7.4.1 Introduction
The purpose of this section is to analyse some of the problems that are encountered dur- ing the application of the coupled model to multimode cavity problems and to demon- strate, by means of an example, the type of effects that may be observed in temperature dependent systems. Several recent papers have considered the problem coupling the thermal and electromagnetic fields for microwave heating problems [Dibben & Metaxas, 1994a; Sundberg, 1994; Huang et al., 1994]. However, very few results are available for the coupled problem in multimode cavities. Ayappaet al. [1991] use a one dimensional finite element model to investigate the heating caused by plane waves incident on a multilayered slab. The results show that the temperature dependence of the material properties can appreciably effect the temperature distribution. The application of this one dimensional model to real problems is, however, extremely limited. Shouzheng &
Davis [1991] and Sekkak et al. [1994] have also considered the effects on the electric field of the temperature dependence of dielectric properties although for a short circuited waveguide loaded with a dielectric material. They have shown that the use of a coupled model results in power density and temperature profiles that are significantly different from those produced by assuming no variation in power density with temperature.
It was seen in Chapter 6 that for low loss materials it was necessary to produce solutions at several frequencies in order to determine the sensitivity of the system.
This remains true for the coupled model which will require repeated calculations for each frequency being modelled: the coupled model cannot produce multiple frequency solutions from a single calculation. This obviously will increase the cost of the calculation enormously.
The power supplied to the cavity determines the temperature rise, which in turn will determine the development of the electric field distribution during heating. Small variations in the power supplied to the cavity may therefore cause the temperature dis- tribution to evolve in a different manner. This leads to the observation that small errors in the reflection coefficient, which through equation (4.21) will determine the power ab- sorbed by the load, may lead to large changes in the field and temperature distributions
7.4 Coupled Electromagnetic and Thermal Models 153
later on in the heating cycle. The results in Chapters 5 and 6 for the reflection coeffi- cient show large variations in the magnitude with frequency. This suggests that it is also necessary to solve the system with a variety of power levels to determine the sensitivity to changes in absorbed power. This will also give an indication of the sensitivity to errors in the reflection coefficient.
Taking the above comments together, and assuming that it is desired to solve the system at, say, 5 frequencies and 5 power levels then this requires 25 repeated solutions.
Each solution will require several field calculations. If, for example, 10 field calculations were required per solution then this leads to a total of 250 field calculations. Even the shorter calculations presented in Chapter 6 took 4 hours to complete, so that the problem described here would require 41 days to complete! These numbers are somewhat arbitrary, but they demonstrate one of the most obvious problems with implementing the coupled model; the fact that the large computational burden of repeat solutions leads to unacceptably large solution times.
Before the temperature feedback scheme, which is outlined below, can be used for real multimode cavity problems there will need to be a considerably more research carried out to understand the electromagnetic phenomena involved in modelling multimode cavities. The scheme presented here is therefore of only theoretical interest at present.
7.4.2 Temperature Feedback
In order to reflect the changes in dielectric properties with temperature the algorithm shown in Figure 7.2 was implemented. The time scales of the thermal and electromag- netic systems are very different. For the electromagnetic system changes take place in nano seconds compared to the several seconds required for the thermal system. This allows the two calculations to be performed separately. During the heat flow calculation the temperature of each element is monitored. When the change in temperature is suffi- cient to produce a significant change in dielectric properties the temperature calculation is suspended and the electric field distribution is recalculated. After each recalulation of the electric field the reflection coefficient is calculated so that the absorbed power can be determined and the values of the power density scaled accordingly. This procedure continues until the load has reached a pre-set temperature or a specified time period
Figure 7.2: Flow chart showing temperature feedback algorithm.
7.4 Coupled Electromagnetic and Thermal Models 155
0 100 200 300 400 500 600
Temperature °C 0
5 10 15 20 25 30
Permittivity (real part)
0 0.5 1 1.5 2
Permittivity (imaginary part)
ε'' ε'
Figure 7.3: Relative permittivity and conductivity of Zirconia against temperature.
(from Araiet al. [1993]) has elapsed.
Conceptually this procedure is very simple, however, there are many issues that are not yet understood. For example, what constitutes a significant rise in dielectric properties? If the electric field is recalculated after too small a change then the solution time will be prolonged. Conversely, if too large a change is allowed before re-calculation then the development of the temperature distribution will be altered. The change in properties that will cause a significant change in field distribution will be very problem- dependent. If the system is operating close to a sharp resonance a small change could produce very large changes in the reflection coefficient and therefore in the absorbed power.
7.4.3 Ceramic Block Example
In order to demonstrate the application of the temperature feedback to a multimode cavity, a block of ceramic was modelled. The ceramic that was chosen was zirconia, which has a permittivity that varies with temperature, as shown in Figure 7.3. A
a) Start of process b) After 20 minutes
c) After 30 minutes d) After 50 minutes
1
0 Insulator
Ceramic
Figure 7.4: Normalised power density distribution in the ceramic block on an x-y plane through the centre of the ceramic.
a) After 12 minutes b) After 24 minutes
c) After 40 minutes d) After 55 minutes
1000
20
°C
°C Insulator
Ceramic
Figure 7.5: Temperature density distribution in the ceramic block on an x-y plane through the centre of the ceramic.