Predictions of eddy-current signals due to a parallel rectangular coil outside a two-layer tube structure

Apart from remaining life and state-of-age assessments, reliable detection of deeply buried cracks in reformer tubes, important for petrochemical industries, is still challenging. In this work, a theoretical investigation was conducted to determine the electromagnetic response signals from a parallel, axially displaced (offset) rectangular coil positioned outside a two-layer tube, simulating an eddy current inspection of a reformer tube. The second-order vector potential (SOVP) formalism was adopted for this purpose, in which the SOVP is expressed using transverse electric (TE) and transverse magnetic (TM) potentials. The modal coefficients are inherently coupled, which leads to complex analytical expressions, especially for multi-layer tubes. To circumvent the complexities, we numerically treated the boundary conditions at each interface. This method enabled the discrete estimation of modal coefficients, which were then used to accurately predict the magnetic field signature and induced eddy currents. The results were validated against a finite element method (FEM) model. This work provides the necessary theoretical framework for a subsequent investigation into the response signals generated by circumferential cracks in the reformer tubes.