# The Development of a Digital Twin to Improve the Quality and Safety Issues of Cambodian Pâté: The Application of 915 MHz Microwave Cooking

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## Abstract

**:**

^{®}as the model food and Cambodian pâté). The model developed with COMSOL

^{®}Multiphysics software can accurately predict both local temperatures and global moisture losses within the pâté sample (RMSE values of 2.83 and 0.58, respectively). The moisture losses of Cambodian pâté at the end of the process was 28.5% d.b (dry basis) after a ramp-up heating activity ranging from 4 to 80 °C for 1880 s and a temperature-holding phase at 80 °C for 30 min. Overall, the accurate prediction of local temperatures within Cambodian pâté is mainly dependent on the external heat-transfer coefficient during the temperature-holding phase, and is specifically discussed in this study. A 3D model can be used, at present, as a digital twin to improve the temperature homogeneity of modulated microwave power inputs in the future.

## 1. Introduction

^{®}Multiphysic. Interestingly, the study by Shen et al. [6] used the Chilton–Colburn analogy to account for the mass transfer from the convective heat-transfer coefficient at the surface.

^{®}) is used as a preliminary experimental validation of the model due to its more uniform composition. The model is then improved and validated further to account for the mass transfer phenomena during the microwave heating of raw Cambodian pâté. The effect of the external natural convective heat-transfer coefficient is also discussed, based on the temperature profiles of the samples.

## 2. Materials and Methods

#### 2.1. Tylose Preparation

^{®}sample was used for the validation of the electromagnetic and heat-transfer models due to its homogenous and simple composition, which allowed us to avoid potential uncertainties concerning the heterogeneity of the pâté and to guarantee good reproducibility. The process of producing Tylose

^{®}was similar to previous research investigations [12] with some modifications. The sample was created by slowly mixing an 11% mass-to-mass ratio of methyl 2-hydroxyethyl cellulose (Tylose MH 1000 YP2, Shin Etsu, Ibaraki, Japan) with water at 90 °C, while continuously stirring it with a magnetic stirrer with some additional help from a small, glass rod. After the mixture was well mixed, it was poured into a polyethylene terephthalate (PET) microbiology sample container cup with a diameter of 5.2 cm and height of 6.5 cm. The cylindrical shape of the sample was selected to simplify the preparation process and reduced the overheated parts of the sample submitted to microwaves as compared to other shapes, such as cubes, cuboids, and spheres [13]. The sample was stored at 4 °C overnight before it was used in the experiment.

#### 2.2. Cambodian Pâté Preparation

#### 2.3. Dielectric Properties’ Measurements

^{®}and raw pâté were placed, respectively, inside a small heating chamber where hot water inside an external jacket coming from a water bath (Julabo model F32, Seelbach, Germany) circulates as a closed loop. The coaxial probe was placed in contact with the sample’s surface and the probe was held in this position. The small heating chamber was covered by a thermal insulator and until thermal equilibrium was attained. Since the Tylose

^{®}sample started to change its aspect and lost its integrity at 70 °C, the dielectric properties were measured at 915 MHz at temperatures of 4, 10, 20, 30, 40, 50, and 60 °C for the Tylose

^{®}sample and an additional 70, 80, and 90 °C for the pâté sample.

#### 2.4. Thermal Conductivity Measurements

^{®}and raw pâté) were measured using the transient-line heat-source method (Thermal analyzer TEMPOS with probe KS-3, Meter, München, Germany). The probe was first calibrated by using a known thermal conductivity glycerin (glycerol) solution to obtain reliable measurements with the reference value in the range of the measured values for the Tylose

^{®}and pâté. The samples were prepared in a 50 mL centrifugal tube and stored at 4 °C. Since the thermal conductivity of the Tylose

^{®}was fairly changed with the changing of the temperature, the thermal conductivity of the Tylose

^{®}was measured at the temperature of 20 °C. The thermal conductivity of the pâté was measured by placing a cylindrical container with raw pâté inside the small heating chamber, as described in the dielectric measurement method. The thermal conductivity of the pâté was measured at temperatures of 4, 10, 30, 50, 70, and 90 °C.

#### 2.5. Heat Capacity Measurements

#### 2.6. Microwave Device Configuration

^{®}Multiphysics by including all the elements, as shown in Figure 1. The waveguides used were standard WR975 single-mode aluminum waveguides (SAIREM, Décines-Charpieu, France), with a 9.4 cm aperture iris (microwave-coupling device), a single-mode microwave applicator, and a sliding short circuit (SSC), as shown in Figure 1A. The iris was used as a metal plate that contained an opening through which the waves could pass to improve the impedance matching of the microwave cavity. The iris was located in the transverse plane of the electric-field strength to act as an inductive iris where the edges were perpendicular to the magnetic field’s orientation within the waveguide. The bolts and cavity rods were included due to their influence on the electric field inside the cavity [14]. The inner surface of the waveguide, antenna, cavity rod, iris, and bolt in the cavity were all composed of aluminum. However, the antenna feeder and SSC surface properties had the properties of copper and brass, respectively. Moreover, the geometry of the PET plastic container was also included to approximate the experiment. The iris was used in this system due to its ability to focus the electromagnetic wave with negligible energy losses. The electromagnetic wave was transmitted from the antenna through port 1 (TEM mode), while the microwaves were transmitted from the antenna through the waveguide transition (TE10 mode).

#### 2.7. Impedance-Matching Characterization

^{®}or pâté sample with an initial ambient temperature (20 °C) was placed at positions A, B, C, and D, as shown in Figure 2. The waveguide transition (port 1) was connected to a vector network analyzer with a coaxial cable, and the reflected voltage (then converted to the S11 parameter) was measured for each sample location following the various positions of the sliding short circuit (from 700 to 900 mm from the center of the cavity with a 10 mm increment step). The microwave power-reflection coefficient for each sliding short-circuit position and each sample position was calculated using the following expression:

#### 2.8. Microwave-Heating Experiments

^{®}or raw pâté) was kept at 4 °C overnight before it was placed at the center of the single-mode cavity, and the waveguide transition was connected to the microwave generator (GLS 1500, SAIREM

^{®}, Décines-Charpieu, France). Three optical fibers connected to a data logger (Rugged monitoring H201, Quebec, PQ, Canada) were placed at positions 1, 2, and 3 within the sample (shown in Figure 3A,B) with the help of a polyethylene sensor holder to insure a precise geometrical position (shown in Figure 3C).

^{®}, respectively, when both samples were placed at position A, and fixed at 563 mm when the Tylose

^{®}sample was placed at position B.

^{®}sample. For the pâté sample, heating consisted of two sequences: a ramp-up heating and temperature-holding phase. During the ramp-up heating phase, 30 W of microwave power was constantly applied until the temperature of point 1 reached 80 °C. Then, the second phase consisted of a microwave power regulation to hold the temperature of point 1 at 80 °C.

#### 2.9. Moisture Loss Characterization

_{0}). The pâté sample was heated up to 80 °C and held at that temperature for 0, 15, and 30 min. After reaching each condition, the microwave power was switched off and the sample was immediately weighed to obtain the final mass (m

_{f}). The recorded microwave power-input profile was used as the input value for the simulation. The moisture loss was calculated as the percentage of dry basis by using Equation (2), where rd was the ratio of dry matter in the wet sample (23%):

#### 2.10. Infrared Image

^{®}sample and 14 min for the pâté sample. The emissivity of the samples was approximated to 0.9 by measuring the ambient surface temperatures with thermocouples and the emissivity value was adjusted with Altair software (Cedip Infrared Systems, Muchen, Germany).

## 3. Modeling

#### 3.1. Assumptions

- Tylose and raw pâté were considered homogenous and isotropic [10].
- The samples had homogenous initial temperatures [10].
- The thermophysical and dielectric properties of Tylose
^{®}were constant within the temperature range of the experiment [15]. - The thermal conductivity and dielectric properties of the pâté were a function of temperature, while the specific heat capacity of the pâté was assumed to be constant within the range of temperatures assessed.
- Thermal conductivity was constant within the variation range of moisture.
- Density was considered constant throughout the process [15].
- The shrinkage of the samples was considered negligible throughout the process [10].
- The bottom of the sample in contact with the PTFE support was considered thermally insulated.
- The sample was in perfect contact with the PET container.
- The ambient air was at a constant temperature (Tair = 293.15 K).
- The absolute humidity of the air surrounding the sample was constant throughout the experiment due to a small amount of evaporation from the sample into the air inside the microwave cavity.

#### 3.2. Modeling of Microwave Propagation

_{0}= 4π· 10

^{−7}H/m for the vacuum and ε is the complex permittivity with ${\epsilon}_{0}=8.854\xb7{10}^{-12}\text{}\mathrm{F}/\mathrm{m}$ for the vacuum. The relative complex permittivity ${\epsilon}_{r}$ is composed of the relative dielectric constant ${\epsilon}_{r}^{\prime}$ and the relative dielectric loss factor ${\epsilon}_{r}^{\u2033}$ with the complex notation “i″ as shown in Equation (7):

- -
- Initial condition E = 0, t = 0.
- -
- The boundary condition for the TE10 mode:
- Ex = 0 at y = 0;
- Ex = 0 at y = 12.38 cm (width of the waveguide);
- Ey = 0 at x = 0;
- Ey = 0 at x = 24.76 cm (height of the waveguide);
- Ez = 0.

- -
- The impedance boundary condition is used at the walls of the waveguides for brass, copper, and aluminum metallic surfaces [17]

- -
- Perfect electrical conductor: apply to the surface of the metallic core inside the PTFE ring of the antenna (n × E = 0).
- -
- Port boundary condition: the percentage values of microwave reflected power (RF) at the input port (coaxial port for antenna) and port 2 were calculated from the squared magnitude values of S
_{11}and S_{21}[16]. The S-parameter values at both ports are:

_{M}) due to dielectric losses within the medium is expressed by the following equation [4,10,18]:

_{ab}is the microwave absorbed power.

#### 3.3. Heat Transfer

_{M}must be included in the heat conservation equation for the heat-transfer study. However, the sample was also exposed to water evaporation from the top surface and to natural convection at both lateral and upper surfaces during the heating process. Taking into account the heat source term due to microwaves, the energy balance equation is expressed as [19,20,21]:

_{s}is the density of the sample, C

_{p}is the specific heat capacity of the sample, and k is the thermal conductivity of the sample.

- -
- Initial condition: T
_{0}= 4 °C, t = 0. - -
- The boundary condition for the top surface of the sample involves both evaporation and natural convection phenomena [22]:

_{ev}is the heat loss due to evaporation, hc is the convective heat-transfer coefficient, T

_{s}is the temperature at the surface of the sample, and T

_{ext}is the temperature of the hot-air film close to the surface of the sample by the effect of natural convection.

- -
- The boundary values for the side surface of PET sample cells are:

#### 3.4. Mass Transfer

_{m}is the moisture diffusivity (m

^{2}/s) [23], λ = ${M}_{w}{h}_{fg}$ is the molar latent heat of vaporization of water (J/mol) (M

_{w}is the molar mass of water kg/mol, h

_{fg}is the latent heat of evaporation (J/kg), C is the concentration of water inside the food (mol/m

^{3}), ρ

_{s}is the density of the food product (kg/m

^{3}), m is the mass of moisture inside the food (kg), and C

_{m}is specific moisture capacity (kg

_{moisture}/kg

_{food}).

_{m}is crucial to determine Q

_{ev}. K

_{m}is the moisture mass conductivity (kg·m

^{−1}·s

^{−1}). Q

_{ev}is calculated using the top-surface boundary condition in Equation (16).

- -
- Initial condition:

- -
- At the top surface of the sample, the boundary equation is expressed as:

_{b}is the equilibrium moisture concentration of the surrounding air (mol/m

^{3}), m

_{e}is the equilibrium moisture content of air (decimal wet basis), C is the moisture concentration on the surface of the food material (mol/m

^{3}), and k

_{c}is the mass transfer coefficient (m/s).

- -
- At the bottom and the lateral surfaces, no evaporation occurred. The equation for this boundary is:

- -
- The moisture loss is calculated from the following expression:

_{c}is an important parameter for the calculation in Equation (23). This parameter is calculated from the following relation [23]:

_{m}is the convective mass transfer coefficient (kg/(m

^{2}·s)) and C

_{m}is the specific moisture capacity (kg

_{moisture}/kg

_{food}).

_{m}is calculated using Lewis’s analogy between the mass- and heat-transfer coefficients [25,26].

#### 3.5. Thermophysical and Dielectric Properties

^{®}are constant throughout the experiments.

**ε**’

_{r}and

**ε**″

_{r}) of pâté as a function of temperature are shown in Figure 4.

^{+}) inside the sample, which increases its electrical conductivity and dielectric loss [37]. The salt also influences the variation in dielectric loss as a function of temperature, which increases the dielectric losses along with the increase in temperature when the salt concentration is greater than 0.5% (for 915 MHz) [12].

Surface Boundary | Electrical Conductivity | Reference |
---|---|---|

Aluminum | 3.774·10^{7} S/m | COMSOL^{®} database |

Brass | 1.59·10^{7} S/m | [38] |

Copper | 5.99·10^{7} S/m | COMSOL^{®} database |

Parameters | Value/Units | References |
---|---|---|

RH | 0.3 | [47] |

% dry matter | 23.52% | Measure |

me | 0.02 | [22] |

h_{0} | 2501·10^{3} J/kg | [29] |

Da | 2.5·10^{−8} (m^{2}·s^{−1}) | [48] |

Km | 1.29·10^{−9} (kg·m^{−1}·s^{−1}) | [22] |

Cp_{d.a} | 1005 J/(kg·K) | [29] |

Cp_{m} | 1870 J/(kg·K) | [29] |

C_{m} | 0.003 kg/kg | [22,49] |

Z | 0.999 | [30] |

α | 1.0062 | [30] |

β | $3.14\xb7{10}^{-8}{\mathrm{P}}^{-1}$ | [30] |

γ | $5.6\xb7{10}^{-7}{\text{}\mathrm{K}}^{-2}$ | [30] |

A | 1.2811805$\xb7{10}^{-5}{\text{}\mathrm{K}}^{-2}$ | [30] |

B | −1.9509874$\xb7{10}^{-2}{\text{}\mathrm{K}}^{-1}$ | [30] |

C | 34.04926034 | [30] |

D | −6.3536311$\xb7{10}^{3}\text{}\mathrm{K}$ | [30] |

## 4. Model Design

#### 4.1. Computational Details

^{®}Multiphysics 6.0. Coupled-heat and -mass transfers were implemented in COMSOL

^{®}and the radio frequency module was used to model the electromagnetic field propagation. The frequency used in the model was 915 MHz, which was the same frequency generated by the solid-state generator during the experiment.

^{®}PrecisionTM Workstation computer equipped with 2 × Intel

^{®}Xeon processors (8 cores), at 2.5 GHz, with 256 GB RAM, running on Windows

^{®}8 Professional, 64 bits. The computation time took around 120 h to finish the whole model.

#### 4.2. Mesh Configuration

#### 4.3. Validation Method

_{1}and X

_{2}are the values for analyzing the parameters of the compared data.

^{®}; the p-values were compared with a significant confidence-level alpha value of 0.05.

## 5. Results and Discussion

#### 5.1. Electromagnetic Validation

^{®}was compared with the results of the simulations, as shown in Figure 6. Although the similar reflection coefficient profile did not indicate the similarity of the electric-field distribution inside the sample, it confirmed a similar level of transmitted microwave power due to the interactions between the electric field and sample (dielectric properties) at each SSC position. Interestingly, both simulations and experiments showed the same SSC positions, which led to the minimum reflection coefficient (impedance-matching), and similar reflection coefficient values at each position. This result confirms the good agreement of the overall interaction between the electromagnetic field and sample for each SSC position. The simulation also showed a similar agreement while predicting the reflection coefficient of the pâté at various SSC positions, as shown in Figure 7. The overall RMSE of the prediction of the reflection coefficient was 0.05. From a statistical point of view, the numerical and experimental results were statistically indistinguishable (p-value > 0.05).

#### 5.2. Temperature Validation

#### 5.2.1. Tylose

^{®}used the heat-transfer coefficient hc = 6 W/(m

^{2}·K), which was within the range derived from Kosky et al. [52]. Figure 8 shows the temperature profiles of three points (T1, T2, and T3) during the heating of Tylose

^{®}at positions A and B. Since the duration of the ramp-up heating was short (only 387 s), the mass transfer was not included in the model prediction, which comprised only the coupling of electromagnetic waves and heat transfer. Only two sample positions were used to validate the robustness of the model for the temperature prediction, because the reflection coefficients for the C and D positions were high, which is not an ideal condition for the operation of microwave heating. Figure 8 shows the temperature distribution profiles of positions A and B, which are similar despite the different positions inside the cavity. Temperature T1 was significantly higher than T2 and T3 (p-value < 0.05), because T1 was located at the center of the sample where the electric-field distribution was greater compared to T2 and T3. Interestingly, the temperatures of T2 and T3 made no significant difference, despite a 0.3 cm-distance difference from their central positions. Statistically, the numerical and experimental results for the three points at both positions A and B were not significantly different (p-value > 0.05). Additionally, the overall RMSE of the temperature prediction at both positions (A and B) was 0.1621. However, the model appears to be slightly better at predicting the temperature profile at position B (RMSE = 0.1023) than at position A (RMSE = 0.2023). Overall, the numerical model exhibited a good agreement with the microwave heating of Tylose

^{®}.

^{®}during the first 5 min of heating, since the model could provide an acceptable result without considering the mass-transfer phenomena.

^{®}deviated from the sample properties from the model’s assumptions. Thus, the microwave-heating experiment for Tylose

^{®}was conducted for only 5 min for the validation of the model.

#### 5.2.2. Cambodian Pâté

^{®}during heating, the temperature profile of T1 was significantly higher than T2 and T3 at the beginning, and both T2 and T3 had similar temperature profiles. The heating rate at each geometrical point was governed by the thermophysical properties of the sample (specific heat and thermal conductivity) and also the evolution of the electric-field strength at that point due to the temperature-dependence of the dielectric properties (Figure 4). Three convective heat-transfer coefficients on the upper surface were tested (hc

_{1}= 3.5 W/(m

^{2}·K), hc

_{2}= 6 W/(m

^{2}·K), and hc

_{3}= 8.4 W/(m

^{2}·K)). For temperatures ranging from 4 to 80 °C, the heat-transfer coefficients were in the range of the natural convection predictions calculated from empirical correlations dedicated to free natural convections in the COMSOL program. During the ramp-up heating, the χ

^{2}analyses of all hc values (3.5, 6, and 8.4 W/(m

^{2}.K)) exhibited insignificant differences, as compared with the experimental results (p-value > 0.05) with RMSE values of 3.5589, 2.3947, and 2.0000, respectively. This means that the simulation provided a better prediction of the temperature profile during the ramp-up heating for all heat-transfer coefficients that were in the studied range. Interestingly, a significant deviation between the temperature profiles of the simulation and experiment occurred during the temperature-holding phase, due to the effect of the evaporation flux that became noticeable, even at a low microwave-power input (around 11 W). The Lewis analogy calculation showed that the moisture-evaporation value calculated for hc

_{1}was lower than the evaporation rate calculated for hc

_{3}, which resulted in reduced evaporative heat-loss values in the case of hc

_{1}, and it ultimately resulted in the temperature profile of hc

_{1}being significantly higher than the temperature profiles of hc

_{2}and hc

_{3}during the temperature-holding phase (RMSE values of 15.8756, 12.6705, and 3.5048 compared to the experimental results, respectively). Therefore, the value of hc

_{3}provided better temperature-prediction results during the temperature-holding phase. Overall, the RMSE between the experimental temperature results and predicted temperatures of hc

_{1}, hc

_{2}, and hc

_{3}, for both heating processes, were 11.3768, 9.0139, and 2.8357, respectively, which confirms that the hc

_{3}value provides the best temperature prediction for both processes. During the temperature-holding phase, a small degree of deformation was observed experimentally due to water loss; however, the RMSE of the temperature prediction for hc

_{3}showed that the simulation could still predict the temperature profile well, without taking into account the sample shrinkage in the model. Although the influence of the sample deformation was demonstrated in the study by Gulati, Zhu, and Datta [8], which showed its influence on the temperature profile during drying, the sample in the study lost a small amount of water that only led to a very small degree of deformation that did not have a significant interference with the model prediction. The RMSE analysis of the temperature prediction during the ramp-up heating phase of the pâté was higher than the RMSE for Tylose

^{®}; however, it still produced a good prediction regarding the duration of the microwave heating of the pâté, which was more than two-times longer.

^{®}cut-line temperature profiles, the initial temperature is fairly different; however, the temperature of the hot and cold spots evolved to be close to the temperature predicted by the simulation. The chi-squared analysis of both results are insignificantly different (p-value > 0.05).

#### 5.3. Mass Transfer Validation

_{1}, hc

_{2}, and hc

_{3}were significantly different (p-value < 0.05). Statistically, the percentage of global moisture loss due to the evaporation of water in the experiments and simulations were similar (p-value > 0.05) for all three hc values (RMSE values of 1.9219, 0.2862, and 0.5829, respectively). On the other hand, the reduced water concentration on the top surface of the pâté may be a concern in relation to the organoleptic quality of the product after cooking.

#### 5.4. Sensitivity Analysis

_{3}value. However, no significant effect was observed for the prediction of the temperature profile during ramp-up heating with a hc

_{1}value for a similar model comparison. This was due to the low magnitude of hc

_{1}leading to the low value of the mass transfer coefficient that was calculated using the Lewis analogy. However, at the temperature-holding phase, the temperature profile predicted by both models (with and without mass transfers) produced significant differences when calculated using hc

_{1}and hc

_{3}values.

## 6. Conclusions

^{®}during ramp-up heating was also in good agreement with the experiment (RMSE of 0.1621). Moreover, the result also shows that the temperature obtained from the simulation using an hc

_{3}value is in agreement with the experimental result with an RMSE value of 2.0000 for the ramp-up heating and 3.5038 for the temperature-holding phase, while the overall prediction’s RMSE value for hc

_{3}is 2.8357. The relevant validation of the developed model perfectly illustrated the use of a digital twin to improve the process. Indeed, it allowed us to show the following setbacks of microwave cooking. The 2D y–z-plan temperature profile during ramp-up heating meant that this cooking method did not entirely cook the sample because there was a large temperature gradient (95 °C) between the hot and cold spots; however, it can be used as a preheating method for further cooking processes. Moreover, the temperature-holding phase could improve the temperature uniformity inside the sample and reduce the temperature gradient to 73 °C. However, the temperature of a small part on the surface of the sample remained insufficiently low. Therefore, other improvements to temperature-uniformity methods, such as sample rotation and thin-film resonators, are also interesting for further exploration.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

${\epsilon}_{r}^{\u2033}$ | Dielectric loss factor |

${\epsilon}_{r}$ | Relative permittivity |

${\rho}_{a}$ | Air density (kg/m^{3}), |

ρ_{s} | Density of the food product (kg/m^{3}) |

µ | Magnetic permeability (H/m) |

µ_{o} | Magnetic permeability of vacuum 4π × 10^{−7} H/m |

C_{b} | Equilibrium moisture concentration (mol/m^{3}) |

C_{m} | Specific moisture capacity (kg_{moisture}/kg_{food}) |

C_{p} | Specific heat capacity of the sample (J · kg^{−1}·K^{−1}) |

Cp_{a} | Heat capacity of the air (J·kg^{−1}·K^{−1}) |

Cp_{d.a} | Heat capacity of dried air 1005 J/(kg·K), |

Cp_{m} | Heat capacity of water vapor 1870 J/(kg·K) |

C_{w} | Moisture concentration at the surface of food material (mol/m^{3}) |

d | Humidity ratio in mass (kg moisture/kg dry air) |

D_{a} | Air-diffusion coefficient of water vapor in the air (10^{−11} m^{2}/s) |

d.b | Dry base matter |

D_{m} | Moisture diffusivity (m^{2}/s) |

E | Electric field (V/m) |

E_{c} | Calculated electric field |

E_{1} | Analytic field of excitation port |

E_{2} | Eigenmode calculated from the boundary mode analysis and normalized concerning the outgoing power flow |

E* | Complex conjugate electric field |

ε | Complex permittivity of the material |

ε_{o} | $\mathrm{Permittivity}\text{}\mathrm{of}\text{}\mathrm{vacuum}\text{}8.854\times {10}^{-12}\text{}\mathrm{F}/\mathrm{m}$ |

f | Microwave frequency (Hz) |

f’ | Correction function |

hc | Convective heat-transfer coefficient (W·m^{−2}·K^{−1}) |

h_{a} | Enthalpy of dry air (J·kg^{−1}) |

h_{0} | Enthalpy of vaporization at 0 °C (J·kg^{−1}) |

h_{m} | Convective mass-transfer coefficient (kg/(m^{2}·s)) |

h_{f} | Enthalpy of hot-air film (J·kg^{−1}) |

h_{v} | Enthalpy of water vapor (J·kg^{−1}) |

${h}_{fg}$ | Latent heat of vaporization of liquid water (J·kg^{−1}) |

i | Complex number |

k_{a} | Thermal conductivity of air (W ·m^{−1}·K^{−1}) |

k_{c} | Mass-transfer coefficient (m/s) |

le | Lewis number |

m_{e} | Equilibrium moisture content of air, decimal wet basis |

mf | Mass of sample after heating process (kg) |

m_{ml} | Amount of moisture loss (% d.b) |

m_{0} | Mass of sample before heating process (kg) |

M_{w} | Molar mass of water (kg/mol) |

p_{sv} | Saturation pressure of vapor (Pa) |

Q_{ev} | Heat loss due to evaporation (W·m^{−2}) |

RH | Relative humidity of surrounding air |

rd | Ratio of dry matter in wet sample (kg dry matter/kg wet product) |

Ta | Temperature of the hot-air film (K) |

T_{air} | Ambient air temperature (293.15 K) |

T_{ext} | External temperature (K) |

T_{s} | Temperature at the surface of food material (K) |

X_{1} | Value of analyzing parameter received from first data comparison |

X_{2} | Value of analyzing parameter received from second data comparison |

x_{sv} | Mole fraction of saturated water vapor |

X_{v} | Mole fraction of moisture inside the air (mol_{water}/mol_{dry air}) |

$\mathsf{\lambda}={M}_{w}{h}_{fg}$ | Latent heat of vaporization of water (J/mol) |

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**Figure 2.**Sample positions within the single-mode microwave cavity (view from the top of the microwave cavity).

**Figure 3.**Position of optical fiber sensors in the sample (

**A**,

**B**) and sensor holder (

**C**) (the red stars are the measured location, which is named 1, 2 and 3).

**Figure 4.**Dielectric properties of Cambodian pâté as a function of temperature at 915 MHz (

**A**): measured value of Ԑr’(T), (

**B**): measured value of Ԑr″(T)).

**Figure 6.**Reflection coefficients of Tylose

^{®}at positions A, B, C, and D following various SSC positions.

**Figure 8.**Temperature profiles of three points (1, 2 and 3) inside the Tylose

^{®}sample during experimental heating at positions A and B (Exp) compared to simulation (Sim).

**Figure 9.**Surface-temperature cartography of Tylose

^{®}during the ramp-up heating period at position A. (The blue line is the temperature sampling line used in Figure 10).

**Figure 10.**Experimental and predicted temperature profiles at Tylose

^{®}sample’s surface along a cut-line during microwave heating during t = 0 to 5 min of microwave heating respectively.

**Figure 11.**Temperature profiles of three points inside the pâté sample during microwave heating at positions A (Exp) compared with simulations.

**Figure 12.**2D Temperature cartography at the end of the ramp-up heating (1880 s) process and temperature-holding phase (3680 s) (each plan passed through the centerline); 3D temperature map of the sample at 3680 s (simulation with hc

_{3}value).

**Figure 13.**Surface temperature cartography of pâté during the ramp-up heating period at position A (simulation with hc

_{2}value). (The blue line is the temperature sampling line used in Figure 14).

**Figure 14.**Experimental and predicted temperature profiles (using hc

_{3}value) on the pâté’s surface along a cut-line during ramp-up heating.

**Figure 15.**Water concentration profiles (% d.b.) along a central cut-line from the top surface to the bottom (

**A**), and global moisture loss during cooking (

**B**) (hc

_{1}= 3.5 W/(m

^{2}·K), hc

_{2}=6 W/(m

^{2}·K), and hc

_{3}= 8.4 W/(m

^{2}·K)).

**Figure 16.**Two-dimensional plan of water concentration levels (% d.b.) at t = 1880 s and t = 3680 s for various heat-transfer coefficients on the surface of the pâté sample.

Name | Equation | Reference |
---|---|---|

Latent heat of vaporization of liquid water | ${h}_{fg}=1.91846\xb7{10}^{6}{\left[\frac{T}{T-33.91}\right]}^{2}$ | [27] |

Lewis number | $le=\frac{{k}_{a}}{{\rho}_{a}C{p}_{a}{D}_{a}}$ | [28] |

Heat capacity of the air around the sample | $C{p}_{a}=C{p}_{d.a}+d.C{p}_{m}$ | [28] |

Humidity ratio in mass | $d=0.622\frac{RH\xb7{p}_{sv}}{{p}_{o}-RH\xb7{p}_{sv}}$ | [29] |

Thermal conductivity of air around the sample | ${k}_{a}=\left(251.626+7.734{T}_{a}+167.6{x}_{v}-7.432\xb7{10}^{-4}{T}_{a}{x}_{v}+8.631\xb7{10}^{-6}{T}_{a}{}^{2}\right)\xb7{10}^{-5}$ | [29] |

The temperature of the hot-air film near the surface | ${T}_{a}=0.5\left({T}_{air}+{T}_{s}\right)$ | [30] |

Mole fraction of moisture in the air | ${x}_{v}=RH\xb7f\prime \xb7{p}_{sv}\xb7{P}^{-1}$ | [30] |

Correction function | ${f}^{\prime}=\alpha +\beta p+\gamma {\left({T}_{a}-273.15K\right)}^{2}$ | [30] |

Saturated pressure | ${p}_{sv}=1Pa\times \mathrm{exp}\left(A{T}_{a}{}^{2}+B{T}_{a}+C+D{T}_{a}{}^{-1}\right)$ | [30] |

Air density | ${\rho}_{a}=3.48353\xb7{10}^{-3}\xb7\left(\frac{p}{Z{T}_{a}}\right)\left(1-0.378\text{}Xv\right)$ | [30] |

Enthalpy of hot-air film | ${h}_{f}={h}_{a}+d\xb7{h}_{v}$ | [29] |

Enthalpy of dry air | ${h}_{a}=C{p}_{a}\left({T}_{a}-273.15K\right)$ | [29] |

Enthalpy of water vapor | ${h}_{v}={h}_{0}+C{p}_{v}\left({T}_{a}-273.15K\right)$ | [29] |

Material | Density | Thermal Conductivity | Heat Capacity | References |
---|---|---|---|---|

Teflon | 2.2$\xb7$10^{3} kg/m^{3} | 0.32 W/(m.K) | 1.02 J/(g.K) | [39,40] |

Polypropylene | 930 kg/m^{3} | 0.3 W/(m.K) | 1.2 J/(g.K) | [41,42,43] |

PET | 1380 kg/m^{3} | 0.2 W/(m.K) | 1.2 J/(g.K) | [44,45] |

Tylose^{®} | 965 kg/m^{3} | 0.5 W/(m.K) | 3859.9 J/(kg.K) | Measurement |

Pâté | 903.6 kg/m^{3} | Figure 5 | 3511.3 J/(kg.K) | Measurement |

**Table 6.**RMSE and p-value of various comparisons of hc values of the model with and without mass transfer.

Ramp-Up Heating RMSE/p-Value | Temperature-Holding Phase RMSE/p-Value | |
---|---|---|

Including mass transfer | ||

hc_{1} and hc_{3} | 2.5603/p-value > 0.05 | 18.8437/p-value < 0.05 |

hc_{2} and hc_{3} | 0.9630/p-value > 0.05 | 15.5992/p-value < 0.05 |

hc_{1} and hc_{2} | 2.0257/p-value > 0.05 | 3.3303/p-value > 0.05 |

Without mass transfer | ||

hc_{1} and hc_{3} | 2.1473/p-value > 0.05 | |

Comparing models with and without mass transfer | ||

hc_{1} | 3.5680/p-value > 0.05 | 18.2840/p-value < 0.05 |

hc_{3} | 8.2854/p-value < 0.05 | 31.0090/p-value < 0.05 |

hc_{1} = 3.5 W/(m^{2}·K), hc_{2} = 6 W/(m^{2}·K), and hc_{3} = 8.4 W/(m^{2}·K)). |

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**MDPI and ACS Style**

Nget, S.; Mith, H.; Boué, G.; Curet, S.; Boillereaux, L. The Development of a Digital Twin to Improve the Quality and Safety Issues of Cambodian Pâté: The Application of 915 MHz Microwave Cooking. *Foods* **2023**, *12*, 1187.
https://doi.org/10.3390/foods12061187

**AMA Style**

Nget S, Mith H, Boué G, Curet S, Boillereaux L. The Development of a Digital Twin to Improve the Quality and Safety Issues of Cambodian Pâté: The Application of 915 MHz Microwave Cooking. *Foods*. 2023; 12(6):1187.
https://doi.org/10.3390/foods12061187

**Chicago/Turabian Style**

Nget, Sovannmony, Hasika Mith, Géraldine Boué, Sébastien Curet, and Lionel Boillereaux. 2023. "The Development of a Digital Twin to Improve the Quality and Safety Issues of Cambodian Pâté: The Application of 915 MHz Microwave Cooking" *Foods* 12, no. 6: 1187.
https://doi.org/10.3390/foods12061187