The Development of a Digital Twin to Improve the Quality and Safety Issues of Cambodian Pâté: The Application of 915 MHz Microwave Cooking
Abstract
:1. Introduction
2. Materials and Methods
2.1. Tylose Preparation
2.2. Cambodian Pâté Preparation
2.3. Dielectric Properties’ Measurements
2.4. Thermal Conductivity Measurements
2.5. Heat Capacity Measurements
2.6. Microwave Device Configuration
2.7. Impedance-Matching Characterization
2.8. Microwave-Heating Experiments
2.9. Moisture Loss Characterization
2.10. Infrared Image
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
- -
- 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 S11 and S21 [16]. The S-parameter values at both ports are:
3.3. Heat Transfer
- -
- Initial condition: T0 = 4 °C, t = 0.
- -
- The boundary condition for the top surface of the sample involves both evaporation and natural convection phenomena [22]:
- -
- The boundary values for the side surface of PET sample cells are:
3.4. Mass Transfer
- -
- Initial condition:
- -
- At the top surface of the sample, the boundary equation is expressed as:
- -
- 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:
3.5. Thermophysical and Dielectric Properties
Surface Boundary | Electrical Conductivity | Reference |
---|---|---|
Aluminum | 3.774·107 S/m | COMSOL® database |
Brass | 1.59·107 S/m | [38] |
Copper | 5.99·107 S/m | COMSOL® database |
Parameters | Value/Units | References |
---|---|---|
RH | 0.3 | [47] |
% dry matter | 23.52% | Measure |
me | 0.02 | [22] |
h0 | 2501·103 J/kg | [29] |
Da | 2.5·10−8 (m2·s−1) | [48] |
Km | 1.29·10−9 (kg·m−1·s−1) | [22] |
Cpd.a | 1005 J/(kg·K) | [29] |
Cpm | 1870 J/(kg·K) | [29] |
Cm | 0.003 kg/kg | [22,49] |
Z | 0.999 | [30] |
α | 1.0062 | [30] |
β | [30] | |
γ | [30] | |
A | 1.2811805 | [30] |
B | −1.9509874 | [30] |
C | 34.04926034 | [30] |
D | −6.3536311 | [30] |
4. Model Design
4.1. Computational Details
4.2. Mesh Configuration
4.3. Validation Method
5. Results and Discussion
5.1. Electromagnetic Validation
5.2. Temperature Validation
5.2.1. Tylose
5.2.2. Cambodian Pâté
5.3. Mass Transfer Validation
5.4. Sensitivity Analysis
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
Dielectric loss factor | |
Relative permittivity | |
Air density (kg/m3), | |
ρs | Density of the food product (kg/m3) |
µ | Magnetic permeability (H/m) |
µo | Magnetic permeability of vacuum 4π × 10−7 H/m |
Cb | Equilibrium moisture concentration (mol/m3) |
Cm | Specific moisture capacity (kgmoisture/kgfood) |
Cp | Specific heat capacity of the sample (J · kg−1·K−1) |
Cpa | Heat capacity of the air (J·kg−1·K−1) |
Cpd.a | Heat capacity of dried air 1005 J/(kg·K), |
Cpm | Heat capacity of water vapor 1870 J/(kg·K) |
Cw | Moisture concentration at the surface of food material (mol/m3) |
d | Humidity ratio in mass (kg moisture/kg dry air) |
Da | Air-diffusion coefficient of water vapor in the air (10−11 m2/s) |
d.b | Dry base matter |
Dm | Moisture diffusivity (m2/s) |
E | Electric field (V/m) |
Ec | Calculated electric field |
E1 | Analytic field of excitation port |
E2 | 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 |
εo | |
f | Microwave frequency (Hz) |
f’ | Correction function |
hc | Convective heat-transfer coefficient (W·m−2·K−1) |
ha | Enthalpy of dry air (J·kg−1) |
h0 | Enthalpy of vaporization at 0 °C (J·kg−1) |
hm | Convective mass-transfer coefficient (kg/(m2·s)) |
hf | Enthalpy of hot-air film (J·kg−1) |
hv | Enthalpy of water vapor (J·kg−1) |
Latent heat of vaporization of liquid water (J·kg−1) | |
i | Complex number |
ka | Thermal conductivity of air (W ·m−1·K−1) |
kc | Mass-transfer coefficient (m/s) |
le | Lewis number |
me | Equilibrium moisture content of air, decimal wet basis |
mf | Mass of sample after heating process (kg) |
mml | Amount of moisture loss (% d.b) |
m0 | Mass of sample before heating process (kg) |
Mw | Molar mass of water (kg/mol) |
psv | Saturation pressure of vapor (Pa) |
Qev | 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) |
Tair | Ambient air temperature (293.15 K) |
Text | External temperature (K) |
Ts | Temperature at the surface of food material (K) |
X1 | Value of analyzing parameter received from first data comparison |
X2 | Value of analyzing parameter received from second data comparison |
xsv | Mole fraction of saturated water vapor |
Xv | Mole fraction of moisture inside the air (molwater/moldry air) |
Latent heat of vaporization of water (J/mol) |
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Name | Equation | Reference |
---|---|---|
Latent heat of vaporization of liquid water | [27] | |
Lewis number | [28] | |
Heat capacity of the air around the sample | [28] | |
Humidity ratio in mass | [29] | |
Thermal conductivity of air around the sample | [29] | |
The temperature of the hot-air film near the surface | [30] | |
Mole fraction of moisture in the air | [30] | |
Correction function | [30] | |
Saturated pressure | [30] | |
Air density | [30] | |
Enthalpy of hot-air film | [29] | |
Enthalpy of dry air | [29] | |
Enthalpy of water vapor | [29] |
Material | ε’r | ε″r | Reference |
---|---|---|---|
PET container | 3.5 | 0.0035 | [31] |
PTFE | 2.1 | 0.002 | [32] |
Air | 1 | 0 | [33] |
Tylose | 69 | 8.9 | Measurement |
Material | Density | Thermal Conductivity | Heat Capacity | References |
---|---|---|---|---|
Teflon | 2.2103 kg/m3 | 0.32 W/(m.K) | 1.02 J/(g.K) | [39,40] |
Polypropylene | 930 kg/m3 | 0.3 W/(m.K) | 1.2 J/(g.K) | [41,42,43] |
PET | 1380 kg/m3 | 0.2 W/(m.K) | 1.2 J/(g.K) | [44,45] |
Tylose® | 965 kg/m3 | 0.5 W/(m.K) | 3859.9 J/(kg.K) | Measurement |
Pâté | 903.6 kg/m3 | Figure 5 | 3511.3 J/(kg.K) | Measurement |
Ramp-Up Heating RMSE/p-Value | Temperature-Holding Phase RMSE/p-Value | |
---|---|---|
Including mass transfer | ||
hc1 and hc3 | 2.5603/p-value > 0.05 | 18.8437/p-value < 0.05 |
hc2 and hc3 | 0.9630/p-value > 0.05 | 15.5992/p-value < 0.05 |
hc1 and hc2 | 2.0257/p-value > 0.05 | 3.3303/p-value > 0.05 |
Without mass transfer | ||
hc1 and hc3 | 2.1473/p-value > 0.05 | |
Comparing models with and without mass transfer | ||
hc1 | 3.5680/p-value > 0.05 | 18.2840/p-value < 0.05 |
hc3 | 8.2854/p-value < 0.05 | 31.0090/p-value < 0.05 |
hc1 = 3.5 W/(m2·K), hc2 = 6 W/(m2·K), and hc3 = 8.4 W/(m2·K)). |
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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
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 StyleNget, 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