Contactless inductive flow tomography: inverse problem and applications

In many industrial applications dealing with liquid metals even a rough knowledge of the flow field of the melt would be of high value. For instance, in continuous casting of steel the flow of the melt in the upper region of the mold is very important for the quality of the produced steel, regarding surface defects or the number of inclusions. The high temperatures and the chemical aggressiveness of liquid melts recommend contactless measuring techniques. Well-established optical methods like particle image velocimetry are not applicable, because of the opaqueness of the melt.
Due to the high electrical conductivity of liquid metals, inductive methods can be used. One of them is the Contactless Inductive Flow Tomography (CIFT) which allows the reconstruction of the mean three-dimensional flow structure of conducting liquids. CIFT works by applying one or more magnetic fields to the melt and measuring the flow induced perturbation of those fields outside the fluid volume. From these measurements the mean three dimensional velocity field can be reconstructed by solving a linear inverse problem similar to magnetoencephalography. In order to handle the non-uniqueness, Tikhonov regularization in combination with the L-curve method is used.
In this paper we will give an overview about the mathematical foundation of CIFT and delineate the linear inverse problem. In order to illustrate the numerical model and the regularization, we will show numerical and physical results of a model of a continuous caster and of a Rayleigh-B ́enard convection setup. Complementary Ultrasound Doppler Velocimetry measurements will be shown to be in good agreement with the reconstructed flow using CIFT. We will conclude with a short overview of the challenges to measure the flow induced magnetic field perturbations, which are usually about 3 to 4 magnitudes smaller than the applied magnetic field.