# Experimental and numerical study of microwave heating in sand columns: Water saturation effects on temperature profiles and heat transfer mechanisms

Maxime Cochennec, Dorian Davarzani, Stéfan Colombano, Jean-Yves Richard, Mohamed Krimissa, Mélanie Lorthioy, Abderrahim Jardani, Michel Quintard

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Maxime Cochennec, Dorian Davarzani, Stéfan Colombano, Jean-Yves Richard, Mohamed Krimissa, et al.. Experimental and numerical study of microwave heating in sand columns: Water saturation effects on temperature profiles and heat transfer mechanisms. Chemical Engineering Science, 2026, 325, 10.1016/j.ces.2026.123508. hal-05528396

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# Experimental and numerical study of microwave heating in sand columns: Water saturation effects on temperature profiles and heat transfer mechanisms

Maxime Cochennec<sup>a,∗</sup> <sup> ID </sup>, Dorian Davarzani<sup>a</sup>, Stéfan Colombano<sup>a ID</sup>, Jean-Yves Richard<sup>b</sup>, Mohamed Krimissa<sup>c</sup>, Mélanie Lorthioy<sup>c</sup>, Abderrahim Jardani<sup>d</sup>, Michel Quintard<sup>e ID</sup>

## ABSTRACT

Subsurface heating finds applications in soil remediation and enhanced resource recovery, with microwave heating offering advantages over conventional methods (conductive heating, hot vapor injection). However, the transition from laboratory-scale studies conducted in microwave ovens to in situ applications using embedded antennas in relevant environments remains poorly studied. This study investigates microwave heating mechanisms of a 2.45 GHz coaxial antenna in sand across varying water saturation levels.

Results reveal that the reported advantages of microwave heating—rapid and homogeneous temperature rise—require a critical water content. For near-dry sand (<5% water saturation), temperature increased but exhibited steep, non-constant radial gradients. Conversely, at higher saturation levels (14% and 22%), temperature profiles rapidly developed near-constant, reduced gradients. Despite these differences, modeling indicates that thermal conduction, rather than direct absorption, remains the dominant heat transfer mechanism across all tests. Furthermore, the study identifies a significant discrepancy between nominal power and actual absorbed power, highlighting a low conversion efficiency. These results suggest that while microwave heating is effective for moderately saturated soils, further optimization of power and heat delivery is essential to promote its applicability to thermal desorption of polluted soil.

## 1. Introduction

Microwave heating (MWH) is widely regarded as one of the most energy-efficient methods for drying various porous materials (Rattanadecho & Makul, 2016). The most common applications of microwave heating with hygroscopic porous media include the drying of food and agricultural products (Guo et al., 2017), wood (Oloyede & Groombridge, 2000), coal (Lu et al., 2022), and concrete or ceramics (Agrawal, 1998; Huang et al., 2023). MWH has been proven efficient in geosciences for thermal decomposition of ores (Li et al., 2020a; Li et al., 2020b), and the extraction of oil (Abdulrahman & Meribout, 2014; Bera & Babadagli, 2015) and natural gas hydrates (Fan et al., 2025). All these industrial microwave heating applications demonstrate advantages in terms of process speed compared to more traditional techniques.

Microwave heating is also applied as a thermal soil remediation technique. While its behavior is well characterized at small scales in closed setups such as microwave ovens (Abramovitch et al., 1999; Cho

et al., 2020; Cochennec et al., 2024; Davarzani et al., 2022; Falciglia et al., 2013; Falciglia et al., 2016a; Falciglia et al., 2016b; Windgasse & Dauerman, 1992; Yuan et al., 2006) its application at larger laboratory scales and field scales remains insufficiently explored (Falciglia et al., 2018a). Among others, Davarzani et al. (2022) attempted to replicate the behavior of a single-slot antenna in situ soil microwave heating system using a sand-packed column directly connected to a microwave oven port. Their results demonstrated that water saturation plays a critical role in microwave heating efficiency, with 12% saturation providing the optimal balance for rapid and uniform heating for the investigated soil material. For this saturation, the dielectric properties are favorable, allowing a quick temperature increase followed by stabilization due to the latent heat of vaporization. Lower water saturations lead to slower heating, while higher saturations imply that a significant part of the energy expenditure is devoted to evaporating the water.

In situ MWH primarily involves installing several antennae emitting microwaves – i.e., electromagnetic waves with frequencies ranging

<footnote><sup>a</sup> BRGM, Orléans F-45060, France</footnote>

<footnote><sup>b</sup> VEOLIA Hazardous Waste Europe, Industrial & Innovation Department, Limay 78520, France</footnote>

<footnote><sup>c</sup> EDF R&D, LNHE, Châtou F-78400, France</footnote>

<footnote><sup>d</sup> Université de Rouen, M2C, UMR 6143, CNRS, Morphodynamique Continentale et Côtière, Mont Saint Aignan F-76821, France</footnote>

<footnote><sup>e</sup> IMFT, Université de Toulouse, CNRS, Toulouse F-31000, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: m.cochennec@brgm.fr (M. Cochennec).</footnote>
from 300 MHz to 300 GHz – into application wells, mainly in the unsaturated zone (Colombano et al., 2020). MWH can also be utilized onsite to treat excavated contaminated soils (Buttress et al., 2016; Falciglia and Vagliasindi, 2014). As one of the thermal remediation methods, MWH aims to elevate the soil temperature to facilitate the removal of volatile or semi-volatile organic compounds such as BTEX or kerosene (Zhang et al., 2024). Organic contaminants tend to be more mobile at higher temperatures, as their viscosity decreases with increasing temperature. Depending on the temperature achieved and the physicochemical properties of the pollutants, they can also be volatilized and/or extracted from the soil by the hot water vapor stream. In this regard, MWH is similar to other “low-temperature” remediation techniques, such as electrical resistance heating and hot steam extraction (Tripplett Kingston et al., 2010). Here, low temperature refers to techniques that differ from techniques such as vitrification or smoldering (J. Li et al., 2023), which operate at temperatures above 300 °C. While remediation yield is typically very high above these temperatures, significant physicochemical parameters of the soil start to be irreversibly altered (O'Brien et al., 2018).

The main difference between MWH and other thermal remediation techniques is that MWH allows for selective and direct heating of the soil material rather than depending on non-selective conductive heat transport. Consequently, MWH is unaffected by common heat insulators, such as air. Advection-based heating with hot fluid is sensitive to soil heterogeneities, particularly permeability contrasts, with injected fluids mostly occupying more permeable zones. Although for MWH, uneven heating can occur due to the heterogeneous distribution of dielectric materials (mainly water), this effect is often temporary, as water changes state during heating. In contrast, the sensitivity of advection-based heat transfer to soil physical heterogeneity, such as permeability contrasts, is constant.

During MWH, when the condition of local electric equilibrium is valid, the spatial average absorbed power on a representative elementary volume of porous media (⟨P⟩), is given by

$$ \langle P \rangle = \frac{1}{2} f \epsilon_{\text{eff}}'' \langle \mathbf{E} \rangle \cdot \langle \mathbf{E} \rangle \quad (1) $$

with ⟨E⟩ the spatial average electric field magnitude, f the wave frequency, and $\epsilon_{\text{eff}}''$ the effective dielectric loss factor of the bulk material (De Río & Whitaker, 2000). When the condition of local electric equilibrium is valid, the effective dielectric loss factor is expressed as $\epsilon_{\text{eff}}'' = \sum_i \phi_i \epsilon_i'',$ with $\phi_i$ and $\epsilon_i''$ the volume fraction and the dielectric loss factor of phase i, respectively. In practice, the effective dielectric loss factor is highly dependent on the water content, because the water dielectric loss factor is higher than most other phases (Metaxas & Meredith, 2008). For example, the dielectric loss factor for dry sand (particle size between 2 to 4.8 mm) is approximately 0.11, while it is about 0.82 for wet sand (Horikoshi et al., 2011). Water saturation is thus essential, and previous studies showed that temperature and/or removal of contaminant is higher when the water saturation is higher (Davarzani et al., 2022; Falciglia et al., 2013; Falciglia et al., 2018b). Based on these points, MWH is generally regarded as a cost-effective soil remediation technique. However, this technology is still rarely used by soils and site remediation professionals (Jones et al., 2002). The reasons may stem from the resistance commonly associated with adopting new techniques. Still, it is important to recognize that MWH is more challenging to understand and master compared to conductive heating or hot steam injection. This is because the heating process, as illustrated by Equation (1), is a highly nonlinear phenomenon that depends on parameters that are difficult to obtain. From a technical perspective, the entire heating setup is more complex, with various applicator/waveguide designs, power levels, and frequencies.

Krouzek et al. (2018) conducted in situ MWH (6 kW, 2.45 GHz, and 25 days of heating). They first used a waveguide with only an applicator at its end (i.e., a waveguide whose end is an aperture). However, the

temperature near the waveguide was excessively high due to limited contact with the dielectric material. They then installed hollow sections along the antenna using a ceramic tube. Various polluted sites were investigated, resulting in different soil types (mixed soil or debris smaller than 1 cm). Although the overall mass of pollutants decreased near the antenna relative to the initial state, the authors acknowledged that the results were inconsistent at a distance of 50 cm or more. They even observed a higher concentration of pollutants in the soil further from the antenna than before the test. This may have occurred due to soil heterogeneity, which could mislead the interpretation of soil sampling. Alternatively, it may be due to vapor transfer of contaminants from the heated area to the cooler area, resulting in the recondensation of contaminants. They proposed that further investigations be based on sophisticated numerical simulations to overcome the complexities of in situ studies. In this regard, they noted that their experimental data could be used to validate this model. However, the task is complicated by the complexity of the soil and the lack of characterization, such as missing dielectric properties. A similar study was conducted previously by Chien (2012), who successfully remediated hydrocarbon-contaminated soils. The author used a rectangular antenna and a nominal microwave generator power of 2 kW. Only one side of the antenna permits the microwave emission, and the temperature in the soil was not monitored. Hence, no conclusion on heating uniformity could be drawn, but the author found a decrease in trichloroethylene concentration in soil after the treatment. Kawala and Atamańczuk (1998) performed the most comprehensive study using a pilot-scale system with a 0.6 kW and 2.45 GHz operating antenna. This antenna radiated into a 1-meter-high, 0.9-meter-diameter column filled with sand through slots made along the antenna. The initial water saturation in the sand tank averaged 5%, and the sand was spiked with tetrachloroethylene. Most of the pollutant was removed after 75 h of treatment. Their study demonstrates that intermediate-scale experiments are necessary to better understand the complex mechanisms involved during microwave heating. However, they did not investigate the effect of water content and heating homogeneity (only a r-z plane was considered), although they are essential parameters for industrial applications. To fill this gap, our setup introduces a methodological novelty by enabling in situ temperature measurements without interrupting microwave heating, avoiding the use of metallic thermocouples. Unlike the field application of the technique, we operated without applying a vacuum, thus preserving the perturbation in the temperature distribution of the soil. This allows for a more accurate investigation of heating phenomena, providing a solid foundation for modeling with simplified boundary conditions.

Finally, the microwave antenna proposed by Pauli et al. (2006) consists of a coaxial waveguide with slots distributed to ensure that the electromagnetic field along the antenna remains uniform. The authors tested the antenna in wet sand. They found that the temperature was relatively constant with depth (e.g., along a vertical line parallel to the antenna) but decreased near the surface because of the heat loss. Although this study was conducted in a 3D sand container, the temperature profile was evaluated only along one dimension, and thus, it does not confirm even temperature distribution. We replicated this antenna design from its original configuration, which was intended to enable axisymmetric microwave propagation.

While microwave heating has demonstrated significant advantages, its application to subsurface environments remains technically challenging. The transition from laboratory-scale studies conducted in close microwave cavities to applications at larger scales using embedded antennas is currently poorly understood. Specifically, the diametrical symmetry of the heating, the impact of water saturation, and the conversion efficiency to heat are understudied in embedded antenna systems. To address this gap, we conducted well-instrumented experiments in homogeneous sand embedded in a 3D metric column. The originality of this study lies in investigating both the 3D heating distribution, specifically validating the thermal symmetry through diametrically opposed measurements. The impact of the water content on MWH
performance was also evaluated. This is conducted at a significantly larger scale than typical laboratory experiments performed in cavities, where the medium dimensions here exceed the waves' penetration depth. This work is significant as a symmetrical temperature distribution is essential for the field implementation of the technique. Furthermore, understanding the impact of water content and energy conversion efficiency is important for the technology's ability to operate across different conditions in the vadose zone, ranging from near-dry conditions to the capillary fringe.

## 2. Materials and methods

### 2.1. 3D column and microwave apparatus

The column consisted of a 220 L polypropylene truncated cone sealed at the bottom. Its height was 120 cm, with a bottom and top diameter of 45 cm and 58 cm, respectively. The microwave apparatus included a 2 kW generator, a 2.45 GHz magnetron, and a 2 m long coaxial antenna, designed by the company IDCO following the design from Pauli et al. (2006). Unlike the designs used by other authors (e.g. Chien (2012)), which may lead to localized heating, the Pauli et al. (2006) design is specifically intended to promote radial heating. Radiation is localized to the lower one-meter section of the antenna, where a series of slots was intentionally machined. The outer conductor has a radius of 35 mm, and the inner conductor radius is 4 mm. The antenna features rectangular slots, each 47 mm wide and 9 mm high. The slots are opened in the outer conductor along the circumference while leaving the inner conductor intact. The slots are spaced longitudinally with a center-to-center spacing of 61 mm, corresponding to half the free-space wavelength at 2.45 GHz. Every 61 mm, the two diametrically opposed slots are rotated by 90°. This choice is motivated by the nominal design requirement to optimize emissions, notably maximizing the S11 parameter, and maintain field homogeneity within the radiating structure (Pauli et al., 2006). To prevent overheating and ensure stable operation, a cooling fan is installed on the magnetron to dissipate the thermal energy generated during microwave emission. The entire external surface of the column was covered with aluminum foil to prevent microwave leakage. The sand was clean quartz sand with a granulometry of 0.7–1.3 mm. Its absolute permeability measured 285 darcy, and its porosity was 0.35. This material was chosen to represent sandy soil, typical of alluvial aquifers, particularly vulnerable to contamination by volatile and semi-volatile organic compounds. A schematic of the experiment is presented in Fig. 1.

### 2.2. Instrumentation

We monitored the temperature in various locations using fiber optic temperature sensors (Optocon AG TS3/2 with a length of the probe tip of 1 cm and an accuracy of ± 0.2 °C). These sensors offer a key advantage as they are transparent to microwave radiation, thereby avoiding potential interference between the microwaves and temperature measurements—a common issue when using conventional metallic thermocouple sensors (Davarzani et al., 2022). The fiber optic sensors were arranged to obtain two radial lines from one side of the antenna to the other side and one vertical line in front of the slots, as illustrated in Fig. 1. The coordinates of each sensor are provided in Table 1, following a cylindrical coordinate system. The vertical line of sensors gives information about the temperature distribution in front of each slot, while the radial line gives information about the radial temperature distribution. The opposite radial line is used to validate the diametrical symmetry. Finally, the initial water saturation level was measured based on a time-domain reflectometer (TDR) sensor (CS655 from Campbell Scientific), with an accuracy of ± 3%.

<figure><img src="image_3.png" /><figcaption>Fig. 1. Schematic of the microwave heating experiments. The setup comprises a metric column (1.2 m high and 0.45 to 0.58 diameter) packed with near-dry or wet sand. Fiber optic temperature sensors are used to assess heating homogeneity and the effects of water content.</figcaption></figure>

<caption>Table 1</caption>

Coordinates of each fiber optic sensor within the column (cylindrical coordinate system). Fiber optics are used instead of thermocouples because they are transparent to microwaves.

<table><thead><tr><td>Sensor name</td><td>r(cm)</td><td>z(cm)</td><td>θ(°)</td></tr></thead><tbody><tr><td>h<sub>0</sub>, h<sub>0</sub>'</td><td>2</td><td>50</td><td>0 / 180</td></tr><tr><td>h<sub>1</sub>, h<sub>1</sub>'</td><td>5</td><td>50</td><td>0 / 180</td></tr><tr><td>h<sub>2</sub>, h<sub>2</sub>'</td><td>10</td><td>50</td><td>0 / 180</td></tr><tr><td>h<sub>3</sub></td><td>22</td><td>50</td><td>0</td></tr><tr><td>v<sub>0</sub></td><td>2</td><td>12.5</td><td>0</td></tr><tr><td>v<sub>1</sub></td><td>2</td><td>37.5</td><td>0</td></tr><tr><td>v<sub>3</sub></td><td>2</td><td>62.5</td><td>0</td></tr><tr><td>v<sub>4</sub></td><td>2</td><td>87.5</td><td>0</td></tr></tbody></table>

### 2.3. Protocol

The antenna was first placed in the column as in Fig. 1, i.e., aligned with the middle axis of the column and about 10 cm from the bottom of the column. Then, a layer of about 5 cm of gravel topped with geotextile was added to homogenize the inlet/outlet conditions. After this first layer, sand was poured and wet-packed by pluvation into the column until it reached the level of a slot where an optical fiber was to be placed. The packing steps were performed only once, after which the residual saturation was established through the imbibition/drainage cycle. Drainage at different flow rates allowed for changes in the initial water saturation, denoted as $S_i$, by subtracting the volume of water injected from the volume of water pumped using a buffer tank. Three initial water saturation values were considered, from near-dry sand ($S_i$ = 4%) to wet sand ($S_i$ = 14%, and $S_i$ = 22%). The near-dry condition was obtained by injecting air for two days after the previous experiment concluded, and the water that drained from the bottom was recovered. The sand was not replaced for all experiments, and residual humidity remains, as demonstrated by a TDR measurement at the top of the column and 10 cm deep (4% water saturation). All experiments last for 180
min.

## 2.4. Numerical simulations

Numerical simulations were performed to help analyzing the experimental results. These numerical simulations were conducted by solving the steady state 2D radial heat conduction equation with a non-homogeneous source term:

$$ \frac{1}{r} \frac{d}{dr} \left( rk \frac{dT}{dr} \right) + S_0 \exp(-\alpha r) = 0 $$

with k the thermal conductivity (W/(K m)), $S_0$ (W/m$^3$) the amplitude of the source term, and $\alpha$ (m$^{-1}$) the attenuation constant, which can be linked to the penetration depth $d_p = 1/\alpha$ of the waves into the porous media. The form of the source term is consistent with the standard modeling approach for microwave heating (Jones et al., 2002). The above equation has been solved by the finite-element method, using FreeFEM (Hecht, 2012). The boundary conditions at the inner circle (antenna) and outer circle (edge of the column) were fixed based on the measured temperature at these points.

# 3. Results and discussion

## 3.1. Temperature distribution

Fig. 2 shows the temperature evolution over time, focusing on the radial lines of sensors ($h_i$ and $h'_i$). Each subplot corresponds to one initial water saturation level, i.e., 4%, 14%, and 22%, from left to right, respectively. After 180 min of heating, the highest temperature recorded is 106 °C, reached by the sensor closest to the antenna ($h_0'$ at 2 cm), and for the near-dry sand (4%). Notably, the symmetrical sensor $h_0$ displays a temperature of less than 100 °C. This 10% discrepancy is not observed for the two other initial water saturation levels. Indeed, for initial water saturations of 14% and 22%, the temperature at a given distance from the antenna is consistent with that on the opposite side. This could be quantified by the relative temperature difference between each pair of diametrically opposed probes:

$$ \text{Relative difference} = \frac{|T_{\theta=0} - T_{\theta=180}|}{\text{mean}(T_{\theta=0}, T_{\theta=180})} \times 100 $$

The relative temperature difference between diametrically opposed probes remains low, with a global mean difference below 5% for the 14% and 22% water saturation levels. The maximum relative difference reached 11%, at $r = 2$ cm for the near-dry sand case. For wet sand, the maximum temperature recorded is 75 °C ($S_i = 22%$), aligning with the temperature reported by Pauli et al. (2006) for a similar water saturation. In their study, the temperature was not quantified but presented by

<figure><img src="image_4.png" /><figcaption>Fig. 2. Evolution of the temperature in time at different radial positions for the three initial water saturations (4%, 14%, and 22%).</figcaption></figure>

the authors as high. While these temperatures do not reach the boiling point of most organic pollutants, the temperature continued to rise at the end of the experiment, and the slopes are positive, indicating that higher temperatures can be reached. Most significantly, the results displayed in Figure 2 suggest that one could expect diametrical symmetry in temperature when employing a cylindrical waveguide with slots, such as the one utilized in this study.

It appears to contradict Equation (1) that the maximum temperature is observed for the near-dry sand heating. Based on this equation, it would imply that the effective dielectric loss factor of the near-dry sand is higher than that of the wet sand, which is not the case according to laboratory measurements (Horikoshi et al., 2011). Because all sensors show higher or equal temperature for the near-dry sand, the most likely explanation is that the magnitude of the incident electric field (i.e., from the slots) is higher when the porous medium is near-dry rather than wet. It is in accordance with the impedance mismatch across the antenna and the surrounding medium, which means that some fraction of the power cannot be delivered to the surrounding medium (voltage standing wave ratio) (Sadeghi et al., 2017). It has been shown with a similar antenna that impedance mismatch increases with higher water saturation (Pauli et al., 2011). However, this remains a hypothesis as no specific instrument, such as a reflectometer or SWR meter, was used to quantify the standing wave ratio during the experiments. Furthermore, this observation lacks support from the existing literature, since most studies use a domestic microwave oven where the porous medium is significantly smaller than the cavity, thus having little impact on the incident electromagnetic field (Davarzani et al., 2022; Falciglia et al., 2013; Horikoshi et al., 2011).

Figure 3 shows the temperature at 5 cm from the antenna across different vertical positions under three saturation conditions (4%, 14%, and 22%, from left to right, respectively). The range of temperature values among the various vertical positions is significantly larger in the near-dry sand, with a 50 °C difference between the upper and lower sensors. The temperature range of the wet sand is significantly smaller, particularly at an initial water saturation of 22%. Consequently, while the vertical temperature profile can be regarded as homogeneous for the wet sand cases, it cannot be for the near-dry sand.

Here again, temperatures are higher in the near-dry sand, confirming that the incident electromagnetic field significantly differs between near-dry and wet sand. The vertical temperature gradient observed for the near-dry sand aligns with a gradient of residual water, with higher saturation at the bottom of the column due to gravity drainage. However, no plateau is observed after the temperature reaches 100 °C on sensor v0. This indicates that the sand is indeed in near-dry conditions at that point; otherwise, boiling water would induce a plateauing temperature around 100 °C. The remaining water is thus most likely bound

<figure><img src="image_5.png" /><figcaption>Fig. 3. Evolution of the temperature in time at different vertical positions for the three initial water saturations (4%, 14%, and 22%). Sensor v0 is at the bottom of the column. The vertical temperature profile shows a gradient from top to bottom for near-dry sand.</figcaption></figure>
water, which has a higher boiling point.

## 3.2. Impact of the water saturation

Figure 4 shows the radial temperature distribution from the antenna toward the column wall at different times (represented by shades of blue) and for the three initial water saturations (i.e., 4%, 14%, and 22%, from left to right, respectively). The temperature gradient along the radial coordinate is steeper for the near-dry sand. Firstly, the temperature close to the antenna is higher, as previously shown. The temperature decreases faster with the radial coordinate than with the wet sand. Finally, the temperatures of all three cases converge to a similar value at 22 cm from the antenna. However, about 10 cm from the antenna, the temperature is slightly higher for the wet sand. It should be noted that previous works using cavity applicators (ovens or waveguides) have shown a different behavior. For example, some studies have reported differences in the heating kinetics between wet and dry sand, even though both reach similar temperatures once the sand is fully dried at the end of the experiments (Davarzani et al., 2022; Horikoshi et al., 2011), or higher temperatures for wet sand (Falciglia et al., 2013). Generally, all works reported higher rates of temperature increase in wet sand than in dry sand. This is not the case in this work; however, all observations also indicate that microwave heating through a coaxial antenna exhibits significantly different behavior between near-dry and wet sand. The difference in the radial temperature profile is one of the most striking differences and can only be observed if the system's dimensions are greater than the microwave penetration depth. The radial temperature profile is now studied more quantitatively.

Table 2 shows the coefficient of determination for the linear regression of $T$ to $r$, $1/r$, and $log(r)$, after 150 min of heating. The logarithmic relation is chosen because it is characteristic of the purely conductive case. The linear and inverse relationship, although not specific to a heat-transport mechanism, indicates either a constant or rapidly changing temperature gradient. The radial temperature profile is well described by a linear relation with $r$ when the sand is wet. On the other hand, a linear relation is not appropriate to describe the temperature profile for near-dry sand, while a profile in

$T \propto 1/r$ reproduces the data well. It indicates a sharp decrease near the antenna, which is not shown for wet sand, and implies a higher radius of influence in the case of wet sand compared to near-dry sand. The reason is that the thermal conductivity is lower for near-dry sand than for wet sand, and consequently, the heat is less efficiently transported further from the antenna.

Interestingly, all temperature profiles can be well reproduced as

$T \propto log(r)$, which is in line with a conductive mode of heat transport. In theory, microwave heating is a volumetric heating method, so the temperature profiles should not be efficiently reproduced by a radial

<figure><img src="image_6.png" /><figcaption>Fig. 4. Radial temperature distribution at different times (shade of blue) and for the three initial water saturations (4%, 14%, and 22%). The radial temperature gradient is higher for near-dry sand.</figcaption></figure>

<caption>Table 2</caption>

Coefficient of determination ($R^2$) for the linear regression of $T$ with three different functional forms of $T(r)$, evaluated at each initial water saturation after 150 min of heating.

<table><thead><tr><th></th><th>4%</th><th>14%</th><th>22%</th></tr></thead><tbody><tr><td>T<sub>αr</sub></td><td>0.72</td><td>0.99</td><td>0.96</td></tr><tr><td>T<sub>α1/r</sub></td><td>0.97</td><td>0.71</td><td>0.79</td></tr><tr><td>T<sub>αlog(r)</sub></td><td>0.95</td><td>0.94</td><td>0.97</td></tr></tbody></table>

conductive heat transport equation. To the best of our knowledge, this is the first report showing that a radial temperature profile resulting from microwave heating under different initial water saturation conditions matches a profile produced by conductive heat transfer. Assuming a steady-state, purely radial, conductive heat transport, the governing equation for temperature $T$ is:

$$ \frac{dT}{dr} = - \frac{\dot{q}}{k2\pi L} \bullet \frac{1}{r}, \qquad (2) $$

where $\dot{q}$ is the radial heat transfer rate, and $k$ is the effective thermal conductivity. Fig. 5 shows the temperature gradient as a function of the reciprocal distance from the antenna. According to Equation (2), the slope is proportional to both the heat transfer rate and the inverse of the thermal conductivity. Assuming the heat transfer rate remains constant across different initial saturation conditions, the variation in slope (from -35.3 °C to -2.3 °C for 4% and 14% initial water saturation, respectively) must be attributed to differences in thermal conductivity. This aligns with the expected changes in thermal conductivity between near-dry and wet sand. However, the ratio of the two slopes gives a factor of 15, which exceeds the reported factor of 6-7 for the variation in thermal conductivity between dry and water-saturated sand (Smits et al., 2010). Furthermore, the temperature should converge to ambient temperature infinitely far from the antenna (i.e., at $1/r = 0$), but the linear regression shows a non-zero temperature gradient at the origin. These two deviations from the canonical conductive heat transfer behavior are likely due to the peculiarities of microwave heating. They may result from the hypothetically stronger incident electromagnetic field for the near-dry sand case, the lack of a steady state at the time of data acquisition, and the spatial variation in thermal conductivity for the wet sand case.

To further analyze the observed results, we have compared the experimental results to numerical simulations. The simulations involved solving the 2D radial heat conduction equation with a non-homogeneous source term. Although not explicitly modelled with multiphysics software, the spatial distribution of temperature suggests that the electromagnetic energy is rapidly attenuated near the antenna-sand interface. It is important to mention that the steady state was not reached over the course of the experiments, and consequently, the

<figure><img src="image_7.png" /><figcaption>Fig. 5. Temperature gradient as a function of the reciprocal distance to the antenna. According to the steady-state radial heat conduction equation, the slope is dependent on the effective thermal conductivity.</figcaption></figure>
modeling is limited in terms of predictive power. Nonetheless, it is used
here as a simplified tool for identifying the dominant heat transfer
mechanism. As shown in Fig. 6 (left), the experimental results for near-
dry sand exhibit a temperature gradient that is even steeper than pre-
dicted by the steady-state heat conduction equation without a source
term. This could be attributed precisely to the absence of a steady state
at the time of the data acquisition. On the other hand, the volumetric
heat source should be considered to accurately reproduce the micro-
wave heating of wet sand. As shown in Fig. 6 (right), a volumetric heat
source of amplitude 9,090 W/m³ (calculated from the volume of the
column and the nominal power of the microwave generator), a pene-
tration depth of 5 cm, and a thermal conductivity of 2 W/(m K), enable a
precise reproduction of the experiment. In contrast, the zero-source
simulation underestimates the temperature inside the column. Inter-
estingly, the fitted penetration depth (5 cm) is of the same order of
magnitude as the one obtained by the Beer-Lambert relation (3 cm).

While the source term appears necessary to reproduce the experi-
mental results for wet sand, its strength can be further assessed by
integrating the source term expression over our geometry and using the
above values for α and S₀. By doing so, the absorbed power per meter of
height amounts to 125 W/m. With respect to the nominal power of the
antenna, it can be concluded that the benefit of the dielectric heating
source term is significant. However, it is significantly lower than what
could be expected from the electrical consumption of the microwave
apparatus.

These experimental and numerical observations highlight why mi-
crowave heating is particularly interesting for soil heating: the volu-
metric heat source provides smaller temperature gradients. However,
and this is the main observation of this work, near-dry sand heating does
not exhibit this interesting property, and even for wet sand, the absorbed
power from the dielectric heating source term is modest compared to the
apparatus's nominal power. Consequently, more conventional tech-
niques, such as conductive heating, should be preferred for thermal
desorption in near-dry sand, and the microwave heating technique
should be improved.

## 4. Conclusions

This work presents intermediate-scale microwave heating experi-
ments that offer novel insights into subsurface applications, with a focus
on the heating uniformity achieved by a coaxial antenna design, the role
of soil water content in influencing the thermal response, and the
strength of the heating source term due to dielectric heating. Specif-
ically, using a leaking coaxial microwave antenna shows a satisfactory
diametrical symmetry in a metric column packed with sand. Then, the

advantages of microwave heating, i.e., fast establishment of a rather
linear radial temperature profile, have been observed for water satura-
tions of 14% and 22%. Furthermore, microwave heating of near-dry
sand showed a more heterogeneous temperature distribution, espe-
cially vertically, and a radial temperature distribution with a steeper and
non-constant gradient. It may be related to the heterogeneity in the
vertical distribution of water because of gravity. These findings advocate
for employing conductive heating techniques in the case of near-dry soil
and confirm the potential of microwave heating in wet media. The radial
temperature for wet sand can be approximated by a linear relation with
the radial coordinate, while it is not the case for the near-dry sand
condition, for which a temperature profile as T ∝ 1/r (fast temperature
decrease near the antenna) is more accurate. A simplified 2D radial heat-
conduction model with a non-homogeneous source term representing
dielectric heating was used to identify the dominant heat transfer
mechanism. The steady state assumption was used to simplify the
modeling exercise, which should rather be performed with a multi-
physics tool to obtain a rigorous solution. Nonetheless, the model with a
non-homogeneous source term fits the measured temperature for wet
sand. On the other hand, the source term should be neglected to
reproduce the near-dry sand experiments. However, the absorbed power
due to the dielectric heating source term appears to be low (125 W/m)
compared to the microwave apparatus's nominal power. It suggests that
improvements to the microwave heating technique are needed (e.g.,
antenna design) and that a careful comparison with the resistance
heating method in identical conditions is pertinent. This study highlights
the importance of conducting experiments in a representative/relevant
environment to better understand microwave heating for subsurface
applications, such as the thermal desorption of polluted soils.

CRediT authorship contribution statement

**Maxime Cochennec:** Writing – original draft, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Dorian Davarzani:** Writing – review & editing, Supervision, Project administration, Methodology, Funding acquisition, Conceptualization. **Stéfan Colombano:** Writing – review & editing, Funding acquisition. **Jean-Yves Richard:** Project administration, Funding acquisition, Conceptualization. **Mohamed Krimissa:** Project administration, Funding acquisition, Conceptualization. **Mélanie Lorthioy:** Project administration, Funding acquisition, Conceptualization. **Abderrahim Jardani:** Writing – review & editing, Supervision. **Michel Quintard:** Writing – review & editing, Methodology, Formal analysis.

<figure><img src="image_8.png" /><figcaption>**Fig. 6.** Numerical simulations and comparison against the experiments with near-dry sand (left) and wet sand (right). The microwave volumetric source term is necessary to reproduce the experimental results for the wet sand (22% initial water saturation).</figcaption></figure>
Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors gratefully acknowledge support from the DEPOLOND2 project, jointly funded by EFD, SUEZ, and BRGM. The authors also acknowledge financial support from the PIVOTS project, the "Région Centre-Val de Loire", the European Regional Development Fund, and the experimental facilities of the PRIME platform.

Data availability

Data will be made available on request.

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