## 2 Materials and Methods
### Numerical Modeling  
The footprint of MW machine was chosen to develop a prototype that processed the product continuously; therefore, it was necessary to design a system consisting of a cylindrical treatment chamber and an internal transport system made up of a metallic helix. The design choice of using the helix as an alternative to the screw is due to its lower resistance to the diffusion of the microwaves in the volume of the treatment chamber. The choices of the geometric parameters of the treatment chamber (diameter and length) arise from the operational needs of the system. A minimum flow rate of the system equal to at least 2 kg min\({}^{-1}\) and a minimal residence time of the almonds inside the treatment chamber equal to at least 90 s were assumed. The flow rate and residence time values were measured on the basis of preliminary tests carried out before the construction of the plant, with a simplified prototype consisting of a feeding hopper, tube (treatment chamber) and internal helix (transport system). Figure 1 shows the technical design of the MW prototype. The treatment chamber consists of a polypropylene tube (green region) with an inner diameter, outer diameter and length of the tube of 90.8 mm, 125 mm, and 3000 mm, respectively. Inside the tube a rotating metallic helix allows the almonds to be mixed and moved from the supply section to the discharge section. On the feeding side is placed a honeycomb filter integrated with the feed hopper. The dust extraction chamber is shown on the opposite side. Figure 1 shows three different setups with 1, 3 and 5 MW sources equidistant from each other along the treatment chamber:  
A1: 1 MW source of 7.5 kW (Figure 1a);  
A2: 3 MW sources of 2.5 kW (Figure 1b);  
A3: 5 MW sources of 1.5 kW (Figure 1c). For the three different arrangements, the numerical simulations were carried out under different processing times (residence time of the almonds inside the treatment tube) to identify the right number, placement, and MW power minimizing hotspots and allowing a uniform heating of the almonds in the treatment chamber. The residence time used for numerical simulation was of 80 s, 100 s, 120 s, 140 s, 160 s, 180 s and 200 s.  
For each arrangement, the whole design was numerically carried out using the 3D simulation software Dassault Systemes CST STUDIO SUITE(r) 2022. The time domain solver was used to perform both the electromagnetic and thermal analyses. The computational domain is an air box containing the MW prototype. In this way, it was possible to quantify the electromagnetic field inside the treatment chamber as well as the one escaping in the external environment. So, the electromagnetic simulations were carried out considering a frequency range of 2.4 to 2.5 GHz and the perfectly matched layer boundary condition surrounding the whole computational domain. The waveguide port setting provided by the electromagnetic solver was used to feed the computational domain with power and to absorb the returning power. Hexahedral meshing was used, characterized by 10 cells per wavelength, 20 cells per max model box edge and a fraction of maximum cell near to model of 15 (giving about 30 million of cells). The thermal simulations were performed by fixing a temperature T\({}_{\text{b}}\) = 16 \({}^{\circ}\)C on the boundary of the air box containing the MW prototype as well as a uniform initial temperature T\({}_{\text{0}}\) = 16 \({}^{\circ}\)C on the whole computational domain. The latter constrain is justified by considering the stationary solution of Equation (10) with no volumetric heat source. Moreover, Equation (10) was applied to the whole computational domain including the bunch of almonds, propylene tube, air volume inside and outside the treatment chamber, metallic walls and helix. Hexahedral meshing characterized by 30 cells per max model box edge and a fraction of maximum cell near to model of 20 (giving about 140,000 cells) was set.