thermal conductivity assessment for the test with a continuous heating cable in Orléans.

Depth (m) | Thermal Conductivity (W·m^{−1}·K^{−1}) | Absolute Error δk (W·m^{−1}·K^{−1}) | Relative Error (%) |
---|---|---|---|

11.6 | 1.08 | 0.02 | 2.13 |

17.6 | 1.27 | 0.03 | 2.13 |

23.6 | 1.18 | 0.03 | 2.13 |

29.6 | 1.23 | 0.03 | 2.13 |

35.6 | 1.4 | 0.03 | 2.13 |

41.6 | 1.35 | 0.03 | 2.13 |

47.6 | 1.47 | 0.03 | 2.14 |

53.6 | 1.57 | 0.03 | 2.14 |

59.6 | 1.52 | 0.03 | 2.13 |

65.6 | 1.72 | 0.04 | 2.14 |

71.6 | 2.18 | 0.05 | 2.15 |

77.6 | 1.55 | 0.03 | 2.13 |

83.6 | 1.53 | 0.03 | 2.14 |

Average | 1.47 | 0.03 | 2.14 |

Parameter | Continuous Heating Cable | Heating Cable Sections | Conventional with FO-DTS |
---|---|---|---|

Power requirement | Low | Low | High |

Test time | +++ | ++ | ++ |

Equipment complexity | + | ++ | +++ |

Cost | $ | $$ | $$$ |

Measured parameters | T_{0}, λ_{s} | T_{0}, λ_{s} | T_{0}, λ_{s}, R_{b} |

Surface disturbances | No | No | Yes |

Analysis method | Infinite line source solution | Finite line source solution | Infinite line source solution |

© 2018 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Vélez Márquez, M.I.; Raymond, J.; Blessent, D.; Philippe, M.; Simon, N.; Bour, O.; Lamarche, L.
Distributed Thermal Response Tests Using a Heating Cable and Fiber Optic Temperature Sensing. *Energies* **2018**, *11*, 3059.
https://doi.org/10.3390/en11113059

**AMA Style**

Vélez Márquez MI, Raymond J, Blessent D, Philippe M, Simon N, Bour O, Lamarche L.
Distributed Thermal Response Tests Using a Heating Cable and Fiber Optic Temperature Sensing. *Energies*. 2018; 11(11):3059.
https://doi.org/10.3390/en11113059

**Chicago/Turabian Style**

Vélez Márquez, Maria Isabel, Jasmin Raymond, Daniela Blessent, Mikael Philippe, Nataline Simon, Olivier Bour, and Louis Lamarche.
2018. "Distributed Thermal Response Tests Using a Heating Cable and Fiber Optic Temperature Sensing" *Energies* 11, no. 11: 3059.
https://doi.org/10.3390/en11113059