# Modeling and Experimental Verification of Induction Heating of Thin Molybdenum Sheets
## 1 Introduction  
Sapphire single crystals are widely used, especially in laser devices. Sapphire is an ideal material for producing so-called structures of silicon on single sapphire crystals. Production of single-crystal sapphire is possible by several vertical technologies such as single crystal growth using Czochralski, Verneuill, or Stepanov methods. The horizontal method for the growth of the single crystal has an enormous advantage compared to the vertical progression with a precisely pre-given orientation of the crystal's optical axis to the surface of the growing crystal [1, 2, 3]. These processes are carried out under a high vacuum and temperature up to 2150 \({}^{\circ}\)C. As mentioned above, the horizontal method is the most promising, even though many engineering-design challenges are presented. One of them is the use of crucible-vessel, where crystallization takes place. The container is exposed to a temperature of 2150 \({}^{\circ}\)C in the longitudinal direction. Currently, container-vessels-crrystallization \"boats\" are made of molybdenum sheets having a thickness of 0.5 mm, made by powder metallurgy technology, i.e., plastic deformation sintering procedure. The shape and dimensions of the crystals currently identified meet the requirements for hand-manual production and, of course, automatic production in the complex mechatronic deformation systems with local resistance heating or induction heating of thin molybdenum sheets. The deformation ability of the various types of molybdenum sheets must be minimal; besides, there is a great affinity of Mo to oxygen. This affinity has a very strong progressive initiating effect on the corrosion happening, which dominantly occurs in the surface layers of molybdenum sheets [4, 5, 6, 7, 8, 9, 10]. A considerable consumption of electrical energy generally characterizes the process of heating. In the case of molybdenum sheets, the material must be heated above its hotforming temperature, usually about 950-2050 \({}^{\circ}\)C. In order to reduce the amount of energy, the process should be optimized appropriately. Nowadays, the cheapest way of doing this is computer modeling [11; 12]. Due to these facts, it is very important to uniformly heat molybdenum sheets during the manufacturing process of containers, if possible, through a contactless approach. The induction heating presents this approach here. Because induction heating is specific due to the proper setting of the operational conditions, it is necessary to identify them. The most uniform thermal distribution during the molybdenum containers' high-temperature forming process is the most important. The approach of the investigation of the proper induction heating settings could be experimental, resulting in time-consuming modifications of laboratory stands. There are currently modern methods based on simulation modeling by finite element methods, enabling the investigation of the mentioned processes quickly and accurately [13; 14]. This article discusses the development of a finite element method (FEM) simulation model of inductive heating. The corresponding experiments are performed for the initial comparisons with the simulation results. Consequently, the simulation model is optimized to achieve precise and comparable results to the physical sample, while a very high accuracy level shall be reached. It considers the possibility of identifying heating processes within sheet volume, while the geometric shape modifications are available (e.g., design of high-performance thermal shells). Moreover, the problem is non-linear due to the temperature dependence of the system's physical parameters. In principle, electrical conductivity, thermal conductivity, and specific heat change nonlinearly within a wide range of temperatures. For this reason, the problem must be solved numerically. Because expectations of fast verification processes are required, the mathematical computer model is being developed to increase the variable solution investigations [15; 16].