# Optimisation of specific absorption heat rate distribution in microwave DIC reactor

Magdalena Kristiawan, Ladislav Klíma, Karim Allaf, Václav Sobolík

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Magdalena Kristiawan, Ladislav Klíma, Karim Allaf, Václav Sobolík. Optimisation of specific absorption heat rate distribution in microwave DIC reactor. Food and Bioproducts Processing, 2025, 154, pp.68 - 78. 10.1016/j.fbp.2025.09.006. hal-05493432

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# Optimisation of specific absorption heat rate distribution in microwave DIC reactor

Magdalena Kristiawan<sup>a, *, ID</sup>, Ladislav Klíma<sup>b</sup>, Karim Allaf<sup>c</sup>, Václav Sobolík<sup>c,d</sup>

<figure><img src="image_2.png" /></figure>

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## ARTICLE INFO

### Keywords:

Microwave heating

Controlled instantaneous pressure-drop (DIC)

Agar gel

SAHR distribution

## ABSTRACT

The distribution of specific absorption heat rate (SAHR) was evaluated across seven configurations of a microwave cavity in a pilot-scale DIC (Détente Instantanée Contrôlée) reactor. The configurations featured two waveguide apertures, one vertical with two wings and the other horizontal with one wing. SAHR values were calculated from temperature evolution in 2 % Agar gel samples heated by microwave radiation at 2450 MHz. The temperature was measured using eight fibre-optic sensors embedded in a Teflon cross, placed successively at eight different heights along a 306 mm-long glass tube. The experiments demonstrated that SAHR homogeneity is strongly influenced by cavity configuration, with the right vertical aperture and back-wing (R_B) configuration achieving the most uniform axial and radial distributions. Furthermore, SAHR profiles at 500 W and 1500 W exhibited consistency, displaying the scalability of the optimal configuration for diverse power levels. These findings offer valuable insights for optimizing microwave cavity designs to enhance heating uniformity and efficiency in industrial applications.

## 1. Introduction

The Instant Controlled Pressure Drop (DIC) process, developed by Allaf and Allaf (2014), is a thermo-mechanical treatment used to improve the structural, functional, and sensory properties of foods and other materials. Unlike puffing (Schwartzberg et al., 1995) or extrusion cooking (Moraru & Kokini, 2003; Arhaliass et al., 2009), DIC is defined by unique steps: brief heating followed by instant cooling due to rapid decompression from high pressure to vacuum. In practice, materials are treated with saturated steam in an autoclave, then suddenly decompressed to vacuum. The resulting explosive vapor expansion strains cell structures, creating porosity that enhances drying, oil extraction, and microbial decontamination while maintaining quality (Rochova et al., 2008; Kristiawan et al., 2008; Mounir et al., 2012; Bouallegue et al., 2019; Sulaiman et al., 2022).

These examples underscore the broad applicability and potential of DIC for innovative uses in food, biorefinery, and other industrial domains. Despite its advantages, steam-based DIC systems face limitations in achieving rapid and uniform heating due to the restricted rate of steam penetration and the time required to reach temperature

homogeneity.

To address these challenges, a pilot-scale microwave-based DIC (MW DIC) reactor operating at a frequency at 2450 MHz was developed, featuring a Pyrex tube in a multimode cavity connected to an expansion chamber via a spherical valve (Klima, 2006). The cavity uses waveguides, circulators, impedance adapters, and baffle wings to improve field distribution, energy absorption, and heating uniformity. The SAHR, which quantifies the rate of MW energy transformed into heat per unit mass, is a critical parameter for evaluating temperature distribution within the reactor.

The MW DIC reactor's first application involved texturizing Brazilian coffee beans, with performance compared to steam DIC (Kamal et al., 2008). Under optimized conditions (600 W magnetron power, 0.6 MPa pressure, 27 % d.b. moisture, 181 s heating), the MW DIC reactor achieved an expansion ratio of 1.39, closely matching the 1.60 ratio obtained with a steam DIC (0.7 MPa, 29.4 % d.b., 112 s). These findings demonstrate the potential of MW DIC as a viable alternative for processes requiring rapid heating and precise energy control.

While steam-based DIC processes, such as swell-drying, have been widely implemented on an industrial scale as continuous operations,

<footnote><sup>a</sup> INRAE, UR 1268 Biopolymers Interactions and Assemblies (BIA), Nantes 44316, France</footnote>

<footnote><sup>b</sup> L. Klíma Automatické Mlýny Kresín, Libochovice S.R.O., Czech Republic</footnote>

<footnote><sup>c</sup> LaSIE, UMR 7356, University of La Rochelle, La Rochelle 17 062, France</footnote>

<footnote><sup>d</sup> Institute of Chemical Process Fundamentals of the CAS, Praha 6, 165 00, Czech Republic</footnote>

<footnote>* Corresponding author.
E-mail address: magdalena.kristiawan@inrae.fr (M. Kristiawan).</footnote>
MW DIC offers new opportunities for niche applications. This includes the processing of high-value products such as medicinal plants, mush-rooms, and ginseng, where precise energy delivery and heating uniformity are critical.

Microwave heating offers significant advantages over conventional methods, including rapid volumetric heating and reduced temperature gradients, as highlighted by Drouzas et al. (1999), Ferhat et al. (2006), and Singh and Singh (1999). However, non-uniform heating remains a challenge due to the heterogeneity of electromagnetic fields in multi-mode cavities, which often causes hot and cold spots (Silveira Felipe et al. 2022). To mitigate these issues, strategies like pulsed microwave heating (Gunasekaran and Yang, 2007) and material movement within the cavity (Chen et al., 2016) have been investigated. Pulsed heating promotes temperature equilibration during off-cycles through heat conduction, improving uniformity. However, this method may not be ideal for applications requiring rapid treatment to preserve structural, sensory, and nutritional properties (Yang and Gunasekaran, 2004). Similarly, mechanical movement of materials, such as using turntables or rotating components, can redistribute electromagnetic energy and enhance heating uniformity. Despite its benefits, this approach is often impractical for processes like DIC due to design constraints (Shen et al., 2022).

Recent advancements in microwave heating technologies, such as frequency variation and dynamic frequency-shifting techniques, address the challenges of heating non-uniformity by dynamically altering electromagnetic field distribution. These methods enhance temperature uniformity and heating performance, as demonstrated by Yang et al. (2022a) through complementary-frequency strategies. Microwave energy interactions with materials in the cavity, including the sample and surrounding components, critically influence energy absorption and distribution (Mishra and Sharma, 2016).

Mathematical modeling and numerical simulations have further enriched our understanding of mitigating heating heterogeneity. Zhang et al. (2001) studied sterilisation of solid foods using numerical modelling completed by experimental verification. Yang et al. (2022b) showed that strategic frequency and power inputs variations improve uniformity in continuous-flow systems, while Tuta and Palazoglu (2017) demonstrated the benefits of helical tube configurations for viscous materials. Salvi et al. (2011) extended this approach by optimizing electric field distribution through the iterative coupling of electromagnetic, thermal, and fluid flow dynamics. Their research underscored the pivotal role of precise adjustments in cavity geometry and flow rates in minimizing inconsistencies, further validating their models with experimental data. All these studies incorporated concepts, such as volumetric heat generation and power density that are linked to SAHR.

Recent advances in combining DIC with microwave technology have demonstrated significant potential for enhancing process efficiency and product quality (Nader et al. 2022). Haddad et al. (2008) outlined a three-stage drying process involving hot air drying, DIC treatment, and microwave-assisted ambient air dehydration. The study demonstrated improvements in drying efficiency and product quality through this combined approach. Mounir et al. (2019) investigated the application of DIC followed by intermittent microwave and airflow drying for producing shrimp snacks. The combined method significantly reduced drying time and enhanced product quality attributes such as texture and flavor.

Waghmare et al. (2023) discussed various food applications of DIC technology, including its combination with microwave drying. The integration was shown to improve drying kinetics and preserve nutritional and sensory qualities of food products. These studies highlight the growing importance of understanding and optimizing microwave field distribution to maximize process efficiency and product quality consistency.

DIC combined with microwave-assisted drying is a notable application for shrimp meat processing. Pech-Almeida et al. (2021) found that the combination DIC with microwave-assisted drying significantly

reduced drying time, while enhancing the texture sensory attributes. However, the successful implementation of this combined technology depends critically on achieving uniform heating throughout the product matrix.

The present study builds on these advancements by systematically analyzing the influence of cavity configurations on SAHR distribution within the MW DIC reactor. By examining modifications to waveguides and baffle wings, this research aims to achieve uniform SAHR distribution and improve the reactor's performance. Temperature measurements recorded using fibre-optic sensors embedded in a 2 % Agar gel model material at multiple radial and axial locations provide a comprehensive three-dimensional analysis of heating uniformity. This study will contribute to optimizing MW DIC technology for broader applications in food and non-food industries.

## 2. Experimental

### 2.1. Microwave DIC reactor

The experimental setup for the MW DIC reactor is shown in Fig. 1. This system integrates key components designed to enable microwave heating as a substitute for traditional steam heating.

The central element of the reactor is the Pyrex tube Grosseron (1), which functions as the microwave heating chamber. The Pyrex material was selected for its ability to resist thermal shocks and endure high pressure, withstanding up to 1 MPa and 180°C. Technical data of PYREX Borosilicate Glass can be retrieved from internet (https://www.berkelcystainless.co.uk/downloads/Glass_Data.pdf,

https://www.pegasus-glass.com/Portals/0/technical_infopyrex.pdf). Despite having a higher dissipation factor than quartz, its superior safety characteristics made it preferable. The Pyrex tube has an internal diameter of 42 mm and a length of 306 mm.

This diameter is indeed less than the wavelength of 2450 MHz

<figure><img src="image_3.png" /><figcaption>Fig. 1. Microwave DIC set-up. 1 - Pyrex tube (heating chamber), 2 - multimode cavity, 3 - piston, 4 - spherical valve, 5 - expansion chamber, 6 - condenser, 7 - filters, 8 - vacuum tank, 9 - vacuum pump, 10 - magnetron, 11, 11' - circulators, 12, 12' - impedance adapters, A1 - pressure air, SV - safety valve, P - pressure gauge.</figcaption></figure>
microwaves in free space ($\lambda_0$ = 122 mm). This design choice was deliberate and based on several considerations. First, for safety reasons, the stresses in a tube increase with its diameter. Second, in dielectric materials such as the 2% Agar gel used in our experiments, the wavelength is significantly reduced compared to free space due to the higher dielectric constant ($\lambda$ = $\lambda_0 / \sqrt{\epsilon'}$ ≈ 14 mm in Agar gel). This shorter effective wavelength allows for more efficient coupling of microwave energy into the sample even with smaller tube dimensions. Finally, the sub-wavelength diameter helps minimize the formation of complex higher order modes within the sample that could contribute to heating non-uniformity.

Microwave energy at 2450 MHz is generated by a magnetron GMP60 Sairem (10) with the maximum power output of 6 kW. The microwaves are transmitted to the multimode cavity (2) through two brass waveguides (12, 12'). These waveguides are equipped with circulators Sairem (11, 11') and automatic impedance adapters AI4S (12, 12') to ensure optimal energy transfer while maintaining a constant pressure within the Pyrex tube during operation.

Additional components include a spherical valve (4), which connects the Pyrex tube to the expansion chamber (5). The vapor generated during material expansion is condensed in a U-shaped cooling tube (6). The pressure in the vacuum tank (8) is maintained at approximately 1 kPa using a vacuum water ring pump Antigua S52 MIL'S (9). The system incorporates a safety valve (SV) and a pressure gauge (P) to ensure safe and precise operation. Key system capacities include a 0.5 L Pyrex tube, a 9 L expansion chamber, and a 300 L vacuum tank.

### 2.1.1. Equipment operation

During operation, material is manually loaded into the Pyrex tube. The piston close the tube. The heating phase begins due to microwave radiation directed into the multimode cavity. At the end of the heating phase, the spherical valve rapidly opens, creating an instantaneous pressure drop. This process propels the material into the expansion chamber. The material undergoes physical and chemical transformations during these phases.

### 2.1.2. Multimode cavity design

The multimode cavity, Fig. 2, enclosing the Pyrex tube is constructed from 15 mm-thick aluminium plates. Its design features two waveguide apertures: a vertical aperture on the right side (85 × 43 mm) and a horizontal aperture on the left side (43 × 85 mm), see Fig. 2. These apertures are crucial for manipulating the electromagnetic field distribution within the cavity.

To control the SAHR distribution, three removable baffle wings are employed: two vertical wings (90 × 147 mm) at the right aperture and a horizontal wing (146 × 50 mm) inclined downward by 15° near the left aperture. The modular design of the cavity allows for configurations ranging from single-sided to double-sided microwave energy transfer, enabling experiments to evaluate how these modulations impact SAHR field and thus material treatment efficiency.

Seven configurations were tested, including variations in aperture and baffle wing arrangements, using a minimum magnetron power of 500 W. These configurations provided insights into optimizing SAHR distribution in the material treated in the heating chamber.

## 2.2. Temperature measurement

To accurately measure the temperature distribution within the microwave DIC reactor, a Teflon cross embedded with fibre-optic sensors was employed (Fig. 3a). This cross design allowed simultaneous temperature measurement at eight radial positions on a horizontal plane within the Pyrex tube. Fibre-optic sensors (FOT-L / 2 m, FISO) were chosen for their inertness and transparency to electromagnetic waves, ensuring precise data collection without electromagnetic interference. Each sensor operated within a temperature range of -40°C to 250°C, with an accuracy of ±2°C.

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 2. MW DIC treatment chamber in multimode cavity with right vertical and left horizontal waveguides (dimensions in mm). a) sketch of cavity, b) photograph of cavity.</figcaption></figure>

For comprehensive spatial temperature mapping, the Teflon cross was installed successively at eight different axial heights along the Pyrex tube: 30, 60, 90, 120, 150, 180, 210, and 240 mm. Each axial height required a separate experimental batch, with the Teflon cross and sensors repositioned for each run with a new Agar gel. This approach ensured detailed three-dimensional temperature profiling throughout the tube.

The system was equipped with a universal fibre-optic signal conditioner (UMI8, FISO), which processed sensor signals with high precision (± 0.5 % of the full scale). The collected data were recorded at a sampling frequency of 1.2 Hz and exported in ASCII format for further analysis using Microsoft Excel. The entire setup was controlled remotely via a USB interface utilizing FISO Commander 2 software, enabling efficient data acquisition and analysis.

In the experiments, Agar gel served as the model material due to its homogeneous dielectric properties, which closely resemble those of water. Agar gel was prepared with an initial temperature below 10°C and carefully molded to fit snugly within the Pyrex tube. During each experimental run, the Agar cylinder was heated using microwave
<figure><img src="image_6.png" /><figcaption>Fig. 3. a) Teflon cross with sensors, b) Agar cylinder with specimen, c) Agar cylinder with sensors in Pyrex tube.</figcaption></figure>

energy. Upon reaching the critical temperature, the material liquefied, necessitating the preparation of a new sample for subsequent measurements. This iterative process ensured consistent and reproducible experimental conditions across all axial heights.

## 2.3. Preparation of agar gel samples

To evaluate the temperature distribution within the MW DIC reactor, 2% Agar gel was selected as the test material due to its uniform dielectric properties and ease of preparation. This material closely mimics the behavior of water while remaining solid below 50°C, thereby eliminating natural convection and ensuring accurate temperature measurements. The relevant thermophysical properties of Agar gel, Teflon and Pyrex glass are shown in Table 1. These properties ensure that the gel effectively simulates the electromagnetic and thermal behavior of water-rich food materials while maintaining structural stability during the heating process.

<table><caption>**Table 1**

Thermophysical properties of selected materials at 2450 MHz.</caption><thead><tr><th>Properties</th><th>Agar gel 2 %<sup>a</sup></th><th>Teflon<sup>b</sup></th><th>Pyrex<sup>c</sup></th></tr></thead><tbody><tr><td>Density ρ (kg.m<sup>-3</sup>)</td><td>1070</td><td>1427</td><td>2230</td></tr><tr><td>Specific heat capacity C<sub>p</sub> (J kg<sup>-1</sup> K<sup>-1</sup>)</td><td>4200</td><td>1372</td><td>835</td></tr><tr><td>Dielectric constant ε'</td><td>73.6</td><td>2.1</td><td>4.5</td></tr><tr><td>Dielectric loss factor ε''</td><td>11.5</td><td>0.00063</td><td>0.005</td></tr><tr><td>Thermal conductivity k (W m<sup>-1</sup> °C<sup>-1</sup>)</td><td>0.5</td><td>0.024</td><td>0.56</td></tr></tbody></table>

The Agar gel was prepared by dissolving Plate Count Agar (Biokar Diagnostics) powder in distilled water, heating the solution to 80°C, and then cooling it at 4°C for 15 h to solidify into cylindrical molds.

Two types of Agar gel cylinders were fabricated for the experiments, as illustrated in Fig. 3b and c. The first cylinder type featured a cross matrix on its top surface, designed to align with the Teflon cross holding the fibre-optic sensors. This alignment ensured precise and stable sensor positioning during the experiments. The second cylinder type included a central hole with a diameter of 7 mm and a longitudinal cut from the perimeter to the hole, facilitating the insertion of sensor optic fibres.

The first cylinder was placed at the bottom of the Pyrex tube (Fig. 3b), serving as the base for sensor alignment. The Teflon cross, equipped with fibre-optic sensors, was carefully positioned to fit the imprinted cross matrix on the top surface of this cylinder. Subsequently, the second cylinder was placed above the first, and the optic fibres were gently inserted through its longitudinal cut into the central hole, ensuring consistent and reproducible sensor placement.

Each experimental setup utilized approximately 420 g of Agar gel. The assembled system (Fig. 3c) consisted of the Pyrex tube containing the stacked Agar gel cylinders, the Teflon cross, and the embedded fibre-optic sensors.

## 2.4. Evaluation of cavity configuration through SAHR analysis

The primary objective of this study was to evaluate and optimize the cavity configuration to improve homogeneity of temperature distribution within the DIC reactor. This was accomplished by analyzing the SAHR, which quantifies the rate of heat absorption per unit mass of material.

<footnote><sup>a</sup> Yang and Gunasekaran, 2004</footnote>

<footnote><sup>b</sup> Zhang et al. 2001</footnote>

<footnote><sup>c</sup> Von Hippel (1954).</footnote>
The SAHR was derived using the energy balance equation for a solid material:

$$ \rho C_p \left(\frac{\partial T}{\partial t}\right) = \text{div}(k \nabla T) + Q \qquad (1) $$

where $\rho$ is the material density (kg.m$^{-3}$), $C_p$ is the specific heat capacity (J.kg$^{-1}$.K$^{-1}$), $T$ is the temperature (K), $t$ is the time (s), $Q$ is the local rate of absorbed energy (W.m$^{-3}$) and, $k$ the thermal conductivity (W.m$^{-1}$K$^{-1}$). Given the short heating duration (<100 s) and the low thermal diffusivity of Agar gel ($\alpha = k/\rho C_p$), heat conduction effects were neglected. The SAHR = $Q/\rho$ was derived from the temperature rise in the solid Agar gel:

$$ SAHR = C_p \left(\frac{\partial T}{\partial t}\right) \qquad (2) $$

The temperature derivative was approximated using two-point difference. The times $t_{20}$ and $t_{50}$ were evaluated for the temperatures 20°C and 50°C measured in the hottest point, i.e. sensor 6 in Fig. 4. For every sensor, the temperatures $T_{min}$ and $T_{max}$ corresponding to these times $t_{20}$ and $t_{50}$, respectively, were used for SAHR calculation:

$$ SAHR = C_p \left( \frac{T_{max} - T_{min}}{t_{50} - t_{20}} \right) \qquad (3) $$

The upper limit of 50°C was selected to ensure that Agar gel remained in the solid state, eliminating natural convection. The sensor reaching the temperature 50°C as the first defined the times $t_{50}$ and $t_{20}$. None of the sensors exceeded 50°C during the SAHR calculation period, ensuring Agar solid state across all measurement points and thus avoided natural convection effects. The initial time $t_{20}$ was chosen to exclude the transient heating period. As SAHR is calculated for this time interval, the corresponding temperatures are different for each sensor. It means also that the SAHR values were evaluated at slightly different conditions, especially $\epsilon''$. All sensors besides the sensor 8 exhibited linear temperature increases within the time interval ($t_{20}$, $t_{50}$). Heat conduction from the hotter place of the sensor 2 to the colder place of the sensor 8 may be one of the explanation of the accelerated temperature increase.

## 2.5. Estimation of measurement uncertainty

To ensure the reliability of the experimental results, potential sources of uncertainty in temperature measurements and SAHR evaluations were carefully assessed.

### 2.5.1. Temperature measurement uncertainty

The fibre-optic sensors used for temperature measurement had an

<figure><img src="image_7.png" /><figcaption>Fig. 4. Temperature history in Agar gel at 30 mm height and magnetron power of 500 W (right aperture with back-wing, R_B). Temperature differences for SAHR evaluation were taken at $t_{50}$ and $t_{20}$. The numbers stand for probes.</figcaption></figure>

intrinsic precision of ±2°C. Additionally, their thermal inertia introduced a response delay (~0.5 s) during rapid temperature changes. Calculated temperature differentials have an error of 5%. Positional variations of sensors within the Agar gel, estimated at ±2 mm, have also contributed to discrepancies in the position of calculated SAHR.

Before each experiment, the heating tube and Teflon cross were equilibrated to a controlled temperature of 10 °C. Nevertheless, the preparation and assembly of the experimental setup at ambient conditions took approximately 15 min, potentially introducing additional errors due to temperature differentials.

### 2.5.2. Systematic factors

System-related factors, such as the response time of the microwave magnetron (~4 s to reach full power at 3 kW) and the automatic impedance adapter (response time: 0.5–6 s), introduced inconsistencies in energy application. These factors could have caused variations in the uniformity of microwave energy distribution during the heating process.

### 2.5.3. Mitigation of uncertainties

To minimize the impact of these uncertainties, the experimental setup was calibrated rigorously, and standardized protocols were followed for sensor placement and sample preparation. The consistent use of Agar gel with well-defined properties further ensured reproducibility. Repeated measurements and careful data analysis mitigated potential errors, providing robust and reliable results. All measurements were carried out in triplicate.

# 3. Results and discussion

## 3.1. Verification of SAHR measurements

Fig. 4 illustrates the temperature history in Agar gel during microwave heating at a magnetron power of 500 W. Initially, in the absence of microwave heating, temperatures remain constant and uniform throughout the material. Upon activation of the magnetron, a temperature rise occurs, transitioning into a near-linear increase. The sensor 6 exhibits the steepest temperature rise, indicating the highest SAHR. Notably, at approximately 70°C (t = 158 s), sensor 6 shows a deceleration in temperature rise. This phenomenon is due to the development of natural convection in the fused Agar gel, which redistributes heat locally. All sensors besides the sensor 8 exhibited linear temperature increases within the time interval ($t_{20}$, $t_{50}$). Heat conduction from the hotter place of the sensor 2 to the colder place of the sensor 8 may be one of the explanation of the accelerated temperature increase.

Fig. 4 provides clear evidence of this behavior through distinct changes in the temperature profiles.

To evaluate SAHR measurement reproducibility, standard deviations were calculated from multiple measurements under a full cavity configuration (All) at magnetron powers of 500 W and 1500 W (Tables 2 and 3). At 500 W, the standard deviations range from 2 % to 21 %, with the majority below 10 %. These results confirm an acceptable reproducibility of SAHR measurements at this power.

In contrast, higher magnetron powers introduce transient heating effects that complicate SAHR evaluation. At 1500 W, SAHR standard deviations range from 2 % to 26 % (Table 3). These higher variabilities are attributable to the transient heating process, which lasted approximately 4 s, coinciding with the emergence of hot spots (temperatures exceeding 50°C) and the initiation of Agar fusion. Despite these challenges, the results demonstrate that the experimental setup provides reproducible SAHR measurements within an acceptable range of variability, even at elevated power levels. Additionally, the increased variability highlights the complexity of accurately capturing SAHR during rapid heating and localized hot spot formation, as the system transitions into a non-linear heating regime.

The relatively high standard deviations observed in some SAHR measurements reflect the inherent challenges in achieving
<caption>**Table 2**</caption>

SAHR mean values and standard deviations of three measurements at different heights for magnetron power of 500 W using two apertures and three wings (configuration All).

<table><caption>**Table 3**</caption><thead><tr><th>Height</th><th>Sensor</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th></tr></thead><tbody><tr><td rowspan="2">30</td><td>SAHR</td><td>1045</td><td>851</td><td>932</td><td>1584</td><td>734</td><td>444</td><td>625</td><td>1300</td></tr><tr><td>s, %</td><td>6.4</td><td>7</td><td>4.8</td><td>2</td><td>9.8</td><td>6.2</td><td>14.9</td><td>1.6</td></tr><tr><td rowspan="2">60</td><td>SAHR</td><td>800</td><td>674</td><td>741</td><td>808</td><td>786</td><td>536</td><td>565</td><td>561</td></tr><tr><td>s, %</td><td>3.8</td><td>8.8</td><td>11.4</td><td>4.4</td><td>8</td><td>7.7</td><td>18.6</td><td>3.4</td></tr><tr><td rowspan="2">90</td><td>SAHR</td><td>1063</td><td>691</td><td>1200</td><td>1363</td><td>938</td><td>447</td><td>779</td><td>953</td></tr><tr><td>s, %</td><td>9.3</td><td>8.3</td><td>6.1</td><td>9</td><td>8</td><td>5.2</td><td>11.3</td><td>11.1</td></tr><tr><td rowspan="2">120</td><td>SAHR</td><td>1542</td><td>1143</td><td>1567</td><td>1899</td><td>1610</td><td>1043</td><td>1062</td><td>1461</td></tr><tr><td>s, %</td><td>4.4</td><td>2.2</td><td>7.8</td><td>8.3</td><td>2.6</td><td>5</td><td>4</td><td>7.5</td></tr><tr><td rowspan="2">150</td><td>SAHR</td><td>1796</td><td>1489</td><td>1695</td><td>1589</td><td>1641</td><td>1338</td><td>1912</td><td>897</td></tr><tr><td>s, %</td><td>13</td><td>12.2</td><td>18.5</td><td>7.9</td><td>12.2</td><td>8.8</td><td>20.1</td><td>7.5</td></tr><tr><td rowspan="2">180</td><td>SAHR</td><td>1990</td><td>1097</td><td>2244</td><td>1173</td><td>2118</td><td>522</td><td>2223</td><td>668</td></tr><tr><td>s, %</td><td>3.5</td><td>8.4</td><td>9.9</td><td>1.7</td><td>9.6</td><td>13.8</td><td>10.6</td><td>5.7</td></tr><tr><td rowspan="2">210</td><td>SAHR</td><td>1746</td><td>1097</td><td>1677</td><td>998</td><td>2023</td><td>494</td><td>1066</td><td>560</td></tr><tr><td>s, %</td><td>8.5</td><td>14.5</td><td>12.7</td><td>10.9</td><td>12.3</td><td>18.1</td><td>11.4</td><td>21.8</td></tr><tr><td rowspan="2">240</td><td>SAHR</td><td>1077</td><td>935</td><td>787</td><td>752</td><td>1524</td><td>452</td><td>647</td><td>395</td></tr><tr><td>s, %</td><td>10.7</td><td>12</td><td>7.8</td><td>5.6</td><td>10.1</td><td>17.9</td><td>3.3</td><td>14.4</td></tr></tbody></table>

SAHR mean values and standard deviations of three measurements at different heights for magnetron power of 1500 W using two apertures and three wings (configuration All).

<table><thead><tr><th>Height</th><th>Sensor</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th></tr></thead><tbody><tr><td rowspan="2">30</td><td>SAHR</td><td>3352</td><td>3408</td><td>2773</td><td>5989</td><td>2473</td><td>1233</td><td>1487</td><td>5405</td></tr><tr><td>s, %</td><td>10.3</td><td>12.1</td><td>14.1</td><td>3.3</td><td>14.6</td><td>13.4</td><td>8.2</td><td>10.9</td></tr><tr><td rowspan="2">60</td><td>SAHR</td><td>1849</td><td>1850</td><td>1547</td><td>1736</td><td>1869</td><td>1495</td><td>1347</td><td>1171</td></tr><tr><td>s, %</td><td>2.7</td><td>4.7</td><td>4.6</td><td>8</td><td>4.7</td><td>6.4</td><td>15.7</td><td>11.9</td></tr><tr><td rowspan="2">90</td><td>SAHR</td><td>2087</td><td>1878</td><td>1969</td><td>2348</td><td>2293</td><td>1383</td><td>1222</td><td>1379</td></tr><tr><td>s, %</td><td>10.4</td><td>12.4</td><td>6.7</td><td>8.2</td><td>8.7</td><td>7.6</td><td>22.3</td><td>10.4</td></tr><tr><td rowspan="2">120</td><td>SAHR</td><td>3439</td><td>3109</td><td>3717</td><td>4460</td><td>3774</td><td>2504</td><td>2411</td><td>2842</td></tr><tr><td>s, %</td><td>8.1</td><td>12.6</td><td>8</td><td>13</td><td>6.9</td><td>16.7</td><td>22.3</td><td>17.3</td></tr><tr><td rowspan="2">150</td><td>SAHR</td><td>4092</td><td>3722</td><td>3542</td><td>4769</td><td>4151</td><td>3649</td><td>4172</td><td>2975</td></tr><tr><td>s, %</td><td>12.2</td><td>12.4</td><td>9.4</td><td>26</td><td>10.8</td><td>8</td><td>8.8</td><td>23</td></tr><tr><td rowspan="2">180</td><td>SAHR</td><td>5114</td><td>2708</td><td>5399</td><td>4168</td><td>4640</td><td>1157</td><td>5054</td><td>2248</td></tr><tr><td>s, %</td><td>11.2</td><td>9.1</td><td>10.4</td><td>15.7</td><td>8.3</td><td>21.9</td><td>13.6</td><td>6.1</td></tr><tr><td rowspan="2">210</td><td>SAHR</td><td>3382</td><td>2046</td><td>3717</td><td>2080</td><td>4030</td><td>649</td><td>2516</td><td>1299</td></tr><tr><td>s, %</td><td>13.3</td><td>15.7</td><td>1</td><td>11.9</td><td>15</td><td>16.8</td><td>10.7</td><td>12.1</td></tr><tr><td rowspan="2">240</td><td>SAHR</td><td>3613</td><td>3272</td><td>2434</td><td>2267</td><td>5124</td><td>1444</td><td>1427</td><td>1363</td></tr><tr><td>s, %</td><td>9.4</td><td>7.5</td><td>9.8</td><td>9.7</td><td>9.3</td><td>9.2</td><td>12.3</td><td>6.1</td></tr></tbody></table>

reproducibility in semi-industrial microwave heating experiments. These deviations stem from multiple sources: (1) Variations in sample positioning (±2 mm) can significantly impact local field intensity due to the wavelength scale standing wave patterns. (2) Fluctuations in magnetron performance (±2% power output) are unavoidable in commercial microwave generators. (3) Differences in sample preparation, despite standardized protocols, can affect dielectric properties. (4) The temperature measurement itself and calculation of temperature differences has an associated uncertainty (±5°C). The variations in field distribution are expected due to the sensitivity of EM wave behavior to boundary conditions and material properties. The changes in load placement, load composition, or temperature can affect the field distribution due to the formation of standing waves and mode shifting within the cavity.

Overall, the findings confirm that the experimental approach is robust across a range of power levels, with acceptable reproducibility under both uniform heating conditions (500 W) and more challenging transient scenarios (1500 W).

## 3.2. SAHR distribution and cavity design optimization

The SAHR distributions as a function of height, with sensors as curve parameters, are illustrated in Fig. 5 for a magnetron power of 500 W across all cavity configurations. The thick lines represent the mean values derived from the eight temperature probes at each height. Among the configurations, the setup featuring the right waveguide with a back-wing (Fig. 5c) exhibits the most consistent SAHR distribution, with

minimal variation along the tube axis. This configuration achieves SAHR values ranging between 1000 and 1400 W/kg, with a peak observed at a height of 150 mm. At 90 mm, the SAHR variation is remarkably low, while near the bottom of the cavity (30 mm height), variability is high. The standard deviation for this setup, calculated over all measurement points, is 32.5% (Table 4), indicating the lowest degree of in-homogeneity among the tested configurations.

The back-wing design significantly enhances SAHR uniformity by modifying the electromagnetic field distribution through multiple mechanisms. First, it disrupts standing wave patterns that typically form in rectangular cavities, reducing the spatial periodicity of field maxima and minima. Second, it enhances mode coupling by creating additional reflection surfaces that redirect microwave energy, promoting more complex wave interactions and field redistribution. Third, the wing’s geometric configuration introduces phase shifts in the reflected waves, leading to more constructive interference patterns throughout the cavity volume. These combined effects result in a more homogeneous energy distribution, particularly along the axial dimension where traditional designs often exhibit significant variations due to wavelength-dependent standing wave patterns.

All—two apertures, three wings, R_B—right aperture, back-wing, R_F—right aperture,

front wing, R_all—right apertures, both wings, R_0—right aperture without wings, L_H—left aperture, horizontal wing, L_0—left aperture without wing. s—standard deviation, P—total power in 420 g Agar.

<figure><img src="image_8.png" /></figure>
<figure><img src="image_9.png" /><figcaption>**Fig. 5.** SAHR distribution in Agar gel as function of height at a magnetron power of 500 W for different cavity configurations. Symbols stand for temperature sensors (radial position).</figcaption></figure>
<table><caption>**Table 4**
SAHR mean value for different cavity configurations for magnetron power 500 W.</caption><thead><tr><th></th><th>All</th><th>R_all</th><th>R_B</th><th>R_F</th><th>R_O</th><th>L_H</th><th>L_O</th></tr></thead><tbody><tr><td>SAHR, W/kg</td><td>1131</td><td>1021</td><td>1166</td><td>1223</td><td>1144</td><td>1218</td><td>1075</td></tr><tr><td>s</td><td>502</td><td>663</td><td>379</td><td>652</td><td>548</td><td>631</td><td>593</td></tr><tr><td>s, %</td><td>44.4</td><td>64.9</td><td>32.5</td><td>53.3</td><td>47.9</td><td>51.8</td><td>55.2</td></tr><tr><td>P, W</td><td>475</td><td>429</td><td>490</td><td>514</td><td>480</td><td>512</td><td>451</td></tr></tbody></table>

mean value.

In comparison, the full configuration, incorporating both waveguides and three wings (Fig. 5a), provides the second-best homogeneity, with a standard deviation of 44.4 % (Table 4). The configuration using the right waveguide with both wings (Fig. 5b) exhibits the least homogeneous SAHR distribution, characterized by a high standard deviation of 64.9 %. This setup also shows a significant outlier, with sensor 5 registering an SAHR value of 3412 W/kg at a height of 120 mm, far exceeding the display range of Fig. 5b.

These results underscore the critical influence of cavity configuration on SAHR distribution uniformity. The configuration with the right waveguide and back-wing not only achieves the most homogeneous energy distribution along the cavity height but also maintains a mean SAHR value of 1166 W/kg (Table 4, R_B), higher than the configuration with the right waveguide and both wings (1021 W/kg, Table 4, R_all). In contrast, the higher variability in the latter configuration and the localized hotspot at 120 mm (3412 W/kg) highlight the challenges in maintaining uniform energy absorption in less optimized setups.

The R_B configuration achieves the lowest SAHR variability (32.5 %) but exhibits lower absorbed power ($P = 490 W$) compared to others (e.g., L_B: $P = 512 W$). The observed trade-off between heating uniformity and energy efficiency presents important considerations for industrial applications. In industrial settings, this balance must be optimized based on specific process requirements. For heat sensitive products where quality degradation occurs at localized hot spots (e.g., certain pharmaceuticals, nutraceuticals, or high-value food products) or cold spots (curing of composites), the enhanced uniformity of R_B would likely justify the modest efficiency sacrifice. Conversely, for energy intensive processes where throughput and operating costs are primary concerns, the L_B configuration might be preferable. Additionally, the uniformity advantage becomes increasingly critical at higher power levels, where temperature differentials are amplified.

Fig. 6 complements these findings by presenting the radial SAHR distribution at different heights. The thick lines indicate the mean SAHR values for each sensor averaged across all heights. The right waveguide with a back-wing again emerges as the optimal configuration, maintaining homogeneity in both vertical and radial directions. Sensor 4 exhibits nearly constant SAHR values across all heights, whereas sensors 2 and 6 show notable variability, indicating localized differences in energy absorption. The axially averaged SAHR for most sensors falls between 1000 and 1400 W/kg, except for sensor 8, which is approximately 700 W/kg. This configuration also achieves the highest efficiency, with an absorbed power of 511.7 W (Table 4), close to the nominal input power of 500 W.

To better characterize localized non-uniformities that might be obscured by global statistics, we conducted sub-regional analyses by dividing the cavity into top (150–240 mm) and bottom (30–120 mm) sections. For the optimal R_B configuration, the top section exhibited a SAHR variability of 22.3 %, while the bottom section showed 30.8 % variability. This reveals that despite the overall improved uniformity, the bottom section still experiences more pronounced heating heterogeneity. In contrast, the R_F configuration showed significantly greater disparity between sections (top: 42.3 %, bottom: 47.6 %).

The original configuration (All) exhibited a variability of 39.7 % and 27.2 % at the top and bottom, respectively, highlighting the superior homogenization capability of the R_B design across sub-regions. Additionally, the analysis for each sensor revealed that the mean coefficient

of variation across axial height ranged from 11.2 % (sensor 4) to 40 % (sensor 6) with a mean value of 23.5 % for R_B, compared to 19.9 % (sensor 4, R_all) to 83.7 % (sensor 7, L_O) for other configurations, further confirming the comprehensive homogenization achieved by the optimal design.

While our initial assessment of radial homogeneity was primarily based on visual interpretation of the SAHR profiles, we acknowledge that this approach insufficiently addresses quantitative non-uniformities such as the notably lower SAHR value (≈400 W/kg) observed at Sensor 8 at 240 mm height compared to the mean value (≈1000 W/kg). To address this limitation, we have calculated the variation across radial positions at each height. For the R_B configuration, the radial CV ranges from 15.3 % (90 mm) to 59.2 % (30 mm), due primarily to the lower values at Sensor 8. This radial non-uniformity, while still representing an improvement over other configurations (16.9 % – 67.8 %), indicates a persistent challenge in achieving complete homogeneity. Comparing All and R_B configurations, the maximum variations were 98.4 % (All, sensor 3, 180 mm) and 131 % (RB, sensor 6, 30 mm). The statistical analysis (global variation 44.4 % for All and 32.5 % for R_B) reveals that our claim of improved homogeneity rather than complete homogeneity more accurately represents the performance of the R_B configuration, which still exhibits specific regions of significant variation despite its overall superior performance.

Further analysis of the interaction between axial and radial non-uniformities reveals important patterns that impact overall heating performance. By mapping the combined axial-radial SAHR distribution, we identified that certain positions consistently exhibit lower heating rates across multiple configurations. Notably in R_B configuration, the region around Sensor 8 at 240 mm height represents a persistent cold zone where SAHR values remain below 50 % of the mean observed values. The SAHR values at sensor 8 remain cooler at all heights. Sensors 3 and 7 often show higher SAHR at many heights. The mid-height axial zone (120–150 mm) coincides with higher SAHR radial positions, amplifying non-uniformity. This phenomenon results from constructive interference between axial standing wave patterns and radial energy distribution characteristics. In the R_B configuration, while this cold zone is less pronounced (SAHR ≈ 400 W/kg compared to ≈200 W/kg in other configurations), it nonetheless represents a limitation in achieving heating uniformity. The practical implication is that even in optimized cavity designs, product rotation or multiple waveguide arrangements may be necessary to eliminate these persistent cold zones. Additionally, the interaction between axial and radial non-uniformities creates complex three-dimensional heating patterns that cannot be fully characterized by independent axial or radial analyses alone, highlighting the importance of comprehensive spatial mapping in microwave cavity optimization.

These results underscore the critical influence of cavity design on SAHR uniformity. The right waveguide with a back-wing configuration demonstrates superior energy distribution compared to other setups, highlighting the importance of precise cavity design in optimizing microwave-assisted processes.

### 3.3. Scalability of SAHR distribution with magnetron power

Fig. 7 presents the normalized longitudinal mean SAHR distributions for 500 W and 1500 W, measured in the full cavity configuration (two waveguide apertures and three wings). This SAHR represents the
<figure><img src="image_10.png" /><figcaption>Fig. 6. SAHR distribution in Agar gel as function of radial (sensor) position for different height. ◆ - 30, ● - 60, ▲ - 90, □ - 120, ○ - 150, △ - 180, ◦ - 210, ■ - 240, ■ - mean value.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>**Fig. 7.** Longitudinal mean SAHR devised by magnetron power for 500 and 1500 W and whole configuration with two apertures of waveguide and three wings. □ –500 W, △ – 1500 W.</figcaption></figure>

average SAHR calculated across all heights within the cavity for each radial position (sensor) divided by the magnetron nominal power. The normalized SAHR provides a comparative measure of energy absorption independent of input power. The profiles for 500 W and 1500 W are remarkably similar, indicating that SAHR distribution remains largely independent of magnetron power.

This observation suggests that conclusions drawn about SAHR homogeneity at 500 W can be reliably extrapolated to higher power levels, such as 1500 W. Such scalability confirms the robustness of the cavity design, ensuring consistent energy absorption across varying power levels. This consistency is significant for industrial applications, as it enhances process efficiency and ensures uniform product quality during microwave treatments.

### 3.4. Synthesis of findings

The findings across Figs. 4–7 emphasize the robustness of the experimental setup and the critical role of cavity design in achieving uniform and efficient energy absorption. The right waveguide with a back-wing configuration consistently demonstrated superior SAHR homogeneity in both vertical and radial directions, with lower variability compared to alternative setups. Furthermore, the scalability of SAHR distribution from 500 W to 1500 W underscores the versatility of this design for industrial applications.

These results align with existing literature (Jeon et al., 2022; Baranowski et al., 2023; Zhou et al., 2022), which highlights the significance of cavity design in achieving uniform SAHR distribution and heating uniformity. While the specific configuration tested here is unique, prior research supports the broader principle that variations in waveguide placement, reflector geometry, and cavity height significantly affect SAHR and heating patterns. Optimizing cavity design enhances electromagnetic field uniformity, thereby improving SAHR homogeneity and ensuring scalability for diverse applications.

The superior performance of the R_B configuration can be explained by fundamental principles of electromagnetic wave propagation and interaction. According to Metaxas and Meredith (1988), the back-wing design effectively disrupts standing wave patterns by creating additional reflection surfaces that promote wave scattering. This mechanism aligns with our observations of improved axial uniformity, as the standing wave patterns typically responsible for axial hot and cold spots are effectively mitigated. Furthermore, Basak (2005) and Mishra and Sharma (2016) demonstrated that strategic placement of metallic reflectors can enhance mode stirring effects without mechanical movement, which explains the improved field homogenization we observed with the back-wing configuration.

The observed trade-off between uniformity and absorbed power

efficiency aligns with findings from Vadivambal and Jayas (2010), who reported similar compromises in industrial microwave systems. Their work suggests that optimal designs must balance these competing factors based on specific application requirements. Our results extend this understanding by quantifying the relationship between cavity configuration and this fundamental trade-off.

The persistent cold zone observed at specific axial-radial coordinates (particularly around Sensor 8 at 240 mm) represents a common challenge in microwave cavity design. Silveira Felipe et al. (2022) identified similar phenomena in cylindrical cavities, attributing them to destructive interference patterns that create persistent low-field regions. Their proposed solution involving multiple feed points could potentially address this limitation in our design.

The scalability of SAHR distribution patterns across power levels (500 W and 1500 W) confirms the linearity of the electromagnetic field distribution with respect to input power, as theoretically predicted by Maxwell's equations. This finding has significant practical implications, as it suggests that optimization performed at lower, more manageable power levels can be reliably scaled to industrial processing conditions.

### 4. Conclusions

This study systematically evaluated the influence of cavity configurations on the SAHR distribution within a microwave-based DIC reactor. Temperature measurements in Agar gel revealed inherent heterogeneity in SAHR distribution, which could be mitigated through optimized cavity design. Among the seven configurations tested, the setup with a right waveguide aperture and a back-wing emerged as the most effective, achieving superior SAHR homogeneity in both axial and radial directions.

The results highlight that SAHR uniformity can be enhanced significantly by simplifying the cavity design: the R_B configuration alone provided comparable or superior homogeneity compared to the All configuration that is more complex. Furthermore, the scalability of SAHR distribution from 500 W to 1500 W demonstrated the robustness of this configuration for diverse power levels, ensuring consistent energy absorption.

These findings underscore the importance of precise cavity configuration in optimizing microwave energy distribution, offering a framework for improving process efficiency and product quality in microwave-assisted industrial applications.

Future research should focus on several promising directions to further enhance MW DIC technology. First, the integration of computational electromagnetic modeling with experimental validation could accelerate the optimization process and provide deeper insights into field distribution mechanisms. Second, the development of dynamic field control systems, such as variable phase shifters or adaptive tuning mechanisms, could potentially address the persistent cold zones identified in this study. Third, investigation of hybrid heating approaches combining microwave energy with other heating modes (e.g., infrared) could further improve uniformity while maintaining energy efficiency. Fourth, the application of these optimized cavity designs to actual food processing trials is essential to validate the practical benefits in terms of product quality, processing efficiency, and energy consumption. Finally, scaling these designs to industrial throughput levels while maintaining heating uniformity represents a critical engineering challenge that warrants dedicated investigation. Addressing these research directions would significantly advance the practical implementation of MW DIC technology in food and biomaterial processing industries.

### CRediT authorship contribution statement

**Magdalena Kristiawan:** Conceptualization, Data curation, Formal analysis, Methodology, Validation, Visualization, Writing – review & editing.
**Ladislav Klíma:** Experiments, Data curation, Formal analysis, Investigation, Software, Validation, Visualization, Writing – original
draft. Karim Allaf: Funding acquisition, Project administration, Resources, Review & editing. Václav Sobolík: Conceptualization, Methodology, Supervision, Validation, Writing – review & editing.

## Informed Consent Statement

Not applicable

## Funding

This work was performed in the frame of Innovation and SME: DIC-Extract by the European Union.

## 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.

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