Integrated Approach for Detecting Convection Effects in Geothermal Environments Based on TIR Camera Measurements

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Infrared thermography can be used for the thermal characterization of geothermal environments, through the monitoring of the surface that allows the computation of the thermal imbalance including the three heat transfer methods, consequently estimating thermal conductivity and convection heat rate.

Abstract

Thermal characterization of soils is essential for many applications, including design of geothermal systems. Traditional devices focus on the computation of thermal conductivity, omitting the analysis of the convection effect, which is important for horizontal geothermal systems. In this paper, a procedure based on the monitoring of the surface of the soil with a thermal infrared (TIR) camera is developed for the evaluation of the global thermal imbalance on the surface and in-depth. This procedure allows for the computation of thermal conductivity and global convection heat rate, consequently constituting a complete thermal characterization of the geothermal system. The validation of the results is performed through the evaluation of the radiometric calibration of the thermal infrared camera used for the monitoring and the comparison of the thermal conductivity values obtained in-depth, with traditional methods, and for the surface of the system.

1. Introduction

Soils are essential elements for life, due to their contribution to Ecosystem Services as providers of food production, water and climate regulation, energy provision and biodiversity [1]. Among many parameters considered as soil quality indicators [2], knowing the thermal conditions of the soil is important for several reasons: (i) soil temperature has an effect on soil biota for plant growth [3], (ii) thermal conductivity is critical in different fields, such as the automotive and aerospace industries, ceramics, glass and building materials sectors and the energy one, especially in geothermal applications, in which this parameter conditions the performance of the underground thermal exchange [4], (iii) the thermal properties of the soil are directly related with soil moisture [5] and (iv) soil thermal properties are the main factors for mass and energy exchange processes on Earth [6]. In the geotechnical context, controlling the thermal conductivity of the ground could act as a support for defining changes in composition, humidity and other influential properties such as porosity, resistance to freezing, etc. When trying to measure the thermal conductivity of the soil, practice problems are mainly linked to difficulties for covering a representative volume of the sample, in which the variations in the composition and properties are reflected.

In this sense, there are many different models for the determination of thermal conductivity of soil, both theoretical and experimental [7]. Regarding theoretical models, the authors of Reference [8] present an empirical model for estimation of soil thermal conductivity, which is based on the model in Reference [9] but including the influence of organic matter content and particle composition. This model is validated for 19 soils, in contrast to other models such as in References [10] and [11], that were specifically developed for sandy and peat soils, and for sand-kaolin clay mixtures, respectively.

Among experimental measurements, the authors of Reference [12] prove the capacity of a thermo-TDR (Time-Domain Reflectometry) probe to measure thermal conductivity of sand, while the authors of Reference [13] measure the thermal conductivity of different samples with a device based on the Guarded Hot Plate method.

However, all the models are based on the performance of measurements of certain parameters on punctual positions of the area under study. This procedure makes the determination of thermal conductivity time-consuming and restricted to certain areas. In order to perform larger measurements, satellite data has shown to be a valid input for the determination of ground thermal conditions at large-scale, as well as the determination of the spatial distribution of the parameter under study [14].

All models and measuring methodologies mentioned so far focus on the determination of the thermal conductivity of the soil, omitting the other heat transfer methods (convection, radiation). Conduction–convection algorithms are used for the estimation of soil apparent thermal diffusivity, since these result in more accurate thermal diffusivity values because of their consideration of the vertical heterogeneity of the soil [15]. Heat transfer through convection is also considered for the estimation of thermal diffusion, which is a key parameter for the design of Ground Source Heat Pump Systems [16]. Heat transfer through radiation has also been considered in the case of studies for horizontal ground-coupled heat pumps [17,18], but radiation was only considered from a theoretical point of view, calculating the input from the sun in the position under study. Thus, the a priori analysis of soil materials would benefit from a laboratory test for the determination of all the heat fluxes involved in the thermal imbalance. In this sense, infrared thermography can be an adequate technique, in its nature of producing thermal (temperature) images of the area under study [19].

The performance of a valid measurement with infrared thermography requires the consideration of certain requirements, since the thermal infrared technique is based on the measurement of the incoming radiation to the camera, and its conversion to temperature values [20], which presents many parameters of influence, mainly: radiation from surrounding objects, the emissivity of the object under study, the distance from camera to object and ambient temperature and humidity [21]. In addition to the consideration of the parameters of interest, the conversion from incoming thermal infrared radiation to temperature values requires the thermal or radiometric calibration of the camera, that is, the equation that inputs radiation and outputs temperature values.

Traditionally, thermal infrared cameras are calibrated using a temperature modulated black body target, under laboratory conditions [22]. This type of calibration reaches an accuracy of 0.06–0.2 K [22] for static measurements, going up to 0.32 K for measurements performed from Unmanned Aerial Vehicles, UAV [23]. It is a time-consuming and complicated process, with a high cost regarding the final price of the camera [24]. The radiometric calibration is usually based on Plank curves for the determination of the correlation between infrared radiation and temperature [25], although other mathematical relations, such as linear and polynomial models, have been tested [26]. Recent advances have focused on image processing techniques and neural networks for the radiometric calibration, in an image-by-image calibration [27] or with specifically trained artificial neural networks [26].

Provided the complexity of the radiometric calibration procedures, efforts are being made towards the simplification of the process, starting from the design of plate radiators in contrast to the more expensive cavity blackbodies [28]. The authors of Reference [29] prove the validity of a three-point calibration for measurements performed from UAV platforms, while the authors of Reference [30] perform a radiometric calibration using images acquired by the thermographic camera of an object at constant temperature.

The radiometric calibration can be completed with a geometric calibration that allows knowing the interior geometry of the camera: dimensions of the pixels, focal length of the camera lens and principal or central point of the image. In the case of thermal infrared cameras, the geometric calibration requires calibration boards specifically designed, either based on a temperature difference [31] or on an emissivity difference [32] between the targets and the background.

This paper presents a novel approach for the thermal characterization of geothermal environments. This approach consists of the monitoring of the soil surface with a thermographic camera, which has been radiometrically calibrated with a novel, low-cost approach. The thermal imbalances within the soil depth and in the soil surface allow for the determination of the thermal conductivity of the soil material, which is complemented with the thermographic monitoring for the computation of the convection effect. In addition, the comparison of thermal conductivity values computed in-depth and at the surface works as a validation of the radiometric calibration of the thermographic camera, together with its comparison with manufacturer values. Thus, the paper is structured as follows: Section 2 describes the devices used for the study, the methodology for the radiometric calibration of the camera and the modeling of the thermal behavior of the system (underground and at the surface). Section 3 shows the results obtained from the thermal imbalances in-depth (thermal conductivity) and at the surface (convection effect). Section 4 includes a discussion of the results, while Section 5 presents the conclusions reached.

2. Materials and Method

This research proposes a novel monitoring methodology using thermal imaging as an innovative technique regarding those traditionally used in geothermal studies. The approach is based on the principles of thermodynamics and its three mechanisms of thermal energy transmission: conduction, convection and radiation. The method has been successfully validated in a borehole physical model thanks to the temperature monitoring carried out by a set of sensors distributed in the borehole physical model, at laboratory scale and under controlled ambient conditions (temperature, humidity and no forced ventilation).

2.1. Devices

2.1.1. Borehole Heat Exchanger

As already mentioned, the thermal analysis included in this research has been performed on a borehole heat exchanger reproduced at laboratory scale [33]. The global device is constituted by a cylindrical polyethylene container (1.20 m of height and 0.77 m of interior diameter) that contains a vertical single-U heat exchanger. The ground is represented by a sandy material with a humidity of 15%, and the space between the ground and heat exchangers is filled with a high conductive grouting material [33]. This material is selected because of its heterogeneity and the security that there will be no changes in its composition during the tests, apart from humidity loss. In this way, the validation of the results will be possible.

As in a real shallow geothermal system, the working fluid (water) is responsible for allowing the thermal exchange between the ground and the installation. This fluid is taken from a glass container equipped with an electric resistance (heats and sets the fluid temperature to a certain value) and a pump that allows the circulation of the fluid inside the single-U heat exchanger. In the case of this setup, heat is obtained from the electric resistance and is dissipated in the ground through the working fluid.

With the aim of monitoring the temperature inside the system, the experimental setup is constituted by 13 sensors set in different areas of the experimental device: 6 temperature sensors placed inside copper tubes control the temperature of the ground (sandy material) in several locations, 3 temperature sensors to measure the temperature of the inlet and outlet fluid and inside the glass container, 2 sensors to control the ambient temperature, 1 laser sensor to measure external temperature of the main container and a flowmeter sensor to control the flow rate during the tests.

Figure 1 shows the distribution of the mentioned components that constitute the experimental geothermal heat exchanger. Additional information is also provided in the already published research of this setup [33].

2.1.2. Infrared Thermal Camera

The monitoring of the surface temperature was performed with an uncooled microbolometer thermal camera from Xenics (Gobi-384, Figure 2) at a frame rate of 1 image every 10 s. Among its main features, it is worth mentioning: a 384 × 288 focal plane array (FPA) sensor equipped with a 10 mm lens, a field of view of 51°(H) × 40°(V), an operating spectral range between 8 and 14 µm and a 16-bits radiometric resolution. For optimum results, this camera should be used in the ambient temperature range between 0 to 50 °C.

The camera is installed on top of the borehole heat exchanger, at a distance of 1.4 m from the surface of the sand. The geometry of the temperature sensors used and the container are shown in Figure 3, and more details are given in Reference [33].

2.2. Methodology

2.2.1. Underground Thermal Behavior

The geothermal heat exchanger reproduced in laboratory can be used to understand the thermal exchange that takes place in depth between the ground and the working fluid. Heat is provided through the working fluid which, in turn, increases the temperature of the surrounding ground (sandy material in the device). At the depth of the sensors located in the sand (in the middle of the total container height), heat is mainly transmitted by conduction. The sand temperature is measured at certain distances from the center of the borehole using the sensors installed in the cupper tubes. As explained in the already published research [33,34], heat transmission is expressed in cylindrical coordinates and the heat transfer relation in a cylinder is defined as:

$$\dot{Q}=-kA\frac{dT}{dr}\to \dot{Q}=2\pi kL\frac{T_1-T_2}{ln\left(r_2/r_1\right)}$$

(1)

where $\dot{Q}$ is the heat transfer rate, $k$ is the material thermal conductivity, $L$ is the borehole length, $dT/dr$ is the temperature differential in relation to the radius differential, $T_1$, $T_2$ are the temperatures in positions 1 and 2 and $r_1$, $r_2$ refer to the radius for positions 1 and 2.

Considering the above Equation (1) as an analogy to electricity circuits, the heat transfer rate could be expressed as the quotient of the thermal gradient in two positions ($\Delta T$) and the thermal resistance between them ($R_T)$. In this way, the thermal resistance between two points can be calculated applying Equation (2) [34]:

$$R_T=\frac{ln\left(r_2/r_1\right)}{2\pi kL}$$

(2)

After the corresponding calibration of the sensors that constitute the device and the validation of the heat transfer model [33], the experimental test consists of the circulation of the heated working fluid that comes from the thermal tank [35]. The system is then working and temperatures in the different locations of the sensors are registered. Once the stationary regime is reached, the calculation of the thermal resistance between two different positions and the ground thermal conductivity is possible. The estimation of this last property requires the calculation of the previously mentioned thermal resistance. When obtaining this resistance, the heat transfer going through the model must be defined, and its determination is made according to the following Equation (3):

$${\dot{Q}}_{cond}=\dot{m}{c}_p\left(T_i-T_o\right)$$

(3)

where ${\dot{Q}}_{cond}$ is the conduction heat rate (W), $\dot{m}$ is the mass flow rate, c_p is the working fluid specific heat and T_i and T_o are the inlet and outlet temperatures. For the solution of Equation (3), the mass flow rate is obtained from the product of the pump flow rate (measured by one of the sensors) and the fluid density. Finally, the thermal conductivity of the surrounding material is deduced as follows (Equation (4)):

$$k=\frac{ln\left(\frac{r_2}{r_1}\right)}{2\pi R_{T}L}$$

(4)

2.2.2. Surface Thermal Behavior

On the surface, all three possible heat transmission mechanisms (conduction, convection and radiation) occur.

• Convection

Experience shows that the convection heat transfer highly depends on certain properties of the fluid considered in the system but also on the geometric configuration and the roughness of the solid surface, and the conditions of the fluid flow (laminar or turbulent). According to Newton’s cooling law, the ratio of heat transferred by this mechanism is proportional to the temperature difference and is conveniently expressed by [35,36]:

$${\dot{q}}_{conv}={\displaystyle \sum }{h}_c·\left(T_s-T_{\infty }\right)$$

(5)

where ${\dot{q}}_{conv}$ is the convection heat rate (W/m²), $h_c$ is the convection heat transfer coefficient (W/m²·°C) and $T_s$ and $T_{\infty }$ are the temperature of the surface and the ambient temperature, respectively (°C).

For applying Equation (5), it is necessary to define the convection cell and the convection heat transfer coefficient that operates in the conditions of the present experimental test. With that aim, Equation (6) is used to determine the aforementioned convection heat transfer coefficient:

$$h_c=\frac{Nu·k_a}{Xc}$$

(6)

where $Nu$ is the Nusselt number, $k_a$ is the thermal conductivity of the air (W/m °C) and $Xc$ is the length of the boundary layer (m).

At the same time, the Nu and Xc are calculated following the expression of Equations (7) and (8):

$$Nu=0.664·\sqrt{Re}·P{r}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$$

(7)

$$Xc=\frac{\nu ·Re}{u_F}$$

(8)

where Re is the Reynolds number, Pr is the number of Prandtl, $\nu$ is the kinematic viscosity (m²/seg) and $u_F$ is the atmospheric air velocity (m/s).

The properties of the air required in the previous equations must be defined for a reference temperature (Equation (9)):

$$T_R=\frac{T_s-T_{\infty }}{2}$$

(9)

Once the air properties [37] are determined for the reference temperature ($T_R$), the values of Nu and Xc are calculated using the corresponding Equations (7) and (8), for then obtaining the convection heat transfer coefficient and the convection heat rate.

• Radiation

The total amount of energy being radiated by an object can be estimated by applying the Stefan-Boltzmann law, which states that for an object of a certain temperature, the radiated heat will be [36]:

$${\dot{q}}_{rad}={\displaystyle \sum }\epsilon ·\sigma ·T_s^4$$

(10)

where ${\dot{q}}_{rad}$ is the radiation heat rate (W/m²), $\epsilon$ is the emissivity constant that depends entirely on the material (sand) and is established at 1 for an ideal blackbody, $\sigma$ is the Stefan-Boltzmann constant and $T_s$ is the temperature of the sand (K).

• Conduction

The quantification of the heat exchanged by conduction at the surface is not as simple as in the above conduction and radiation effects. In the conditions of the experimental setup, according to the line source model, the heat transfer rate does not depend on the depth considered, in such a way that it remains constant in the horizontal slides of the model ($Q_{surface}=Q_{depth})$. From that equality, and given that heat is only transmitted in depth by conduction, the heat transferred by conduction at the surface can be obtained from the following expression [38]:

$$Q_{cond}=Q_{surface}-Q_{conv}-Q_{rad}$$

(11)

where $Q_{cond}$ is the heat transferred by conduction at the surface (W), $Q_{surface}$ is the total heat transferred at the surface (W) and $Q_{conv}$ and $Q_{rad}$ are, respectively, the heat transmitted by convection and radiation at the surface (W).

The estimation of each heat transfer mechanism was made with the help of the data collected by the thermal camera. While thermal cameras do not directly derive temperature values, they record the radiance emitted by surfaces that is directly related to the object surface temperature. To extract temperature measurements, a proper calibration of the thermal camera (both radiometrically and geometrically) must be previously performed. The Gobi-384 camera used in this study provided a manufacturer radiometric calibration dated from 2012, which benefited from an update, performed with a new low-cost radiometric calibration methodology. For its part, the geometric calibration of the camera had been previously performed through the methodology described in Reference [32] and for analyzing temperatures in other thermal studies [19].

2.2.3. Low-Cost Radiometric Calibration of the Thermal Camera

• Thermal data acquisition

The low-cost radiometric calibration was performed under controlled laboratory conditions with a constant ambient temperature of 25 °C and through an ad-hoc black body simulator. Specifically, an aluminum container painted with 0.9–0.95 emissivity matte black paint was used (Figure 4). It was filled with distilled water and heated in a laboratory oven to 53 °C. Its cooling down to 3.5 °C was monitored by both the Gobi-384 and a set of thermocouples fixed to the container surface, used to obtain the reference temperature for calibration. The cooling to 3.5 °C was achieved by gradual addition of distilled water ice cubes as the temperature tended to stabilize. As a result of the experiment, a wide temperature range was achieved (3.5–53 °C) in order to apply the calibration model and derive temperature measurements from objects that meet this temperature range.

With the thermal camera installed in a fixed position at 1.5 m distance to the black body simulator, a total of 258 thermal images were acquired at a rate of 1.5 images/min, so that the calibration lasted approximately 3 h.

• Calibration algorithms

To aid in the camera calibration process, the authors developed a script in Matlab^® to select a representative area of the aluminum container (64 pixels) in all thermal images. The area selected is close to the set of thermocouples, and the mean value of that set of pixels is stored, in digital levels.

Regarding the calibration model, both the linear and polynomial models (Equations (12) and (13)) were tested since they are widely used to perform temperature adjustments [24].

RT = a + b × DL

(12)

RT = a + b × DL + c × DL²

(13)

where RT is the reference temperature of the black body simulator measured by the thermocouples, DL is the mean value of the pre-selected pixels in digital levels from the thermal images and a, b and c are the fit regression coefficients.

In order to compare both model adjustments (Figure 5), the coefficient of determination (R²) was analyzed.

For the evaluated temperature range, a better fit was obtained through the polynomial model, in which a coefficient of determination of 0.9974 was obtained, compared to the 0.9908 obtained from the linear model.

2.3. Experimental Procedure

The objective of integrating the experimental borehole heat exchanger with the infrared thermal camera is obtaining a global thermal model that allows evaluating all the thermal mechanisms (usually extremely difficult to define) that take place in a geothermal system. As already explained, the temperature monitoring in the borehole heat exchanger enables the calculation of the ground thermal conductivity (sand) since only conduction effects appear in depth. However, when measuring with the infrared thermal camera (surface temperature), the three main thermal transfer mechanisms are detected (Figure 6).

As shown in Figure 5, the methodology included in this work can be summarized as follows:

• Monitoring of the surface temperature using the infrared thermal camera.
• Monitoring of the temperature of the ground in-depth through the set of sensors implemented in the experimental borehole heat exchanger.
• Calculation of the thermal conduction (in-depth) and the sand thermal conductivity from the data obtained from the sensors of the experimental setup.
• Determination of the convection and radiation effects that occur in the surface of the setup during the performance of the test.
• Establishment of the general thermal exchange balance and obtaining of the thermal conductivity of the material in the surface.

The approach consisted of the simultaneous acquisition of temperature data every 5 s from the temperature sensors [39] installed at a depth of 0.5 m (Figure 7a), and every 10 min from thermal images for 3.5 days. As a result, 530 thermal images were acquired and processed. The sector analyzed with the thermal images was limited to a quarter of the surface area of the borehole heat exchanger. Specifically, the sector on the left (Figure 7b) where the temperature sensors were located, and furthest from the surface heat source (pipes) to minimize its thermal influence on the surface temperature measurements. Figure 7b shows the radial temperature attenuation with the distance from the heat source (in the center).

3. Results

The experimental geothermal setup was working for a total period of 3.5 days. During this time, data were simultaneously registered by the temperature sensors installed in the borehole heat exchangers and by the infrared thermal camera. The following subsections present the results of the radiometric calibration of the thermal camera but also the results of the heat transfer analysis in depth and surface.

3.1. Validation of the Low-Cost Radiometric Calibration through the Manufacturer Information

One of the ways to validate the low-cost radiometric calibration of the thermal camera is by comparing the calibration results with those from the existing manufacturer calibration.

The radiometric calibration model obtained through the low-cost calibration is the one represented in Equation (14), with T being the surface temperature values in Celsius, while the manufacturer’s calibration model responds to Equation (15).

T = −11.620 + 0.0036 × DL − 4 × 10⁻⁸ × DL²

(14)

T = −17.049 + 0.0036 × DL − 3 × 10⁻⁸ × DL²

(15)

Figure 8 shows that the low-cost calibration results in a homogeneous absolute error, with a maximum value of 3.46 °C. However, the manufacturer´s calibration results in an increase of the absolute error for lower temperatures, with a maximum error value of 8.34 °C. For both calibrations, the greatest uncertainty occurs in the temperature range between 16 and 18 °C, probably due to the interference of the temperature of the environment of the camera in this temperature range. However, (i) the greatest error in the low-cost calibration is below the nominal precision of the camera (±2 °C), and (ii) the thermal conductivity test is performed for temperatures between 28 and 40 °C, where the error in the low-cost calibration is stable.

With respect to the calibration validation in the temperature range evaluated (3.5–53 °C), it can be concluded that: (i) less uncertainty is obtained when using the low-cost radiometric calibration model, and that (ii) the largest discrepancy between both models (the provided by the manufacturer and the one from the low-cost calibration), was ±5.25 °C in terms of absolute error.

3.2. Thermal Conduction in Depth

Temperature sensors S2, S3 and S4 were the ones used for the estimation of the heat conduction in depth and the thermal conductivity of the filling material. The selection of these sensors, located in the copper tubes of the experimental setup at the depth of 0.5 m, was made with the aim of avoiding thermal influences from the heat source and covering the central area of the sandy material (dismissing the registers in the middle and the opposite end of the setup). The results of the measurements of the sensors can be observed in Figure 9.

From all the data presented in Figure 8, only the registers belonging to the stationary regime are valid to be included in further calculations, since it is in this period when the equalities presented in Equations (1)–(4) can be used. The absolute error of the calibration of the thermographic camera in this temperature range (28–40 °C) is below the nominal precision of the camera (2 °C), in such a way that the error in the thermal conductivity introduced by the temperature measurement is minimized.

After applying the corresponding correction factors to each sensor, the stationary regime was selected considering the moment in which temperature values remain constant over time.

Once the stationary part of the test is identified and the temperature values are corrected, the heat transferred by conduction was calculated by applying Equation (3). With that aim, the mass flow rate was firstly defined by multiplying the flow rate of the pump (registered by one of the setup sensors) and the density of the working fluid (water).

Table 1. Determination of the heat transferred by conduction.

$\dot{\mathit{m}}$	${\mathit{T}}_{\mathit{i}}$ (°C)	${\mathit{T}}_{\mathit{o}}$ (°C)	${\dot{\mathit{Q}}}_{\mathit{c}\mathit{o}\mathit{n}\mathit{d}}\left(\mathbf{W}\right)$
38.78	59.14	58.85	69.30

presents the calculated average mass flow rate and the average inlet and outlet fluid temperatures required in the estimation of the global heat transfer.

The next step consists of the calculation of the ground thermal conductivity from the expression presented in Equation (4). After defining the thermal resistance between each couple of sensors from the quotient of the temperature difference and the heat transfer rate, the thermal conductivity of the sandy material was calculated for the portion of material between sensors S2–S3, S3–S4 and S2–S4. The results of these calculations are included in

Table 2. Average temperature for the sensors included in the study (during the stationary regime) and thermal conductivity of the sandy material calculated for the different sets of sensors.

Average Temperatures (°C)			Thermal Conductivity (W/mK)
S2	S3	S4	K2–3	K3–4	K2–4
38.78	34.69	30.27	0.590	0.445	0.515

, which also presents the average temperature values of sensors 2–4 used for the estimation of the thermal conductivity parameter.

3.3. Thermal Exchange at the Surface

As already mentioned in the methodological section, when analyzing the heat transmission in the surface, it is mandatory to consider that the three heat transmission mechanisms are possible.

Starting with the convection process, as stated in Equation (5), the quantification of the convection heat rate requires defining a series of parameters concerning the convection cell. First of all, the reference temperature (Equation (9)) was set (by considering the average sand and ambient temperatures) at the value of 44.40 °C. This value constitutes the starting point to establish the properties of the air in the conditions of the experimental test.

Table 3. Properties of the air for the reference temperature of the study.

$\mathit{\rho }(\mathbf{kg}/{\mathbf{m}}^3)$	$\mathit{v}\left({\mathbf{m}}^2/\mathit{s}\right)$	$\mathit{\mu }\left(\mathbf{P}\mathbf{a}·\mathbf{s}\right)$	Ka (W/mK)	$\mathit{\alpha }\left({\mathbf{m}}^2/\mathbf{s}\right)$	Pr	Cp (kJ/kg·°C)
1.092	17.6 × 10−6	19.12 × 10−6	0.0265	24.8 × 10−6	0.71	1.014

presents the values of these properties for the reference temperature [40,41].

Considering the information of Table 3, a Re = 5 × 10⁵ (for a turbulent regime) and $u_F$ = 0.5 m/s (measured during the tests) are established, and $Xc$ and Nu are then respectively calculated for obtaining the convection heat transfer coefficient $h_c$ (Equations (6)–(8)).

Table 4. Calculation of Xc, Nu and $h_c$ required for the calculation of the global convection heat rate.

$$\mathit{X}\mathit{c}\left(\mathbf{m}\right)$$	Nu	$${\mathit{h}}_{\mathit{c}}$$ (W/m2·°C)
1.76	418.86	6.18

shows the aforementioned parameters.

When trying to determine the total convection heat rate in the case under study, it is important to consider the variation of the air temperature. For including this variation in the calculation, the region between the sensors under study (S2–S4) was divided into a series of rings (Figure 10). As shown in Figure 9, each ring selected in the area between sensors corresponds to one pixel of the camera.

In this way, the convection heat rate is extended to all the pixels for which temperature is evaluated by the thermal chamber corresponding to all the radial rings from S2 to S4.

Table 5. Determination of the convection heat rate at the surface.

d (m)	TS (K)	${\dot{\mathit{q}}}_{\mathit{c}\mathit{o}\mathit{n}\mathit{v}}(\mathbf{W}/{\mathbf{m}}^2)$	${\dot{\mathit{Q}}}_{\mathit{c}\mathit{o}\mathit{n}\mathit{v}}\left(\mathbf{W}\right)$	${\dot{\mathit{q}}}_{\mathit{r}\mathit{a}\mathit{d}}(\mathbf{W}/{\mathbf{m}}^2)$	${\dot{\mathit{Q}}}_{\mathit{r}\mathit{a}\mathit{d}}\left(\mathbf{W}\right)$
0.168	320.63	139.845	---	358.621	---
0.171	320.27	137.623	0.535	357.927	1392
0.175	320.04	136.194	0.553	356.895	1450
0.179	319.71	134.184	0.557	355.446	1475
0.183	319.96	135.709	0.575	356.545	1511
0.186	319.96	135.687	0.587	356.529	1542
0.190	319.82	134.827	0.595	355.908	1570
0.194	319.76	134.454	0.605	355.640	1600
0.198	319.71	134.180	0.615	355.443	1630
0.201	319.68	133.998	0.626	355.312	1661
0.205	319.55	133.193	0.634	354.734	1689
0.209	319.38	132.153	0.641	353.986	1716
0.212	319.27	131.470	0.649	353.497	1745
0.216	319.09	130.344	0.655	352.690	1772
0.220	318.99	129.702	0.663	352.232	1800
0.224	318.77	128.331	0.667	351.253	1826
0.227	318.64	127.586	0.674	350.721	1854
0.231	318.53	126.906	0.682	350.238	1882
0.235	318.44	126.303	0.690	349.808	1910
0.239	318.35	125.752	0.698	349.417	1939
0.242	318.27	125.289	0.706	349.088	1967
0.246	318.26	125.196	0.716	349.022	1997
0.250	318.25	125.157	0.727	348.995	2028
		Σ${\dot{Q}}_{conv}$	14.050	Σ${\dot{Q}}_{rad}$	37,955

shows the calculation of the convection heat rate for the distances from the center to each pixel and the total heat exchanged by convection in the surface. Each value of ${\dot{q}}_{conv}$ is obtained from the average sand temperature registered by the thermal infrared camera and using Equation (5).

Focusing on the radiation effect, its quantification is made following the same procedure, which considers the average temperature data per pixel from sensors S2 to S4 in the stationary regime. Applying Equation (10) and considering the sand emissivity ($\epsilon$ = 0.6) and the Stefan-Boltzmann constant ($\mathsf{\sigma }=5.67\times {10}^{-8}\mathrm{W}\cdot {\mathrm{m}}^{-2}·{\mathrm{K}}^{-4})$, the summations of the heat energy exchanged by radiation are presented in Table 5.

Finally, the heat conduction at the surface is estimated from the theoretical assumptions based on the finite line source model (Equation (11)). Applying Equation (11), the total heat transmitted by conduction at the surface is 17.29 W.

4. Discussion

4.1. Experimental Validation of the Low-Cost Radiometric Calibration

An additional procedure to validate the low-cost radiometric calibration of the thermal camera is by comparing the thermal conductivity derived from the calibrated thermal images against the thermal conductivity resulting from the borehole heat exchanger geothermal model, which can be considered as the reference value.

The temperature monitoring approach proposed by comparing thermal data from (i) the set of sensors introduced in the matrix of the borehole physical model at a depth of 0.5 m, and (ii) from the thermal images (Figure 10) has allowed:

-

The characterization of thermal models by comparing internal and surface temperature data.

-

The determination of the spatial distribution of radial temperature attenuation from a central heat source.

-

The validation of the low-cost radiometric calibration based on thermal conductivity data.

The 3.5-day monitoring test for both internal and surface temperature can be seen in Figure 11, in which the change of transient to stationary thermal regime has been established after 46.7 h of experiment. In addition, a greater difference temperature is observed between sensors in the interior of the borehole heat exchanged than the derived temperature from the surface. In this case, a representative surface area for the depth location of each of the 3 temperature sensors (S2, S3 and S4), within the analyzed sector (Figure 12), was evaluated.

With the register of the thermal infrared camera, the thermal conductivity of the sand at the surface can be calculated and compared with that obtained for the section at 0.50 m using Equation (16):

$$k=\frac{qln\left(\frac{r_2}{r_1}\right)}{2\pi \Delta T}$$

(16)

The thermal conductivities obtained (using Equation (16) and data from previous sections) can be compared to get an idea of the consistency of the results obtained.

Table 6. Thermal conductivities obtained.

$\mathbf{Section}$	Thermal Conductivity (W/m·K)
Surface	0.52 *
Sensor depth (0.5 m)	0.46

* Mean of conductivities obtained from the data in Table 2.

shows the different results obtained.

The slight decrease can be explained by a lower compaction of the material at the surface in comparison with the compaction at 0.5 m depth (where the sensors are), as well as a reduction of moisture content in the pores.

4.2. Thermal Balance Model (Applications and Future Developments)

The combination of tests developed in this work can be used in a way adapted to different situations of measurements. As an example, we are going to cite a series of future research cases where the research group has the prospect of including these results:

• The possibility pointed out by this work of using non-destructive tests for the thermal characterization of materials can be useful, especially considering those with difficult or no access for the installation of thermocouples that measure the evolution of temperatures during the measurement process.
• Once the thermal characteristics of a material are known, the possibility of knowing the conditions of the convection cell that is established in the surface temperature exchanges could also be interesting. This would allow, for example, obtaining data related to the behavior of the evapotranspiration phenomenon in soils and other surfaces. This is also carried out non-destructively and even without the need for contact with the medium to be analyzed.
• Given a known thermal characteristic of a material, its state can be monitored by temporal analysis of the evolution of its thermal conductivity, which can change depending on certain processes that are happening in it.
• The installation of an external heat source, such as a halogen lamp, with the role of sunlight would allow for a more realistic simulation of the scenario, in such a way that the methodology developed could also be applied to the calculation of the thermal conductivity due to radiation.

It is possible that new applications derived from the global test presented in this work will appear, especially in the field of evaluating materials with difficulties to access them in a physical way to perform the temperature measurements required.

As an example, a general sensitivity analysis of the thermal conductivity related to changes in the porosity is included to evaluate the geotechnical conditions of a certain material, in this case felsic rocks. As can be seen in

Table 7. Thermal conductivities for felsic rocks with different quartz content and porosity [42].

Rock Type	Solidity (1-Porosity)	Thermal Conductivity (W/mk)
Felsic rock (20% quartz content)	0.5	1.50
	0.6	1.65
	0.7	1.85
Felsic rock (30% quartz content)	0.5	1.80
	0.6	1.90
	0.7	2.05
Felsic rock (40% quartz content)	0.5	1.95
	0.6	2.15
	0.7	2.34
Felsic rock (50% quartz content)	0.5	2.05
	0.6	2.25
	0.7	2.55

, the thermal conductivity data for different types of felsic rocks decrease when their porosity increases, that is, when their fracturing increases. Therefore, a decrease in thermal conductivity can mean a decrease in its mechanical capabilities.

As a way to control the stability in the structural capacities of this type of rocks, the thermal conductivity can be measured periodically with a procedure like the one described in this work.

5. Conclusions

This research work presents a global test about thermal properties in a heat exchange process. Different thermal processes are measured under different conditions and using different measurement techniques. The coherence of the results obtained (such as the thermal conductivity obtained through different approaches and for different areas of the material) allows thinking that the techniques described are valid for their use in the characterization of the thermal processes studied. In addition, considering the validity of the data obtained through the observations of the thermal camera, this route can be proposed as an alternative to the direct measurement on the material to characterize it thermally at a given time (assuming steady state in that process).

The advantage of the techniques used derives from the inclusion of the thermal camera in the acquisition of temperature and position data, and makes some processes described in this work suitable for their application as non-destructive tests and also when there is no physical access to the material under evaluation (for example in radioactive environments or hazardous atmospheres). In some types of processes, it can be useful to know the variation of the thermal conductivity of a material in real time to relate its changes with possible variations on its structure. Derived from these particularities, some lines of research may emerge from this work (detailed in Section 4.2).