Model Setup

The simulations were solved using the Electromagnetic Waves Frequency Domain physics interface which is found in COMSOL under the RF Module. The governing equation of the electric field wave is given by:

∇ × 𝜇𝑟−1(∇ × 𝐸) − 𝑘𝑜2(𝜀𝑟−

𝑗𝜎

𝜔𝜀𝑜) 𝐸 = 0 6-2

Where

𝜇𝑟 is the relative permeability of the material (for biomass 𝜇𝑟= 1);

148 𝑘0 is the wave number which is given by 𝑘0= 𝜔 𝑐⁄ 𝑜, where 𝜔 is the wave

angular frequency (rad·s-1_{) and 𝑐}

𝑜 is the speed of light in vacuum (3×108

m·s-1₎

𝜀𝑟 is the relative permittivity of the material;

𝜎 is the electric conductivity of the material, S·m-1_; 𝑗 is the √−1;

𝜀𝑜 is the permittivity of free space (8.854×10−12 F·m-1).

The power loss density (absorbed microwave power by the material) is determined through Equation 3.2 which was discussed in Chapter 3.

The model was solved at 2.45 GHz frequency. As stated in Section 5.3, sycamore of 212 – 850 µm particle size was used as the basis for the design. Therefore, the parameters listed in Table 4-2 were used for the simulation. It is to be noted that the fluidised bed was simulated as a bulk solid using the bulk density of the bed at the minimum fluidisation velocity. The model was built and solved following the steps below.

6.4.1.1 Simulation geometry

The general geometry used for the simulation is shown in Figure 6-13 which is mainly a quartz column (reactor) with a gas distributor inside a rectangular metallic box. The quartz reactor is surrounded by a metallic tube all the way outside the cavity. Quartz was chosen because of its good dielectric and thermal properties for the present case. It has a loss factor less than 0.001 at 2.45 GHz (Meredith, 1998). It has a typical operating temperature of around 1000 o_{C. The inner and outer} diameter of the column were 0.05 and 0.06 m respectively. The inner diameter of the metallic tube surrounding the column was assumed to have the same value as the outer diameter of the quartz column. This is lower than the critical cut-off

149 diameter*_{; the diameter above which microwave power would propagate through} the tube (Meredith, 1998). The length of the column was set at 0.6 m. Based on the fluidisation experiments this length is enough for the bed expansion and disengagement.

Figure 6-13: Geometry used to simulate the fluidised bed process in a multimode cavity.

6.4.1.2 Materials

The materials involved in the simulation were the biomass material, quartz, and air. The walls of the cavity and the waveguide were assumed to be perfect conductors. The complex permittivity of the biomass material was determined from the quadratic functions which were developed in Section 4.4 for the relationship between both parts of the permittivity and the packing density. The relationships for sycamore were as follows:

𝜀′_{= 0.69𝜌}2_{+ 1.376𝜌 + 1; 𝜀}′′_{= 0.117𝜌}2_{+ 0.197𝜌}

* _{The cut-off radius, 𝑎, for the lower-order mode (TE11) through circular choke is 𝑎 = 𝜆}

0⁄(3.413√𝜀′). Where

𝜆0 is the free-space wavelength, 𝜀′ is the dielectric constant of the medium. At 2.45 GHz frequency and

150 Where 𝜌 is the packing density in g·cm-3_{. Substituting the bed bulk density of} sycamore at the minimum fluidisation velocity which is 0.24 g·cm-3 _{as displayed in} Table 4-2 gives a permittivity of: 𝜀 = 1.37 − 0.055𝑗.

The space above the bed and that outside the reactor was assumed to be air. The COMSOL library permittivities for air and quartz were used which are 1.0 and 4.2 respectively.

6.4.1.3 Boundary conditions

A port boundary condition was added to feed the microwave power into the system. The port is a rectangular TE10 placed at the start of a standard WR430 waveguide as indicated in Figure 6-13. The microwave power input through the port was set at 6 kW. The obtained power loss density will be compared to the minimum power density required for the pyrolysis which is 54 kW·m-3_{. A perfect conductor boundary} condition was also added to the model to simulate the internal walls of the waveguide, the cavity and the metallic tube.

6.4.1.4 Mesh

The finite element method divides the model into small and geometrically simple elements. The accuracy of the solution is affected by the resolution of the mesh. Fine mesh size improves the computational accuracy but increases the solving time and memory requirement. For the current model, the mesh type selected was physics controlled mesh and an extremely fine element size was selected. This generated 248,251 to 412,739 tetrahedral elements depending on the dimensions of the cavity in each case. The simulation was run on an Intel workstation with eight Dual Core 2.5 GHz Xeon processors, 192 GB RAM (Random Access Memory), with a Microsoft Windows 7, 64 bit operating system. With these memory and processing capabilities, the time needed to solve each case was 4 to 8 minutes.

151

6.4.1.5 Study

There are many parameters related to the cavity dimensions, load position and microwave power feed position needed to be optimised. In order to limit the number of the variables, some practical choices/assumptions were made as follows:

 The cavity depth (D) equals the width (W)  The column is at the centre of the cavity.

 The waveguide is attached to the side of the cavity, and not at the top or the bottom.

 The tapered section below the gas distributor is 10 cm long, and all of it is inside the cavity.

This leaves four variables: the cavity height (H); the cavity width (W); the waveguide Y position; and the waveguide Z position. These variables were defined as parameters in the simulation to allow them to be changed during the simulations through the parametric sweep function. The parametric sweep function in COMSOL instructs the software to change the value of a parameter within a specified range and solve for each of these values instead of changing it manually. The function allows for more than one parameter to be changed, and the simulation is then solved for all the combinations of the specified parameters. The cavity width, W, was set to vary from 20 to 34 cm at steps of 2.0 cm which covers a number of cavity modes* _{slightly less than three to more than four. As the load is placed at} the centre of the cavity, two modes would create an area of a low electric field within the load. The cavity height, H, was changed from 18 to 30 cm with step changes of 2.0 cm. The lower limit is determined by the height from the bottom of the tapered section to the top of the load. The waveguide Z position was changed to cover all the cavity height while the Y position was changed to cover only half of

* _{The term “modes” refers to the number of the half-wavelengths generated by the reflection of the}

waves by the walls inside the cavity. The number of modes could be estimated by dividing the cavity length by the guide half-wavelengths, 𝜆𝑔⁄2.

152 the depth due to the symmetry. Step changes of 2 cm were used for both Y and Z positions.

The aim was to run the simulations for a range of cavity dimensions and waveguide positions to find the optimum values that minimise the loaded cavity Q-factor (maximise the percentage absorbed power) while providing uniform heating.