The Laplace transform dual reciprocity boundary element method for electromagnetic heating problems

There are many situations in applied science and engineering where materials are heated electrically via the so-called ohmic heating, or Joule heating, process. In this process the heating occurs throughout the volume as compared with the surface heating in conventional processes. The technique is frequently used as a method of food sterilisation in the food processing industry. It is important to know both that the food material itself is not degraded and that the temperatures reached are sufficient to kill bacteria. These problems exhibit significant non-linearities since, for food materials, the electrical and thermal properties are dependent on the temperature. When the electrical and the thermal conductivity depend on the temperature the resulting model of the ohmic heating process comprises a pair of non-linear coupled partial differential equations. The time dependence can be handled using the Laplace transform by first linearising the equations and solving the system iteratively. The resulting linearised equations may, after taking the Laplace transform, be written in Poisson-type form and as such are ideally-suited to a solution using the dual reciprocity boundary element method. The Laplace transform has been shown to be well-suited to the solution of problems, such as diffusion-type problems, which are parabolic in time. They are transformed to elliptic problems in the space variables and any suitable solver may be used. We shall use the dual reciprocity boundary element method. A numerical inversion of the transform is required and the Stehfest method has been shown to be robust, easy to implement and accurate and that is the approach adopted.